Whittle : EXTRAGALACTIC ASTRONOMY

 
       
 

1 : Preliminaries	6 :   Dynamics I	11 : Star Formation	16 : Cosmology
2 : Morphology	7 :   Ellipticals	12 : Interactions	17 : Structure Growth
3 : Surveys	8 :   Dynamics II	13 : Groups & Clusters	18 : Galaxy Formation
4 : Lum. Functions	9 :   Gas & Dust	14 : Nuclei & BHs	19 : Reionization & IGM
5 : Spirals	10 : Populations	15 : AGNs & Quasars	20 : Dark Matter

(1) Introduction

Cosmology is in a truely golden era: now is a great time to learn it!
• Some recent observational developments have been:
  • the ¾-century long uncertainty in H_o is now over:   H_o = 72 5 km s^-1 Mpc^-1
  • the long search for a decent standard candle is also over:   SNIa seem to work well
  • COBE & WMAP have now measured the CMB spectrum & anisotropy with high accuracy
  • 2dFGRS & SDSS have recently mapped cosmologically significant local structures
  • observations at z ~1-6 are almost routine, and probe a significant slice of cosmic history

• In close partnership, theoretical developments and their applications are increasingly robust:
  • The cosmic constituents and their relative proportions have been ascertained
       = 0.73 0.02 : dark energy dominates the current universe
    _DM = 0.26 0.02 : dark matter is very important, particularly in structure formation
    _b     = 0.04 0.01 : baryons are a trace (though vital!) component
        = 5.0 × 10^-5 : CMB photons reveal an early hot phase
        = 3.4 × 10^-5 : CNB neutrinos are predicted, but have not yet been observed

  • An "ordinary" FRW cosmological world model has emerged as being completely adequate
    The FRW parameters have been measured     "The Concordance Model"
    This yields a reliable framework for charting cosmic history -- ie we now know t(z)
    A number of puzzles are removed by invoking an early period of inflation.

  • A detailed theory exists tracing the average conditions from very early times
    a fairly full description of t < 1s exists, though it is not yet well tested
    nucleosynthesis at t ~ 1-5 mins nicely recovers the observed light element abundances
    (in fact, this measures conditions at t ~ 1s, when the neutron/proton ratio was fixed)
    the theory accurately describes recombination at ~½Myr and the origin of the CMB

  • A detailed theory now exists which describes the growth of perturbations
    Starting from inflation, a natural spectrum of fluctations can be followed to z~1000
    here it matches the CMB anisotropies in great detail
    it can then be followed to z=0, where it accurately matches local structure
    

• After thousands of years of wild speculation, the true story of creation is finally emerging
    we are living through (and participating in!) a historic period of intellectual growth
    in the future, our time will be recalled much like that of Copernicus, Newton, or Darwin.

• Before we get too dizzy with euphoria, let's regain some humility by recalling:
  • we have no idea what the dark matter or dark energy are actually made of (ie 96% !)
  • although inflation is a promising idea, it is far from proven and its cause is unknown
  • the origin of the baryon asymmetry is only guessed at
  • why particular cosmological values are favoured is unknown beyond anthropic arguments.
  • why there is something and not nothing is as unknown now as it was in Aristotle's day.

• Of course, these (and many other) puzzles are not really bad news at all:
  they signify a rich subject in good health, with more fascinating work to be done.
  In anycase, it would be unseemly to fully understand Creation after only a couple of generations.

• Understandably, "The New Cosmology" has attracted enormous interest and effort
    The subject is now mature and sophisticated -- much is well beyond our/my range
    Our aim will be to outline the overall framework, while ignoring details

• Here is our path through the subject:
  1. Following tradition, we begin with a discussion of some global cosmic properties:
     
     isotropy, homogeneity, expansion, structure, composition
  2. We need a General Relativistic framework, comprising two inter-dependent parts:
     
     geometry: metrics define the spacetime
     dynamics: contents define coefficients of the metric and hence its time evolution
  3. We clarify some confusing aspects of geometry on an expanding coordinate grid:
     various types of distance and horizon
  4. We introduce a toolkit for measuring intrinsic properties at cosmological distances.
     these allow us to convert redshift (our prime observable) to distance; time; etc.
  5. Such relations lead to the classic tests of the world model parameters.
  6. As you may know, several deep conundrums/puzzles emerge from the standard picture:
     the so-called flatness; horizon; structure; anti-matter; problems
  7. These are "explained" by introducing an early period of rapid inflation.
     we note how inflation solves some of these problems, and how it might be caused
  8. We briefly outline the evolution of the hot early universe:
     the various "eras" : quark; hadron; lepton; radiation; matter; dark energy
     a brief treatment of cosmic nucleosynthesis, and recombination
  9. We see recombination directly as the Cosmic Microwave Background (CMB)
     
     its primary characteristics are its remarkable isotropy and purity of black body form
     it provides the strongest evidence for the hot big bang
     it concludes our review of homogeneous Cosmology.

• Following homogeneous Cosmology, we are ready to start discussing inhomogeneities
  These provide the starting point for our next Topic: structure formation.

(2) Global Properties

• To humans, the universe seems highly anisotropic
  down is solid, up is the sky, with the sun, moon, & stars in specific directions
  even statistically, stars prefer the milky way while bright galaxies cluster in Virgo and Coma

• Only at much fainter levels and much greater distances does isotropy begin to emerge
  Here are some nice examples: [images]
  2 million faint (m_J < 17) galaxies cover ~1 sr with only slight structure visible
  31,000 radio sources (typical z ~ 1) uniformly cover the northern hemisphere
  the CMB with the galaxy & dipole removed is isotropic to one part in 10⁵

• At faint levels (i.e. large scales) the Universe seems remarkably isotropic

(b) The Cosmological Principle

• Ever since Copernicus, we are loath to assign Earth a special location
  Going one step further, we promote egalitarianism to a "Cosmological Principle"
    The Universe looks, statistically, the same from all locations
  go to any galaxy -- you will witness an istotropic universe.

• One consequence of this is that the Universe has no edge or center
  for a Euclidean space this also means the universe is infinite
  Note, this need not be true for other, closed, geometries (see later).

• There is also a "Strong Cosmological Principle"   =   Cosmological Principle   +   "at all times"
    in addition to uniformity, it explicitly states the universe does not evolve
  this underpins the Steady State Cosmology advocated by Hoyle, Bondi & Gold in the 50s & 60s
  The evidence for evolution first came in the 1960s and has grown stronger ever since   [ song ].
    we won't consider the Steady State cosmology further.

• It is easy to show that:   isotropy   +   cosmological principle   =   homogeneity
    all locations are (statistically) equivalent (e.g. have the same mean density)
  

• This is sometimes extended to postulate that the laws of physics are also global
  The observations of familiar spectral features in distant galaxies certainly supports this

• Homogeneity only sets in on large enough scales, somewhere between 100 Mpc & 1 Gpc   [images]
  On smaller scales, of course, one encounters much inhomogeneity (see below)

• This leads to a heart warming conclusion:
  Right "now", a civilization 1000 Gpc away (ie, well beyond our horizon) would:
  
  • see a microwave background; high-z QSOs; and distant young galaxies
    
  • be surrounded by sheets & voids of mature galaxies
    
  • find local galaxies with their stars and planets to be much like ours.

  Far from being bizarrely remote; the distant Universe would be remarkably familiar.

• The Universe's small scale inhomogeneity is much more obvious than its global homogeneity.
  At first sight this is a rather puzzling fact: why isn't the Universe just fully one or the other?
  And why is there a special length scale that marks the boundary between the two?

• The inhomogeneity appears as a heirarchy of structures: stars; galaxies; clusters; tapestry
  Out of almost perfectly uniform gas comes all these rich forms, each with its own character.
  This is a remarkable and profound property of our Universe.
  

• Our cosmology must explain this origin and development of structure.

(i) The Hubble Law

• As soon as galaxy spectra were measured it became clear that most were redshifted
  In 1929 Hubble found a "roughly linear" relation between redshift and distance: cz d
  as data improved, this relation was confirmed and has strengthened ever since [image]

• Don't confuse establishing the linear relation with measuring its gradient:
  It took ~75 years to achieve 10% uncertainty in the gradient, H_o (see below)
  This is primarily because calibrating the distance scale is notoriously difficult.
  The current best estimate for H_o is 72 5 km/s/Mpc   (72 × 10^-6 Myr^-1 in psm units)
  Note, it is still customary to quote measurements scaled to h = 100 km/s/Mpc
  For example, "The luminosity of M87 is 2.3 × 10¹¹ h^-2 L"   [Topic 1.3k]

• Note also that there is always some small scatter (~few 100 km/s) in the Hubble relation
    galaxy redshifts have two components:
  1. the global "Hubble Flow" (arising from cosmic expansion)
  2. a small "peculiar" velocity (arising as the galaxy responds to the gravity of its neighbors).
  ["peculiar" comes from Latin: "peculium", meaning "private property" - specific to it alone]

  (ii) Everyone Sees the Same Law

• At first glance the Hubble Law seems to violate the Cosmological Principle:
    galaxies appear to move radially away from us suggesting we are somehow central
  However, its linear form ensures that all locations witness the same relation:

  Consider two (vector) locations k and p and the vector field: v = H r centered on us (at O)
  We see p move at v_p = H p;   so how does an observer located on k see p move?
  use primes to denote values measured by k [image]:

  p' = p - k   and   v'_p = v_p - v_k = Hp - Hk = H(p - k) = Hp'

  Hence, for any point p, k sees a Hubble Law: v'_p = H p'
    if we see v = Hr then so does everyone else!
    the cosmological principle still holds true

• Since everyone witnesses the same Hubble Law, we conclude that:
    the Universe itself is undergoing isotropic expansion with form v = H r

  This is a remarkable and profound result.

• The linear nature of the Hubble law has a second fascinating consequence:
  Tracing the expansion backwards, there was a moment when all galaxies were together
  This singular event is called "The Big Bang"

  The term was coined in 1952 by Hoyle to gently mock opponents of his Steady State Cosmology.

• If no forces act to accelerate or decelerate the expansion, then v(t) = const,
  In that case, the singular event ocurred at a time t_H,o = r/v = 1/H_o in the past
  t_H,o = H_o^-1 = 10 h^-1 Gyr   is called The Current Hubble Time

  If the average expansion rate isn't too different from its current value, then t_H,o current age.
  For H_o = 72 km/s/Mpc, t_H,o = 14.0 Gyr     a ballpark figure for the age of the Universe.

• Being physics graduates, we are often nervous when time is introduced:
  does it depend on our location or motion, i.e. is it different for different observers?

  In this case, you can relax: the time we've introduced is a proper time
    it is measured by inertial observers, and can be agreed upon by everyone.

• In fact, the cosmological principle, coupled with global expansion, guarantees a universal time:
    fundamental observers may synchronize clocks when the local density reaches some value
    since these observers are at rest w.r.t. their local frame, they measure proper time

• If we start the clocks at the Big Bang (ie set t_BB to 0 at = ), we have cosmic time:
    we can all agree on the age of the universe, and the time/age at a given redshift: t(z).

• Olbers (~1820) is credited with first noticing that a dark night sky has cosmological implications.
  Imagine an infinitely large, old, static, homogeneous universe.
  Every sight line intersects a star's surface, whose surface brightness is independent of its distance
    hence the entire sky should shine with the surface brightness of stars!
  However it doesn't, so one or more of the assumptions must be wrong.

  (Absorption doesn't help: dust ultimately heats to reach equilibrium with the radiation field).

• Let's use modern numbers to quickly assess how this paradox might constrain cosmology.
  The mean local cosmic luminosity density is   _L ~ 10⁸ L_B, Mpc^-3 with M/L ~ 10   [Topic 4.3]
    _L ~ 10^-32 erg s^-1 cm^-3;   n ~ 10⁸ Mpc^-3;   ~ 10^-31 gm cm^-3  ( ~ 0.01);   ~ R²

• There are several ways to cast these numbers, all confirming the Universe's emptiness & darkness:
  1. mfp = 1 / n ~ 10¹⁸ Mpc :   a typical sight line terminates at an enormous distance.

  2. the sky brightness, J, out to distance D is   _L r² d dr / 4r² = _LD/4   erg s^-1 cm^-2 sr^-1
     for D_CMB (now) ~ 40 Glyr we find J ~ 10^-4.5 erg s^-1 cm^-2 sr^-1   µ ~ 24^m arcsec^-2 ie very dark!
     to get J ~ 10¹⁰ erg s^-1 cm^-2 sr^-1 (5000K black body) we need D ~ 10¹⁸ Mpc (as before).
     Even _L ~ 0.07 L_V, pc^-3 for the MW disk, requires D ~ 10⁹ Mpc to reach Olbers' bright sky.

     Clearly, given its current emptiness, a dark sky only rules out gigantic old static Universes.

  3. A 5000K radiation field has energy density u ~ 30 erg cm³.   Can stars generate this?
     At _L ~ 10^-32 erg s^-1 cm^-3 it would take them 10²⁶ years !

     A bright sky requires our static universe to be both gigantic and immensely old.

  4. But real stars cannot shine that long anyway, nor can they create so much light:
     Converting all baryons to light:  u = 0.04 _crit c² ~ 10^-8 erg cm^-3, well below what is needed
     Stated differently: in a bright sky universe the radiation alone would contribute   _rad ~ 10⁹ !

     Stars alone cannot yield Olbers' bright sky, even in an old enough and big enough Universe.

• There is, of course, another source of radiation: the 3000K photosphere of the CMB.
  The omnidirectional CMB is uncannily like the classic Olbers' bright sky, except for one detail....
  Redshift kills its surface brightness by a factor (1 + z)⁴ ~ 10¹²   [sec 8d]
  In the absense of redshift, we would indeed witness a lethal sky with J ~ 10⁶ Watt m^-2 sr^-1
  (roughly equivalent, of course, to filling the sky with solar disks, as Olbers' originally conceived)

  For this light source, then, it is expansion which keeps the sky dark.

• Much of this boils down to the fact that the Universe is not in thermodynamic equilibrium.
  Hardly surprising, since:
  
  1. adiabatic expansion coupled to graviational instability leads away from LTE.
  2. this yields a highly inhomogeneous distribution of matter, with huge temperature variations;
  3. interaction/equilibration times are often much longer than the Hubble (expansion) time

  If Olbers had lived in the first 500 kyr, when the Universe was in LTE, there would be no paradox!

(3) More on Expansion

(a) Expanding Coordinate Grids

• When thinking about cosmic dynamics, it helps to think of an expanding coordinate grid
  Imagine a (flat) sheet of graph paper which is slowly getting bigger, carrying the grid lines with it [image].

• One can imagine three kinds of motion:
  1. Points (ink dots) in the paper participate in its motion. They are fixed on the grid.
     Such points are called fundamental observers: their motion is the Hubble flow.
     As already shown in [sec 2eii] they all witness the same Hubble law.
     In introductory texts, this is the expanding raisin bread analogy.

  2. Points (ants, if you like) can also "walk" over the paper, crossing grid lines.
     This motion is the peculiar velocity: it is a true space velocity.

  3. Now imagine a speedy ant which always runs quickly at a constant rate in a straight line
     this represents a photon, crossing the expanding space, always moving locally at c.
     

• From General Relativity, we know that in Cosmology we may need to consider a curved grid.
  For this, one imagines our dots and ants on an expanding sphere with longitude/latitude lines.
  Fortunately, it turns out that expanding flat graph paper is all that's needed for our Universe.

• First notice that the Hubble law preserves shapes:   patterns of galaxies becomes bigger patterns
    for a set of i points, cosmic expansion gives:   r_i(t) = a(t) r_i(t_o), for a fiducial time t_o
  Here, a(t) is a universal scalar function of time, and is called the scale factor
  a(t) simply tracks how the separation of galaxies changes over time

    Finding the form for a(t) is a holy grail in cosmology.

• Sensibly, we assign the current time special status: t_o = now;   a(t_o) = 1   and r(t_o) = r_o.
  Hence:
  1. the current values, r_o, provide the coordinate grid, and are called comoving coordinates
     as the grid expands, the comoving coordinates do not change
     
  2. at any time, the physical coordinate of an object is:   r = a(t) r_o
     
  3. by setting a(t_o) = 1, we ensure that in the past, a < 1 while in the future a > 1.

  For example, at the time of recombination, a 0.001
  The comoving distance to proto-M87 is still 15 Mpc, but its physical distance is only 15 kpc.

• Notice that r and r_o are both proper distances:
  they tell us how many non-expanding rulers fit end-to-end from here to the galaxy.

  Later we introduce several pseudo-distances: eg luminosity & angular diameter distance; D_L, D_A.
  these are not true (proper) distances, but convenient functions of distance.
  

• Warning:  symbol conventions for physical/comoving/pseudo distances varies greatly
  You must be careful when reading texts: "is this r or d physical or comoving?"

    In these notes I will try to be consistent: r = physical; r_o = comoving; D = pseudo.
  

• With our new expanding coordinate grid, how does H fit in?
  To find out, take time derivatives of r(t) = a(t) × r(t_o):
  

  dr/dt = v(t) = da/dt × r(t_o) = (1/a) da/dt r(t)

  But this is simply:   v(t) = H(t) r(t)   with H(t) = (1/a) da/dt
    we have found that the Hubble relation applies at all times

  H(a)    H(t)  =  1/a da/dt     and     dr/dt  =  v  =  H r

• In general, H(t) and a(t) both vary with time
  For the current epoch, we write H(t_o) H_o and it has units of inverse time
  Our best current estimate for H_o is [sec 3g] 2.34 × 10^-18 Hz (or 72 km/s/Mpc)
  

• At this point, we need to clarify something: there are two apparently similar relations.
  1. the theoretical "proper" velocity-distance relation:   v = H r
  2. the observational redshift-distance "Hubble Law":   cz = H D
  These are, in fact, somewhat different.

