THE AGE–REDSHIFT RELATIONSHIP OF OLD PASSIVE GALAXIES
Jun-Jie Wei1,2, Xue-Feng Wu
1,3,4, Fulvio Melia
1,5, Fa-Yin Wang
6,7, and Hai Yu
6,7
1 Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China; [email protected], [email protected] University of Chinese Academy of Sciences, Beijing 100049, China3 Chinese Center for Antarctic Astronomy, Nanjing 210008, China
4 Joint Center for Particle, Nuclear Physics and Cosmology, Nanjing University-Purple Mountain Observatory, Nanjing 210008, China5 Department of Physics, The Applied Math Program, and Department of Astronomy, The University of Arizona, AZ 85721, USA; [email protected]
6 School of Astronomy and Space Science, Nanjing University, Nanjing 210093, China7 Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210093, China
Received 2014 August 15; accepted 2015 May 28; published 2015 July 7
ABSTRACT
We use 32 age measurements of passively evolving galaxies as a function of redshift to test and compare thestandard model (ΛCDM) with the =R cth universe. We show that the latter fits the data with a reducedc = 0.435dof
2 for a Hubble constant = -+H 67.20 4.0
4.5 km -s 1 -Mpc 1. By comparison, the optimal flat ΛCDM model,
with two free parameters (including W = -+0.12m 0.11
0.54 and = -+H 94.30 35.8
32.7 km s−1 -Mpc 1), fits the age-z data with areduced c = 0.428dof
2 . Based solely on their cdof2 values, both models appear to account for the data very well,
though the optimized ΛCDM parameters are only marginally consistent with those of the concordance model(W = 0.27m and H0 = 70 km -s 1 -Mpc 1). Fitting the age-z data with the latter results in a reduced c = 0.523dof
2 .However, because of the different number of free parameters in these models, selection tools, such as the Akaike,Kullback and Bayes Information Criteria, favor =R cth over ΛCDM with a likelihood of ∼66.5%–80.5% versus∼19.5%–33.5%. These results are suggestive, though not yet compelling, given the current limited galaxy age-zsample. We carry out Monte Carlo simulations based on these current age measurements to estimate how large thesample would have to be in order to rule out either model at a ~99.7% confidence level. We find that if the realcosmology is ΛCDM, a sample of ∼45 galaxy ages would be sufficient to rule out =R cth at this level ofaccuracy, while ∼350 galaxy ages would be required to rule out ΛCDM if the real universe were instead =R cth .This difference in required sample size reflects the greater number of free parameters available to fit the datawith ΛCDM.
Key words: cosmology: observations – cosmology: theory – early universe – Galaxy: general
1. INTRODUCTION
In recent years, cosmic evolution has been studied using adiversity of observational data, including cosmic chronometers(Melia & Maier 2013), Gamma-ray bursts (GRBs; Weiet al. 2013), high-z quasars (Melia 2013, 2014a), stronggravitational lenses (Wei et al. 2014; Melia et al. 2015), andSNe Ia (Wei et al. 2015). In particular, the predictions ofΛCDM have been compared with those of a cosmology werefer to as the =R cth universe (Melia 2007; Melia &Shevchuk 2012). In all such one-on-one comparisons com-pleted thus far, model selection tools show that the data favor
=R cth over ΛCDM.The =R cth universe is a Friedmann–Robertson–Walker
(FRW) cosmology that has much in common with ΛCDM, butincludes an additional ingredient motivated by severaltheoretical and observational arguments (Melia 2007; Melia& Abdelqader 2009; Melia & Shevchuk 2012; see alsoMelia 2012a for a more pedagogical treatment). Like ΛCDM,it adopts an equation of state r=p w , with = + +p p p pm r de(for matter, radiation, and dark energy, respectively) andr r r r= + +m r de, but goes one step further by specifying that
