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 |
Our cosmology must explain the origin, development, and maximum scale of all this structure.
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'
This is a remarkable and profound result.
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.
(Absorption doesn't help: dust ultimately heats to reach equilibrium with the radiation field).
A bright sky requires our static universe to be both gigantic and immensely old.
Stars alone cannot yield Olbers' bright sky, even in an old enough and big enough Universe.
For this light source, then, it is expansion which keeps the sky dark.
Finding the form for a(t) is a holy grail in cosmology.
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.
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.
In these notes I will try to be consistent: r = physical; r_{o} = comoving; D = pseudo.
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 |
^{ } r_{H,o} = c/H_{o} 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].
_{o} / _{e} = 1 + z = a(t_{o}) / a(t_{e}) = 1 / a(t_{e})^{ } |
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:
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_{V}: -2 to -6) and hence can be seen to considerable distances (~25 Mpc by HST)
However, they are also rare: the nearest is 250pc away, and only 10 have parallax distances measured to 10%.
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 mission 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 = 18.44 0.03 48.7 0.7 kpc (uses E(B-V) = 0.1)
The HST Key Project has now measured 800 Cepheids in ~18 galaxies out to ~25 Mpc.
These galaxies were then used to calibrate the following methods:
This is a luminosity-linewidth relation for spirals [Topic 5.5b]
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)
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)
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)
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)
H_{o} = 72 7 km s^{-1} Mpc^{-1} (eg Friedman et al 2001) [image]
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.
So far, only one good example of this method exists: NGC 4258, Miyoshi et al 1995 [ Topic 14.4e]
a compact (~1pc) molecular disk orbits central black hole
VLBI of H_{2}O masers gives (Keplerian) velocities and proper motions
distance, by comparing linear and angular velocities.
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
Hot electrons in galaxy cluster ICMs do two things:
For several components, and p are both additive: _{tot} = _{i} and p_{tot} = p_{i}
(1/a^{2})(da/dt)^{2} = H^{2} = (8G/3) _{c} giving _{c} = 3H^{2}/(8G) |
_{c} = 2.65 × 10^{-7} h^{2} M_{}pc^{-3}
= 1.80 × 10^{-29} h^{2} gm cm^{-3} = 10.8 h^{2} m_{p} m^{-3}
Component | _{o} / _{c,o} | w | x = 3(1+w) = _{o} a^{-x} |
x = 2/(3+3w) a t^{x} |
1 + 3w sign accel |
Dark Energy | 0.73 | -1 | 0 | a e^{t} | -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 |
we live in a universe with "flat" spatial geometry
This is one of the most important discoveries in recent cosmology.
I feel this is an unnecessarily obtuse approach.
Here is a more straightforward approach (which amounts to the same thing).
But in cosmology, x 3 is also possible.
In such cases energy is either leaving or entering our expanding box.
One can consider this as "work done" by a pressure.
d( c^{2} a^{3}) = -p d(a^{3})
d c^{2} a^{3} + c^{2} 3a^{2} da = -p 3a^{2} da now divide by 3a^{2}c^{2} da
a/3 d/da + = -p/c^{2}
-x/3a + = -p/c^{2}
p = (x/3 - 1) c^{2} wc^{2}
= _{o} a^{-x} = _{o} a^{-3(1+w)}
Basically, the thermal energy is << rest energy
so cooling during expansion doesn't contribute to the change in
Basically, when a photon gas expands its pressure does work on the surroundings
Energy is lost from the expanding box because the photons are redshifted.
Basically, you must provide energy to create more vacuum
Negative pressure is tension: imagine a strange piston with a little "strange water" in it.
You pull extremely hard, with force F = p × A, with p = 9 × 10^{20} dyne cm^{-2} (~10^{15} atm) and A = 1 cm^{2}
the piston slowly moves out by d = 1 cm -- you have spent F × d = 9 × 10^{20} erg of energy.
To your surprise, the piston now contains an additional cm^{3} of water!
Your 9 × 10^{20} erg were converted to 9 × 10^{20}/c^{2} = 1 gm of new water.
(Note: since dark energy is 6.8 × 10^{-30} gm cm^{-3}, it only requires 6 × 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!).