• The theoretical relation v = H r is globally exact, though it is observationally inaccessible:
  • both v and r are "proper" quantities, ie as measured in a local rest frame. For example
    for r:   how many non-expanding rulers must be laid down between us and the galaxy?
    after 1 second, v additional rulers must be laid down, where v = H r.
  • notice that the values are all measured at the same cosmic time:
    we deal with distant galaxies as they are, right now, not as we see them.

• The Hubble Law is strictly observational:
  • 1 + z = _obs/_em; and cz rather than v(z) is sometimes taken as a "Doppler velocity"
  • D is usually a luminosity distance, which matches the proper distance only at low z.
  • both z and D apply to the time when the light set out, not the current time
    during the light's journey, the galaxy moved further away and, possibly, slowed down
  At high z, several factors break linearity, indeed this deviation is used to measure q_o.

• At low redshift the Hubble Law and the velocity-distance relation look the same
  Cosmic expansion is best described by v = Hr; it is exact and holds everywhere at all times
  the Hubble Law, cz = H D, only provides imperfect observational access to this cosmic expansion.

• Let's push the velocity-distance law to great distances:
  For H_o = 100 km/s/Mpc, we have :
  at r = 10 Mpc,       v = 10³ km/s
  at r = 10 Gpc,       v = 10⁶ km/s
  at r = 1000 Gpc,   v = 10⁸ km/s

  Yes, these velocities are faster than light:   special relativity does not apply to this motion:
    it arises from expansion of space, not motion through space.

• Can we see the galaxies which recede faster than light?
  The light they emit moves through space at speed c towards us
  but over time the wavefronts get further from us, at speed v - c > 0     we will never see them!
  There is a critical distance r_H,o = c/H_o = 3.0 h^-1 Gpc which is now receeding at v = H_o r_H,o = c

  rH,o  =  c/Ho  is called the Hubble distance;   where, right now, galaxies recede at c

  For constant rate of expansion, we will ultimately see everything inside a sphere of radius r_H,o
  Only if v slows down significantly will we be able see beyond r_H,o.
  These, and other potentially confusing things, should become clearer later [sec 7].

• When we first encounter redshift, we view it as a Doppler shift due to a galaxy's motion
  while this is appropriate for peculiar velocities, it is not for expansion "velocities"
  

• A more mature view sees the light embedded in space, getting stretched on its way to us
  This kind of redshift is called a Cosmological Redshift   [image]

• With this view, redshift clearly measures the stretch factor between emission and observation

  o / e   =   1 + z   =   a(to) / a(te)   =   1 / a(te)

  since the current scale factor a(t_o) = 1
  This is a fundamental relation & globally exact
  It tells us the relative change in size since the light set out
  

  Some examples:

  at cz = 30,000 km/s, z = 0.1     a(t_e) = 1/1.1 = 0.909     90%
  at z = 1, a = 0.5   all galaxies were half their current separation (8 × the current density)
  at z = 6, a = 0.14   (currently, the most distant QSOs)
  at z = 1000, a = 0.1%   (at recombination, everything is 10³× closer; 10⁹× denser)

• Since redshift is so important, its interpretation has occasionally been questioned.
  e.g. "tired light"   -   photons simply lose energy during their long (and tiring) journey to us
  

  A nice observation verifies that the redshift is indeed caused by expansion:
  If a distant event has true duration , the object is v further away at the end of the event
  The light therefore arrives v/c late and we observe a duration t = + v/c (1 + z)
  This is called cosmological time dilation   (a detailed analysis changes to = )
  It has been observationally verified: distant SNIa have slower light curves by 1+z [image L 5.6]

The struggle to measure reliable astronomical distances and H_o has a long history   [image].
Rather than consider that history, we shall briefly summarize the current (1995 - 2005) situation.
A good review is Mould's (1997) EAA article (quoted H_o values are taken from this article).
• To derive H_o one must measure both distances and redshifts
  • Distance measurements take two rather different forms:
    • ladder methods which depend on nearby calibration such as parallax & Cepheids
      
    • one shot methods which need no calibration and rely only on known physical properties

  • Redshifts, while easily measured, need correction for peculiar motion:
    
    • our motion includes: the sun's orbit; MW motion in local group; Virgocentric infall [Topic 15]
      
    • the target galaxy is chosen to be a group/cluster member, & the group's mean redshift is used.

  (i) Ladder Method

  In elementary texts, the distance ladder is often presented with many rungs [image]
  In practice, there are really only three:
  1. Use the Hipparcos satellite to get trigonometric parallaxes of nearby Cepheids
       calibrate Period-Luminosity (PL) relation

  2. Use HST to get Cepheid distances to nearby (25 Mpc) galaxies
       calibrate Tully-Fisher (TF) & Fundamental-Plane (FP) (& other) methods

  3. Use TF, FP (& other) distances to groups where peculiar velocities are unimportant.
       group mean redshifts & distances now give H_o

  Let's outline these various steps & techniques.

• Cepheid variables

  This class of pulsating stars defines a tight period-luminosity(-color) relation [images]
    measure period to get luminosity and hence distance

  they are luminous stars (M: -8 to -12?) and hence can be seen to considerable distances (~25 Mpc by HST)
  however, they are also rare:   so there are only ?? within ??pc, which tend to be of low luminosity.

  Historically, the PL relation was calibrated by Main Sequence fitting to open clusters containing Cepheids
  Now, Hipparcos provides direct trigonometric calibration (eg Perryman et al 1997)
  however, this calibration still needs to be improved   (eg using future astrometric missions SIM, GAIA).

  The distance to the LMC plays a very important role     (and also still needs to be improved)
    it contains enough Cepheids to define the PL relation in m (not M)
    hence extragalactic Cepheids yield relative distances to the LMC
  the current best estimate for the LMC is:  m-M  =  18.50  0.13    50  3.2 kpc   (uses E(B-V) = 0.1)

  the HST Key Project has now measured ~?? Cepheids in ~?? galaxies out to ~25 Mpc.
  These galaxies were then used to calibrate the following methods:

• TF:  Tully-Fisher Relation

  This is a luminosity-linewidth relation for spirals   [Topic 5.4b]
  scatter is minimum in the near IR (I, H), hence the method is often referred to as "IRTF"
  about 20 spirals now have Cepheid distances
  about 25 groups/clusters out to 10,000 km/s have TF distances
    H_o = 71  8   (eg Sakai et al 1999)

• FP:  Fundamental Plane Relation

  This is a refinement of the luminosity-linewidth (Faber-Jackson) relation for ellipticals   [Topic 7.4b]
  Either D_n- (isophotal diameter/dispersion) or surface brightness/radius/dispersion relations
  since no Cepheids in Es, calibration uses Es in groups with Cepheid distances (eg Virgo, Fornax, Leo)
  many groups/clusters out to 10,000 km/s now have FP distances
    H_o = 78  10   (eg Mould et al 1996; Kelson et al 1999)

• SNIa:  WD binary thermonuclear detonation

  These are very luminous, so well suited to q_o studies (high z), but also useful for H_o (lower z)
  the light curves aren't all the same; but peak luminosity correlates with fading rate (and color) [image]
  unfortunately, very few SNIa have ocurred in galaxies with Cepheid distances     calibration not ideal
    H_o = 68  6   (eg Gibson et al 1999)

• SBF:  Surface Brightness Fluctuations

  Consider a set of CCD pixels recording the light from an E galaxy, each one with perfect S/N ratio
  there is still variation between the pixels because of N fluctuations in # stars
  Although the mean surface brightness is independent of distance, the variation is not
    nearer galaxies have fewer stars per pix     larger variation.

  difficulties: contamination by globular clusters; color/population dependency; calibration.
  HST can use this method out to about 7000 km/s
    H_o = 69  7   (eg Ferrarese et al 1999)

• HST Key Project  combines all these methods (plus GC & PN luminosity function methods)

    H_o = 72  5?   km s^-1 Mpc^-1   (eg Friedman et al 2002?)   [image]

  (ii) Direct Methods

• EPM:  Expanding Photospheres Method   (originally from Baade & Wesselink)

  This method applies to all pulsating/expanding photospheres -- particularly Type II (core collapse) SN
  angular size is derived from flux, temperature and emissivity (black body = 1)
  linear size is derived from integrating velocity (linewidth) over time
    distance, by comparing angular and linear sizes.

  Fortunately (!), EPM distances agree, statistically, with Cepheid distances
  EPM distances now available for SN II out to 14,000 km/s (check)
    H_o = 73  11   (eg Schmidt et al 1994)

• VLBI Masers in Nuclear Gas Disks

  So far, only one good example of this method exists:   NGC 4258,   Miyoshi et al 1995   [ Topic 13.4e]
  a compact (~1pc) molecular disk orbits central black hole
  VLBI of H₂O masers gives (Keplerian) velocities and proper motions
    distance, by comparing linear and angular velocities   (??  ?? Mpc)

  this method has good potential for future (more distant) objects (eg at z > 0.5 it would give H_o and q_o).

• Gravitational Lensing Time Delays

  2 QSO images have different light paths with different physical lengths
  this path difference is given by the time delay between QSOs light curved (via cross-correlation).
  the calculated path difference depends on projected mass density and linear scale
    distance by comparing observed angular scale and calculated linear scale
  About 10 now done
    H_o  -  60 - 65   (puzzlingly low).

• SZ:  Sunyaev-Zeldovich Effect in Clusters

  Hot electrons in galaxy cluster ICMs do two things:

  1. they generate X-rays via bremsstrahlumg:   L_x    n_e² r_c³ T_x^1/2
  2. they Compton scatter CMB photons:   (T/T)_CMB    n_e r_c T_x

  you can solve for r_c and compare with _c to get a distance
    H_o = 60  - 65   (eg Birkinshaw 1998) also puzzlingly low.

• Possible Concern:

  Why do the more distant (lensing & SZ) methods seem to give systematically low values for H_o?
  Perhaps we live in a void with higher local H_o than the global value?

  The answer is "probably not", for several reasons:
  

  • the most distant TF work is now out to 15,000 km/s  (200 Mpc) which is hardly local
  • the Hubble relation is linear from 100 to 1000 Mpc
  • from CMB anisotropies, the incidence of voids of size 10⁴ km/s is quite rare

    The local value is probably within a few percent of the global value.
  Why the more distant estimates seem to yield low values is not yet understood.

• Currently Favoured Value of H_o :   72  5   km s^-1 Mpc^-1

  Spergel et al (2003) used this HST Key Project value for their WMAP concordance model.
  Many people now adopt this as the (currently) favoured value.

  [images: D1.6; D1.5; OU7.2]
  also need to say whether virgoflow infall corrections are used or not...

(4) Components, & Their Densities & Pressures

• The current universe contains (we think) just five components:
  1. Dark Energy
  2. Dark Matter
  3. Baryonic Matter
  4. Photons
  5. Neutrinos

• You should not think of these as "merely" components: like furniture sitting in a room
  In general relativity, the contents help to define the space-time in which they reside
  (the furniture actually influences the shape and size of the room!)
  

• To understand the global properties, we therefore need a complete itinerary of the components
  
  • Most important, we need to know their densities:
    
  • In moving from Newtonian to Einsteinian gravity, we also need to know their pressures: p
  For several components, and p are both additive: _tot = _i   and   p_tot = p_i

• Although it looks like we're introducing two quantities here, we aren't:
  An equation of state (EOS) tells us how pressure depends on density: p = p()

• As the universe expands, the density and pressure of each component changes
  Fortunately, we know how and p vary, using local conservation of energy + the EOS.

• All we need, therefore, is to measure the density and pressure at one time
  Obviously, the most convenient time is the present.     The rest, as they say, is history.

• Let's first see how plays its role
  We will need a few equations which we justify later on [sec 6a], so just accept them for now.
  Here's the cosmic energy (Friedmann) equation, using scale factor "a"   [sec 6a iv]
  (da/dt)²   - (8G/3)a²   =   -kc²/R₀

  This matches the basic form KE + PE = E_tot, with E_tot appearing as a spatial curvature:
  k = +1     -ve E_tot     closed geometry   (bound system)
  k = -1     +ve E_tot     open geometry   (unbound system)
  k =   0     zero E_tot     flat geometry
  

• Hence, we can define a critical density _c which yields a flat geometry (k=0):

  (1/a2)(da/dt)2   =   H2   =   (8G/3) c     giving   c   =   3H2/(8G)

  _c   =   2.65 × 10^-7 h² Mpc^-3   =   1.80 × 10^-29 h² gm cm^-3   =   10.8 h² m_p m^-3
  

• Notice _c varies with time through H(t) (or h), and we'll write its current value as _c,o
  However, once set, k (-1,0,+1) is fixed for all time.
  In particular, if k = 0 now (flat geometry) then we always have a flat geometry
    if _o = _c,o now, then = _c always   (worth stressing, since this seems to be the case).

• With such tiny values, it is convenient to express densities relative to _c using / _c

• The table lists the measured current relative densities for the various cosmic components
  (values are for the concordence model, ~10% uncertainties)
  (other functions are given which we'll come to in a moment)

  Component	/ c,o	w	x = -3(1+w) ax	x = 2/(3+3w) a tx	1 + 3w sign accel
  Dark Energy	0.73	-1	0	a et	-2
  Dark Matter	0.23	0	-3	2/3	1
  Baryons	0.04	0	-3	2/3	1
  Photons	5.0 × 10-5	1/3	-4	1/2	2
  Neutrinos	3.4 × 10-5	1/3	-4	1/2	2

• The current best estimate for the total density is _tot = _i = 1.00 0.02

    we live in a universe with "flat" spatial geometry

  This is one of the most important discoveries in recent cosmology.

• Warning: please don't continue to view _tot as defining the future of the Universe,  it doesn't.
  this was appropriate in the pre-lambda days ( = 0), when it was common to state:
  if _tot > 1, the Universe will turn around, collapse, and end in a big crunch.
  if _tot < 1, the Universe will expand forever.
  with 0, one cannot infer the future simply from _tot.
  Instead, _tot only fixes the spatial geometry (open/flat/closed), not the future.

• Why is pressure relevant? Not because "pressures push", they don't, only pressure gradients push.
  It is because pressure is fundamentally momentum flux density: it arises from internal motion
  just as motion of charges make B fields, so motion of masses make gravitomagnetic fields
  if the motion is omnidirectional (e.g. a gas), the result is simply an additional gravitational force
  if the motion is coherent (a rotating mass) the force has direction     frame dragging.

  In slightly more technical terms:
  Einstein's G_µ = -8G/c² T_µ states: space-time geometry = energy-momentum distribution.
  the µ indices are 0,1,2,3 (ct,x,y,z), and T_0,0 alone covers the energy (including rest mass)
  the other three components are for momentum of which pressures are a form

• Equations of State (EOS) tell us how pressure depends on density
  Here on earth, equations of state can be quite complex, although gases are simple: p = kT/µ
  In cosmology we need a relativistic equation of state, where contains both rest mass and energy
  Fortunately, the cosmic fluids all have simple forms:   p = w × c²
  (the constant "w" is itself often referred to as "the equation of state")
  w tells us what fraction of the total energy density is in the form of pressure
  (recall, the units of pressure are energy density: J m^-3 or erg cm^-3)

• It is useful to recall that sound speed c_s² = dp/d = w c²   (true for relativistic & non-rel. fluids)

• There are some limits on w
  • w 1 keeps sub-luminal sound speed: c_s < c   (w = 1 refers to "stiff matter", with c_s = c)
  • all known matter satisfies the "dominant energy condition" :     |p|
  Together, these limit plausible values of w to   -1 w 1
  

• As we shall see, there are three relevant values of w:

  w  =  0	this is called "dust" (for historical reasons), matter with zero pressure.
  w   0	non-relativistic matter (present day baryons, CDM)
  w = 1/3	relativistic matter (photons, neutrinos)
  w = -1	vacuum energy (the cosmological constant is also described by w = -1)

• Do these jive with our prior experience of "normal" equations of state?
  A perfect gas has equation of state p = nkT and adiabatic index
  its internal energy density u = nkT/( - 1) and its pressure p = ( - 1) u  = nkT
  (recall, is closer to 1 when particles themselves store internal, rot/vib, energy)
  e.g. a monatomic gas has = 5/3 with u = 3/2 nkT and p = nkT
              a diatomic gas has = 9/7 with u = 7/2 nkT and p = nkT

  Let's quickly recover pV = const.   and w    0   for our perfect (non-relativistic) gas
  Consider U = uV and adiabatic expansion dQ = 0 = dU + p dV, so that dU = -pdV.
  Multiply p = ( - 1) u by V to get pV = ( - 1) U, and then differentiate:
    pdV + V dp = dU( - 1) = -pdV( - 1)     dp/p = -dV/V     pV = const

  p = nkT   and   3/2 kT = ½ m<v²>   so using (rest) density _o   we have p = (_o/m) (m <v²>/3)
    p = _oc² × <v²>/3c²   so w = <v²>/3c²   0 for a non-relativistic gas.

