---
abstract: |
    We place constraints on the redshift-averaged, effective value of the equation of state of dark energy, $w$, using [*only*]{} the absolute ages of Galactic stars and the observed position of the first peak in the angular power spectrum of the CMB. We find $w<-0.8$ at the 68% confidence level. If we further consider that $w \geq -1$, this finding suggests that within our uncertainties, dark energy is indistinguishable from a classical vacuum energy term.

    We detect a correlation between the ages of the oldest galaxies and their redshift. This opens up the possibility of measuring $w(z)$ by computing the relative ages of the oldest galaxies in the universe as a function of redshift, $dz/dt$. We show that this is a realistic possibility by computing $dz/dt$ at $z \sim 0$ from SDSS galaxies and obtain an independent estimate for the Hubble constant, $H_0 = 69 \pm
    12$ km s$^{-1}$ Mpc$^{-1}$. The small number of galaxies considered at $z>0.2$ does not yield, currently, a precise determination of $w(z)$, but shows that the age–redshift relation is consistent with a Standard $\Lambda$CDM universe with $w=-1$.
author:
- 'Raul Jimenez, Licia Verde, Tommaso Treu, and Daniel Stern'
title: |
    Constraints on the equation of state of dark energy\
    and the Hubble constant from stellar ages and the CMB
---

Introduction
============

There is now solid observational evidence that the universe is accelerating. The absolute ages of high-redshift galaxies [@D+96; @Spinrad+97] and the value of the Hubble constant $H_0$ [@Freedman+01], the Hubble diagram of Type Ia supernovae [@Riess+98; @Perlmutter+99], the angular power spectrum of the cosmic microwave background [CMB; @Benoit+02; @Ruhl+02; @Goldstein+02] in combination with galaxy surveys [@Efstathiou+2dF02; @Verde+2dF02] and lensing [@WTJZ02], all indicate that the universe is flat and dominated at present by some form of dark energy with negative pressure. Strong confirmation for this comes from the newly released results from the [*Wilkinson Microwave Anisotropy Probe*]{} ([*WMAP*]{}) first year data [@SpergelWMAP03].

The equation of state of the dark energy, $p=w \rho$, expresses the ratio between the pressure, $p$, and the mass density, $\rho$, of the dark energy in terms of the parameter $w$ (in units of $c=1$). The value of $w$ could either be constant, as in the case of a cosmological constant ($w=-1$), or be time-dependent, as in the case of a rolling scalar field or “quintessence” [@PeeblesRatra88; @CDS98]. Any such behavior would have far-reaching implications for particle physics.

In this paper we use the ages of the oldest stellar populations between $z=0$ and $z=1.5$ to derive new and independent constraints on two key cosmological parameters, $w$ and $H_0$. Both parameters are relatively poorly constrained by CMB data alone. For example, in order to get accurate constraints on $w$, @SpergelWMAP03 combined the [*WMAP*]{} data with six external data sets: two additional CMB data sets [CBI and ACBAR; @Pearson+02; @Kuo+02], large scale structure measurements from the 2dF galaxy redshift survey [2dFGRS; @Percival+01; @Verde+2dF02], measurements of the Lyman $\alpha$ forest power spectrum [@Croft+02; @Gnedin+02], the luminosity distance–redshift relation from Type Ia supernovae measurements [@Garnavich+98; @Riess+01], and $H_0$ from the [*Hubble Space Telescope*]{} ([*HST*]{}) Key Project [@Freedman+01]. Therefore it is particularly important to obtain accurate and independent measurements of cosmological parameters relying on different physics.

In Section 2 we focus on $w$, summarizing the basis of our method and deriving tight constraints on the equation of state using the ages of globular clusters in combination with the location of the first acoustic peak in the CMB power spectrum. Then, in Section 3, we determine stellar ages for a sample of early-type galaxies between $z=0$ and $z=1.5$ and use the age–redshift relationship defined by the old envelope to determine $H_0$. The prospects of using this method to derive a measurement of $w(z)$ are also discussed. Section 4 summarizes and discusses the results.