r r r= + = -w w( 3 ) 1 3r de de at all times. Here, p is thetotal pressure and ρ is the total energy density. One might comeaway with the impression that this Equation of state cannot beconsistent with that (i.e., r r r= - Lw [ 3 ]r ) in the standardmodel. But in fact if we ignore the constraint = -w 1 3 andinstead proceed to optimize the parameters in ΛCDM by fittingthe data, the resultant value of w averaged over a Hubble time
is actually -1 3 within the measurement errors (Melia 2007;Melia & Shevchuk 2012). In other words, though
r r r= - Lw ( 3 )r in ΛCDM may be different from -1 3from one moment to the next, its value averaged over the age ofthe universe equals what it would have been in =R cth allalong (Melia 2015).In this paper, we continue to compare the predictions of
ΛCDM with those in the =R cth universe, this time focusingon the age–redshift relationship, which differs from oneexpansion scenario to another. Though the current age of theuniverse may be similar in these two cosmologies, the ageversus redshift relationship is not, particularly at high redshifts(Melia 2013, 2014b). Previous work with the age estimates ofdistant objects has already provided effective constraints oncosmological parameters (see, e.g., Alcaniz & Lima 1999;Lima & Alcaniz 2000; Jimenez & Loeb 2002; Jimenez et al.2003; Capozziello et al. 2004; Friaça et al. 2005; Simonet al. 2005; Jain & Dev 2006; Pires et al. 2006; Dantaset al. 2007, 2009, 2011; Samushia et al. 2010). For example,using the simple criterion that the age of the universe at anygiven redshift should always be greater than or equal to the ageof the oldest object(s) at that redshift, the measured ages wereused to constrain parameters in the standard model (Alcaniz &Lima 1999; Lima & Alcaniz 2000; Jain & Dev 2006; Pireset al. 2006; Dantas et al. 2007, 2011). Cosmological parametershave also been constrained by the measurement of thedifferential age D Dz t, where Dz is the redshift separationbetween two passively evolving galaxies having an age
difference Dt (Jimenez & Loeb 2002; Jimenez et al. 2003).And the lookback time versus redshift measurements for galaxyclusters and passively evolving galaxies have been used toconstrain dark energy models (Capozziello et al. 2004; Simonet al. 2005; Dantas et al. 2009; Samushia et al. 2010). This kindof analysis is therefore particularly interesting and comple-mentary to those mentioned earlier, which are essentially basedon distance measurements to a particular class of objects orphysical rulers (see Jimenez & Loeb 2002, for a discussion oncosmological tests based on relative galaxy ages).
Age measurements of high-z objects have been valuable inconstraining the cosmological parameters of the standardmodel even before dark energy was recognized as an essentialcomponent of the cosmic fluid (see, e.g., Bolte & Hogan 1995;Krauss & Turner 1995; Dunlop et al. 1996; Alcaniz &Lima 1999; Jimenez & Loeb 2002). Of direct relevance to theprincipal aim of this paper is the fact that, although thedistance–redshift relationship is very similar in ΛCDM and the
=R cth universe (even out to -z 6 7; Melia 2012b; Weiet al. 2013), the age–redshift dependence is not. This differenceis especially noticeable in how we interpret the formation ofstructure in the early universe. For example, the emergence ofquasars at z 6, which are now known to be accreting at, ornear, their Eddington limit (see, e.g., Willott et al. 2010; DeRosa et al. 2011). This presents a problem for ΛCDM becauseit is difficult to understand how ~ M109 supermassive blackholes could have appeared only 700–900Myr after the BigBang. Instead, in =R cth , their emergence at redshift ∼6corresponds to a cosmic age of 1.6 Gyr, which was enoughtime for them to begin growing from ~ - M5 20 seeds(presumably the remnants of Pop II and III supernovae) atz 15 (i.e., after the onset of re-ionization) and still reach a
billion solar masses by ~z 6 via standard, Eddington-limitedaccretion (Melia 2013).