• For a photon gas, ₀ = 0 and = 4/3, so -1 = 1/3 and we find:
  p  =  ( - 1) u  =  1/3 u = 1/3 (c² - ₀c²)  =  1/3 c²     w = 1/3   as we had before.

• Finally, for the expansion of a vacuum with constant density & pressure, we have:
  dU = -p dV         _vc² dV = -p dV         p = - _vc²         w = -1

  Basically, you must provide energy to create more vacuum; ie do work to increase the volume
  Normally, of course, gas does work on the surroundings, reducing its internal energy.

Let's now look briefly at the 5 cosmic components

(i) Dark Energy

• This component has had a checkered history:
  • Famously introduced in 1917 by Einstein to balance matter & yield a static universe
  • Dumped in 1929 when Hubble discovered cosmic expansion
  • Resurrected in 1930s-40s when large H_o implies t_H < t   (loitering model)
  • Dumped in 1960s-70s when H_o revised down, giving t_H > t
  • Resurrected in 1980s when t_H < t_GC; & _M ~ 0.3 but inflation suggests _tot = 1
  • Discovered in 1998 with accelerating expansion: q₀ = ½_M - _v < 0
  • Here to stay (2005)? Now 3 independent & consistent confirmations.

• Dark Energy is generic term; there are three main possibilities:
  A "cosmological constant" appearing in Einstein's equations as an integration constant, .
  Quantum mechanical energy of the vacuum. Both vacuum energy and have w=-1
  Quintessence: a strange new form of matter with -1 < w < -1/3, possibly evolving over time.
  We dont yet know which of these dark energy is:
  Acceleration only demands w < -1/3, while current observations constrain w < -0.8
    vacuum energy is currently favoured (which can be treated as a term)

• Current theoretical understanding is ~zero
  a "natural" energy density has _vc² = m⁴/(hc)³   for "natural mass" m
  if m = Planck mass = (hc/G)^½     _vc² ~ 3 × 10¹²⁶ eV cm^-3   (~10⁹³ gm cm^-3   !!)
  Observed _vc² ~ 10³ eV cm^-3, or ~10¹²³ × smaller; maybe the worst guess ever.
  The particle mass which does give the correct value is m ~ 10^-3 eV. No known candidates.
  Studying DE in the lab is very difficult - only energy differences can be measured
  (The Casimir effect, for example, shows the vacuum is "live" but not by how much).

  (ii) Dark Matter

• Introduced by Zwicky in 1930s to explain high galaxy cluster dispersions. Then forgotten.
  Re-introduced in the 1970s to explain spiral rotation curves at large radii
  Increasingly important on larger scales: galaxies; binaries; clusters; large scale flows.
  Necessary to create large scale structure rapidly from CMB (baryons alone are inadequate).
  Combined methods yield _DM = 0.23 0.04

• Theoretical understanding ~zero, though some properties are constrained.
  • non-baryonic (ie neither protons nor neutrons); BBNS rules out high _b
    
  • non-relativistic after T ~ 1 MeV     cold dark matter (CDM)     clusters efficiently
  • interacts only via weak and gravitational forces (dark matter)
  • possibly made of "WIMPS" -- Weakly Interacting Massive Particles
  • WIMP candidates?   not known; maybe lightest supersymmetric particle.

• Current laboratory searches underway. Nothing yet found.

  (iii) Baryons

• In this context, baryons refers to neutrons, protons, and electrons
  Big Bang NucleoSynthesis (BBNS) and CMB power spectra together give: _b = 0.044 0.005

• Primordial H/He abundances are 75/25 by mass or 12/1 by number
  Since then, stars have converted ~2% (by mass) into heavier elements.

• Only 10% of baryons are luminous, ie are found in stars
  The other 90% is mosly in diffuse ionized gas surounding galaxies.
  The fraction of baryons in neutral form is tiny (TBD)

• Mass to light ratios help inter-convert:   _m = j_X × (M/L)_X   in band X   [Topic 1.3j]
  For the local universe, j_B 2.0 × 10⁸ h L_,B Mpc^-3   ( 9 Watt AU^-3)
  This yields several fiducial values:
  using _crit we have (M/L)_crit = 1400 h (M/L)_,B     7 million kg/Watt (very dark!)
  using _m = 0.27 _crit we have (M/L)_m = 375     close to some clusters
  using _b = 0.04 _crit we have (M/L)_b = 55     significantly more than most galaxies
  
  The universe is optically quite dark     on average, ~7000 tonnes/Watt.

  (iv) Photons

• The CMB is by far the strongest of all the cosmic background radiation fields [images]
  This is true for both number density, n ~ 410 cm^-3, and energy density, u ~ 0.26 eV cm^-3.
  It's energy density is even comparable to the photon and thermal components of the local ISM.

• The CMB spectrum spans 500µ to 1 cm, and peaks at 1.0 mm (B) or 1.9 mm (B)
  It is an exceedingly accurate black body, with T_CMB = 2.725 ( 0.002 K = 0.07%) [image]

• Summarizing the integrated energy and number densities for a BB radiation field, we have:
  [note: here a = 8⁵ k⁴/15h³c³ = 4 /c is the radiation constant and NOT the scale factor].

  Energy density	u  =  a T4  =  7.56 × 10-15 T4 erg cm-3      ucmb  =  4.17 × 10-13 erg cm-3  =  0.26 eV cm-3
  Energy flux	J  =  uc/4  =  caT4/4  =  T4/  =  1.80 × 10-5 T4 erg s-1 cm-2 sr-1       Jcmb  =  9.94 × 10-4 erg s-1 cm-2 sr-1
  Number density	n  =  a T3/2.7kB  =  20.3 T3 cm-3     ncmb  =  410 cm-3
  Number flux	N  =  nc/4   =  4.84 × 1010 cm-2 s-1 sr-1     Ncmb  =  9.78 × 1011 cm-2 s-1 sr-1

  (v) Neutrinos

• Like all particles, neutrinos were created out of energy in the early Universe.
  Since they are stable, they survive to today in great, though undetectable, numbers.
  There are three families, _{e   µ}   , each with particle/anti-particle pairs   six in all.

• They decoupled at t ~ 1s, z ~ 10¹⁰, kT ~ 1 MeV, and have travelled unscattered ever since
    Cosmic Neutrino Background (CNB) similar to, but much younger than, the CMB

• Since kT >> mc² ( 0?) at decoupling, they were relativistic "hot" dark matter.
  They behave like a relativistic gas ( = 4/3; w = 1/3) similar to the CMB.

• Two facts make the CNB slightly different from the CMB
  
  1. Shortly after decoupling the e⁺e^- pairs annihilate (at kT ~ 0.5 MeV)
     this puts energy & entropy into the CMB but not the CNB     T > T
     one can show [sec 12] that T = (11/4)^1/3 T   giving T = 1.94 K

  2. Neutrinos obey Fermi-Dirac statistics, whereas photons obey Bose-Einstein statistics.
     Summing over all six species, one can show [sec 12]:
     
     u_,tot   =   0.68 u
     n_,tot   =   9/11 n     =   335 cm^-3   (currently)

• The total energy density in 's and 's (after e⁺e^- annihilation) is:   u_rel = 1.68 u = 1.68 a T⁴
  The total relativistic contribution to is:   _rel   =     +     =   8.4 × 10^-5   (today)
  This is the value to use when evaluating the expansion rate during the radiation era.

• What if neutrinos don't have zero rest mass?
  Currently, kT = 1.67 × 10^-4 eV, so only neutrino masses larger than this change things
  For each species, one finds a current = m(eV) / 94 h^-2
  Hence, a mean mass of 8 eV per species will give _tot = 1 and close the Universe.
  Laboratory limits are presently well above this, though structure formation suggests is small.
  (note: neutrino oscillation experiments only give mass differences ~ 0.05eV, not masses)
  

• As the universe expands, how do the densities of the components change?
  The (too) quick answer is simply a^-3 (1 + z)³   ie, densities drop as the volume increases.
  However, since pressure also adds to density (_p = p/c²), we must include this too.

• Let's consider the expansion of a volume V a³
  it is filled with a component with energy density c² and equation of state p/c² = w
  The expansion is clearly adiathermal: as much energy enters as exits (homogeneity)
  It is also adiabatic: the expansion is ~isentropic because the CMB dominates the total entropy.

  Consider conservation of energy:  dQ = dU + pdV = 0 (since adiabatic)

  dU = -pdV     d(c²a³) = -p d(a³)     c²a³ d   +   3c²a³ da   =   -3pa² da
  

  d/da   =   (-3/a)( + p/c²)   =   -3(1 + w) /a     which has solution:

  = ₀ a^-3(1+w)   =   (1 + z)^3(1+w)     where once again we set a₀ = a(t₀) = 1

• From the above table we find:

  matter:	m(a)  =   m,o a-3  =   m,o (1 + z)3	as expected by "conservation of mass"
  radiation:	r(a)   =   r,o a-4  =   r,o (1 + z)4	since n a-3 and E a-1 from redshift
  vacuum:	v(a)   =   v,o  =   const	space is space

• Going to earlier times (smaller a), _m increases quickly, but not as quickly as _r
    radiation dominates over matter at earlier times

  Since _v is constant while the other densities are increasing
    the vacuum rapidly becomes irrelevant and one can ignore it for the first Gyr or so.

  More generally, the component with most positive w dominates at early times (radiation)
  the component with most negative w dominates at late times (vacuum)     [images]

• Quantitatively, we can identify times when the various component densities are equal:
  (a_eq and z_eq are easily evaluated, while t_eq requires the integral relations from [sec 6c v] )

  density match	condition	a @ equality	z @ equality	t @ equality
  v = m	0.73 = 0.27 a-3	0.72	0.39	9.43 Gyr
  v = rel	0.73 = 8.4 × 10-5 a-4	0.103	8.3	615 Myr
  b =	0.04 a-3 = 5.0 × 10-5 a-4	1.25 × 10-3	800	620 kyr
  m = rel	0.27 a-3 = 8.4 × 10-5 a-4	3.11 × 10-4	3200	57 kyr
  m =	0.27 a-3 = 5.0 × 10-5 a-4	1.85 × 10-4	5400	22 kyr

  Note that here _rel refers to the sum of photons and neutrinos (relativistic matter)
  Likewise _m refers to the sum of baryons and CDM (non-relativistic matter)

• In general time dependencies are complicated [sec 6c].
  However, for flat geometries with a single component, we find simple forms for a(t)
  From the cosmic energy equation, with k = 0, _o = _c,o and = _o a^-3(1+w), we have
  (da/dt)²  =  (8G/3)a²  =  (8G/3)a²_o a^-3(1+w)  =  H_o² _o/_c,o a^-3(1+w)  =  H_o² a^-(1+3w)

   da/dt   =   H_o a^-(1+3w)/2     a^(1+3w)/2 da   =   H_o dt     a   =   [(3 + 3w)/2 . t/t_H]^2/(3+3w)

• Now, as it happens the universe is flat, and there are times when just one component dominates:

  during the radiation era:	a     t1/2
  during the matter era:	a     t2/3
  during the vacuum era:	a     et   (from da/dt a)

• The function a(t) from zero to the present indeed matches these simple relations [ see image and sec 6c iii ]

• In Newtonian gravity, Poisson's equation relates the gravitational field to the matter:
  
  ²   =   -4G

  In General Relativity, a similar relation gives the acceleration:

  (1/a)d²a/dt²   =   -(4/3)G ( + 3p/c²)   =   -(4/3)G (1 + 3w)

  Notice that the source term for gravity is not just , but it includes a pressure term.
  This comes from internal momentum associated with particle motion, and it adds to gravity [sec 4c].
  

• Since acceleration is (1 + 3w), then if w < -1/3 we find that gravity is repulsive!
  Current observations find an accelerating expansion, requiring a component with w = -1.0 0.1.
  There is presently a strong preference for w = -1 : vacuum energy or a cosmological constant, .
  

• Note that _v is +ve and adds to gravity, while p_v is -ve and dominates, giving net acceleration.

• What is -ve pressure?   It is tension: e.g. stretched rubber; I pull on it to increase its volume
  For +ve pressure, expansion does work on the surroundings internal energy decreases
  For -ve pressure, expansion requires work from the surroundings internal energy increases
  The vacuum, with constant energy density _vc² behaves the same way
  To expand a vacuum, I must provide the energy for the new volume.

  An extreme example:   imagine a strange piston with a little "strange water" in it.
  You pull extremely hard, with force F = p x A, with p = 9 x 10²⁰ dyne cm^-2   (~10¹⁵ atm)  and A = 1 cm²
  the piston slowly moves out by d = 1 cm -- you have spent F x d = 9 x 10²⁰ erg of energy.
  To your surprise, the piston now contains an additional cm³ of water!
  Your 9 x 10²⁰ erg were converted to 9 x 10²⁰/c² = 1 gm of new water.
  (Note: since dark energy is 6.8 x 10^-30 gm cm^-3, it only requires 6 x 10^-9 dyne cm^-2 tension to create.
  However, you can't verify this experimentally because there is vacuum on both sides of the piston!).

• We can (roughly correctly) intuit why this leads to accelerated expansion in the following way:
  First, forget your tendency to think that "tension" should "pull" -- there are no gradients here.
  Instead, consider a large volume which is increasing, and focus on the difference between:
        the energy required to make more volume (because there is now more vacuum), and
        the energy released by the gravitational binding energy, U_grav = - GM²/R , which gets more negative.

  For normal matter, expansion spreads mass out, raising the gravitational energy
  The loss of energy to the gravitational field comes out of the kinetic energy of expansion -- it decelerates
  For radiation, the situation is even worse, since expansion affects both M and R in the gravitational term.
  But for vacuum, the M² term beats out the R term, and the gravitational energy is more negative
  (one detail: this only becomes true for regions larger than the Hubble volume).

  The whole thing is not unlike a ball falling downwards:
        by moving down, R is smaller, so U_grav is more negative.
        to conserve energy, the ball must pick up positive kinetic energy, and hence it accelerates.
  For a vacuum dominated universe, a larger universe has more negative binding energy (G M²/R increases)
  Some of this gravitational energy is used to make the additional vacuum.
  Where does the rest go? Into the kinetic energy -- the fluid accelerates
  Thus, a vacuum dominated universe "falls downwards" by expanding and accelerating.

(h) Cosmic Cooling

Consistent with your intuition, cosmic contents cool with cosmic expansion
Ultimately, this occurs because particles move into receding regions so seem to have lower energy
There are two situations to consider: photons & matter (strictly: relativistic & non-relativistic).

(i) Photon (& Neutrino) Cooling

• The relativistic content includes both photons and neutrinos: both have a black body spectrum
  Expansion preserves the black body shape but at a new temperature: T_z   =   T_o(1+z)   =  T_o / a
  Let's quickly show that this is the case:
  • Currently, we have a = 1; T_o = 2.725K; & BB photon number density at frequency _o:
    n(_o) d _o   =   (8/c³) _o² [exp(h_o/kT_o) - 1]^-1 d _o

  • At a time when a 1, the corresponding frequency is = _o / a,   giving  d = d _o / a
    conserving photon numbers gives their density change: n() d   =   n(_o) / a³ d _o
    Inserting these into the current BB spectrum, we get:
    a³ n() d   =   (8/c³) ² a²[exp(ha/kT_o) - 1]^-1 a d ,   giving
    n() d   =   (8/c³) ² [exp(h/kT_oa^-1) - 1]^-1 d

  • This is another black body spectrum, with:

    T = To / a   =   To (1 + z)

• This has been verified using absorption in the CI fine structure lines in damped Ly systems:
  e.g. at z = 1.8, the CI line ratios imply T_CMB ~ 7.6 K,   as expected.

  (ii) Matter (& Galaxy) Cooling

• A similar analysis can be applied to the cooling of non-relativistic matter particles
  There are two key differences:
  1. The distribution is Maxwell-Boltzmann, in which T < v² >, while for BB: T < >
     
  2. The number density is not set by the temperature, but enters as a separate variable.

• First let's derive the matter equivalent of the redshift relation = _o / a,   ie v = v_o / a
  In time dt, a peculiar velocity v traverses a distance r = v dt
  at this new location, the peculiar velocity is reduced below v by the Hubble flow:
  
  dv = -Hr = -H v dt = (-1/a)(da/dt) v dt = -(v/a)da     dv/v = -da/a   with solution:
  v a^-1 or, for v = v_o at a = 1, we have   v = v_o/a
    as time passes a increases & v decreases; hence random motions decrease and T drops

• A gas of particles in thermal equilibrium has a Maxwell-Boltzmann (MB) velocity distribution:
  n(v_o)dv_o = 4N_o v_o² (m/2kT_o)^3/2 exp(-mv_o² / kT_o) dv_o     (N and n in number per cm³)

  at time when a 1, we have v = v_o/a   and   dv = dv_o/a
  particle conservation also requires   n(v)dv = n(v_o)/a³ dv_o and N = N_o/a³
  Substituting these into the MB relation, we get:

  a³ n(v)dv   =   4 a³N a²v² (m/2kT_o)^3/2 exp(-mv²a² / kT_o) a dv
  n(v)dv   =   4 N v² (m/2k(T_o/a²))^3/2 exp(-mv² / k(T_o/a²)) a dv
  which is another MB distribution with temperature:

  T  =  To / a2  =  To (1 + z)2

• Thus, a gas undergoing passive cosmic expansion cools according to T   a^-2
  which is faster than the CMB photons for which T   a^-1
    At z ~ 1100 both have T 3000K; at z = 0, T 2.725 K, while T_matter 2.48 mK.