Throughout the paper we adopt the following notation. The Hubble constant is $H_0=100\,h\,$ km s$^{-1}$ Mpc$^{-1}$ and $c$ is the speed of light. The density of baryons, matter, radiation, photons, neutrinos, and dark energy, at present, in critical units, are indicated as $\Omega_b$, $\Omega_m$, $\Omega_{\rm rad}$, $\Omega_{\gamma}$, $\Omega_{\nu}$, and $\Omega_{\Lambda}$ respectively ($\Omega_{\rm rad}\equiv\Omega_{\gamma}+\Omega_{\nu}$). The physical quantities, independent of $h$, are indicated with $\omega$; e.g., $\omega_m=\Omega_mh^2$. When needed, a standard, flat, reference $\Lambda$CDM model with $\Omega_m=0.27$, $h=0.71$, $\Omega_b h^2=0.023$, and $w=-1$ is assumed [@SpergelWMAP03].

The value of $w$ from stellar ages
==================================

In this section we use the ages of the oldest globular clusters in combination with the location of the first acoustic peak of the CMB power spectrum [@PageWMAP03] to constrain the value of $w$. The method is briefly described in Section 2.1 and the results are given in Section 2.2.

Method
------

If the universe is assumed to be flat, the position of the first acoustic peak ($\ell_1$ in the standard spherical harmonics notation) depends primarily on the age of the universe and on the effective value[^1] of $w$ [@CDS98; @HFZT01; @KCS01]. As noted by these authors, for a fixed $w$ value, a change in the physical density parameter $\omega_m=\Omega_mh^2$ that keeps the characteristic angular scale of the first acoustic peak fixed will also leave the age approximately unchanged. Thus an independent estimate of the absolute age of the universe at $z=0$ combined with a measurement of $\ell_1$ yields an estimate of $w$ largely independent of other cosmological parameters (e.g., $\Omega_m$ and $h$).

This can be better understood by considering that for a constant $w(z)$, the age of a flat universe is given by $$t_0= H_0^{-1} \int_{0}^{\infty} (1+z)^{-1}\left[\Omega_{\rm
rad}(1+z)^4+\Omega_m(1+z)^3+\Omega_{\Lambda} (1+z)^{3(1+w)}\right]^{-1/2}
dz\;.
\label{eq:ageflatwconst}$$ The position of the first acoustic peak is fixed by the quantity $$\theta_A=r_s(a_{dec})/d_A(a_{dec})\;,
\label{eq:thetaa}$$ where $a_{dec}$ is the scale factor at decoupling, $r_s(a_{dec})$ is the sound horizon at decoupling, and $d_A(a_{dec})$ is the angular diameter distance at decoupling. For a flat universe, $$r_s(a_{dec})=\frac{c}{H_0\sqrt{3}}\int_0^{a_{dec}}\left[\left(1 +
\frac{3\Omega_b} {4\Omega_{\gamma}}\right)(\Omega_{\Lambda}x^{1-3w} +
\Omega_mx + \Omega_{\rm rad})\right]^{-1/2}dx
\label{eq:rs}$$ and $$d_A(a_{dec})=\frac{c}{H_0}\int_{a_{dec}}^1\left[\Omega_{\Lambda}x^{1-3w}
+ \Omega_mx + \Omega_{\rm rad}\right]^{-1/2}dx\;.
\label{eq:dA}$$ See @VerdeWMAP03 for more details. We use equations \[eq:ageflatwconst\] to \[eq:dA\] and fix $\theta_A$ to be consistent with the [*WMAP*]{} determination [$\ell_1\simeq
220$; @PageWMAP03]. Figure 1 shows the allowed region in the age-$w$ plane obtained for $\Omega_b h^2=0.02$, $0.05<\Omega_m<0.4$, and $0.5<h<0.9$. It is clear from the plot that an independent and accurate age determination of the universe will provide a measurement of $w$. Thus by using only the [*WMAP*]{} observation of $\ell_1$ and an independent estimate of the age of the universe, we can place constraints on $w$, largely independent of other cosmological parameters.

Results from the ages of globular clusters
------------------------------------------

The ages of the oldest globular clusters (GCs) provide a lower limit to the total age of the universe. Numerous star-forming galaxies have now been observed up to redshift $z=6.6$ [e.g., @Kodaira+03]. Since the oldest GCs contain the oldest stellar populations in galaxies, it is reasonable to assume that GCs too have formed at redshift $z>6$. Furthermore, since small halos are easily destroyed after reionization [e.g., @Bullocketal01], GCs likely formed prior to the reionization epoch. Under this assumption, the [*WMAP*]{} recent estimate of the reionization redshift $z_{\rm reion}\sim
17$ [@KogutWMAP03] implies that the ages of the oldest globular cluster do not significantly underestimate the age of the universe.