In this paper, we will broaden the base of support for thiscosmic probe by demonstrating its usefulness in testingcompeting cosmological models. Following the methodologypresented in Dantas et al. (2011), we will use 32 agemeasurements of passively evolving galaxies as a function ofredshift (in the range ⩽ ⩽z0.117 1.845) to test the predictedage–redshift relationship of each model. From an observationalviewpoint, because the age of a galaxy must be younger thanthe age of the universe at any given redshift, there must be anincubation time, or delay factor τ, for the galaxy to form afterthe Big Bang. In principle, there could be a different ti for eachobject i since galaxies can form at different epochs. However,the simplest approach we can take is to begin with theassumption made in earlier work (see, e.g., Dantaset al. 2009, 2011; Samushia et al. 2010), i.e., we will adoptan average delay factor á ñt and use it uniformally for everygalaxy. But we shall also consider cases in which the delayfactors ti are distributed, and study the impact of this non-uniformity on the overall fits to the data.
We will demonstrate that the current sample of galaxy agesfavors the =R cth universe with a likelihood of~ -66.5% 80.5% of being correct, versus ~ -19.5% 33.5% forΛCDM. Though this result is still only marginal, it nonethelesscalls for a significant increase in the sample of suitable galaxyages in order to carry out more sophisticated and higherprecision measurements. We will therefore also construct mockcatalogs to investigate how big the sample of measured galaxy
ages has to be in order to rule out one (or more) of thesemodels.The outline of this paper is as follows. In Section 2, we will
briefly describe the age–redshift test, and then constrain thecosmological parameters—both in the context of ΛCDM andthe =R cth universe, first using a uniform (average) delayfactor á ñt (Section 3), and then a distribution of ti values(Section 4). In Section 5, we will discuss the model selectiontools we use to test ΛCDM and the =R cth cosmologies. InSection 6, we will estimate the sample size required from futureage measurements to reach likelihoods of ~99.7% and ~0.3%(i.e., s3 confidence limits) when using model selection tools tocompare these two models, and we will end with ourconclusions in Section 7.
2. THE AGE–REDSHIFT TEST
The theoretical age of an object at redshift z is given as
ò=¢
+ ¢ ¢
¥p
pt z
dz
z H z( , )
(1 ) ( , ), (1)
z
th
where p stands for all the parameters of the cosmologicalmodel under consideration and pH z( , ) is the Hubble parameterat redshift z. From an observational viewpoint, the total age of agiven object (e.g., a galaxy) at redshift z is given by
t= +t z t z( ) ( )Gobs , where tG(z) is the estimated age of itsoldest stellar population and τ is the incubation time, or delayfactor, which accounts for our ignorance about the amount oftime elapsed since the Big Bang to the epoch of star creation.To compute model predictions for the age pt z( , )th in
Equation (1), we need an expression for pH z( , ). As we haveseen, ΛCDM assumes specific constituents in the density,written as r r r r= + +r m de. These densities are often written
in terms of today’s critical density, r pº c H G3 8c2
02 , repre-
sented as r rW º cm m , r rW º cr r , and r rW º cde de . H0 is theHubble constant. In a flat universe with zero spatial curvature,the total scaled energy density is W º W + W + W = 1m r de .When dark energy is included with an unknown equation-of-state, r=p wde de de, the most general expression for the Hubbleparameter is
= éëêW + + W +
+ W + + W + ùûú
+
pH z H z z
z z
( , ) (1 ) (1 )
(1 ) (1 ) , (2)( )k
w
0 m3
r4
2de
3 11 2
de
where Wk is defined similarly to Wm and represents the spatial
curvature of the universe. Of course, Wr (~ ´ -5 10 5) is knownfrom the current temperature (2.725 K) of the CMB, and thevalue of H0. In addition, we assume a flat ΛCDM cosmologywith W = 0k , for which W = - W - W1de m r, thus avoidingthe introduction of Wde as an additional free parameter. So forthe basic ΛCDM model, p includes three free parameters: Wm,wde, and H0, though to be as favorable as possible to thismodel, we will also assume that dark energy is a cosmologicalconstant, with = -w 1de , leaving only two adjustable para-meters. However, when we consider the concordance modelwith the assumption of prior parameter values, the fits havezero degrees of freedom from the cosmology itself.The =R cth universe is a flat FRW cosmology that strictly
adheres to the constraints imposed by the simultaneous
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The Astronomical Journal, 150:35 (13pp), 2015 July Wei et al.