  In practice, this cooling for the baryonic gas never occurs:
  

  • prior to recombination, the gas is coupled to the radiation and follows its temparature
    
  • after recombination other processes (virialization shocks, stars, AGN) heat the gas.

• However, one can consider the peculiar motion of galaxies in a similar manner
  Cosmic expansion is continually "cooling" their peculiar velocities, while
  gravitational interaction is continually heating them (e.g. cluster virial motions).

(5) Cosmic Geometry: Robertson-Walker Metric

With all these preliminaries now in place, we are ready to begin a full description of the Universe.
• General Relativity provides a framework for describing both geometry & dynamics
  Einstein's equations of GR can be written in a deceptively compact form:   G_µ = -8G/c² T_µ

• G_µ & T_µ are both 4 × 4 matricies, suggesting 16 equations, though symmetries reduce this to 10.
  The indices µ, = 0,1,2,3 denote 1 time & 3 space coordinates (e.g. ct, x, y, z   or   ct, r, , ).
  G describes the space-time geometry; it incorporates the metric.
  T describes the distribution of mass-energy (µ, = 0), and momentum (µ, = 1,2,3) in the system
  Hence the famous saying: "Mass tells space how to curve, while space tells mass how to move".

• The derivation and solution of the GR equations is a huge subject unto itself.
  Here we will extract only what we need.

  Loosely speaking, this section looks at the geometry part, G.
  The next section looks at the dynamical part, T, and how it relates to G.

• Mankind's attempt to formally describe Nature began with geometry: the properties of space
  Not surprisingly, Euclid's description arose out of our local "flat" space, characterised by:
  parallell lines never meet
  interior angles of a triangle sum to 180^o
  a circle has area r²; a sphere has area 4 r² and volume 4/3 r³
  This is the geometry we first encounter in grade school, with rulers and flat graph paper.

  Adding Newton's intuitively plausible independent time, t, yields a 4 coordinate space-time

• Since we evolved within a "flat" space-time, Euclid's/Newton's descriptions feel deeply TRUE
  However, they are not.
  Like cultural norms, their self-evident correctness merely reflects our parochial experience.
  Measure with great accuracy, visit a black hole, or traverse the Universe, and things are different.

• How do we describe other possibilities?   Using metrics
  A metric defines the separation, ds, of two nearby points (or events if we include time)

  (i) Example 1: "Flat" Geometry

• We call the Euclid space "Flat" because in 2-D it holds true on a flat surface.
  For 2-D and 3-D, we have metrics:
  ds²   =   dx² + dy²   =   dr² + r² d²     where ds² (ds)², etc.

  ds²   =   dx² + dy² + dz²   =   dr² + r² (d² + sin² d²)   =   dr² + r² d²

  Note that the choice of cartesian or polar (or any other) coordinate system is unimportant
  They are equivalent and define the same space.
  

• With these metrics, you can recover all the results of Euclidian geometry
  E.g. find the shortest lines between three points (minimize ds) to construct a triangle
    its interior angles sum to 180^o;   and so on.

  (ii) Example 2: Uniform Positive Curvature

• In 2-D, curved surfaces are everywhere (wineglasses, chairs, lampshades,....)
  Drawing triangles on them quickly reveals interior angle sums different from 180^o
  In general, sums greater/smaller than 180^o define positive/negatively curved surfaces [table 1].
  Such surfaces have metrics which are different from those above.

• Anticipating an isotropic universe, consider the simplest isotropic surface: a sphere of radius R.
  Take coordinates r, from a pole, where r is measured along the surface   [image]
  ds² = dr² + R² sin²(r/R) d²
  Close to the pole, (r << R) we recover the "flat" 2-D metric (ds² = dr² + r²d²)
  Further from the pole, curvature reduces the d² contribution to ds².

• Since "straight" lines provide the shortest distance between two points, they are all great circles
  A "straight edged" triangle is therefore a spherical triangle made of great circles.
  For such a triangle of area A, the sum of interior angles is ang = + A/R²
    any measured triangle allows you to obtain the radius of curvature, R
    for the sphere, all triangles give the same R: the space is homogeneous and isotropic
    the sphere has finite circumference (2R) and finite surface area (4R²).

• Now consider extending this metric to 3-D giving a (un-imaginable) hypersphere of "radius" R
  ds² = dr² + R² sin²(r/R) d²   where d² = d² + sin² d²,   r is a "straight" line from origin.

  To get a feel for the odd nature of this space, imagine holding a laser pointer with visible beam (r)
  Turn it through 1^o (d = 1^o):
  

  at r = 100m it sweeps sideways by 100 × (1/57.3) = 1.74m  (ds at fixed r = R sin(r/R)d)
  at r = n × 100m it should sweep n × 1.74m, right?   Wrong!
  as n increases, the sweep drops below the predicted linear relation   (ie R sin r/R < r)
  in fact, beyond r = R × /2 the sweep begins to decrease for larger r
  waving the laser wildly, its rays always pass through the remote location r = R.
  A friend located at this "antipode" would see a laser "star" zooming around the sky!

  A second mind-bending (ho ho) phenomenon:
  Imagine blowing up a huge balloon; starting in one direction it gets bigger and bigger.
  After some time, it fills half the volume, appearing to extend flat in all directions
  Keep blowing, it begins to bend back behind you, walls now approaching , getting more bent
  Finally, the balloon walls close in on you from behind, and you are inside the balloon!
  (To help with this, think about drawing ever bigger circles on the surface of a sphere).

  If you explored geometry living in this kind of space, you would not recover the Euclidean results
  In 1840 Gauss actually tried to measure the curvature of space by surveying big triangles   [image].
  Of course, his modestly accurate measurements only recovered the Euclidean value of 180^o
  But in principle he could have discovered the non-Euclidean terrestrial Schwarzschild metric.
  (in fact, R ~ 1AU, and a 30 km triangle deviates from 180^o by ~10^-8 arcsec).

• Like the 2-D analog, the spherical 3-D space also has finite extent ( R) and volume (2 ²R³)

  (iii) Example 3: Uniform Negative Curvature

• Replacing sin(r/R) by sinh(r/R) in the above metrics yields spaces of uniform negative curvature.
  Unlike the 2-D sphere, it isn't posible to construct a 2-D surface with this property,
  however, a saddle does have a region of isotropic negative curvature in its center.
  On this surface, "straight" edged triangles have ang = - A/R² which is less than 180^o
  Furthermore, in 3-D: two points can be infinitely far apart, and the total volume is infinite.

• This table summarizes the geometrical properties of the three spaces described above   [image] :

  Curvature	k	d2 coefficient	Parallel Lines	Triangle Angles	Sphere Area	Sphere Volume	Global Form
  Positive	+1	R2 sin2(r/R)	Converge	> 180	< 4r2	< (4/3)r2	Closed
  Flat	0	r2	Never meet	180	4r2	(4/3)r2	Open
  Negative	-1	R2 sinh2(r/R)	Diverge	< 180	> 4r2	> (4/3)r2	Open

  (iv) Adding Time

• It is straightforward to add an independent, Newtonian, time t, as additional coordinate.
  The 3-D + 1-t metrics for positive, flat, and negative space-times become:
  ds²   =   -c² dt²   +   dr²   +   R² sin²(r/R) (d²   +   sin² d²)

  ds²   =   -c² dt²   +   dr²   +   r² (d²   +   sin² d²)     (Minkowski space-time).

  ds²   =   -c² dt²   +   dr²   +   R² sinh²(r/R) (d²   +   sin² d²)

  One can also replace (d² + sin² d²) with d², the angle between the two events on the sky,
  and group them into a single expression, using S_k(x) = sin(x),  x,  sinh(x) for k = +1, 0, -1:

  ds²   =   -c² dt²   +   dr²   +   R² S_k²(r/R)   d²
  

  [ Note: I've adopted Peacock's notation for S_k, rather than Ryden's ]

• Notice that ds²     -c²d²,   where d is the Lorentz invariant proper time interval.
  Indeed, the second is the spacetime of special relativity, and is called Minkowski spacetime.

• Notice also that the flat metric involves sums of quadratics of coordinate differentials
    flat geometry is rooted in the Pythagoraean relation   (e.g. triangles have ang = 180^o).
  Although more general geometries are curved, many are locally flat   (e.g. in 3-D: a sphere)
    expanding the metric to first order at any location gives a quadratic form.
  Such geometries are called Reimannian and the spacetimes of GR are all of this kind.   Why?
  Because locally the Equivalence Principle demands a Minkowski spacetime of Special Relativity.

  As you can see, the three metrics above are all of this kind: as r 0, they are locally flat.

  Of course, the second derivatives do not vanish, and it is these that define the curvature.
  Physically, these give tidal forces which are apparent across regions of finite size.

  (v) Geodesics

• In plane geometry, straight lines have special status.   Why is this?
    because straight lines are the shortest distance between two points.

• In curved geometries, the shortest path between two points is called a geodesic
  E.g. on the 2-D surface of a sphere, geodesics are great circles
  (hence: "aphrodisiacs" are great circles passing through Africa, ho ho).

• In curved 3-D spaces, how do we find these "straight lines".
  1. One (impractical) way would be to stretch a thin piece of thread between the two points:
       it's minimum energy state would occupy the shortest distance.

  2. Alternatively (and more importantly) light beams travel in "straight lines".
     Fermat's principle ensures they take the path of least time the shortest path
      this occurs because longer paths interfere destructively (cf Feynman's little book: QED).

  3. Yet another way would be to follow the path of an object experiencing no external forces
     Recall Newton's 1st law:
       "objects free from external forces remain at rest or in uniform motion in a straight line"
     (its wavefunction behaves just like the light waves: longer paths interfere destructively).

• In the curved 4-D space-time of GR the same methods apply.

  Light moves along special null geodesics, so named because ds = cd = 0   (light never feels time)
  

  We can also use a freely moving object to define a "straight line".
  Recall, in GR gravity is not a force; instead mass curves space, and objects "follow" the space
  Einstein's version of Newton's 1st law becomes:
    free falling objects move along "straight lines" (geodesics) in the curved 4-D space-time.
  e.g. Earth's orbit is curved in 3-space, but follows a geodesic in the 4-D space-time near the sun.

• In FRW cosmology, there are two important contexts for geodesics:
  1. Fundamental observers (i.e. galaxies) "move" along geodesics in the expanding spacetime.
     These form a diverging non-intersecting bundle with common origin in the big bang.
     (This is known as Weyl's postulate (1923), and it ensures a legitimate cosmic proper time).
  2. Light crossing the cosmic space-time always moves along null geodesics (with ds=0).
     In practice, this means we only see things which lie on our past light cone
     (a light cone is a conic surface in a 3-D spacetime, in which the 4th space axis is omitted)

• More generally, the path through space-time an object takes is called its world line
  World lines need not be geodesics; for example if they are acted on by a force.
    you, for example, held up by the earth's surface, follow a world line, not a geodesic.
  Conversely, geodesics are all world lines: the path through space-time of a freely falling object.

• In the 1930s, Robertson and Walker (independently) considered metrics suitable for the Universe.
  The required metrics need to describe a space-time which:
  
  1. is always isotropic & homogeneous
  2. expands (or contracts) with time
  These two conditions limit the possibilities enormously

• Note that in 1930s, the assumption of homogeneity & isotropy was fairly bold
  But the algebraic advantages were so great it was the obvious/only option to pursue.
  The degree to which these assumptions are valid depends on scale:

  On large scales (> few Mpc): these assumptions are excellent [see sec 2a-c]

  the RW-metric works well, and the treatment is valid.

  On intermediate scales, where |/<>|   <   1 they are still useful:

  the RW-metric can be applied locally with parameters slightly different from the global ones
  this approach is integral to the analysis of the early stages of galaxy formation.

  On small scales, where /<>   >>   1 they are poor assumptions:

  local matter dominates the metric which does not expand (planets; stars; galaxies; clusters)
  electrostatic or quantum forces can also overcome the expansion (atoms, people, rulers)

• The form of the RW-metric is:   [warning: other equivalent forms exist; this one is common].

  ds2   =   -c2 dt2   +   a(t)2 [ dro2   +   Ro2 Sk2(ro/Ro)   d2 ]

  where r_o is the comoving proper distance (ie as measured today) to an object.

  This looks very familiar!

  it has the spatial form introduced above
  it has a scale factor, a(t), which tracks expansion (usually, a(t_now) is set to 1)

• The primary parameters of the RW-metric are:
  1. k = +1, 0, -1   specifies +ve, flat, -ve curvature, and the corresponding S_k
  2. R_o = current radius of curvature; units of length   (cf. Gaussian curvature: 1/R_o²) .
  3. a(t) = a time varying scale factor, a dimensionless function of cosmic time, t.
  Much of 20^th century cosmology was a struggle to find these three things.

• As the universe expands, its radius of curvature increases (just like a sphere).
  So why does the RW-metric only include the current radius of curvature, R_o, and not R(t)?
  In fact, R(t) is present, implicitly:
  1. one can show [6biii] that R(t) = a(t) R_o, and this factor is indeed present in the d coefficient
     
  2. r_o is a comoving radius, ie as measured today, hence r_o/R_o in S_k is correct at all times.

  Please don't think of R(t) as "the radius of the universe"; it is a measure of spatial curvature
  although for k = +1 it yields roughly the correct total volume, for k = -1 it is negative.

  Also, the limiting condition near k = 0 with R     is well behaved, since R sin(r/R)     r.

• It is important to recognise that the RW-metric is independent of General Relativity.
    it comes just from requiring isotropy & homogeneity at all times.
  GR enters by specifying how the curvature and scale factor depend on the universe's contents
  We'll turn to this topic in a moment, after briefly getting familiar with the RW-metric.

Let's just check a few simple results which we derived intuitively in sec 3:

(i) Proper Time

• What time do fundamental observers witness?
  For them,   dr_o = d = 0   and so   ds = -c²d² = -c²dt²   giving   t =
  As expected, fundamental observers experience a proper time, on which all can agree.

  (ii) Proper Distance

• At time t, how far away is a galaxy with current (comoving) coordiante r_o?
  by "how far away", we mean "what's the proper distance": how many rulers would span the gap?
  Between us and it is purely radial, so d = 0, and "at time t" means dt = 0, so the RW-metric gives
    ds = a(t) dr_o   and the total proper distance is r = ds = a(t) dr_o = a(t) dr_o = a(t) r_o
  As we suspected, the proper distance, r, is simply the current (comoving) distance scaled by a(t).

  (iii) Velocity-Distance Law

• Let's recover the velocity distance (expansion) relation:
  proper velocity, v, is the rate at which r increases:   dr/dt.   So, using r = a(t) r_o, we have:
  v   =   dr/dt   =   da/dt × r_o   =   da/dt × r/a   =   (1/a) (da/dt) × r

  which is our velocity distance law   v = H × r   with Hubble parameter   H = (1/a) (da/dt)

  (iv) Time Dilation & Redshift

• A photon emitted from a distant galaxy at r_e, travels to us with:   d = 0   and   ds = 0
  (it travels along a radial null geodesic);   since r_e is an unchanging comoving coordinate, we have:
  0  =  -c²dt²  +  a² dr_o²     c dt/a  =  dr_o     c dt/a  =  dr_o  =  r_o   which is constant

  Two photons are emitted at t_e and t_e + t_e and arrive at times t_o and t_o + t_o
  During the time t_e + t_e to t_o both photons are in flight and so dt/a for this interval is the same.
  But for the full trip dt/a is also the same, so the small start/finish contributions must be equal:

  t_e / a(t_e)   =   t_o / a(t_o)     t_o / t_e   =   a(t_o) / a(t_e)   =   1 / a(t_e)     (since a(t_o)     1)

  This tells us that the duration of any event we witness is dilated by a factor a(t_e)^-1
  A heuristic explanation of this dilation (simiar to a normal Doppler effect) was given in [sec 3e]
  As mentioned there, the dilation has been seen in the lengthened light curves of high-z supernovae
  

• Taking t_e and t_o to be the time between wavecrests of light, we have the cosmological redshift:

  t_o / t_e   =   _e / _o   =   _o / _e   =   (1 + z)   =   a(t_e)^-1

  Once again, it is best to think of redshift as a change in scale factor during the photon's journey.
  

  (v) Total Cosmic Volume

• What's the total volume of a closed universe?
  At comoving coordinate r_o and time t, we have a radial vector of proper length:   r = a(t) r_o
  however, swinging this radial vector by d only sweeps out a distance:   a(t) R_o sin(r_o/R_o) d
  the full circumference is:   2 a(t) R_o sin(r_o/R_o),   and the full area is:   4 a(t)² R_o² sin²(r_o/R_o)
  to get the volume, we must integrate this area over dr from 0 to the antipole at r = a(t) R_o
  V   =   4 a(t)² R_o² sin²(r_o/R_o) dr   =   2 ² a(t)³ R_o³

  as expected, the volume grows with a(t)³ and is close to the value for a 2-sphere of radius R
  for an open universe the integral diverges because (i) the area diverges, and (ii) r    

(6) Cosmic Dynamics: the Friedmann Equations

Homogeneity, isotropy and the Cosmological Principle alone have singled out the RW-metric as the only credible cosmological spacetime. However, the RW-metric still contains some arbitrary quantities: a(t), R_o, k. These quantities are determined by whatever generates the spacetime geometry. Einstein's General Relativity tells us exactly this:
G_µ   =   8G/c² T_µ   :   spacetime is curved by the distribution of cosmic energy & momentum.