Globular cluster ages can reliably be estimated in several different ways: from the luminosity of the main-sequence turn-off (if the distance is known), from detached binaries, from the termination of the white-dwarf sequence, and from the luminosity function method which provides a joint constraint on the age and distance of a GC. @R+96 determine the distance to NGC 6752 by means of accurate photometry of GC white dwarfs and obtained an age of $14.5 \pm 1.5$ Gyr. Using the same method they find 47 Tucanae to have an age of $13
\pm 2.5$ Gyr [@Zoccali+01]. @Gratton+97, using the new [*Hipparcos*]{} distance scale to determine main-sequence turn-offs, estimate an age of $11.8^{+2.1}_{-2.5}$ Gyr (95% confidence level) for the oldest Galactic GCs. [@JimenezPadoan97] apply the luminosity function method to M55 and obtain an age of $12.5 \pm 1.0$ Gyr (95% confidence level). @Hansen+02, using very deep [*HST*]{} images, find the white dwarf cooling sequence for M4 and estimate its age to be $12.7 \pm 0.7$ Gyr at the $95$% confidence level. Using the theoretical age–mass relation and applying it to the detached eclipsing binary OGLEGC-17, [@ChaboyerKrauss02] determine Omega Centauri to have an age of $11.2 \pm 1.1$ Gyr (95% confidence level). These age estimates are all remarkably consistent. @KraussChaboyer03 perform the most careful analysis to date of the effects of systematics in GC age determinations. They estimate the age of the oldest Galactic GCs using the main-sequence turn-off luminosity and evaluate the errors with Monte-Carlo techniques, paying careful attention to uncertainties in the distance, to systematics, and to model uncertainties. They find a best-fit age for the oldest Galactic GCs of $12.5^{+3.4}_{-2.2}$ Gyr (95% confidence limits). We adopt their probability distribution for the oldest GC age, which we find can be accurately described by $$P(t)=\frac{A}{\sigma (t-T)} \exp \left[- \frac{\ln \left[ (t-T)/m \right
]^2}{2 \sigma^2} \right ]
\label{eq:kraussagecorr}$$ where $t$ denotes the age of the oldest GCs, $A=1.466$, $\sigma=0.25$, $T=6.5$ and $m=5.9$. Since reionization took place at $z\sim 17$ [@KogutWMAP03] and we assume that oldest GCs must have formed before then, for all reasonable cosmologies we only need add $0.2-0.3$ Gyr to the GC ages to obtain an estimate of the age of the universe. We conservatively add 0.3 Gyr[^2].

In order to constrain $w$, we assume a flat universe and Monte Carlo simulate the distribution of $\ell_1$ subject to only weak constraints on the other cosmological parameters. For different values of $w$, we generate $10^5$ models, randomly sampling the cosmological parameters $\Omega_m$, $\Omega_bh^2$ and $h$ with uniform priors, $0.5< h < 0.9$, $0.05 < \Omega_m < 0.45$, and $0.01 < \Omega_bh^2 < 0.03$. We then impose an age of the universe constraint by randomly sampling these models so that the age of the universe has a probability distribution whose shape is given by equation \[eq:kraussagecorr\], but offset by $0.3$ Gyr. We use the publicly available code [CMBFAST]{} [@SeljakZaldarriaga96] to compute the resulting distribution of $\ell_1$. This is shown in Figure 2 where the two solid lines are the 68% and 90% [*joint*]{} confidence levels.

As expected, the age alone does not constrain $w$; there is a degeneracy between $w$ and $\ell_1$. If we now use the [*observed*]{} position of the first acoustic peak as recently measured, in a model independent way, from [*WMAP*]{} [@PageWMAP03], we can constrain $w$ with high accuracy. The filled contours in Figure 2 are marginalized values for $w$ at the 68% and 90% confidence levels. Thus we find $w<-0.8$ ($w < -0.67$) at the 68% (90%) confidence level. If we drop the assumption that the oldest GC must have formed before reionization, but still impose that they must have formed by $z=6$, this constraint is slightly weakened. In this case the age distribution of equation \[eq:kraussagecorr\] needs to be offset by $\la 1$ Gyr and we obtain $w<-0.5$ at the 90% confidence level. We have also repeated the calculation using Gaussian priors for the cosmological parameters, $h=0.7 \pm 0.2$, $\Omega_m=0.25
\pm 0.2$, $\Omega_b h^2=0.02 \pm 0.01$, and find similar, but slightly more stringent constraints on $w$.