application of the cosmological principle and Weyl’s postulate(Melia 2012b; Melia & Shevchuk 2012). When theseingredients are applied to the cosmological expansion, thegravitational horizon =R c Hh must always be equal to ct.This cosmology is therefore very simple, because µa t t( ) ,which also means that + =z t1 1 , with the (standard)normalization that =a t( ) 10 . Therefore, in the =R cthuniverse, we have the straightforward scaling
= +pH z z H( , ) (1 ) . (3)0
Notice, in particular, that the expansion rate H(z) in this modelhas only one free parameter, i.e., p is H0. From Equation (3),the age of the =R cth universe at redshift z is simply
=+
( )t z Hz H
,1
(1 ). (4)th
00
To carry out the age–redshift analysis of ΛCDM and=R cth , we will first attempt to fit the ages of 32 old passive
galaxies distributed over the redshift interval⩽ ⩽z0.117 1.845 (Simon et al. 2005), listed in Table 1 of
Samushia et al. (2010), assuming a uniform value of the timedelay τ for every galaxy. Following these authors, we will alsoassume a 12% one-standard deviation uncertainty on the agemeasurements (Dantas et al. 2009, 2011; Samushiaet al. 2010). The total sample is composed of three sub-samples: 10 field early-type galaxies from Treu et al. (1999,2001, 2002), whose ages were obtained by using the SPEEDmodels of Nolan et al. (2001) and Jimenez et al. (2004); 20 redgalaxies from the publicly released Gemini Deep Deep Survey,whose integrated light is fully dominated by evolved stars(Abraham et al. 2004; McCarthy et al. 2004); and the 2 radiogalaxies LBDS 53W091 and LBDS 53W069 (Dunlopet al. 1996; Spinrad et al. 1997). These data were first collatedby Jimenez et al. (2003).
3. OPTIMIZATION OF THE MODEL PARAMETERSUSING A UNIFORM τ
In subsequent sections of this paper, we will study the impactof a distributed incubation time on the overall fits to the data.However, in this first simple approach (see, e.g., Dantaset al. 2009, 2011; Samushia et al. 2010) we will assume anaverage delay factor á ñt and use it uniformally for everygalaxy, so that we may compare our results to those ofprevious work.
For each model, we optimize the fit by finding the set ofparameters (p) that minimize the c2, using the statistic
åá ñ
á ñ á ñ
c tt
s
t t
=éëê - - ù
ûú
º - +
=
ppt z t z
A B C
( , )( , ) ( )
2* * * , (5)
i
i G i
tage2
1
32 th 2
2
2
G i,
where sº åéëê - ù
ûúpA t z t z( , ) ( )i G i tth 2 2
G i,, º åé
ëê - ùûúpB t z t z( , ) ( )i G i
th
st2G i,, and sº åC 1 t
2G i,. The dispersions stG i,
represent theuncertainties on the age measurements of the sample galaxies.Given the form of Equation (5), we can marginalize á ñt by
minimizing cage2 , which has a minimum at á ñt = B C , with a
value c = -A B Cˆage2 2 . Note that this procedure allows us to
determine the optimized value of á ñt along with the best-fitparameters of the model being tested.
