Introduction

With the consolidation of the cold dark matter scenario for structure formation (e.g., Blumenthal et al. 1984), the study of galaxy evolution is entering an era of high precision, such that crucial questions can only be answered on the basis of accurate data and models. For instance, roughly half of all stellar mass in today's universe inhabits early-type galaxies (Fukugita, Hogan & Peebles 1998) yet a definitive picture of the history of star formation in these systems is still lacking. That is one of the chief motivations for the construction of stellar population synthesis models. High accuracy is required for such models because the spectrophotometric evolution of stellar populations proceeds at a very slow pace after the first Gyr or so, which makes it very hard to extract reliable age information from the integrated light of galaxies when most of the stars are old.

Stellar population synthesis aims at discerning the stellar mix in galaxies from their integrated spectral energy distributions. With that intent, models are computed which predict the evolution of magnitudes, colors, and absorption line indices of stellar populations. Comparisons of these models with the observations should constrain the age and metal abundance distribution of stars in galaxies, thus yielding constraints on their histories of star formation and chemical enrichment. The problem is complicated, however, as the spectral energy distributions of stellar populations respond to variations of different parameters in degenerate ways. One popular example is the age-metallicity degeneracy (e.g., Faber 1972, 1973, O'Connell 1980, Rose 1985, Renzini 1986, Worthey 1994), whereby stellar population colors and most absorption line strengths respond similarly to variations of age and metallicity. Major improvement was brought by the introduction of the Lick/IDS system of equivalent widths (Burstein et al. 1984, Gorgas et al. 1993, Worthey et al. 1994), which systematized the measurements of absorption line strengths in the spectra of stars and galaxies. Later on, with the development of models to predict the strengths of these indices as a function of stellar population parameters (e.g., Worthey et al. 1994, Worthey 1994, Bressan, Chiosi & Fagotto 1994, Weiss, Peletier & Matteucci 1995, Borges et al. 1995), key aspects of the evolution of early-type galaxies were unveiled. Worthey, Faber & González (1992) showed that giant ellipticals are characterized by enhancement of the abundances of light elements (see also Peterson 1976, O'Connell 1980, Peletier 1989), which possibly indicates that the bulk of their stars were formed in a rapid ( $\mathrel {\copy \simlessbox }$ 1 Gyr) star formation event. Later on, Worthey (1994) showed that the $H\beta$ index is more sensitive to age than to metallicity, thus allowing to break the age-metallicity degeneracy. Further extension of the models towards higher-order Balmer lines was accomplished by Worthey & Ottaviani (1997). Despite some controversy as to how clean an age indicator a given Balmer line is, this sparked a world-wide industry to estimate mean-ages and metallicities of stellar populations in galaxies. Until very recently, however, most of the work has been focussed on the ``green'' Lick indices: $H\beta$, Fe5270, Fe5335, Mg$_2$, and Mg $b$. The blue indices ( $\lambda \mathrel{\copy\simlessbox} 4500 {\rm\AA}$) were for a while relegated to a relative ostracism due to problems in the calibration of original Lick/IDS data in the blue and to intrinsic modeling difficulties related to the higher crowding of lines in that spectral region, which renders a clean absorption line strength measurement extremely difficult.

But the integrated spectra of early-type galaxies in the blue contain a wealth of information for those who take the challenge, as demonstrated by the pioneering work of Rose (1985, 1994). Moreover, combining accurate models in the blue to those currently available for red indices adds the benefit of a wider baseline, which proves to be extremely advantageous for stellar population studies (O'Connell 1976). Another important benefit of constructing consistent models within a large baseline that includes the blue spectral region lies in the need to interpret the integrated spectra of remote galaxies. Ongoing surveys based on 8-10 m class telescopes are obtaining large amounts of high-quality spectroscopic data for galaxies at z $\sim$ 1 (e.g., DEEP survey, Davis et al. 2003; VIRMOS-VLT Deep Survey, Le Févre et al. 2001, K20, Cimatti et al. 2002, Gemini Deep Deep Survey, Abraham et al. 2004). Because of strong telluric emission lines in the far red/near infrared, only the blue spectral region is accessible from the ground for galaxies at the involved redshifts, using current instrumental capabilities. Therefore, models which are consistent from the blue to the red are crucial, so that the mean ages and metallicities of remote systems, which are necessarily based on blue spectra, can be safely tied to those measured in nearby galaxies.