We stress here that this determination depends solely on the GC determination of the age of the universe and on the [*observed*]{} position of the first acoustic peak in the CMB power spectrum. Our constraint is slightly less stringent than that obtained by @SpergelWMAP03 from a joint likelihood analysis of [*WMAP*]{} with six external data sets ([*WMAP*]{} + CBI + ACBAR + 2dFGRS + Lyman$\alpha$ forest power spectrum + Type Ia supernovae + $H_0$ constraint from the [*HST*]{} key project), but is tighter than the CMB-only ([*WMAP*]{} + CBI + ACBAR) determination and comparable to the [*WMAP*]{} + ACBAR + CBI + [*HST*]{} constraint.

$H_0$ and $w(z)$ from the ages of unresolved stellar populations at $z=0-1.5$
=============================================================================

Method
------

We now turn our attention to the age of the oldest objects in the universe at $z \ge 0$ as a function of redshift. As recently suggested by @JL02, these can be used to determine $H(z)= - \frac{1}{1+z}
\frac{dz}{dt}$ and therefore to measure $w(z)$ and $H_0$. The Hubble parameter is given by $H^2(z) = H^2_0 [\rho_{\rm tot}(z) / \rho_{\rm tot}(0)]$, where $\rho_{\rm tot}$ is the total density of the universe. For a flat universe composed of matter and dark energy, $H(z)$ is related to the equation of state by $$H_0^{-1}{dz\over dt} = - (1+z) {H(z)\over H_0}= -(1+z)^{5/2}
\left [\Omega_m +\Omega_{\Lambda} \exp \left \{3 \int_{0}^{z}
\frac{w(z^\prime)}{(1+z^\prime)} dz^\prime  \right \} \right ]^{1/2} ,
\label{eq0}$$ where we have used the conservation equation for dark energy density, $\dot{\rho_\Lambda} = -3 H(z) [1 + w(z)] \rho_\Lambda$. Thus a measurement of $dz/dt$ is also a measurement of $w(z)$; for further details, see @JL02.

To obtain $dz/dt$, @JL02 proposed to use the old envelope of the age–redshift relation of E/S0 galaxies. We assume that E/S0 at different redshifts are drawn from the same parent population with the bulk of their stellar populations formed at relatively high redshift [e.g., @Bower+92; @Stanford+98]. Then, at relatively low redshift, they are evolving passively and may be used as “cosmic chronometers”.

In the rest of this section we will apply this method to a sample of spectra (described in §3.2) to identify the age–redshift relation defined by the old envelope (§3.3) and derive an independent measurement of $H_0$ (§3.4). We show that at the level of accuracy of our measurement, the use of the age–redshift relation allows us to apply this method even if a fraction of the stars in E/S0 galaxies is assembled at $z < 1$ [e.g., @TSCMB02]. We also briefly discuss the prospects of using this method to derive $w(z)$.

Sample and measurement
----------------------

In order to apply this method we need high-quality spectra of a sample of old stellar populations covering the largest possible range in redshift. To this aim we combine the following data sets: (i) the luminous red galaxy (LRG) sample from the Sloan Digital Sky Survey (SDSS) early data release [@Eisenstein+01]; (ii) the sample of field early-type galaxies from @TSCMB99 [@TSMCB01; @TSCMB02 hereafter the Treu et al. sample]; (iii) a sample of red galaxies in the galaxy cluster MS1054$-$0321 at $z=0.83$; and (iv) the two radio galaxies 53W091 and 53W069 [@D+96; @Spinrad+97; @NDJH03 Dey et al., in preparation].