3.1. ΛCDM
In the concordance ΛCDM model, the dark-energy equationof state parameter, wde, is exactly −1. The universe is flat,W = - W - W1de m r, so there remain only two free para-meters: Wm and H0. SN Ia measurements (see, e.g., Garnavichet al. 1998; Perlmutter et al. 1998, 1999; Riess et al. 1998;Schmidt et al. 1998), CMB anisotropy data (see, e.g., Ratraet al. 1999; Podariu et al. 2001; Spergel et al. 2003; Komatsuet al. 2009, 2011; Hinshaw et al. 2013), and baryon acousticoscillation peak length scale estimates (see, e.g., Percivalet al. 2007; Gaztañaga et al. 2009; Samushia & Ratra 2009),strongly suggest that we live in a spatially flat, dark energy-dominated universe with concordance parameter valuesW » 0.27m and »H 70.00 km s−1 Mpc−1.
In order to gauge how well ΛCDM and =R cth account forthe galaxy age–redshift measurements, we will first attempt to fitthe data with this concordance model, using prior values for allthe parameters but one, i.e., the unknown average delay timeá ñt .We will improve the fit as much as possible by marginalizingá ñt ,as described above. Fitting the 32 age–redshift measurementswith a theoretical t z( )th function using W = 0.27m and
=H 70.00 km s−1Mpc−1, we obtain an optimized delay factorá ñt = 1.36 Gyr, and a c = =16.17 31 0.523dof
2 , rememberingthat all of the ΛCDM parameters are assumed to have priorvalues, except for á ñt .If we relax the priors, and allow both Wm and H0 to be free
parameters, we obtain best-fit values W( m, H0)=(0.12, 94.3 kms−1 Mpc−1), as illustrated in Figure 1. The delay factorcorresponding to this best fit is á ñt = 1.62 Gyr, with ac = =12.42 29 0.428dof
2 . Figure 1 also shows the 1– s3constraint contours of the probability function in the Wm-H0
plane. Insofar as the ΛCDM model is concerned, thisoptimization has improved the quality of the fit as gauged bythe reduced cdof
2 , though the parameter values are quitedifferent from those of the concordance model. Nonetheless,these two sets of values are still marginally consistent with each
Figure 1. 1– s3 constraint contours for the flat ΛCDM model, using the 32age–redshift data. The cross indicates the best-fit pair (Wm, H0)=(0.12, 94.3).
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The Astronomical Journal, 150:35 (13pp), 2015 July Wei et al.
other because the data are not good enough yet to improve theprecision with which Wm and H0 are determined. It is alsopossible that treating á ñt as a uniform variable for all galaxiesmay be over-constraining, but we will relax this condition insubsequent sections and consider situations in which ti may bedifferent for each galaxy in the sample. We shall see that theseoptimized values change quantitatively, though the qualitativeresults and conclusions remain the same. The contours inFigure 1 show that at the s1 -level, we have < <H58.5 127.00km s−1 Mpc−1, and < W <0.01 0.66m . The cross indicates thebest-fit pair. For the sake of a direct one-on-one comparisonbetween ΛCDM and the =R cth universe, the current statuswith these data therefore suggests that we should use theconcordance parameter values, which are supported by manyother kinds of measurements, as described above.