It is now time to introduce GR so we can derive expressions for a(t), R_o and k
Fortunately (for you and me) the treatment will be heuristic, brief, and goal oriented.

(i) From Scalars to Vectors to Tensors

• There is a helpful progression of field equation analogs:   scalar vector tensor.
  1. Newton's theory of gravity describes how a scalar field arises from a distribution of mass:

     ²   =   -4G

     Clearly, GR aims to find a more mature "relativistic" version of this.

  2. A similar equation applies to a static distribution of charge, _q:

     ² V   =   -_q / _o

     However, if charge moves (current), then the Lorentz boost yields new (magnetic) forces.
     (Note, that the boost does not increase the charge itself)
     We now need a vector potential, A_µ and 4-current, j_µ to describe this.

     (c²² - ² /t²) A_µ   =   -j_µ / _o

     here, j₀ is the charge density _q while j_1,2,3 are the x, y, z currents

  3. Now return to mass/gravity:
     When mass moves (momentum) the Lorentz boost does two things:
     1. it makes a gravitomagnetic force
     2. it increases the mass density:   m = m_o

     This second aspect undermines a vector treatment, and a tensor treatment is necessary.
     Instead of a 4-vector mass/momentum current, we need a 4 × 4 mass/momentum matrix
     Of course, relativistically "mass" is superceeded by energy and we speak of
     T_µ:   the "energy-momentum" tensor (or "stress-energy" tensor).

     Here, µ and are four (1 time, 3 space) coordinate indices, (eg   ct, x, y, z;   or   ct, r, , )

  (ii) T_µv :   The Energy-Momentum Tensor

• T_µ has 4 × 4 = 16 elements, arranged in a square.
  For an isotropic fluid, all the off-diagonal (cross coordinate) elements are zero:   T_µ is diagonal.
  T_0,0 = E = c²   is the total energy density

  T_1,1, T_2,2, T_3,3 = < p_xp_x >c² / E   is the x-momentum density p_x the x-pressure (y, z etc)
  

  Now, pressure is isotropic, so p_x = p_y = p_z = p     (don't confuse momentum p with pressure p)
  So T_µ = diag (c², p, p, p)     (all off-diagonal elements are zero).

  (iii) G_µv :   The Curvature Tensor

• So much for T_µ, what about G_µ?
  On the LHS, it provides a description of the spacetime curvature in various "directions".
  In general, its components are complex combinations of 2nd order coordinate partial differentials
  (it is, afterall, replacing ² in the Newtonian Poisson equation).
  [The general curvature is the Reimann tensor of rank 4,   R_µ,,   with 4⁴ = 256 elements.
  Symmetry reduces this to the rank 2 Ricci tensor, R_µ  , and its scaler trace, the Ricci curvature, R.
  Finally, the curvature is expressed as:   G_µ = R_µ - ½ g_µ R  where g_µ are the metric coefficients]

• There are two very important constraints for the cosmological spacetime:
  1. Everywhere the spacetime is locally flat     the geometry is Reimannian.
     
     recall, this is ultimately rooted in the Equivalence Principle
  2. Isotropy ensures all off-diagonal elements are zero:   G_µ is diagonal
     
     furthermore, spatial isotropy demands   G_1,1 = G_2,2 = G_3,3

  Evaluating the elements for the RW-spacetime, one obtains:

  G_0,0   =   3/a² [k c²/R_o² + (da/dt)²]

  G_1,1 = G_2,2 = G_3,3 = -1/a² [ 2 a (d²a/dt²)   +   k c²/R_o²   +   (da/dt)²]

  (iv) Two Cosmological Equations

• Now equate these elements in G_µ to the same elements of T_µ & add the proportionality constant:
  G_0,0   =   3/a² [k c²/R_o² + (da/dt)²]   =   8G   =   8G/c² T_0,0

  G_j,j   =   -1/a² [ 2 a (d²a/dt²)   +   k c²/R_o²   +   (da/dt)²]   =   8G p/c²   =   8G/c² T_j,j

  Combining these, we arrive at two fundamentally important cosmic equations

  (da/dt)2   =   (8G/3) a2   -   k c2/Ro2	The Friedmann Equation, or Energy Equation
  d2a/dt2   =   -(4G/3) a (   +   3p/c2)	The Acceleration Equation

  where k = +1, 0, -1 follows the sign of the curvature radius R_o; and a(t) is the scale factor.

• Rather than treat these as two simultaneous equations, one can effectively reduce them to one:
  differentiate the Friedmann equation, substitute for d²a/dt², and rearrange:
  d/dt   =   -3/a da/dt ( + 3p/c²)     now divide by da/dt, insert p = wc² and rearrange:

  1/3(1 + w)  ×  d /   =   da / a     which integrates to give:

   =  _o a^{-3(1 + w)}     with = _o at a = 1

  This expression for (a) can now be inserted in the Friedmann equation, which then yields a(t)

  You will recall deriving (a) in [sec 4e] using conservation of energy dU = -pdV
  Einstein's equations implicitly contain energy conservation, which therefore also provide (a)
  Of course, it helps to choose an equation of state, p = wc², of particularly simple form.

  (v) A Newtonian Analog

• Often a Newtonian analysis is used to give insight into the Friedmann equation.
  So, let's follow tradition and repeat that briefly here:

  Consider a huge sphere uniformly but sparsely filled with rubble (technically: "dust") [image]
  Focus on a single rock at radius r_o:   it feels only interior mass:   M = 4/3 r_o³_o
  To follow the rock's motion, define radial coordinate   r = a(t) r_o   (track using a scale factor)
  Set the whole sphere expanding, with dr/dt = r_o da/dt   for our rock.
  Finally (see below) assume the whole sphere expands uniformly, maintaining uniform density.
  in this case the mass interior to our rock is constant:   and M = 4/3 r³

• What is the equation of motion for the rock?
  Its total energy (per unit mass) is the sum of its kinetic and potential energies:
  ½(dr/dt)²   -   GM/r   =   E   where E is the rock's energy at r

  This is just the equation for throwing a stone vertically upwards:
  if E > 0, then v > v_esc and the rock escapes; if E < 0, then v < v_esc and the rock returns.

  Rewriting this equation using the changing scale factor and density, we get:
  

  (da/dt)²   =   (8G/3) a²   +   E / r_o²
  which is basically the Friedmann equation with E / r_o² standing in for -curvature:   -k c²/R_o²

• Continuing; the cosmic acceleration equation is immediately apparent:
  the rock's acceleration is:   d²r/dt²   =   -GM/r²   =   -(4G/3) r
  incorporating the scale factor:   d²a/dt²   =   -(4G/3) a

  This would be identical to the GR equation if were replaced by ( + 3p/c²).
  Newton, obviously, misses the fully relativistic energy and pressure contributions to gravity.

• Let's now consider all the rocks, not just our test rock:
  At t_o set them all moving outward with speed v_o = H_o r_o :  further out moving faster.
  Rocks starting at smaller r_o have smaller E   =   ½v_o² - GM/r_o   =   ½ H_o² r_o² - 8/3 G r_o² _o

  Inserting E back into the Friedmann equation and cancelling the r_o² terms we get:
  

  (da/dt)²   =   (8G/3) a²   +   ½(H_o² - 8/3 G _o)

  which implies the following:

  1. For all rocks, a(t) is independent of r_o, and hence the entire sphere expands uniformly
     
  2. The velocity law, v = H r, is preserved at all times.
  3. The condition for critical expansion is:   H_o² = 8/3 G _o  or critical density _c = 3H_o² / 8G
  4. The time evolution is identical to an FRW Universe filled with simple pressureless matter.
  5. Extrapolating backwards, we infer a singular "explosive" origin for the expanding sphere.

• What about the total energy of the system?
  If we set the expansion parameter, H, by the critical density, we have a marginally bound system
  this should have zero energy when summed over all rocks.

  For PE, we just perform the standard integration for the PE of a uniform density sphere:

  PE   =   -G (4/3)r³ × 4r² × r^-1 dr   = - (3/5) GM²/R,   for R = outer radius.

  The KE we also get by integrating over the Hubble law, and substituing for H² = 8G/3:

  KE   =   ½ 4r² (Hr)² dr   =   (3/5) GM²/R

  Hence, the total energy of the sphere began and remains zero throughout.
  Later [sec 7d] we find a similar energy conservation result if we take R to be the horizon distance.

• The Newtonian picture also helps us understand the "flatness problem":
  it shows just how close to V_esc we must launch rocks if they are to "get up high" (ie make a big universe).
  However, let's postpone this analysis until we specifically look at the flatness problem in [sec 10a i].

• Finally, the Newtonian framework is extremely useful in studying the growth of perturbations
    denser patches expand more slowly than average so the density contrast increases
    these ultimately act as mini closed universes, and collapse to make stars & galaxies.
  We'll return to this analysis in Topic 19.

• Overall, then, the parallel with GR is is really quite close, though it does lack important details:
  
  The sphere is bounded, and hence not isotropic: there is a special location (the center)
  Making the sphere arbitrarily large still requires an even larger space in which to embed it.
  Since expansion is truely kinematic, velocities approach (and exceed?) c at large radii
  At very early times (small size) one finds v ~ c and black hole conditions.
  Gravity as seen as a force, which misses the link to the spacetime in which expansion occurs
  Nevertheless, the analogy can be helpful as a conceptual starting point.

Let's now draw together some useful cosmological parameter relations:

(i) Hubble Parameter:   H

• Scale factor:   a   =   (1 + z)^-1     (a = 1 now)

• Hubble Parameter:   H   =   (1/a) (da/dt)     (varies with time)
  Hubble Constant:     H_o   =   (da/dt)_o     =   72 × 10^-6 Myr^-1   (today's value, psm units)
  

  (ii) Density Parameters:   ,

• Critical density:   _c   =   3H² / 8G     (varies with time)
  Today's value:     _c,o  =   3H_o² / 8G   =   1.37 × 10^-7 M pc^-3

• Density parameters:     =   / _c   =   (8G/3H²)   (vary with time)
  Example of today's value:   _m,o   =   _m,o / _c,o   =   (8G/3H_o²) _m,o

• Summing gives the total density:   _t   =   _m   +   _r   +   _v
  Alternatively:   _t   =   _m   +   _r   +   _v     (all these vary with time)

• Including their dependence on scale factor [sec 4e] and inserting today's (known!) values gives:

  t(a) / c,o   =   m,o a-3   +   r,o a-4   +   v,o

  This is an exceedingly important relation:
  it gives the full density evolution as a function of a or z using today's measured values.
  [warning:  _t(a) / _c,o   is not the evolving density parameter,   _t(a)    _t(a) / _c ,   see sec 6cvii]

  (iii) Curvature Parameters:   R_o & _k

• What about an expression for the curvature radius, R_o?
  Start with the Friedmann equation and divide by a²:

  (1/a²)(da/dt)²   =   H²   =   (8G/3)   -   kc²/a²R_o²

  substitute for   =   3H²_t / 8G   to get     H²   =   H² _t   -   kc² / a²R_o²   which yields:

  R_o  =  (c/H_o) / [ k (_t,o - 1) ] ^½     [notice k (_t,o - 1) is always positive]

  This is as expected:   R_o       for _t,o     1
  Current estimates put _t,o -1 < 0.02, so   R_o   >   7 c/H_o = 30 Gpc.
  At earlier times, R = a R_o is smaller.

• It is also useful to express the curvature term as an effective density:

  _k     -kc²/R²H²   =   -kc²/a²R_o²H²     and     _k,o     -k c²/R_o²H_o²

  Inserting into the Friedmann equation we find: H²   =   H²_t   +   H²_k     1   =   _t   +   _k
  which is simply our Newtonian energy identity, normalized to KE:   KE = PE + E
  [Notice, _k and k have opposite sign: +ve _k means open geometry]
  

• Using _k allows us to recast the Friedmann equation in compact and soluble(!) form [sec 6ci] :
  1/a² (da/dt)²   =   H_o² [_t(a)/_c,o +  _k,o a^-2 ]   =   H_o² [ _m,oa^-3 +  _r,oa^-4 +  _v,o +  _k,oa^-2 ]

  where k,o is simply 1 - t,o   =   1 - (m,o +  r,o +  v,o)

• Finally, we can re-express R_o in terms of _k,o and the Hubble distance r_H,o = c / H_o :
  

  Ro = rH,o / | k,o | ½

  (iv) Deceleration Parameter:   q

• What about a parameter for acceleration/deceleration?
  A dimensionless parameter is:   q   =   -a (d²a/dt²) / (da/dt)²   =   - (d²a/dt²) / (aH²)

  To find an expression for this, start with the acceleration equation & divide by a:

  1/a (d²a/dt²)   =   -H² q   =   -(4G/3) ( + 3p/c²)   =   -(4G/3) (1 + 3w)

  so:
  q   =   (4G/3H²) (3H²/8G) (1 + 3w)   =   ½ (1 + 3w)

• In practice, we should sum over the various components, each with their own w:

  q   =   ½ [ _m  +  2_r  -  2_m ]   =   ½_m  +  _r  -  _v     at any time, or

  qo   =   ½m,o  +  r,o  -  v,o     today.

• Notice that q is defined to be +ve for deceleration
  In our pre-_v cosmology days (1970s-90s), this was always +ve, with q_o = ½ _o   (_r,o is negligible)
  But with a vacuum term, q can be either -ve or +ve depending on _m and _v
  Currently, with _m = 0.26 and _v = 0.73, we have q_o = -0.6 :   we are accelerating.

  (v) Cosmological Constant:  

• What about the cosmological parameter:  
  Recent fasion (& these notes) treat this term by simply adding a vacum component
  Historically, however, it first appeared as a possible term in G_µ :
  
  G_µ   =   R_µ   -   ½R g_µ   +   g_µ

• This propagates into the Friedmann & Acceleration equations:
  (da/dt)²   =   (8G/3) a²   -   k c²/R_o²   +   1/3 a²

  d²a/dt²   =   -(4G/3) a (   +   3p/c²)     +   1/3 a

  Since the new term appears with the same power of a, we can simply incorporate it in and p:
  this is done by defining in terms of _v, and discovering that w = -1:

  =   8G v   =   3 H2 v     (units of time-2)   and   v   =   -pv/c2     (i.e.   w = -1)

• Let's now use the Friedmann equation to study the time development of the scale factor:   a(t).
    basically, derive those expansion/contraction graphs we've seen since high school.

  The form of a(t) depends on two things:

  1. the curvature:   closed, critical, open,   which depends on _t
  2. the form of the density relation, (a), which depends on the fluid.
  A specific solution is called an FRW world model (FRW = Friedmann, Robertson, Walker)

  Let's begin with the general case, then look at some special cases.

  (i) The General Case

• Start with the Friedmann equation and re-express the density & curvature using 's   [sec 6biii]:

  1/a2(da/dt)2	=   (8G/3)  -  (1/a2) kc2/Ro2
  	=   Ho2 (8G/3Ho2)  +  (1/a2) Ho2 k,o
  	=   Ho2 [ / c,o +   k,o a-2 ]
  	=   Ho2 [ m,o a-3 +  r,o a-4 +  v,o +  k,o a-2 ]
  	=   Ho2 E2(a)     which gives:

  1/a (da/dt)  =  H(a)  =  Ho E(a)       for the evolution of the Hubble parameter

  where we use Peeble's notation for E(a) [or E(z)], and the curvature "density"  _k,o  :

  E(a)     [ m,o a-3 +  r,o a-4 +  v,o +  k,o a-2 ] 1/2

  E(z)     [ m,o (1+z)3 +  r,o (1+z)4 +  v,o +  k,o (1+z)2 ] 1/2

  k,o     1 - t,o   =   1 -  m,o -  r,o -  v,o

  These are exceedingly important functions, and are central to all cosmological calculations.
  Notice that they involve current (hence observable) values of , and so can be evaluated directly.

• We are now set to evaluate a(t), the time evolution of the scale factor, by simple integration:
  da/dt   =   H_o a E(a)   giving   da / a E(a)   =   H_o dt   =   H_o t     (da from 0 to a; dt from 0 to t)

  In general, this and related integrals need to be done numerically
  Since E(a) and its integral are often needed, it's good to have a working subroutine handy.
  (IDL, Mathematica, Matlab all have ready integrators; as does Numerical Recipes, eg qromb).

• Notice we have three related variables here: cosmic time, t; scale factor, a; and redshift, z.
  you can move between them with the following important relations between their differentials:

  dt   =   tH,o da / aE(a)   =   -tH,o dz / (1+z)E(z)     along with     da = -dz/(1+z)2 = -a2dz

  where t_H,o = 1 / H_o is the Hubble time; one can also use the Hubble radius   c dt = r_H,o da / aE(a)   etc.
  For example, the relation for current cosmic age is simply:

  dt   (0 to now)   =   -t_H,o dz / (1+z)E(z)   ( to 0)   =   t_H,o da / aE(a)   (0 to 1)

  Let's now look at the time evolution of some specific FRW world models.

  (ii) The Concordance Model

• The real Universe has several components; with current densities:

  v,o = 0.73;   m,o = 0.27;   r,o = 8.4 × 10-5;   t,o = 1.00     (k,o = 1 - t,o = 0)

  Integrating the general relation gives a current age 0.988 t_H,o and a(t) curve shown here: [image]

• In practice, each term is dominant over a certain range in a, giving three eras :
  during the radiation era:         E(a)    _r,o^½   a^-2
  during the matter era:             E(a)    _m,o^½   a^-3/2
  during the dark energy era:    E(a)    _v,o^½

  Taken individually, these yield reasonable approximations for a(t) over most of cosmic history.
  This [ image ] shows the approximations and their errors, while the table gives the functional forms.
  [The Toolbox includes a more extensive set, including Hubble radius, particle and event horizons.]