The LRG sample is the most heterogenous, since it includes red galaxies in general, regardless of morphology. The Treu et al.sample contains high quality spectra with average $S/N=16$ per Å. The spectra in the Treu et al. sample have higher $S/N$ than the typical LRG spectrum, but the wavelength coverage is better for the LRG sample (see example in Figure 3). To select potentially passively evolving galaxies, we include only galaxies with $S/N > 9$ per Å and for which passively evolving synthetic stellar population models [@JPMH98] provide a good fit to the continuum. This selection criterion discards dusty or star-forming objects. The spectra of seven red galaxies in the cluster MS1054$-$0321 were obtained by one of us (DS) on UT 11 March 2002 using the dual-beam Low Resolution Imaging Spectrometer [LRIS; @Oke+95] at Keck Observatory. On the blue side, we used the 300 lines mm$^{-1}$ grism ($\lambda_{\rm blaze} =
5000$ Å; spectral resolution $\Delta \lambda_{\rm FWHM} \approx
13.5$ Å) and on the red side, we used the the 400 lines mm$^{-1}$ grating ($\lambda_{\rm blaze} = 8500$ Å; spectral resolution $\Delta
\lambda_{\rm FWHM} \approx 8$ Å). These observations, which were obtained at the parallactic angle, totaled one hour of integration. In order to ensure no gaps in the spectra, half the data were obtained with a D560 dichroic and the remainder were obtained with a D680 dichroic. These spectra have a $S/N$ of 8. We also add the red radio galaxies 53W091 and 53W069 for which accurate ages have already been determined [@D+96; @Spinrad+97; @NDJH03 Dey et al., in preparation]. Examples of the spectra used are given in Figure 3.

The age of the dominant stellar population in the galaxies is obtained by fitting single stellar population models [@JPMH98; @JDMPP03] to the observed spectrum. These models have been extensively tested [@KFJ02; @JFMG03] and have two free parameters: age and metallicity. By construction, star formation is assumed to occur in a single burst of duration much shorter ($< 10$%) than the current age of the galaxy [@JFDTPN99]. Therefore, the derived ages are single-burst equivalent ages. The best fit model is found by standard $\chi^2$ minimization. Then, stellar ages and their errors are obtained by marginalizing the resulting likelihood surface over metallicity. Some examples for the parameter confidence regions, at the 1 and 2$\sigma$ level, are shown in Figure 3. Note that for galaxies with high $S/N$, like the Treu et al. sample, the constraints in both age and metallicity are very tight and thus the so-called age-metallicity degeneracy is dramatically reduced and does not affect the age determination at a significant level.

Before we can use the estimated ages to constrain $H_0$ we have to estimate the effects of the potential systematic uncertainties related to adopting a single-burst stellar population model and a single-metallicity model. As far as the first uncertainty is concerned, we know that even massive field E/S0 galaxies can experience some level of star formation activity at $z>0.2$ [@MAE01; @TSCMB02; @WHWL02]. The presence of a young and luminous stellar population superimposed on a dominant old stellar population can bias the single-burst equivalent ages of the spectrum to be much younger than the mass-weighted average age. Although this is a serious concern for the interpretation of individual cases, our method relies on the old envelope of the age distribution at any given redshift. Therefore, the effects of this bias on the determination of $H_0$ and $w$ are strongly reduced, as we discuss further in § 3.2. Concerning single metallicity, the uncertainty connected to metallicity gradients is negligible with respect to other sources of uncertainty. In fact, our spectra typically cover $6$ kpc (for the LRG) to $10$ kpc (for the higher redshift samples). Given that typical metallicity gradients in E/S0s follow $d {\rm log} [{\rm Fe}/{\rm H}]/
d {\rm log} r = -0.2$ [@Davies+93], metallicity only changes by 0.2 dex within this region. This is fairly small and the single-metallicity approximation affects our results well within the 1$\sigma$ value of the error in the recovered metallicity.

An age–redshift relation and $w(z)$
-----------------------------------

Figure 4 shows the derived single-burst equivalent ages of the galaxies as a function of redshift. The circles correspond to galaxies in the SDSS LRG sample, triangles to the Treu et al. sample, diamonds are galaxies in MS1054$-$0321, and crosses are 53W091 and 53W069. For clarity, the SDSS LRG points have not been plotted for ages $<7$ Gyr. Typical errors on the ages of LRG galaxies are 10% and are not plotted. An age–redshift relation (i.e., an “edge” or “envelope” of the galaxy distribution in the age–redshift plane) is apparent from $z=0$ to $z=1.5$.