3.2. The =R cth Universe
Regardless of what constituents may be present in thecosmic fluid, insofar as the expansion dynamics is concerned,the =R cth universe always has just one free parameter, H0.The results of fitting the age–redshift data with this cosmologyare shown in Figure 2 (solid line). We see here that the best fitcorresponds to = -
+H 67.20 4.04.5 km s−1 Mpc−1 ( s1 ). The corre-
sponding delay factor is á ñt = 2.72 Gyr. With - =32 2 30degrees of freedom, we have c = =13.05 30 0.435dof
2 .To facilitate a direct comparison between ΛCDM and=R cth , we show in Figure 3 the galaxy ages (i.e.,á ñt+tG ), together with the best-fit theoretical curves for the
=R cth universe (with =H 67.20 km s−1 Mpc−1 andá ñt = 2.72 Gyr), the concordance model (with á ñt = 1.36Gyr and prior values for all the other parameters), and for theoptimized ΛCDM model (with =H 94.30 km s−1 Mpc−1,W = 0.12m , and á ñt = 1.62 Gyr). As described above, á ñt isthe average incubation time or delay factor, which accounts forour ignorance concerning the amount of time elapsed since theBig Bang to the initial formation of the object. At the veryminimum, á ñt must be greater than ∼300Myr, this being thetime at which Population III stars would have established thenecessary conditions for the subsequent formation of Popula-tion II stars (see, e.g., Melia 2014b and references cited
therein). The galaxies could not have formed any earlier thanthis, based on the physics we know today.In this figure, we also show +t 300G Myr versus z (circles)
to illustrate the minimum possible ages the galaxies could have,given what we now know about the formation of Population IIand Population III stars. We note that the best fit value of á ñt ,in both ΛCDM and =R cth , is fully consistent with thesupposition that all of the galaxies should have formed after thetransition from Population III to Population II stars at ~t 300Myr. Strictly based on their cdof
2 values, the concordanceΛCDM model and the =R cth universe appear to fit thepassive galaxy age–redshift relationship (i.e., á ñt+tG versusz) comparably well. However, because these models formulatetheir observables (such as the theoretical ages in Equations (1)and (4)) differently, and because they do not have the samenumber of free parameters, a comparison of the likelihoods foreither being closer to the “true” model must be based on modelselection tools, which we discuss in Section 5 below. But first,we will strengthen this analysis by considering possibly morerealistic distributions of the delay time τ.
Figure 2. Constraints on the Hubble constant, H0, in the context of =R cth .
Figure 3. The complete age–redshift sample (solid points), and the best-fittheoretical curves: (dotted–dashed line) the concordance model, with its soleoptimized parameter á ñt = 1.36 Gyr; (dashed line) the standard, flat ΛCDMcosmology, with optimized parameters W = 0.12m , = -
+H 94.30 35.832.7 ( s1 ) km
s−1 Mpc−1, and á ñt = 1.62 Gyr; (solid line) the =R cth universe, with= -
+H 67.20 4.04.5 ( s1 ) km s−1 Mpc−1 and á ñt = 2.72 Gyr. The empty circles
show the minimum ages the galaxies could have using +t z( ) 300G i Myr.
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4. OPTIMIZATION OF THE MODEL PARAMETERSUSING A DISTRIBUTED τ
We now relax the constraint that the time delay should havethe same value á ñt for every galaxy, and instead consider tworepresentative distributions: (i) a Gaussian
tt t
sµ -
-
t
P ( ) exp( )
2, (6)c
2
2
where the mean value tc is to be optimized for each theoreticalfit, given some dispersion st; and (ii) a top-hat
tt s t t s
=ìíïïîïï
- < < +t tP ( )const ( )
0 (otherwise),(7)c c
with st now representing the width of the distribution.To isolate the various influences as much as possible, we
begin with the concordance ΛCDM model, for whichW = 0.27m and = -w 1de (i.e., dark energy is assumed to bea cosmological constant), though we optimize the Hubbleconstant to maximize the quality of the fit. For each distribution
tP ( ) and assumed value of st, we randomnly assign the timedelay ti to each galaxy and then find the best-fit values of H0
and tc by minimizing the c2 using the statistic
åc tt t s
s=
éëê - - ù
ûút
=
ppt z t z
( , )( , ) ( ) ( , )
. (8)ci
i G i i c
tage2
1
32 th 2
2G i,
The left-hand panels of Figure 4 show the ensuing distributionsof tc and H0 values for a Gaussian tP ( ), and three differentassumed dispersions st. In this case, the optimized Hubbleconstant (∼77.5 km s−1 Mpc−1) is effectively independent of st,while the best-fit value of the mean delay time tc is restricted to∼0.86 Gyr. Not surprisingly, the scatter about the best-fittheoretical curve worsens as st increases, resulting in larger
values of cmin2 .
Assuming a top-hat tP ( ) with ΛCDM produces the resultsshown in Figure 5. The best-fit values of H0 and tc are verysimilar to those associated with Figure 4. In this case, both H0
and tc are effectively independent of the assumed distributionwidth st.