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 |
So the relative densities of the three components changes with expansion: [image]
Depending on which component dominates the density (hence gravity), we find three different eras:
_{ }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)
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)}
during the radiation era: | a t^{1/2} |
during the matter era: | a t^{2/3} |
during the vacuum era: | a e^{t} (from da/dt a) |
Energy density | u = a T^{4} = 7.56 × 10^{-15} T^{4} erg cm^{-3}_{ } u_{cmb} = 4.17 × 10^{-13} erg cm^{-3} = 0.26 eV cm^{-3} |
Energy flux | J = uc/4 = caT^{4}/4 = T^{4}/ = 1.80 × 10^{-5} T^{4} erg s^{-1} cm^{-2} sr^{-1}_{ } J_{cmb} = 9.94 × 10^{-4} erg s^{-1} cm^{-2} sr^{-1} |
Number density | n = a T^{3}/2.7k_{B} = 20.3 T^{3} cm^{-3} n_{cmb} = 410 cm^{-3} |
Number flux | N = nc/4_{ } = 4.84 × 10^{10} cm^{-2} s^{-1} sr^{-1} N_{cmb} = 9.78 × 10^{11} cm^{-2} s^{-1} sr^{-1} |
T = T_{o} / a = T_{o} (1 + z) ^{ } |
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})/a^{3} dv_{o} and N = N_{o}/a^{3}
Substituting these into the MB relation, we get:
T = T_{o} / a^{2} = T_{o} (1 + z)^{2} |
In practice, this cooling for the baryonic gas never occurs:
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.
Adding Newton's intuitively plausible independent time, t, yields a 4 coordinate space-time
ds^{2} = dx^{2} + dy^{2} + dz^{2} = dr^{2} + r^{2} (d^{2} + sin^{2} d^{2}) = dr^{2} + r^{2} d^{2}
Note that the choice of cartesian or polar (or any other) coordinate system is unimportant
They are equivalent and define the same space.
circumference = 2 R sin(r/R)
This is < 2r, goes through a maximum at r = R/2 (equator) and goes to zero at r = R (antipode) [image]
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}):
Curvature | k | d^{2} coefficient |
Parallel Lines |
Triangle Angles |
Circle Circumf |
Sphere Area |
Sphere Volume |
Global Form |
Positive | +1 | R^{2} sin^{2}(r/R) | Converge | > 180 | < 2r | < 4r^{2} | < (4/3)r^{2} | Closed |
Flat | 0 | r^{2} | Never meet | 180 | 2r | 4r^{2} | (4/3)r^{2} | Open |
Negative | -1 | R^{2} sinh^{2}(r/R) | Diverge | < 180 | > 2r | > 4r^{2} | > (4/3)r^{2} | Open |
ds^{2} = -c^{2} dt^{2} + dr^{2} + r^{2} (d^{2} + sin^{2} d^{2}) (Minkowski space-time).
ds^{2} = -c^{2} dt^{2} + dr^{2} + R^{2} sinh^{2}(r/R) (d^{2} + sin^{2} d^{2})
One can also replace (d^{2} +
sin^{2} d^{2}) with
d^{2}, 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:
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.
On large scales (> few Mpc): these assumptions are excellent [see sec 2a-c]
On intermediate scales, where |/<>| < 1 they are still useful:
On small scales, where /<> >> 1 they are poor assumptions:
ds^{2} = -c^{2} dt^{2} + a(t)^{2} [ dr_{o}^{2} + R_{o}^{2 }S_{k}^{2}(r_{o}/R_{o}) d^{2} ] |
where r_{o} is the comoving proper distance (ie as measured today) to an object.
This looks very familiar!
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.
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:
This tells us that the duration of any event we witness is dilated by a factor a(t_{e})^{-1}
This has been nicely confirmed in the longer apparent duration of high-z SNIa light curves: [image]
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.
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
Spacetime is curved by the distribution of cosmic energy & momentum.
Here, µ and are four (1 time, 3 space) coordinate indices, (eg ct, x, y, z; or ct, r, , )
T_{1,1}, T_{2,2}, T_{3,3} = < p_{x}p_{x} >c^{2} / 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 (c^{2}, p, p, p) (all off-diagonal elements are zero).