Although an envelope can be identified, it is important to notice that there is a large spread of ages at any given redshift. In particular, the measured age for many galaxies makes them significantly younger than the age–redshift envelope. This is in part due to a real spread in the star formation histories of early-type galaxies, especially in the field, but mostly a result of using single-burst equivalent ages. Single-burst equivalent ages effectively measure a luminosity-weighted age. Therefore, even a relatively minor episode of star formation is sufficient to significantly underestimate the age of the bulk of (old) stars. Indeed, many of the galaxies in the Treu et al. sample show \[\] $\lambda$3727 emission, consistent with minor ongoing star formation activity [@TSCMB02]. Is this contamination enough to bias our results, or is the use of the relative ages of the old-envelope sufficient to reduce this bias to useful levels?

To answer this question we consider a scenario where the bulk of stars in E/S0 galaxies formed at high redshift, while a relatively small fraction (up to 10% in mass) of stars is formed at later times ($z<1.5$). This scenario is approximately consistent with the observed evolution of the number density [@Im+02], colors [@JFDTPN99; @MAE01], and spectral properties of early-type galaxies [@Trager+00; @TSCMB02]. Within this scenario, we use Monte Carlo simulations to estimate the effects of this bias on the age–redshift relation and ultimately on $H_0$ and $w(z)$ determinations. We generate galaxy spectra assuming that after the initial burst of star formation at $z>3$ an extra 10% of the stellar mass is formed at a time $t_1$ between redshift $0 < z < 1.5$ with uniform probability for $t_1$. For simplicity, we use discrete values for the ages with spacing 0.1 Gyr and assume a flat $\Omega_m=0.27$, $w=-1$ and $h=0.71$ cosmology to convert time to redshift. The metallicity for a given galaxy is chosen to be uniformly distributed in the range $0.1 <
Z/Z_{\odot} < 4$. Finally, we add random noise to the spectra so that $S/N=15$ and determine the age with the same procedure used for the data.

The results of our Monte Carlo simulations are shown in Figure 5. The top panel shows the simulated age–redshift distribution if only six early-types are found per redshift interval[^3]. The edge is blurred and smeared due to recent star formation contamination and because of the error in the age determination; nevertheless, it is clearly recognizable. The galaxies observed within a few 100 Myr of bursts have very small single-burst equivalent ages, as observed for some galaxies in the LRG and Treu et al. sample, but they do not interfere with the location of the old envelope.

The lower panel shows the distribution for the case where 60 early-types are observed per redshift bin (this is less than the number of galaxies in the SDSS LRG sample). The solid line shows the theoretical position of the edge, which is remarkably close to observed edge. Note also that increasing the number of galaxies has improved the agreement between the observed and theoretical edge. Thus we conclude that small, recent bursts of star formation do not introduce a significant bias on the edge position; our primary uncertainty in determining $w(z)$ to high redshift is the sparse size of the sample of distant, early-types considered here. We can now attempt to recover $dz/dt$ from this simulation by looking at the shift in the edge position as a function of $z$ [see § 3.4 and @JL02] and thus deduce $w(z)$. We recover $w(z)=-1 \pm 0.2$ for the whole redshift range. This shows that, with a sample similar to the LRG sample extended to higher redshift, we should be able to measure $w(z)$ within a few 10%. This is similar to what was already concluded in @JL02.

Having established that the age–redshift relation for the old envelope is not significantly affected by the bias caused by later episodes of star formation, we now consider the observational limits that we can impose on $w(z)$. As a consistency check we note that the age at $z=0$ obtained here is in good agreement with GC ages (see §2) and that the Treu et al.galaxy ages agree with those from of SDSS LRG sample where they overlap. The solid line in Figure 4 corresponds to the age–redshift relation for our flat, reference $\Lambda$CDM model: $\Omega_m=0.27$, $h=71$, and $w=-1$. This is consistent with the observed age–redshift envelope, and it seems to indicate that we live in a universe with a classical vacuum energy density and that, on average, stars in the oldest galaxies formed about 0.7 Gyr after the Big Bang (e.g., at redshifts $z \sim 7-10$). To make the observed age–redshift relation consistent with a model with widely different $w(z)$ behavior, we would have to infer that early-type galaxies at different redshifts have very different formation epochs for their stellar populations. For example, the dotted line in Figure 4 indicates the age-redshift relation for an evolving population in a universe where $w(z)=-2$ for $z > 1$ and grows linearly from $w=-2$ at $z=1$ to $w=0$ at $z=0$. For this model to work, we would have to conclude that the oldest galaxies at $z\sim 1.5$ formed about 0.7 Gyr after the Big Bang, but that the oldest galaxies at $z\sim
0$ formed $\sim 3$ Gyr after the Big Bang. Since the oldest galaxies in the LRG sample are as old as the oldest GCs, we consider this to be an unlikely explanation. Furthermore, not only would this scenario require that [*a*ll]{} local galaxies either formed their stars more recently than galaxies at high redshift or had a recent episodes of recent star formation, but would also require a remarkable fine-tuning for them all to obtain exactly the same age. We therefore conclude that this extreme model for $w(z)$ is unlikely given the data.