We follow the same procedure for the =R cth universe, firstconsidering a Gaussian distribution of ti values (Figure 6),followed by the top-hat distribution (Figure 7). The compar-ison between these two cosmological models may besummarized as follows: the best-fit results are very similarfor both the Gaussian and top-hat tP ( ) distributions, for thesame cosmological model; strictly based on their minimum c2
values, the concordance ΛCDM model and the =R cthuniverse appear to fit the passive galaxy age–redshift relation-ship (i.e., t+tG i versus z) comparably well, independent ofwhat kind of time-delay distribution is assumed.
5. MODEL SELECTION TOOLS
Several model selection tools used in cosmology (see, e.g.,Melia & Maier 2013, and references cited therein) include theAkaike Information Criterion, c= + nAIC 22 , where n is thenumber of free parameters (Liddle 2007), the KullbackInformation Criterion, c= + nKIC 32 (Cavanaugh 2004),and the Bayes Information Criterion, c= + N nBIC (ln )2 ,
where N is the number of data points (Schwarz 1978). In thecase of AIC, with aAIC characterizing model a, theunnormalized confidence that this model is true is the Akaikeweight - aexp( AIC 2). Model a has likelihood
=-
- + -a
a( ) ( )( ) ( )
Pexp AIC 2
exp AIC 2 exp AIC 2(9)
1 2
of being the correct choice in this one-on-one comparison.Thus, the difference D º -AIC AIC AIC2 1 determines theextent to which 1 is favored over 2. For Kullback andBayes, the likelihoods are defined analogously.For the case of the average delay factor á ñt , with the
optimized fits we have reported in this paper, our analysis ofthe age-z shows that the KIC does not favor either =R cth orthe concordance model when we assume prior values for all ofits parameters. The calculated KIC likelihoods in this case are»51.5% for =R cth , versus »48.5% for ΛCDM. However, ifwe relax some of the priors, and allow both Wm and H0 to beoptimized in ΛCDM, then =R cth is favored over the standardmodel with a likelihood of »66.5% versus 33.5% using AIC,»76.6% versus »23.4% using KIC, and »80.5% versus»19.5% using BIC.
For the distributed time delays discussed in Section 4, themodel selection criteria result in the likelihoods shown inTable 1. Note that in this case, both the concordance ΛCDMand =R cth models have the same free parameters (i.e., H0
and tc), so the information criteria should all provide the sameresults. For the sake of clarity, we therefore show only the AICresults in Table 1, where we see that the AIC does not favoreither =R cth or the concordance model, regardless of whichdistribution is adopted for the incubation time.
6. NUMERICAL SIMULATIONS
The results of our analysis suggest that the measurement ofgalaxy ages may be used to identify the preferred model in aone-on-one comparison. In using the model selection tools, theoutcomeD º -AIC1 AIC2 (and analogously for KIC and BIC)is judged “positive” in the range D = -2 6, and “strong” forD > 6. As we have seen, the adoption of prior values for theparameters in ΛCDM produces comparable likelihood out-comes for both models, regardless of whether we assume auniform time delayá ñt , or a distribution of values. Of course, aproper statistical comparison between ΛCDM and =R cthshould not have to rely on prior values, particularly since theoptimized cosmological parameters differ from survey tosurvey. If we do not assume prior values for the parametersin ΛCDM, and optimize them to produce a best fit to thegalaxy-age data, the corresponding Δ using the currentlyknown 32 galaxy ages falls within the “positive” range in favorof =R cth , though not yet the strong one. These results aretherefore suggestive, but still not sufficient to rule out eithermodel. In this section, we will therefore estimate the samplesize required to significantly strengthen the evidence in favor of
=R cth or ΛCDM, by conservatively seeking an outcome evenbeyond D 11.62, i.e., we will see what is required toproduce a likelihood ~99.7% versus ~0.3%, corresponding tos3 .Since the results do not appear to depend strongly on
whether one chooses a uniform á ñt for all the galaxies, or adistribution of individual ti values, we will first use the sameaverage value á ñt in our simulations, and then discuss how
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these results would change for a distributed incubation time.We will consider two cases: one in which the backgroundcosmology is assumed to be ΛCDM, and a second in which it is
=R cth , and we will attempt to estimate the number of galaxyages required in each case in order to rule out the alternative(incorrect) model at a ~99.7% confidence level. The syntheticgalaxy ages are each characterized by a set of parametersdenoted as (z, t z[ ]), where á ñt= +t z t( ) G . We generate thesynthetic sample using the following procedure.