In this paper, we present a new set of model predictions for line indices and UBV colors of single stellar populations. Our goal is to produce models that are accurate and consistent throughout the spectral range going from $\lambda\lambda$ 4000 to 5400 ${\rm\AA}$. This is the fourth paper of a series dedicated to the study of stellar populations in the optical, with emphasis in the blue spectral region. In Schiavon et al. (2002a,b, hereafter Papers I and II), we studied the integrated spectrum of the moderately metal-rich Galactic globular cluster 47 Tuc and in Schiavon, Caldwell & Rose (2004, hereafter Paper III) we constructed and analyzed the integrated spectrum of the metal-rich, intermediate-age Galactic open cluster, M 67.

Our models are based on a new set of fitting functions for indices in the Lick system. We omit on purpose the ``IDS'' part of the usual nomenclature of this system of equivalent widths because, as it will be seen in Section 2.2, our models are not in the Lick/IDS system, as they are not based on index measurements taken in the standard Lick/IDS stellar library (Burstein et al. 1984, Gorgas et al. 1993, Worthey et al. 1994). Instead, the index measurements that form the backbone of our models were taken in a much more recent spectral library, by Jones (1999). The spectra from that library are flux-calibrated, thus being unaffected by the response curve of the old Lick Image Dissector Scanner. However, for reasons that will become clear in Section 2.2, our line indices are measured at the relatively low resolution of the original Lick/IDS system. We are aware of the ongoing work on the construction of better spectral libraries with higher resolution than that of the Lick system, and spanning a wide range of stellar parameters (e.g., Le Borgne et al. 2003, Valdes et al. 2004). However, when this project started, these libraries were not available publicly. Moreover, our main goal is to apply these models to study distant giant early-type galaxies, whose spectra are irretrievably smoothed by their high velocity dispersions to resolutions that are comparable to that of the Lick system ($\sim$ 8 ${\rm\AA}$).

It is also important to justify here our reasons to adopt fitting functions, even though in Papers I and II we produced synthetic spectra of single stellar populations. The main reason is that the degree of accuracy needed for such model predictions cannot be achieved on the basis of the Jones (1999) library, because of its relatively limited coverage of stellar parameter space. Moreover, fitting functions are very convenient for a number of reasons. For instance, they can be easily implemented in any evolutionary synthesis code. In addition, models based on fitting functions can also be corrected to yield line index predictions for varying abundance patterns.

A key feature of our models is related to the stellar parameters adopted for the library stars. They have to be homogeneous, internally accurate, and free of important systematic effects. Accuracy is very important, for it allows a precise assessment of the behavior of stellar observables as a function of fundamental parameters. This aspect of our models was very carefully crafted.

One of the chief applications of stellar population synthesis is the estimation of mean elemental abundances of stars in remote systems from absorption line indices measured in their integrated spectra. Ratios of elemental abundances, such as that between magnesium and iron, hold important clues for the history of star formation and chemical evolution of galaxies (e.g., Matteucci & Tornambè 1987; Wheeler, Sneden & Truran 1989; Peletier 1989, Worthey, Faber & González 1992; Edvardsson et al. 1993; McWilliam 1997; Worthey 1998). Therefore, models that are able to convert, for instance, Mg $b$ and Fe4383 measurements into a mean [Mg/Fe] for a given galaxy are highly desirable.

This is unfortunately not very easy to achieve because of two reasons. First, the integrated spectra of galaxies are smoothed due to the intrinsic dispersion of the velocities of their member stars along the line of sight. As a result, all the absorption lines are blended, making it impossible to isolate absorption features that cleanly indicate the abundance of a given chemical species. Second, detailed abundance patterns for the majority of the stars used in the construction of the models--hence the abundance pattern of the models themselves--are unknown.

A method to address these difficulties was devised by Trager et al. (2000) and further developed by Proctor & Sansom (2002), Thomas, Maraston & Bender (2003a), and Thomas, Maraston & Korn (2004), and Korn, Maraston & Thomas (2005). The core of this method resides in the use of sensitivities of Lick indices to variations in the elemental abundances of all the relevant chemical species with absorption lines included in each index passband and pseudo-continuum windows. Trager et al. used the Tripicco & Bell (1995) tabulations of index sensitivities computed from spectrum synthesis adopting model atmospheres of stars with representative stellar parameters. The sensitivities were used to integrate the effect of abundance ratio variations of the main Lick indices in the green region. That allowed them to estimate by how much the mean [Mg/Fe] of the stellar populations of elliptical galaxies in their sample depart from that of the spectral library used as input in their models (which they assumed to be equal to solar). The method was extended by Thomas et al. (2003a) to include all the Lick/IDS indices in Worthey et al. (1994). Later on, Korn et al. (2005) computed new sensitivities that include also the higher order Balmer lines defined by Worthey & Ottaviani (1997).