Evaluating the elements for the RW-spacetime, one obtains:
G_{1,1} = G_{2,2} = G_{3,3} = -1/a^{2} [ 2 a (d^{2}a/dt^{2}) + k c^{2}/R_{o}^{2} + (da/dt)^{2}]
G_{j,j} = -1/a^{2} [ 2 a (d^{2}a/dt^{2}) + k c^{2}/R_{o}^{2} + (da/dt)^{2}] = 8G p/c^{2} = 8G/c^{2} T_{j,j}
Combining these, we arrive at two fundamentally important cosmic equations
(da/dt)^{2} = (8G/3) a^{2} - k c^{2}/R_{o}^{2} | The Friedmann Equation, or Energy Equation |
d^{2}a/dt^{2} = -(4G/3) a ( + 3p/c^{2}) | The Acceleration Equation |
where k = +1, 0, -1 follows the sign of the curvature radius R_{o}; and a(t) is the scale factor.
Substitute for d/da and rearrange:
d^{2}a / dt^{2} = (4G/3) [ 2a - 3(1+w)a ] = (-4G/3) a [ + 3p/c^{2} ]
Consider a huge sphere uniformly but sparsely filled with rubble [image]
Focus on a single rock at radius r_{o}: it feels only interior mass: M =
4/3 r_{o}^{3}_{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.
This is just the equation for throwing a stone vertically upwards:
if TE > 0, then v > v_{esc} and the rock escapes; if TE < 0, then v < v_{esc} and the rock returns.
Rewriting this equation using the changing scale factor and density, we get:
which is the Friedmann energy equation with 2 TE / r_{o}^{2} standing in for curvature: -k c^{2}/R_{o}^{2}
positive TE open geometry
negative TE closed geometry
_{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]
(1/a^{2})(da/dt)^{2} = H^{2} = (8G/3) - kc^{2}/a^{2}R_{o}^{2}
substitute for = 3H^{2}_{t} / 8G to get H^{2} = H^{2} _{t} - kc^{2} / a^{2}R_{o}^{2} which yields:
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.
_{k} -kc^{2}/R^{2}H^{2} = -kc^{2}/a^{2}R_{o}^{2}H^{2} and _{k,o} -k c^{2}/R_{o}^{2}H_{o}^{2}
Inserting into the Friedmann equation we find: H^{2} = H^{2}_{t} + H^{2}_{k} 1 = _{t} + _{k}
Notice, _{k} and k have opposite sign: +ve _{k} means open geometry
We can also re-express R_{o} in terms of _{k,o} and the Hubble distance r_{H,o} = c / H_{o} :
R_{o} = r_{H,o} / | _{k,o} | ^{½} |
To find an expression for this, start with the acceleration equation & divide by a:
1/a (d^{2}a/dt^{2}) = -H^{2} q = -(4G/3) ( + 3p/c^{2}) = -(4G/3) (1 + 3w)
q = ½ [ _{m} + 2_{r} - 2_{m} ] = ½_{m} + _{r} - _{v} at any time, or
q_{o} = ½_{m,o} + _{r,o} - _{v,o} today. |
d^{2}a/dt^{2} = -(4G/3) a ( + 3p/c^{2}) + 1/3 a
= 8G _{v} = 3 H^{2} _{v} (units of time^{-2}) and _{v} = -p_{v}/c^{2} (i.e. w = -1) |
The form of a(t) depends on two things:
Let's begin with the general case, then look at some special cases.
(da/dt)^{2} - (8G/3) a^{2} _{t} = (da/dt)_{o}^{2} - (8G/3) _{t,o}
(da/dt)^{2} | = (H_{o}^{2}/_{c,o}) a^{2} _{t} + H_{o}^{2} - (H_{o}^{2}/_{c,o}) _{t,o} |
= H_{o}^{2} a^{2} [ _{m,o} a^{-3} + _{r,o} a^{-4} + _{v,o} + (1 - _{t,o}) a^{-2} ] | |
= H_{o}^{2} a^{2} [ _{m,o} a^{-3} + _{r,o} a^{-4} + _{v,o} + _{k,o} a^{-2} ] | |
= H_{o}^{2} a^{2} E^{2}(a) giving: |
(da/dt) = H_{o} a E(a) for the evolution of the scale factor^{ } |
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.
In general, this and related integrals need to be done numerically.
Alternatively, one can solve the ODE with boundary conditions: (a = 1 ; da/dt = H_{o}) at t = t_{now}.
Note: for numerical work, it is sensible to express time in units of t_{H,o} = 1/H_{o}
dt = t_{H,o} da / aE(a) = -t_{H,o} dz / (1+z)E(z) and da = -dz/(1+z)^{2} = -a^{2}dz |
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.