  Approximations to the Concordance model

  Radiation era:       E(a)    r,o½   a-2

  t   =   tH,o da / a E(a)   (0 to a)     ½ tH,o r,o-½ a2     741 a2 h72-1 Gyr   =   2.34 x 1019 a2 h72-1 sec

  a     1.16 x 10-6 h72½ tyr½     2.09 x 10-10 h72½ ts½

  Matter era:       E(a)    m,o½   a-3/2

  t   =   tH,o da / a E(a)   (0 to a)     2/3 tH,o m,o-½ a3/2     17.4 a3/2 h72-1 Gyr

  a     1.49 x 10-7 h722/3 tyr2/3

  Dark Energy Era:       E(a)    v,o½

  t - tnow   =   t   =   tH,o da / a E(a)   (1 to a)     tH,o v,o-½ ln(a)     15.9 ln(a) h72-1 Gyr

  a     exp[ tGyr h72 / 15.9 ]       (tnow = 0.988 tH,o)

  Two details:
        Formally, the exponential term has infinite past, so it is sensible to integrate from a=1, the current time.
        The matter solution is improved slightly by adding 39,000 years to correct the integral over the radiation era.

  (iii) Velocity Histories

• An alternative and possibly more illuminating diagram than a(t), is the velocity history:  v(t)
  In our Newtonian analog [sec 6av]:  how does the rock's speed change after the initial throw?
  To find this, we simply use the Hubble relation:   v = H r = H a r_o = da/dt r_o = H_o a E(a) r_o
  By choosing r_o = 1 Mpc, we get the velocity history for a galaxy now at 1Mpc

• The full velocity history for the concordance model is shown here [image]
  notice the minimum at a 0.65 when _{m,o   v,o}   and we change from de- to ac-celeration
  notice also the ever increasing velocity as t 0, a feature of all non-zero _m or _r models
  The initial expansion is essentially infinitely fast, with a(t) curves all starting out vertical
  This parallels the Newtonian case where v v_esc = (GM/r)^1/2     as R 0.
  For the universe, v(a) increases even faster since radiation dominates giving v a^-1 as a 0.

• Don't confuse v(a) with H(a), though they are related: H(a) = v(a) / a = (1 + z) v(a)
  H(a) gives the velocity at 1 Mpc real distance, not 1 Mpc comoving distance
  Thus, H(a) increases even more quickly as a 0.

  (iv) Flat Models: Single Component

• The special case of flat geometries with a single component are good pedagogical models.
  They also illustrate the behaviour of the various "eras" in the multi-component models.

• We use   t   =   t_H,o da / aE(a)   (from 0 to a); with E(a) containing just one term:

  1.   Pure matter:	m,o	= 1;	a E(a) = [a-1]½	t = (2/3) tH,o a3/2	a = [3/2 t/tH,o]2/3
  2.   Pure radiation:	r,o	= 1;	a E(a) = [a-2]½	t = (1/2) tH,o a2	a = [2 t/tH,o]1/2
  3.   Pure vacuum:	v,o	= 1;	a E(a) = [a2]½	t - to = tH,o loge a	a = e[(t - to)/tH,o]
  4.   Pure curvature:	k,o	= 1;	a E(a) = [k,o]½	t = tH,o a	a = t / tH,o

  These recover the forms from [sec 4f]:   a     t^2/3 , t^1/2 , e^t   for matter, radiation, vacuum.
  The pure curvature model is open (not flat), with simple linear expansion at all times.
  Pure matter & curvature models are also called: Einstein-de Sitter & Milne   [sec 6cvi]
  Note: the coefficients are different from the concordance approximations because the current 's aren't unity.
  

• For a general equation of state parameter w, we have [sec 4f]:

  a E(a) = [a^-(1+3w)]^½     t = 2/(3+3w) t_H,o a^(3+3w)/2     a   =   [ (3 + 3w)/2 (t / t_H,o) ]^2/(3+3w)

  you can quickly check this gives 1 & 2 above, for w = 0 and 1/3.
  Note also that the flat model with w = -1/3 behaves like an open pure curvature (Milne) model.

• It is easy to find the current age of all these models: just set a = 1 and solve for t:

  tage   =   2/(3 + 3w) tH     2/3 tH ,   1/2 tH ,   ,   tH     for 1 - 4 above.

  Here I chose to use t_H in place of t_H,o since the relation for t_age is true at all times.
  As we suspected from the outset, t_age   t_H to within factors ~unity.
  The exception is pure vacuum, with exponential expansion and infinite age (t -  as  a 0)

  Note that w = -1/3 separates ac-/de-celerating models, and hence t_age greater/less than t_H,o
  For example, for flat matter (Einstein-de Sitter; w = 0) we have   t_age = 2/3 t_H,o,   a famous result.
  By including a component with w < -1/3 (eg vacuum/lambda), then t_age can get longer than t_H,o
  This was a key motivation for including to solve t_H,o < t   (1930s), or t_H,o < t_GC (1990s).

• Some more a(t) curves for various models are given here [images]

  (v) Flat Models: Matter + Vacuum

• As it happens, it is possible to derive an analytic form for a flat universe with both _m and _v
  Apart from a brief (~50 kyr) radiation period, this is basically the Universe we live in.

• As an excercise in integration, follow the analysis through to derive the following result:
  
  a(t)   =   (_m,o / _v,o)^1/3   sinh^2/3 [ (3/2) _v,o^1/2 (t / t_H,o) ]   =   0.712 × sinh^2/3 (1.28 × t / t_H,o)

  t_age   =   (2/3) t_{H,o v,o}^-1/2 sinh^-1 [ (_v,o / _m,o)^1/2 ]   =   0.78 × 1.27 t_H   =   0.993 t_H,o

  which recovers an age conveniently close to t_H,o     H_o^-1
  i.e. the decelerating & accelerating portions "average out" to approximate a constant expansion.
  (the numbers here use:  _v,o = 1 - _m,o = 0.73)

  (vi) Curved Models: Matter Only

• For a long time (1960s-90s), curved matter models were favoured, justifying their inclusion here:
   open with infinite future; closed ending in a big crunch; or "critical", balanced between the two
  

  There is, however, a more compelling reason: they are crucial in theories of galaxy formation
    while the real Universe seems to be globally flat, this is not true locally:
    voids form from open regions; while stars/galaxies/clusters form from closed regions
  FRW curved matter models apply since local evolution is independent of the surroundings.

• For a single component:   _s   =  _s,o a^-3(1+w)  and  a E(a)  =  [_s,o a^-(1+3w)  +  (1 - _s,o)]^1/2
  For w = 0 (matter) we have   t(a)   =   t_H,o [ _m,o a^-1 + 1 - _m,o ] ^-1/2 da   (from 0 to a)

  This has a parametric solution which uses a "development angle",
  The form of the solution depends on whether _m,o > 1 (closed) or _m,o < 1 (open):

• For closed models with = 0 to 2, the Universe expands, halts, and recollapses in a cycloid:

  a()   =   ½ amax (1 - cos )

  t()   =   ½ tH amax ( - sin ) / (m,o - 1)1/2

  which turns at a() = a_max = _m,o / (_m,o - 1)   and collapses at t(2) = t_H,o a_max / (_m,o - 1)^1/2
  for example, if _m,o = 1.5, we have a_max = 3 and t_crunch = 6.7 t_H,o

• For open models, the Universe expands forever with similar solution, with = 0 to
  

  a()   =   ½ apar ( cosh - 1)

  t()   =   ½ tH,o apar ( sinh - ) / (1 - m,o)1/2

  where a_par   =   _m,o / (1 - _m,o)   plays the role of a_max

• Graphs of these classic solutions (for _m,o = 0.9,   0.0,   1.1) are shown here [BR 6.1]

  (vii) Specific & Named World Models

• This [figure] illustrates most classes of FRW world models.

• The two key parameters are
  1. curvature: closed (k = +1, _k -ve), flat (k = 0, _k = 0), open (k = -1, _k +ve)
  2. lambda: _v: -ve (odd),   0 (simplest),   +ve (what seems to be true).

• Full histories in linear time usually consider pressure-less matter along with vacuum.
  This is because radiation quickly becomes irrelevant when _r < _m at t ~ 50 kyr.

• Referring to the figures, the overall forms are:
  negative :   unusual; simply adds to matter; all models recollapse
  zero :   most discussed until recently; all decelerate; future depends only on curvature.
  
  closed recollapses; flat halts @ ; open expands forever.
  Examples for _m = 0.9, 1.0, 1.1 are shown [here, L7.6; BR 6.1]
  positive :   early, matter dominates   decelerates;   late, vacuum dominates   accelerates.

  For a closed geometry, additional possibilities include (see below):

  For _v < _v,E both collapse and "bounce" solutions exist
  For _{v v,E}   vacuum repulsion and closed geometry nearly balance long ~static periods.

  Ages and futures for all these models can be nicely seen in these plots   [image]

• Some of these models go by the names of those who advocated/studied them:
  

  • Einstein-de Sitter:  pure matter, flat; "Critical" classical model, between open & closed.
    Ironically, neither Einstein nor de Sitter advocated this.

  • de Sitter:  pure vacuum, flat   exponential expansion: a e^t; infinite past; H(a) = const
    
    This model has renewed importance, since it is believed to occur during inflation.
    It also describes the Steady State model, which needs H(a) = const.

  • Einstein:   His original model; curvature/matter/vacuum all balance to give static Universe.
    pre-dated Hubble's discovery of expansion; doesn't fit above analysis since H_o = 0.
    Instead, set da/dt = d²a/dt² = 0 to find balance when:  _m+ _v -1 = 3/2 (_vm²)^1/3
    The model is unstable to small deviations from balance.

  • Eddington-Lemaitre:   Two additional solutions to Einstein's model
    Infinitely long static history before starting expansion
    big bang in remote past, then expansion comes to gradual halt at infinite future.
    

  • Lemaitre:   close to Einstein model, with _v (_m) slightly below (above) the critical values.
    
    Big bang     expansion   "loitering"     exponential expansion.
    The motivation for loitering was the too short (incorrect) Hubble time, t_H,o = H_o^-1
    The age can be increased by introducing ;   see this example: [images]
    Lemaitre preciently also wanted a big bang to make elements

  • Milne:  pure curvature (empty, not even vacuum energy); _k = +1, open, (k = -1).
    Similar to _m/r/v << 1     no de- ac-celeration: linear expansion a = t / t_H; t_age = t_H
    Milne's (incorrect) motivation: moving galaxies subject only to special relativity.

• Some photos of many of the characters we've mentioned so far are shown here: [images]

(7) Distances & Horizons

• Discussing distances on an expanding, possibly curved, coordinate grid can be tricky.
  So, first let's distinguish between three different kinds of distance measurement

1. proper distances:   r   are "true" distances - the number of non-expanding rulers between objects
   r increases due to expansion according to the velocity-distance relation v = dr/dt = H r
   the current proper distance, r(t_o), to an object is also called its comoving distance, r_o.
   the proper distances all change with the scale factor: r = a(t) r(t_o) = a(t) r_o

2. pseudo (my term) distances:   D   are derived from certain measurements....
   eg luminosity (D_L) or angular diameter (D_A),  as if space-time were static and Euclidean.
   they are not "real" distances so much as convenient functions of distance
   they are closely tied to observations and play a crucial role in establishing the world model

3. redshift:   z   though not an explicit distance, it is our primary observable
   it is derived from spectra and gives the scale factor when the light set out: a(t_e) = (1 + z)^-1
   Given a world model, it is relatively easy to derive all the other distances using z.

• This section deals mainly with proper distances:   r
  The following two sections consider , among other things, the pseudo distances:   D.

• Before we start, let's recall two useful/sensible units of time and distance for cosmology:

  Hubble time:           tH,o   =   Ho-1   =   10.0 h-1 Gyr   =   13.9 h72-1 Gyr

  Hubble distance:   rH,o   =   c / Ho   =   c tH,o   =   13.9 h72-1 G lyr   =   4.26 h72-1 Gpc

  they are units comparable to the current age and visible size of the Universe.

• Because the Universe is expanding, the proper distance to an object depends on time:   r(t)
  In fact, there are three distances one might consider: [image]
  
  1. The proper distance to the object when the light set out:   r(t_e)   (e = emit)
     if the light has travelled for a long time, r(t_e) could be quite small, but certainly less than:

  2. The proper distance to the object when the light arrives:   r(t_o)   (o = observed = now)
     recall, r(t_o) = r_o, is also called the comoving distance
     Since t_e < t_o then r(t_e)  <  r(t_o);  or using the scale factor:  r(t_e)  =  a(t_e) × r(t_o)  =  a(t_e) × r_o

  3. The distance light travelled during its journey:  r_lbt  =  c(t_o - t_e)   (lbt: look-back-time)
     expansion guarantees that r_lbt is intermediate in length between r(t_e)  and  r(t_o)
     the light travel time, (t_o - t_e), is also the look-back time, as invoked in "popular" statements:
     "That galaxy is 5 billion light years away, hence we see it as it was 5 billion years ago"
     

• What is r(t_o) as a function of z?   Here are two approaches:
  Imagine a photon at time t on its way to us.
  In interval dt it crosses c dt which then expands to become c dt / a(t) at time t_o = t_now
  We can also use the various forms for dt given in [sec 6ci] to give several equivalent integrals:

  r(to)   =   c dt / a(t)   (te to to)   =   rH,o da / a2 E(a)  (a to 1)   =   rH,o -dz / E(z)   (z to 0)

  Alternatively, from the RW metric: photons move on radial null geodesics, so d = ds = 0, giving:

  c² dt²   =   a²(t) dr_o²     c dt / a(t)  =  dr   =   r(t_o)   (t_e to t_o and 0 to r_o)   as before.

• What about the lookback time?   Again, using the alternatives [sec 6ci] for dt:

  LBT = (to - te)  =  dt  (te to to)  =  tH,o da / a E(a)  (a to 1);  =  tH,o -dz / (1+z) E(z)  (z to 0)

  Which are all straightforward to evaluate (numerically).

• Examples of r(t_o),  r(t_e) and  r_lbt = c(t_o - t_e) are shown here: [images]
  A different presentation, using "The Astronomer's Universe", is shown here [image]

  (i) Two Examples: z = 6 & 1000

• Let's do a couple of examples using the Einstein-de Sitter Universe (_m,o = 1,  _v,o =  _k,o = 0)
  In this case, E(a) = a^-3/2  and  E(z) = (1 + z)^3/2

1. For a z = 6 QSO:   what are   r(t_o),   r(t_e),   &   c (t_o - t_e)
   There are several ways to treat this; let's use the z relations:
   r(t_o)  =  r_H,o -(1 + z)^-3/2 dz  (z to 0)  =  2 r_H,o [1 - 7^-1/2]  =  1.24 r_H,o

   r(t_e)  =  r(t_o) / (1 + z)  =  1.24 r_H,o / 7  =  0.18 r_H,o

   (t_o - t_e)  =  t_H,o -(1 + z)^-5/2 dz  (z to 0)  =  2/3 t_H,o [1 - 7^-3/2]  =  0.63 t_H,o

   The QSO was 0.18 r_H,o when the light set out; it is now 1.24 r_H,o; the light travelled for 0.63 t_H,o.

2. Pushing further: consider the CMB at z = 1000, we have
   r(t_o)   =   2 r_H,o [1 - 1000^-1/2]   =   1.94 r_H,o

   r(t_e)   =   1.94 r_H,o / 1001  =  0.0019 r_H,o   or only ~ 8 Mpc!

   We are seeing the CMB when it was closer than the Virgo cluster!
   That is the main reason physical scales appear so large on the CMB   [sec 8b].

• Current estimates suggest _tot = 1.0     we are in a spatially infinite Universe.
  Can we see all of this universe?   No!,   for three (somewhat overlapping) reasons:
  1. Right now, galaxies beyond r_H,o = c/H_o = 4.26 Gpc are receding faster than light
     As a result, light is actually getting further from us, not closer!

  2. The universe is only ~ 14 Gyr old: light cannot travel infinitely far in that time

  3. Our view of the universe is inevitably restricted to events on our past light cone
     this is a 3-d slice within a 4-d spacetime; like a sheet in space
     There is much in the Universe not on that sheet!

• On earth, our view is limited by the horizon
  Hence, some of these cosmic limitations are referred to as horizons

  (i) The Hubble Sphere: r_H   (where expansion velocity = c)

• Consider point 1: the velocity distance relation, v = Hr, defines a critical radius r_H = c/H = c t_H
  Beyond this distance, objects recede faster than light and wavefronts actually get further from us
  If the expansion remained constant, then ultimately we can/cannot see objects inside/outside r_H.

• Of course, the rate of expansion changes and so, therefore, does the size of the Hubble sphere.
  For any time:   c = H(a) r_H(a) = H_o E(a) r_H(a)   giving:
  

  rH(a)  =  rH,o / E(a)  =  rH,o / E(z)

  or, re-expressing r_H(a) in its larger current comoving size, r_o,H(a), we have:

  ro,H(a)  =  rH(a) / a  =  rH,o / a E(a)  =  rH,o (1 + z) / E(z)

  In physical coordinates, r_H increases from zero at t = a = 0 (z = ) and continues to increase.
  In comoving coordinates, r_o,H grows/shrinks in de/ac-celerating universes.