It is vital for the success of this method that the abundance pattern of the models be well-known. As emphasized by Thomas et al. (2003a), models that are based on empirical stellar libraries are characterized by an abundance pattern that is equal to that of the stars that make up the adopted spectral library. Therefore, we decided to survey the literature for elemental abundance determinations of the stars in the spectral library adopted in our models. We provide mean abundance ratios as a function of [Fe/H] for several important chemical elements. This information is used in Section 4.3, in combination with our fitting functions and the Korn et al. (2005) sensitivity tables, to produce model predictions for varying abundance patterns. We present a detailed study of the response of line indices to age and elemental abundance variations, in order to explore ways in which our models can be used to constrain those parameters for intermediate-age and old stellar populations. On the basis of the insights gained in this study, we develop a new method to estimate the mean luminosity-weighted age of stellar populations, as well as their abundances of iron, magnesium, calcium, carbon, and nitrogen.

In a degenerate problem like that of stellar population synthesis, knowing the answers that are to be expected for a given set of input parameters is priceless. Therefore, a detailed comparison of our model predictions for single stellar populations with accurate data for well-known Galactic clusters is performed. Our goal is to match all 16 indices for a small sample of clusters which spans the entire range of stellar population parameters of interest. We hope to convince the reader that significant advance has been made towards meeting this initial goal. At the end of this exercise, we show that the ages and metal abundances derived from application of our models to the integrated light of stellar populations is meaningful in an absolute sense, i.e., they are consistent with metallicity and age scales defined by known local systems, such as Galactic clusters and stars in the solar neighborhood.

Once we are convinced that the comparison of the models with cluster data gives satisfactory results, we turn our attention to galaxies. In this paper, we refrain from pursuing a detailed analysis of the galaxy data and instead make more qualitative comparisons between data and models and discuss what can be learned therefrom. We first compare our models to the data from Trager et al. (2000a,b, henceforth simply Trager et al. 2000) in order to make sure that we reproduce some well established results, such as the $\alpha$-enhancement characteristic of massive early-type galaxies and the spread in the mean ages of their stellar populations. Next, we take advantage of the wide baseline covered by our models to compare them to measurements taken on stacked spectra of early-type galaxies from the Sloan Digital Sky Survey (Eisenstein et al. 2003). We determine the mean abundances of iron, magnesium, calcium, and, for the first time, those of carbon and nitrogen for the stars in early-type galaxies. The behavior of these abundances as a function of galaxy luminosity is studied. Of all abundance ratios studied, [N/Fe] is the one that seems to be the most strongly correlated with galaxy luminosity, perhaps indicating an important secondary source of nitrogen enrichment in these galaxies. If confirmed, this result may be telling us that there is a minimum duration for the star formation in early-type galaxies, which is set by the lifetimes of the stars contributing secondary nitrogen. If the characteristic masses of these stars, as proposed by Chiappini, Matteucci & Ballero (2005), range between 4 and 8 $M_\odot$, these timescales are of the order of 50-200 Myr. A more accurate prediction can be obtained on the basis of detailed chemical evolution modeling.

Regarding stellar ages, unlike what we found in the case of clusters, models do not match the data consistently throughout the spectral region considered. Specifically, bluer Balmer lines tend to indicate younger mean ages than $H\beta$. This might be one of the many instances in the field of stellar population synthesis where, when the models do not match the data, there might be something interesting to be learned. We argue that we are detecting an age spread in the stellar content of early-type galaxies.

This paper is organized as follows. In Section 2 we describe the stellar library used in the models and the determination of the stellar parameters and abundance pattern of its constituent stars. The fitting functions are presented in Section 3. Model construction is presented in Section 4. In Sections 5 and 6 the models are compared with cluster and galaxy data, respectively. Our conclusions are summarized in Section 7. The reader who is not interested in model construction details should go directly to Section 5.