_{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]
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,o}^{½} a^{-2} |
t = t_{H,o} da / a E(a) (0 to a) ½ t_{H,o} _{r,o}^{-½} a^{2} 741 a^{2} h_{72}^{-1} Gyr = 2.34 x 10^{19} a^{2} h_{72}^{-1} sec |
a 1.16 x 10^{-6} h_{72}^{½} t_{yr}^{½} 2.09 x 10^{-10} h_{72}^{½} t_{s}^{½} |
Matter era: E(a) _{m,o}^{½} a^{-3/2} |
t = t_{H,o} da / a E(a) (0 to a) 2/3 t_{H,o} _{m,o}^{-½} a^{3/2} 17.4 a^{3/2} h_{72}^{-1} Gyr |
a 1.49 x 10^{-7} h_{72}^{2/3} t_{yr}^{2/3} |
Dark Energy Era: E(a) _{v,o}^{½} |
t - t_{now} = t = t_{H,o} da / a E(a) (1 to a) t_{H,o} _{v,o}^{-½} ln(a) 15.9 ln(a) h_{72}^{-1} Gyr |
a exp[ t_{Gyr} h_{72} / 15.9 ] (t_{now} = 0.988 t_{H,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.
Follow the above analysis, starting from the first line:
(da/dt)^{2} - (8G/3) a^{2} _{t} = (da/dt)_{o}^{2} - (8G/3) _{t,o}
(da/dt)^{2} - (H_{o}^{2}/_{c,o}) a^{2} _{t} = H_{o}^{2} - (H_{o}^{2}/_{c,o}) _{t,o}
Introduce the velocity factor, v/v_{o} = Hr / H_{o}r_{o} = a H/H_{o} = (da/dt) / H_{o}
and express the time-varying density _{t} in terms of today's Omegas, we get:
(v/v_{o})^{2} + [ -_{m,o}/a -_{r,o}/a^{2} -_{v,o}a^{2} ] = 1 - _{t,o}
"KE" "PE" ^{ } "TE" (const) |
Or using the previous notation: v/v_{o} = a E(a).
Of course, to convert v(a) to a(t) we still need to integrate, just as before:
v/v_{o} = (da/dt) / H_{o} = a E(a) giving:
da /^{ } aE(a) (0 to a) = H_{o} dt (0 to t) = t / t_{H,o}
Since the total energy term (1 - _{t,o}) is fixed, the expansion velocity must mirror the gravitational terms.
1. _{ } Pure matter: | _{m,o} | = 1; | _{ } a E(a) = [a^{-1}]^{½} _{ } | t = (2/3) t_{H,o} a^{3/2} | a = [3/2 t/t_{H,o}]^{2/3} |
2. _{ } Pure radiation: | _{r,o} | = 1; | _{ } a E(a) = [a^{-2}]^{½} | t = (1/2) t_{H,o} a^{2} | a = [2 t/t_{H,o}]^{1/2} |
3. _{ } Pure vacuum: | _{v,o} | = 1; | _{ } a E(a) = [a^{2}]^{½} | t - t_{o} = t_{H,o} log_{e} a | a = e^{[(t - to)/tH,o]} |
4. _{ } Pure curvature: | _{k,o} | = 1; | _{ } a E(a) = [_{k,o}]^{½} | t = t_{H,o} a | a = t / t_{H,o} |
These recover the forms from [sec 4e]: 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 6cvii]
Note: the coefficients are different from the concordance approximations
because the current 's aren't unity.
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.
t_{age} = 2/(3 + 3w) t_{H} 2/3 t_{H} , 1/2 t_{H} , , t_{H} 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)
In general, deceleration gives t < t_{H} while acceleration gives t > t_{H} [image]
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).
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)
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.
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):
a() = ½ a_{max} (1 - cos ) ^{ } |
t() = ½ t_{H} a_{max} ( - 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}
a() = ½ a_{par} ( cosh - 1) ^{ } |
t() = ½ t_{H,o} a_{par} ( sinh - ) / (1 - _{m,o})^{1/2} |
where a_{par} = _{m,o} / (1 - _{m,o}) plays the role of a_{max}
For a closed geometry, additional possibilities include (see below):
Ages and futures for all these models can be nicely seen in these plots [image]
Hubble time: t_{H,o} = H_{o}^{-1} = 10.0 h^{-1} Gyr = 13.9 h_{72}^{-1} Gyr |
Hubble distance: r_{H,o} = c / H_{o} = c t_{H,o} = 13.9 h_{72}^{-1} G lyr = 4.26 h_{72}^{-1} Gpc |
they are units comparable to the current age and visible size of the Universe.
r(t_{o}) = c dt / a(t) (t_{e} to t_{o}) = r_{H,o} da / a^{2} E(a) (a to 1) = r_{H,o} -dz / E(z) (z to 0) |
Alternatively, from the RW metric: photons move on radial null geodesics, so d = ds = 0, giving:
c^{2} dt^{2} = a^{2}(t) dr_{o}^{2} c dt / a(t) = dr = r(t_{o}) (t_{e} to t_{o} and 0 to r_{o}) as before.