• For the concordance model, ignoring the early period of inflation, we find for comoving r_o,H:
  the Hubble sphere initially grew outwards: eg at z = 9 (a = 0.1)   r_o,H     0.61 r_H,o
  r_o,H then grew to a maximum size when _m = _v at a = 0.72, when   r_o,H = 1.15 r_H,o     4.9 Gpc
  Since then, r_o,H has been shrinking and is currently at r_o,H(a=1) = r_H,o = 4.26 h₇₂^-1 Gpc
  It is currently shrinking at a rate dr_o,H/da × da/dt = H_o dr_H/da = 0.18 Mpc/Myr or about 0.6 c
  Objects currently at the Hubble sphere are not, of course, visible to us (nor will they ever be).

  (ii) The Particle Horizon: r_p-hor   (furthest currently visible)

• Consider point 2 above, and ask: what is the furthest object currently visible in the sky?
  Its light has travelled non-stop since the big bang, so that t_e = 0 and LBT = (t_o - t_e) = t_age
  We want the comoving version of this distance, r(t_o).
  From the above [sec 7a] relations, just take limits t (0 to t_o) or a (0 to 1) or z ( to 0):

  rp-hor(to)  =  c dt / a(t)   (0 to to)  =  rH,o da / [a2E(a)]   (0 to 1)  =  rH,o -dz / E(z)   ( to 0)

• For non-zero _m or _r, the integral is finite, and there is indeed a most distant object visible.
  For example, take the Einstein-de Sitter case (flat, matter only):   E(z) = (1 + z)^3/2, giving:
  r_p-hor(t_o)  =  r_H,o -(1 + z)^-3/2 dz   ( to 0)  =  2 r_H,o

  so, our three proper distances are:   r(t_e) = 0;   r_lbt = c t_o = 2/3 r_H,o;   r(t_o) = r_p-hor(t_o) = 2 r_H,o

  Using the concordance values in E(z) gives r_p-hor = 2.55 r_H,o = 10.9 h₇₂^-1 Gpc   (35.5 Gly)

• It seems puzzling that light cannot cross the "entire universe" when it has "no size" at the BB.
  But at t = 0, for non-zero _m or _r, all locations expand with infinite speed [sec 6cii]
  Hence, at t = 0 all points are profoundly isolated; even the closest location is outside the horizon.
  Only after some deceleration does light make progress and crosses comoving coordinate space.

  Only if the initial expansion is zero can light cross everywhere in the first instant.
  This occurs in two FRW cases:
  

  empty open (Milne;   linear expansion) and
  pure vacuum (de Sitter;   exponential expansion)
  In these cases, there is a period when the entire universe is causally connected
  If the conditions are right, it can establish coherent (global) properties during this time.
  As we shall see [sec 11] an early period of inflation may have done just this in our universe.

• The particle horizon is important in understanding structure formation.
  By defining the region visible at any time, it also defines the region in causal contact
  Even when optically opaque, gravity still travels at light speed and sets the sphere of influence.
  Thus, the particle horizon defines the largest possible self-interacting region.
  Everything on larger, "super-horizon" scales is fundamentally disconnected.

• So, how does the particle horizon size grow with time?   Simply change the upper limit to t / a / z:
  In comoving coordinates we have for the particle horizon at any time:

  rp-hor(t)  =  c dt / a(t)   (0 to t)   =   rH,o da / a2E(a)   (0 to a)   =   rH,o -dz / E(z)   ( to z)

  Giving 2 a^1/2 r_H,o for Einstein-de Sitter;   a r_H,o for flat radiation;   for flat vacuum (de Sitter)

• The time defined by r_p-hor / c goes by another name, conformal time:
  (t)     dt / a(t)   (0 to t)     da / a²E(a)   (0 to a)     -dz / E(z)   ( to z).

  This is a useful surrogate for time, in part because it gives the horizon size at a given t, a, z.
  It is the time variable of choice for studying growth of perturbations.
  Space-time diagrams (see below) can also look much simpler when plotted using conformal time.

  (iii) The Event Horizon: r_e-hor   (furthest ultimately visible)

• If an event happens now in a distant galaxy (eg a supernova), when will we actually see it?
  If the answer is "never", then the galaxy is said to lie beyond our event horizon
  Like the other horizons, it can change with time (and even be infinite)

• In comoving coordinates, for any time, it is given by:

  re-hor(t)  =  c dt / a(t)   (t to )  =  rH,o da / a2E(a)   (a to )  =  rH,o -dz / E(z)   (z to -1)

  Which is the same as r_p-hor but with different (complementary) integration ranges
  Not surprisingly, if an FRW model has finite r_p-hor it often has infinite r_e-hor, and visa-versa
  Einstein-de Sitter & flat radiation have r_e-hor = ;   while flat vacuum (de Sitter) has r_H,o /2a²
  Vacuum in the concordance model ensures we will never see remote parts of the Universe.
  Indeed, as time passes, we will see less and less as the event horizon shrinks (in comoving radius).

• The event horizon is important during vacuum driven inflation [sec 11]:
  A small causally connected region expands and is pushed outside the (shrinking) event horizon
  After inflation, we have a huge smooth super-horizon region, previously causally conntected
  As normal expansion resumes, the horizon moves out and the region re-enters the horizon.

  It also plays a crucial role generating and amplifying quantum fluctations.
  Ultimately, these fluctations provide the seeds for future galaxy formation [sec 11].

• The three horizons (r_H   r_p-hor   and   r_e-hor) for the concordance model are shown here [image].

• In special relativity, space-time diagrams are often useful in clarifying a situation.
  They (usually) plot ct vertically and a spatial coordinate horizontally, so light rays move at 45^o

  Such diagrams are more complex on curved expanding space-times in cosmological GR:

  • locally, light cones are "tipped" to align with tilted world-lines.
    
  • globally, light rays can follow curved paths as they move across expanding coordinate grids.

  It is, however, often possible to clean up these diagrams & reestablish 45^o light paths:

  • change the spatial axis to comoving distance,   so objects have vertical world lines.
  • change the time axis from ct to c(t)   =   conformal time
    

• These [images] show space-time diagrams with light cones for the concordance model.
  There are three of them:
  • normal distance vs normal time -- light cone appears like an avocado seed
  • comoving distance vs normal time -- light cone appears like a spikey shield volcano
  • comoving distance vs conformal time -- light cone appears like a 45 degree cone.
  They are quite instructive: take a little time to figure them out.

• Although we used local energy conservation to legitimately derive = _o a^-3(1+w),
  discussion of global cosmological energy is much more tricky/impossible, not least because:
  the Universe is ~infinite in extent
  there is no global inertial frame
  expansion kinetic energy is ill-defined for r > r_H where v > c
  gravitational potential energy is ill-defined with no zero-level at infinity.

  Ignoring these concerns (!), we proceed to estimate the total energy within a Hubble sphere: r_H = c / H.
  Our aim is merely to suggest that the Universe's total energy might actually be ZERO

• Let's push the simple Newtonian analysis [sec 6a v] a little further:
  for a critically expanding sphere of radius R, density = 3 H² / 8G and velocity law v = H r   :

  PE  =  -(3/5) GM²/R  =  -(3/20) R⁵ H⁴ / G

  KE  =  ½ 4r²dr (H r)²  =  +(3/20) R⁵ H⁴ / G

  and we recover the Newtonian result:   E_tot  =  PE  +  KE  =  E  =  0   for all R.

• Relativistically, however, we mustn't forget the (positive) rest mass energy:
  RE  =  M c²  =  (4/3)R³ c²  =  (1/2) R³ H² c² / G  =  (1/2) R³ r_H² H⁴ / G

  Hence:   |PE| / RE    (R / r_H)²   so that on small scales rest mass utterly dominates cosmic energy.

  However, when R    r_H the -ve gravitational energy grows to match the +ve rest mass energy.
  e.g. for R ~ r_H,o we have   RE ~ |PE| ~ r_H,o⁵ H_o⁴ / G  =  c⁵ / H_o G    10⁷⁶ erg    10²² M   10¹¹ M_gal
  This crudely illustrates how on cosmic scales the total energy, including rest mass, could be zero.

• A more professional analysis of inflation does indeed suggest the Universe might have zero net energy.

  Everything  may have arisen from Nothing

  This is surely one of the most astonishing of Nature's possible properties.
    All matter in the Universe is borrowed against the negative gravitational energy of space-time.

• A similar notion arises in the context of black holes:
    the mass of a black hole can be found entirely in the energy of its gravitational field.
  

  In the cosmic context, the Hubble radius provides a kind of Schwarzchild/gravitational radius :

  R_S  =  2GM_H / c²  =  2G(4/3) r_H³ / c²  =  2G(4/3) (c³/H³) 3H² / 8G / c²  =  c / H  =  r_H

  which is why the Hubble sphere falls on the black-hole line in a mass/radius diagram [image]

(8) Observables vs Redshift: A Toolkit

• Objects in the Universe have intrinsic properties:   luminosity, size, velocity, space density
  From far far away, we witness observed properties:   flux, angle, proper motion, etc.
  The relationships between these two are straightforward in a static Euclidean space
  However, the real Universe has an expanding possibly curved space-time:
    how does this affect the simple Euclidean relationships?

• Well, things get a bit more complex.
  As usual, we need to keep our eye on two new aspects:
  1. The fact that the geometry might be curved
     
  2. The fact that during light's journey from object to us, the Universe has expanded.

  With these in mind we can derive close analogs to the Euclidean relationships.
  Indeed, one usually expresses them in Euclidean form, using a pseudo-distance, D, in place of r.
  The most famous of these are luminosity distance, D_L, and angular diameter distance, D_A
  I'll try to keep the convention of labelling pseudo-distances with capital D.

So far, we've only used the radial part of the RW-metric. Now we need to use its angular part.
• In a static Euclidean space, brightness f (W m^-2) is derived by spreading luminosity L (W) over a spherical shell with radius equal to the observer-object distance d (m):   f   =   L / 4d²

• Moving to the RW-metric, we need to make four modifications:
  1. the appropriate distance is r(t_o) = r_o = r_H,o dz / E(z)   (z to 0), the current proper distance
  2. the surface area is reduced (increased) relative to 4d² for closed (open) geometries.
  3. the energy of each photon is reduced by a factor (1 + z)
  4. the rate at which photons arrive is also reduced by a factor (1 + z)

• Consider points 1 & 2; the RW-metric for intervals on the comoving sphere is (dt = dr = 0; a = 1):
  ds²   =   R_o² S_k²(r_o/R_o) (d² + sin² d²)

  Integrating over and for the total spherical shell area, we get:

  A(r)   =   4 R_o² S_k²(r_o/R_o)   =   4 D(r_o)²     [ where D(r_o)     R_o S_k(r_o/R_o) ]

  D(r_o) is a comoving distance measure and is our first pseudo-distance:
  think of a × D as giving the correct d if we placed physical area dA at proper distance a × r_o

  dA(r,t)   =   [ a(t) × D(r_o) ]² d

  it is smaller (larger) than the proper distance, a × r_o, for a closed (open) geometry.
  for flat (k = 0) geometry, D(r_o) = r_o and spherical shells have Euclidean area: A = 4D²

• For flux, we set a = 1 and include the redshift factors (points 3 & 4 above):

  f   =   L / 4 D2 × 1 / (1 + z)2   =   L / 4 DL2     [ where DL (1 + z) D ]

  D_L is the luminosity distance and is our second pseudo distance
  it gives the correct (bolometric) luminosity using the Euclidean formula.

• Making the equations more explicit:

  D(ro)   =   Ro Sk(ro/Ro)	effective angular comoving distance
  ro(z) = rH,o -dz / E(z)     (from z to 0)	true comoving proper distance
  Sk(x) = sin(x)   x   sinh(x)   (k = +1  0 -1)	corrects for curvature
  Ro = rH,o / | k,o | ½	the curvature radius [sec 6b iii]
  k,o = 1 - t,o	the curvature parameter

  Notice that for k = 0, D = r_o and apart from the (1 + z)^-2 term, we recover the Euclidean relation.
  Likewise, for r_o << R_o and z << 1, we recover the full Euclidean relation.
  Here is a figure showing D_L(z) for several world models [image]

• There is an important detail we've ignored: the above relation works for bolometric fluxes.
  In practice observations are usually in a spectral band, which introduces two additional effects:
  1. the bandwidth is stretched (compressed) in wavelength (frequency) by a factor (1 + z)
  2. the rest frame spectral region is bluer than the bandpass, by a facor (1 + z); (K correction)
  which lead to the following modifed relations:
  f   =   L / 4 D_L² × [ L((1 + z)^-1) / L() ] × (1 + z)^-1       (units: erg/s/cm²/A)

  f   =   L / 4 D_L² × [ L((1 + z)) / L() ] × (1 + z)       (units: erg/s/cm²/Hz)

  f   =   (1 + z)^-1 L((1 + z)^-1) / 4 D_L²       (units: erg/s/cm²)

  f   =   (1 + z) L((1 + z)) / 4 D_L²       (units: erg/s/cm²)

  Usually, continuum fluxes require these relations, while emission lines are bolometric.

• In a static Euclidean space, an object at distance d of size ds subtends and angle d = ds / d
  In moving to the RW-metric, we have two modifications
  
  1. the object was closer when the light set out
  2. curvature affects the linear distance swept out by an angle

• All rays we see have been travelling on radial null geodesics (d = d = dt = 0)
  for a proper transverse length, ds, which subtends angle d, the RW-metric gives:
  ds   =   a(t_e) R_o S_k(r_o/R_o) d   =   a(t_e) D(r_o) d   =   D(r_o) / (1 + z)   d

  note a(t_e) is included since the light ray geodesics start at emission time, t_e   [image R&T 6.1]

• Thus, we have for the angular diameter:

  d = ds / a(te) D = ds / DA     [ where DA    D / (1 + z)   =   DL / (1 + z)2 ]

  D_A is the angular diameter distance and is our third pseudo distance
  it gives the correct angular diameter using the Euclidean formula.

• D_A(z) curves are shown here [image BR 7.4]: they are famous for turning over
    place galaxies further away:   they first look smaller, then stay the same, then look bigger!
  the reason they look bigger is not due to curvature, but their proximity when the light set out.

• The above analysis applies to an object of specific physical size, ds.
  How does this change for a (large) comoving size dS which expands with the universe?
  For example, how big would a 100 Mpc SDSS supercluster appear at z = 0.2,  5,  1000?

  A comoving (ie current) size dS was smaller at redshift z:   ds = dS/(1 + z).
  Using this in the relation d = ds / D_A we get:

  d   =   dS/(1 + z) / [D/(1 + z)]   =   dS/D   =   dS/DEA     [ where DEA    D ]

  D_EA = D is the angular diameter distance for an object expanding with the Hubble flow.

  Let's do our example of 100 Mpc at z = 0.2,  5,  1000,   choosing Einstein-de Sitter (flat, matter):
  for this, we have D = r_o = r_H,o dz / (1 + z)^3/2 = 2 d_H,o [1 - (1 + z)^-½]   =   [0.17, 1.18, 1.94] × r_H,o
  Using r_H,o = 4.26 Gpc, we have d = 7.9^o,  1.14^o,  0.69^o   for z = 0.2,  5,  1000.
  These angular sizes don't get bigger at high z, because our object was smaller back then.
  Notice that our supercluster is ~8^o in the SDSS, it is still ~1^o at z = 5 and ~0.7^o on the CMB

• Consider an object with transverse velocity,   v_t
  the object takes time dt' to travel ds = v_t dt', but we witness this as time dt = dt'(1 + z).
  Combining the angular diameter and time dilation relations, the observed proper motion is:

  d/dt   =   ds/DA / dt' (1 + z)   =   vt / D   =   vt / DM     [ where DM    D ]

  D_M = D is the proper motion distance, and is seen to be our original pseudo distance, D
  it gives the correct transverse velocity from a proper motion, assuming the Euclidean relation
  This is the appropriate distance to use when measuring projected jet speeds in radio galaxies.

  (note, David Hogg's classic "cheat sheet" uses D_M, D_C and D_H for my D, r_o and r_H,o)

• In a static Euclidean space, surface brightness is famously independent of distance:
  
  SB   =   f / (d)²   =   L/(4d²) / (s/d)²   =   L / 4s²     (independent of d)

  Moving to the RW-metric:   f decreases (1 + z)^-2, and (d)² increases (1 + z)²

  SB   =   f / (d)2   =   L/(4DL2) / (s/DA)2   =   L / 4s2 (1 + z)-4

  This is the (almost) equally famous (1 + z)^-4 rapid drop in surface brightness with redshift.
  note it is independent of curvature and, of course, is Euclidean at low-z.

• The relations for continuum fluxes are: SB     (1 + z)^-3,   and SB     (1 + z)^-5

• the steep drop in surface brightness with redshift is a mixed blessing:
  
  it severely hinders observation of high redshift galaxies, however
  without it we would all be dead.
  (1 + z)^-4 takes the utterly lethal black body radiation at recombination and renders it harmless.

• As we look out to a given redshift, the volume witnessed increases.
  Let's first consider the comoving volume, ie the volume it will expand to today.
  How does this depend on redshift?