Unfortunately, the small number of galaxies in our samples at $z>0.2$ does not allow us to compute $dz/dt$ with enough accuracy to constrain $w(z)$, and therefore this measurement will have to await better data. However, the $z < 0.2$ region is well populated by LRG galaxies (Figure 4) and we can determine $dt/dz$ at $z\sim 0$. This is illustrated in the bottom panel of Figure 4 by the clear shift in the upper envelope of the age histogram between the two redshift ranges $0 < z < 0.04$ and $0.08 <
z < 0.12$. In the next section we will investigate this phenomenon in more detail and use it to determine $H_0$.

The value of $H_0$
------------------

By concentrating on the age–redshift relation for SDSS LRG galaxies at $z \la 0.2$, we now determine the Hubble constant $H_0$. This procedure relies upon determining the “edge” of the galaxy age distribution in different redshift intervals.

If the ages of the galaxies were known with infinite accuracy, for each galaxy$_i$ at redshift $z_i$, one could associate an age $a_i$. The probability for the age of the oldest stellar population in galaxy$_i$, $P(t|a_i)$, would be given by a step function which jumps from 0 to 1 at $a_i$. An age–redshift relation edge could then be obtained by dividing the galaxy sample into suitably-large redshift bins and multiplying the $P(t|a_i)$ for all the galaxies in each bin.

In practice, the age of each galaxy is measured with some error $\delta a_i$. We thus assume $P(t|a_i)=1$ if $t>a_i+\delta_{a_i}$ and $\ln P(a_i)=-x^2$ where $x=(a_i+\delta_{a_i}-t)/(\sqrt{2}\delta_{a_i})$ otherwise. We have divided the $z < 0.2$ portion of the LRG sample into 51 redshift bins. For each bin, we obtain $P(z,t)$ by multiplying the $P(t|a_i)$ of the galaxies in that bin. We define the “edge” of the distribution $t(z)$ to be where $\ln P(z,t)$ drops by $0.5$ from its maximum (i.e. $\Delta \ln P=0.5$. We associate an error to this determination $\delta_{t(z)}$ given by $t_{2}(z)-t(z)$ where $t_{2}(z)$ corresponds to where $\ln P(z,t)$ drops by $2$ from its maximum (i.e. $\Delta \ln
P=2$, this approximately corresponds to the 68% confidence level). This procedure makes the determination of the “edge” less sensitive to the outliers (see also @RBBG97). Figure 6 presents the resultant $t(z)$ relation.

For $z<z_{\rm max}$, we fit $t(z)$ with a straight line whose slope $dt/dz$ is related to the Hubble constant at an effective redshift by $H(z_{\rm eff})= - \frac{1}{1+z_{\rm eff}} (\frac{dt}{dz})^{-1}$. This fit is performed by standard $\chi^2$ minimization. We also compute $P_{\ge \chi^2}$, the probability of obtaining equal or greater value of the reduced $\chi^2$ if the $t(z)$ points were truly lying on a straight line. Values of $P_{\ge \chi^2}<0.1$ means that a straight line is not a good fit to the points.

For $w = -1$, we obtain $H_0=H(z_{\rm eff})\left[\Omega_m(1+z_{\rm
eff})^3+\Omega_{\Lambda}\right]^{-\frac{1}{2}}$ where $\Omega_m=0.27$, $\Omega_\Lambda=0.7$. Figure 7 shows how the $H_0$ measurement depends on $z_{\rm max}$. For $z_{\rm max} > 0.17$ not only the value for $H_0$ drifts but also a straight line is not a good fit to the points ($P_{\ge \chi^2}$ suddenly drops below $0.1$). For our $H_0$ determination, $z_{\rm max}=0.17$, $P_{\ge \chi^2}=0.32$, $z_{\rm
eff}=0.09$, and the correction from $H(z_{\rm eff})$ to $H_0$ is a $4\%$ effect. We obtain $H_0=69\pm 12$ km s$^{-1}$ Mpc$^{-1}$.