LBT = (t_{o} - t_{e}) = dt (t_{e} to t_{o}) = t_{H,o} da / a E(a) (a to 1); = t_{H,o} -dz / (1+z) E(z) (z to 0) |
Which are all straightforward to evaluate (numerically).
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}.
r(t_{e}) = 1.94 r_{H,o} / 1001 = 0.0019 r_{H,o} or only ~ 8 Mpc!
r_{H}(a) = r_{H,o} / E(a) = r_{H,o} / E(z) ^{ } |
or, re-expressing r_{H}(a) in its larger current comoving size, r_{o,H}(a), we have:
r_{o,H}(a) = r_{H}(a) / a = r_{H,o} / a E(a) = r_{H,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.
r_{p-hor}(t_{o}) = c dt / a(t) (0 to t_{o}) = r_{H,o} da / [a^{2}E(a)] (0 to 1) = r_{H,o} -dz / E(z) ( to 0) |
Using the concordance values in E(z) gives r_{p-hor} = 2.55 r_{H,o} = 10.9 h_{72}^{-1} Gpc (35.5 Gly)
Only if the initial expansion is zero can light cross everywhere in the first instant.
This occurs in two FRW cases:
r_{p-hor}(t) = c dt / a(t) (0 to t) = r_{H,o} da / a^{2}E(a) (0 to a) = r_{H,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)
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.
r_{e-hor}(t) = c dt / a(t) (t to ) = r_{H,o} da / a^{2}E(a) (a to ) = r_{H,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} /2a^{2}
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).
It also plays a crucial role generating and amplifying quantum fluctations.
Ultimately, these fluctations provide the seeds for future galaxy formation [sec 11].
Such diagrams are more complex on curved expanding space-times in cosmological GR:
It is, however, often possible to clean up these diagrams & reestablish 45^{o} light paths:
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
PE = -(3/5) GM^{2}/R = -(3/20) R^{5} H^{4} / G
KE = ½ 4r^{2}dr (H r)^{2} = +(3/20) R^{5} H^{4} / G
and we recover the Newtonian result: E_{tot} = PE + KE = E_{} = 0 for all R.
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,o}^{5} H_{o}^{4} / G = c^{5} / H_{o} G 10^{76} erg 10^{22} M_{} 10^{11} M_{gal}
This crudely illustrates how on cosmic scales the total energy, including rest mass, could be zero.
In GR the condition of flat geometry is equivalent to our Newtonian "zero energy".
Everything may have arisen from Nothing |
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.
Integrating over and for the total spherical shell area, we get:
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}
f = L / 4 D^{2} × 1 / (1 + z)^{2} = L / 4 D_{L}^{2} [ where D_{L} (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.
D(r_{o}) = R_{o} S_{k}(r_{o}/R_{o})^{ } | effective angular comoving distance |
r_{o}(z) = r_{H,o} -dz / E(z) (from z to 0)^{ } | true comoving proper distance |
S_{k}(x) = sin(x) x sinh(x) (k = +1 0 -1)^{ } | corrects for curvature |
R_{o} = r_{H,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]
f_{} = L_{} / 4 D_{L}^{2} × [ L_{}((1 + z)) / L_{}() ] × (1 + z) (units: erg/s/cm^{2}/Hz)
f_{} = (1 + z)^{-1} L_{}((1 + z)^{-1}) / 4 D_{L}^{2} (units: erg/s/cm^{2})
f_{} = (1 + z) L_{}((1 + z)) / 4 D_{L}^{2} (units: erg/s/cm^{2})
Usually, continuum fluxes require these relations, while emission lines are bolometric.
note a(t_{e}) is included since the light ray geodesics start at emission time, t_{e} [image]
d = ds / a(t_{e}) D = ds / D_{A} [ where D_{A} D / (1 + z) = D_{L} / (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.