  The proper comoving area at comoving distance r_o is A(r_o) = d D², where D = R_oS_k(r_o/R_o).
  The proper comoving volume of a shell of depth dr_o is dV_C = A(r_o) dr_o = d D² dr_o
  Now, since r_o = r_H,o dz/E(z),   then we have   dr_o = r_H,o dz/E(z)   which gives:

  dV_C = d r_H,o D² dz/E(z),   which can be integrated between redshifts:

  VC(z1 to z2)   =   d rH,o -D(z)2 dz/E(z)   (from z1 to z2)

  Notice that D(z) is itself an integral, since D = R_oS_k(r_o/R_o) and r_o = r_H,o -dz/E(z)   (z to 0)

  The total (d = 4) comoving volume out to redshift z turns out to be:

  V_C(z) = 2 r_H,o³/ _k,o [ D/r_H,o(1 + _k,oD²/r_H,o²)^½   -   | _k,o |^-½ AS_k(_k,o^½ D/r_H,o) ]

  where AS_k(x)  =  arcsin(x) & arcsinh(x) for k = +1 & -1, and V_C(z)  =  4/3 D³   for k = 0

• Let's now consider the actual (not comoving) volume, which is much less as z increases
  Repeating the analysis, but keeping non-comoving quantities, we have A(r,a) = d a² D²
  dr_o becomes a dr_o = a r_H,o dz/E(z)   giving dV = d r_H,o D² (1 + z)^-3 dz/ E(z)  =  dV_C (1 + z)^-3
  which can also be integrated to yield a significantly smaller volume.

• Let's gather many of these relations together.
  Imagine observing an object at redshift z, with (bolometric) luminosity L, and physical diameter S

  First some auxiliary functions & definitions we'll need:

  E(z)   =   [ _m,o (1 + z)³  +  _r,o (1 + z)⁴  +  _v,o  +  _k,o (1 + z)² ] ^1/2

  _k,o   =   1  -  (_m,o  +  _r,o  +  _v,o)   =   1  -  _t,o

  D   =   R_o S_k(r_o/R_o)       with   S_k(x)  =  sin(x) ;  x ;  sinh(x)   for k = +1 ;  0 ;  -1

  R_o   =   r_H,o / | _k,o |^½     with   r_H,o  =  c/H_o  =   c t_H,o

  dr_o   =   c dt / a   =   r_H,o dz / E(z)   =   r_H,o da / a²E(a)

• Proper distance at time of observing:
  r(t_o)  =  r_o  =  c dt/ a   [t_e to t_o]  =  r_H,o -dz/ E(z)   [z to 0]  =  r_H,o da/ a²E(a)   [a to 1]

• Proper distance at time of emission:
  r(t_e)   =   a(t_e) r(t_o)   =   r(t_o) / (1 + z)

• Look-back time (LBT):
  (t_o - t_e)   =   dt   [t_e to t_o]  =  t_H,o -dz/ (1 + z)E(z)   [z to 0]  =  t_H,o da/ a E(a)   [a to 1]

• The physical and comoving Hubble spheres at redshift z:
  r_H(z)   =   c / H(z)   =   r_H,o / E(z)

  r_o,H(z)   =   c / aH(z)   =   r_H,o (1 + z) / E(z)

• The comoving particle horizon at time / redshift / scale factor   t / z / a:
  r_p-hor  =   c dt/ a   [0 to t]  =  r_H,o -dz/ E(z)   [ to z]  =  r_H,o da/ a²E(a)   [0 to a]

• The comoving event horizon at time / redshift / scale factor   t / z / a:
  r_e-hor  =  c dt/ a   [t to ]  =  r_H,o -dz/ E(z)   [z to -1]  =  r_H,o da/ a²E(a)   [a to ]

• Observe a bolometric flux (W m^-2):
  f   =   L / 4D_L²       D_L  =  D(1 + z)     is the luminosity distance

• Observe an angular diameter (radians):
    =   S / D_A       D_A  =  D/(1 + z)     is the angular diameter distance

  _c   =   S_c / D     for an object expanding with the universe, with comoving size S_c

• Bolometric surface brightness, SB (W m^-2 sr^-1):
  SB   =   L / S² × (1 + z)^-4

• For transverse velocity v_t (km s^-1) we observe a proper motion, PM (radians s^-1):
  PM   =   v_t / D_M       D_M  =  D   is the proper motion distance.

• Comoving proper volume, dV_c (m³) in a shell from z to z + dz in solid angle d :
  dV_c   =   d D² dr_o   =   d D² r_H,o dz/ E(z)

  dV   = dV_c × (1 + z)^-3     is the physical (non-comoving) volume in the same shell

• I have made a number of charts and graphs which show concordance model cosmic properties
  These should help you to get a feel for the relations given above.
  In particular, they all use linear time since this is closest to our experience.
  To cover everything, we need three epochs:
  • Full history :            Parameters   and   Events   and   Graphs
    
  • The first Gyr :         Parameters   and   Events
    
  • The first 500 kyr :   Parameters   and   Events

  Yes, these did take me a long time to make.

(9) The First Three Minutes

On the limits of extragalactic astronomy, so this will be brief

(10) Recombination & the CMB

(11) Measuring Cosmological Parameters

• Early geology was pre-occupied with measuring the size, shape, composition and age of the Earth.
  So too with early (ie recent!) cosmology: it has focused on measuring the basic cosmic parameters.
  Indeed, a cliche in 1960-90 was that cosmology was "the hunt for two numbers":   H_o and q_o
  Today, we recognize several more:  H_o,   _dm,o,   _b,o,   _r,o,   _v,o   w_v   plus a few others
  But the 1990-2010 period is no less concerned with measuring these fundamental parameters.

• How does one go about measuring them?     There is an overall strategy:
  The relationsips defined in the previous section include many of the basic parameters
  Hence, one needs to compare intrinsic and observed properties of distant objects.
  The main difficulty is knowing the intrinsic properties
    one needs good "standard candles" or "standard rulers", as we shall see.

  Some useful figures: [images]

  The Primary Cosmological Parameters
  Measuring Mean Component Densities
  Number Counts
  Luminosity Distance Measurements
  Angular Diameter Measurements
  The Structure Parameters
  The CMB Power Spectrum
  The Concordance Model
  Future Work
  

(12) Problems With the Standard Model

The Flatness Problem
The Horizon Problem
The Structure Problem
The Antimatter Problem
The Entropy Problem
The Existance Problem

• The Standard Hot Big Bang model described so far seems remarkably cogent.
  Recall, its basic assumptions are:
  • The Cosmological Principle   --   large scale homogeneity & isotropy, yielding the RW metric
  • General Relativity   --   relates contents to dynamics and geometry
  • Five components with equations of state   --   radiation; neutrinos; baryonic matter; dark matter; dark energy
  • An initial perturbation spectrum close to "scale invariant"   --   P(k) k.
  • Adiabatic cooling from a hot early phase
  • The Standard Model of particle physics   --   QED, QCD, etc

  This framwork manages to account for an extremely wide range of observations,
  It also provides several independent estimates of its basic parameters .
  Apart from the unknown nature of dark matter & energy, the framwork seems quite robust.

• However, there exist a number of problems with this standard picture
  "Problems", here, does not refer to an error or mismatch to data.
  Rather, they are deeper concerns about the seemingly unlikely nature of the initial conditions
    why was the expansion launched exactly the way it needed to be to yield our current universe?

• This section summarizes these cosmic peculiarities (7 of them)
  The section following shows how an early burst of inflation can explain how most of them come about.
  

• The flatness problem starts our by noting the Universe is close to a rather special condition:
  • The universe's geometry is within a few % of flat (zero curvature).
  • its density is within a few % of the critical density.
  • its expansion speed is within a few % of the (Newtonian) escape speed.

• As we'll see, the standard model rapidly evolves away from this special state, not towards it.
  Having expanded over many factors of 10, its early state must have been exceedingly close to the special case.
  In the absence of an explanation for this fine tuning, we seem to have a "problem".

  (i) Newtonian Escape Speed

• Let's return to our Newtonian sphere of expanding rocks to gain an intuitive picture:   [review sec 6a v ].
  Imagine a large sparse spherically expanding ensemble of rocks, and focus on an outer rock moving upwards.
  recall that no rocks pass eachother     whatever its radius, this rock always has the same mass, M, beneath it.
  We will imagine that the whole ensemble was once at one location and was "launched" in an explosive manner.

• Since M is constant, near the origin our rock must initially move very close to v_esc to get "up high".
  If it starts at r_in moving at v_in and turns around high up at r_turn, then we have for v_in and v_esc:
  
  ½ v_in² - GM/r_in = E = -GM/r_turn     and     ½v_esc² = GM/r_in
  

  Hence, ½ v_in² = GM (1/r_in - 1/r_turn)   giving   (v_in/v_esc)² = (1/r_in - 1/r_turn) / (1/r_in)     1 - r_in/r_turn

  So, in order to reach r_turn, it must start within v of v_esc where:   v / v_esc     ½ r_in/r_turn
  As r_in gets smaller and smaller, the initial velocity must get closer and closer to v_esc.
  This is the Newtonian analog of the "flatness problem":
    in order to get so BIG, the Universe must have been launched incredibly close to v_esc.

• Let's briefly illustrate with the Earth, radius ~6000 km (where v_esc ~ 11 km/s)
  Imagine starting when R ~ 1 km, where v_esc ~ 850 km/s   (<>    10⁶ tonnes cm^-3)
  To reach 6000 km, it must be launched within 0.07 km s^-1  (~10^-4) of 850 km s^-1.
  From 1 m, it must be launched within 2 m s^-1  (~10^-8) of 27000 km s^-1 to reach 6000 km.
  These times correspond to redshifts of 6000 and 6 million (~20 kyr & 1 wk), not all that remote.

• In this terrestrial context, such precision would require a bizarrly controlled initial explosion.
  In the cosmological context, of course, inflation provides precisely this kind of "explosive launch".

  (ii) Curvature/Omega Evolution

• This topic is usually cast in terms of curvature, which is of course related to density.
  Measurements suggest the current total density, _t,o, is within 5% of unity   (equivalently, _k,o < 0.05).
  Now, although an exactly flat universe is always flat, what if the current Universe is in fact slightly curved?
  Let's derive how the curvature changes as the Universe evolves -- does it get more or less curved?

• The current total density parameter is   _t,o  =  _t,o / _c,o  =  8G_t,o / 3H_o²
  and its deviation from 1 specifies the spatial curvature:   _t,o - 1  =  - _k,o  =  kc²/R_o²H_o²     [see sec 6b iii]
  At epochs other than the present:   _t(a)  =  8G_t / 3H²,   where H and _t are now both functions of a.
  Let's substitute:   H(a) = H_oE(a)   and   _t / _c,o  =  _m,oa^-3 + _r,oa^-4 + _v,o  =  E²(a) - _k,oa^-2
  
  _t(a)  =  8G_c,o (_t / _c,o) / 3H_o²E²(a)  =  [ E²(a) - _k,oa^-2] / E²(a)  =  1  -  _k,o / a²E²(a)

  which is the relation we need.

• More specifically, since   a²E²(a)  =  _m,oa^-1 + _r,oa^-2 + _v,oa² + _k,o
  then for any currently non-flat universe (_k,o 0)   we find going backwards in time (ie   a    0),
  the geometry was more flat:   _t    1 ,   _k    0 ,   as long as either matter or radiation dominates.
  

• There is a more transparent way to express this, using:   a²E²(a)  =  a²H²(a) / H_o²  =  [ v(a) / v_o]²
  This is the (normalized) velocity history discussed in sec 6c iii, with its [ image ] shown here.
  Summarizing:

  t(a)  =  1  -  k,o / a2E2(a)  =  1  -  k,o / [ v(a) / vo]2

  giving a simple rule:

  • Decelerating expansion (matter/radiation) makes the Universe less flat
  • Accelerating expansion (vacuum) makes the Universe more flat

  Hence, for the standard model, the Universe was flatter in the past, and will get flatter in the future

• Let's look at the past more quantitatively.   In the radiation era we have [sec 6c ii ] :

        a²E(a)²    _r,o / a²    and    t  =  t_H,o da / a E(a)  (0 to a)     ½ t_{H,o r,o}^-½  a²    2.33 x 10¹⁹ h₇₂^-1  a²   sec.

  so the curvature becomes:

  k(a)     k,o a2 / r,o     104 a2 k,o     5.0 x 10-16 tsec h72-1 k,o

  
  At the time of the CMB (a ~ 10^-3)   the curvature _k is only ~1% of the current value, _k,o ;
  at He synthesis (a ~ 10^-9)   _k    10^-14 _k,o ;   and at the GUT era (a ~ 10^-28)   _k    10^-52 _k,o !
  Clearly, the fact that we are within 5% of flat today implies the early universe must have been extremely flat.

  Of course, inflation's early acceleration can generate just this kind of extreme flatness.

• One final potentially confusing point:   although the Universe was flatter in the past (ie   _k 0  &  _t 1),
  its curvature radius,   R = a R_o ,  was smaller in the past, and indeed R 0 as a 0 !
  How do we understand this apparent contradiction?   Easy:  R and _k depend differently on a.
  Recall [sec 6biii] the definitions of _k and _k,o which use R and R_o:  

  _k  =  -kc² / R²H²  =  -kc² / [ a²R_o² H_o²E²(a) ]  =  _k,o / a²E²(a)

  So R²H² follows a²E²(a), and although R 0 as a 0,  R × H increases and drives _k to 0.
  Of course, a × H(a)    v(a), the velocity history (our previous result), and we have v as a 0.

• Look around you -- the objects, the trees, the earth, moon, sun and stars.
  Where did all that matter come from?
  As physicists, we are triply perplexed:
        (a) more than most people, we appreciate just how MUCH matter exists in the universe;
        (b) we know that matter is concentrated energy: multiply by c² and its a HUGE amount of energy;
        (c) we have deep respect for conservation of energy -- again:   where did it all come from?

• From sec 7d, we get half the answer... appearances are misleading:
    integrated over a Hubble sphere, the positive mass-energy is balanced by an equal negative gravitational energy
  The total energy is ZERO!   In a sense, the Universe "SUMS TO NOTHING!".
  Your matter (and everything around you) is "on loan", borrowed against a huge intergalactic gravitational debt.

• Our puzzle doesn't vanish, however -- it changes to something a little more tractable:
    what mechanism can start with nothing, and create arbitrary amounts of matter and gravity?
  You guessed it...   inflation can....   read on.

(13) Inflation & Its Solutions

• As promised, we now turn to examine inflation and its implications for solving the above problems.
  "Inflation" here refers to any framework that includes an early period of accelerated expansion.
  It does not replace or contradict the standard hot big bang model, who's virtues all remain.
  Rather, inflation is inserted at the beginning and provides a way to launch the standard model.

• As you are probably aware, it adds the following new features:
  • it ensures that the observable universe was once well inside a horizon,
    this allows time to establish homogeneity and isotropy, which we now observe on large scales.
  • it launched the expansion at exactly the "escape velocity", allowing the universe to become large.
    -- equivalently, it drove the geometry exceedingly close to flat (Euclidian)
    -- equivalently, it drove the total density exceedingly close to the critical density.
  • it amplified quantum fluctuations into classical perturbations, allowing subsequent structure formation.
  • after inflation, "reheating" provides an exceedingly hot relativistically dominated early phase
  • it dilutes to ~zero any relic particles left over from the earliest times (eg magnetic monopoles).
  • it miraculously converts nothing into arbitrarily large amounts of something
    actually, exactly equal amounts of positive (mass) energy and negative gravitational field energy.

• This field is very sophisticated, and most is beyond our (read my) expertise.
  So, we'll approach this at a suitably unsophisticated level.
  We'll start with the implications of acceleration, and then look at what might be driving the acceleration.

(i) Preliminaries

• Fortunately, we have already encountered accelerated expansion in the context of dark energy [see Topics 4di, 4f, 4h]
  We simply need a dominant component with equation of state parameter: w < -1/3   ie p < -c²/3.
  This follows directly from the Friedmann acceleration equation:   d²a/dt²  =  -(4G/3) a (  +  3p/c²)
  for w < -1/3,  d²a/dt² is positive, gravity is repulsive, and expansion accelerates
  (review [sec 4h] for an intuitive description of why this happens.)

• For a flat geometry we have [sec 6c iii]   = _o a^-3(1+w)   and the Friedmann energy equation gives:
  da/dt  =  (8G/3)^½ a  =  (8G_o/3)^½ a^{-(1 + 3w)/2}  =  H_o a^{-(1 + 3w)/2}

  which has solutions:

  w > -1:     a(t)  =  [(3 + 3w)/2 . t/t_H]^2/(3+3w)     power law expansion; index > 1 (acceleration) for -1 < w < -1/3.
  w = -1:     a(t)  =  a(t_o) e^{(t - t_o)/t_H}     pure exponential expansion, with e-folding time t_H = H^-1.
  w < -1:     a(t)  =  [1  +  (3 + 3w)/2 . (t - t_now)/t_H]^2/(3+3w)     big rip (a ) at   t = t_now + 2t_H / |3+3w|.

  The last two solutions have no clear big bang,   ie  a 0 only as t -

• Inflation is often assumed to be a vacuum energy, with w -1, giving approximately exponential growth.
  In this case, the number of expansion e-foldings, N, is given by (recall e^N = 10^N/2.3)
  N  =  ln (a_end / a_start)  =  (t_end - t_start) / t_H     where
  t_H  =  (3 / 8G)^½  =  1.34 ₆^-½   sec     (with ₆ in units of 10⁶ gm cm^-3)

  (ii) Expansion Factor Required

  (iii) Convergence to Flatness

  (iv) Shrinking Event Horizon

  (v) Dilution of Relics