Our $H_0$ determination is in good agreement with the Hubble Key Project measurement [$h = 0.72 \pm 0.03 \pm 0.0 7$; @Freedman+01], with the valued derived from the joint likelihood analysis of [*WMAP*]{} + 2dFGRS + Lyman-$\alpha$ forest power spectrum [$h =
0.71^{+0.04}_{-0.08}$; @SpergelWMAP03], with gravitational lens time delay determinations [$h = 0.59^{+0.12}_{-0.07} \pm
0.03$; @TreuKoopmans02], and Sunyaev-Zeldovich measurements [$h =
0.60^{+0.04+0.13}_{0.04-0.18}$; @Reese+02]. We emphasize that our method is fully independent from any of the above measurements; it is reassuring that so many different methods yield such a consistent value for $H_0$.

Conclusions
===========

We have obtained tight constraints on the effective value of $w$ using solely the determination of the ages of the oldest Galactic globular clusters and the observed position of the first acoustic peak of the CMB power spectrum. Our constraint on $w$ is in good agreement with that of @SpergelWMAP03, but is subject to different possible systematics and is very weakly model dependent. Our constraint is slightly less stringent than that obtained by @SpergelWMAP03 from a joint likelihood analysis of [*WMAP*]{} with six external data sets ([*WMAP*]{} + CBI + ACBAR + 2dFGRS + Lyman$\alpha$ forest power spectrum + Type Ia supernovae + [*HST*]{} Key Project $H_0$ determination), but is tighter than the CMB-only ([*WMAP*]{}+CBI+ACBAR) determination and comparable to the [*WMAP*]{} + ACBAR + CBI + [*HST*]{} constraint [@SpergelWMAP03]. We speculate that the joint constraint from [*WMAP*]{} and all the external data sets considered in @SpergelWMAP03 could be further improved by adding stellar populations age determination as an additional external data set. We have restricted our analysis to values of $w > -1$ since theoretical models for $w < -1$ seem difficult to construct [@Carroll+03]. The use of external data sets (see e.g. @KLSW02 and references therein for other approaches to measure $w$) could help to remove this restriction and constrain $w$ for values $< -1$.

We have dated a large sample of galaxies and shown that a clear age–redshift envelope exists. By computing the relative ages of SDSS red galaxies at $z < 0.2$ we obtain an estimate of $H_0$ independent of and in good agreement with determinations from the Hubble Key Project, gravitational lensing time delays, Sunyaev-Zeldovich, and [*WMAP*]{}. We also show that when surveys with a substantial number of high $S/N$ spectra of old galaxies at $z>>0$ become available, a determination of the possibly time-dependent equation of state parameter $w(z)$ will be obtainable.

We thank David Spergel for discussions that stimulated this investigation. We gratefully thank Fiona Harrison and Megan Eckart for providing the time during which the spectra of MS1054$-$0321 were obtained. We also thank Peter Eisenhardt and Charles Lawrence for useful discussions and Hiranya Peiris for insightful discussion on the position of the peak and $w$. LV thanks David Schelegel for help with the SDSS early data release catalogue. The work of RJ is supported in part by NSF grant AST-0206031. LV is supported by NASA through [*Chandra*]{} Fellowship PF2-30022 issued by the [*Chandra X-Ray Observatory*]{} Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of NASA under contract NAS8-39073. The work of DS was carried out at Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA. The work of TT is partially supported by STScI grant AR-09222. Finally, the authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.

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\[fig:wageth\]

\[lwplane\]

\[spectra\]

\[agered\]

\[mcsimul\]

\[fig3\]

\[fig4\]

[^1]: I.e., the average value over redshift. We use $w(z)$ to indicate when $w$ is allowed to vary over time.

[^2]: The uncertainty in this correction is negligible compared to the uncertainty in the determination of GC ages.

[^3]: We have drawn galaxies ages randomly from a grid that spans the age range 0.1 to 14 Gyr, with a spacing 0.1 Gyr. For our standard, baseline cosmology, this age spacing translates to the redshift interval seen in Figure 4.