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/D_{EA} [ where D_{EA} 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
d/dt = ds/D_{A} / dt' (1 + z) = v_{t} / D = v_{t} / D_{M} [ where D_{M} 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})
Moving to the RW-metric: f decreases (1 + z)^{-2}, and (d)^{2} increases (1 + z)^{2}
SB = f / (d)^{2} = L/(4D_{L}^{2}) / (s/D_{A})^{2} = L / 4s^{2} (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 proper comoving area at comoving distance r_{o} is A(r_{o}) = d D^{2}, where D = R_{o}S_{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 D^{2} 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:
V_{C}(z_{1} to z_{2}) = d r_{H,o} -D(z)^{2} dz/E(z) (from z_{1} to z_{2}) |
The total (d = 4) comoving volume out to redshift z turns out to be:
First some auxiliary functions & definitions we'll need:
_{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 / a^{2}E(a)
r_{o,H}(z) = c / aH(z) = r_{H,o} (1 + z) / E(z)
_{c} = S_{c} / D for an object expanding with the universe, with comoving size S_{c}
dV = dV_{c} × (1 + z)^{-3} is the physical (non-comoving) volume in the same shell
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 framework seems quite robust.
Hence, ½ v_{in}^{2} = 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}.
which is the relation we need.
_{t}(a) = 1 - _{k,o} / a^{2}E^{2}(a) = 1 - _{k,o} / [ v(a) / v_{o}]^{2} |
giving a simple rule [image]. :
Hence, for the standard model, the Universe was flatter in the past, and will get flatter in the future
a^{2}E(a)^{2} _{r,o} / a^{2} and t = t_{H,o} da / a E(a) (0 to a) ½ t_{H,o} _{r,o}^{-½} a^{2} 2.33 x 10^{19} h_{72}^{-1} a^{2} sec.
so the curvature becomes:
_{k}(a) _{k,o} a^{2} / _{r,o} 10^{4} a^{2} _{k,o} 5.0 x 10^{-16} t_{sec} h_{72}^{-1} _{k,o} |
Of course, inflation's early acceleration can generate just this kind of extreme flatness.
_{k} = -kc^{2} / R^{2}H^{2} = -kc^{2} / [ a^{2}R_{o}^{2} H_{o}^{2}E^{2}(a) ] = _{k,o} / a^{2}E^{2}(a)
So R^{2}H^{2} follows a^{2}E^{2}(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.
For example: the CMB is (almost) identical on opposite sides of the sky [image].
But the light from each side has only just reached us, in the middle
Hence the two sides have not yet been in contact -- so why are they the same?
Smoothing requires communication: things must be in contact or have a common origin to be be similar
You might think: Everything was "together/touching" at the big bang, so what's the problem?
However, in the standard cosmology, with a t^{1/2} we have v t^{-1/2} and the initial expansion (a = 0) is infinitely fast.
These [images] show how all points, no matter how close, are causally disconnected at the big bang.
In practice, smoothing can only occur at light speed:
we only expect things to be similar within a horizon distance.
At 400 kyr the horizon is roughly 400 kly across, which is ~1 degree on the CMB [image]
we only expect uniformity within 1 degree regions, which is clearly not the case at all.
The fundamental origin of the horizon problem is that the standard cosmology begins with infinite expansion and decelerates.
Inflation solves the problem by introducing accelerating expansion [image].
Two possible sources of fluctuations:
statistical fluctuations of particle numbers
quantum fluctuations in all things.
However, in a decelerating expansion, it transpires that such fluctuations decrease with expansion.
Basically, if the Universe is born smooth, it stays smooth, even given it's particulate/quantum nature.
Again, we need an accelerating expansion to amplify these seed fluctuations.
Solution: relax the condition that radiation (or matter) dominated the early universe
Instead, suggest that vacuum energy dominated
This ensures (a) the natural evolution is one of expansion
and (b) since it is accelerating expansion, it's launching speed will be exactly the escape speed.
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 - to)/tH}
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 -
PDF of powerpoint lecture slides on Inflation (4.4 Mb): here.
PDF of powerpoint lecture slides on Early Universe (4.7 Mb): here.
PDF of powerpoint lecture slides on CMB (11.4 Mb): here.
PDF of powerpoint lecture slides on Growth of Structure (4.8 Mb): here.
On the limits of extragalactic astronomy, so this will be brief
Some useful figures: [images]