1. Since the current 32 old passively evolving galaxies aredistributed over the redshift interval ⩽ ⩽z0.117 1.845,we assign z uniformly between 0.1 and 2.0.
2. With the mock z, we first infer t(z) from Equations (1)and (4)corresponding either to a flat ΛCDM cosmologywith W = 0.27m and =H 700 km s−1 Mpc−1 (Section
6.2), or the =R cth universe with =H 700 km s−1 Mpc−1
(Section 6.1). We then assign a deviation (Dt) to the t(z)value, i.e., we infer ¢t z( ) from a normal distributionwhose center value is t(z) and s = 0.35 is its deviation(see Bengaly et al. 2014). The typical value of s = 0.35is taken from the current (observed) sample, which yieldsa mean and median deviation of s = 0.38 and 0.33,respectively.
3. Since the observed error st is about 12% of the agemeasurement, we will also assign a dispersions = ¢t z0.12 ( )t to the synthetic sample.
This sequence of steps is repeated for each galaxy in thesample, which is enlarged until the likelihood criteriondiscussed above is reached. As with the real 32-ages sample,we optimize the model fits by minimizing the c2 function
Figure 4. ΛCDM with a Gaussian distribution of τ values. Left-hand panels: fitted values of tc and H0 using the complete age–redshift sample (right-hand panels;solid points). The theoretical curves (right-hand panel; dotted–dashed curves) correspond to the parameter values that minimize the c2 (shown on the left). ForΛCDM, a Gaussian distribution in τ results in an optimized value of the Hubble constant (∼77.5 km s−1 Mpc−1) only weakly dependent on st , and a mean delay timet ~ 0.86c Gyr.
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c s= å éëê - ¢ ù
ûú{ }pt z t z( , ) ( )i i t i2 th 2
,2 . This minimization is
equivalent to maximizing the likelihood function cµ -( )exp 22 . We employ Markov-chain Monte Carlotechniques. In each Markov chain, we generate 105 samplesaccording to the likelihood function. Then we derive thecosmological parameters from a statistical analysis of thesample.
6.1. Assuming =R cth as the Background Cosmology
We have found that a sample of at least 350 galaxy ages isrequired in order to rule out ΛCDM at the ~99.7% confidencelevel. The optimized parameters corresponding to the best-fitΛCDM model for these simulated data are displayed inFigure 8. To allow for the greatest flexibility in this fit, we relaxthe assumption of flatness, and allow Wde to be a freeparameter, along with Wm. Figure 8 shows the 1D probability
distribution for each parameter (Wm, Wde, H0), and 2D plots ofthe s1 and s2 confidence regions for two-parameter combina-tions. The best-fit values for ΛCDM using the simulatedsample with 350 ages in the =R cth universe are W = 0.011m ,W = -
+0.37de 0.320.25 s(1 ), and = -
+H 79.90 12.710.7 s(1 ) km s−1 Mpc−1.
Note that the simulated ages provide a good constraint on Wde,but only a weak one on Wm; only an upper limit of ∼0.025 canbe set at the s1 confidence level.In Figure 9, we show the corresponding 1D probability
distribution of H0 for the =R cth universe. The best-fit valuefor the simulated sample is = -
+H 70.20 0.50.5 s(1 ) km s−1 Mpc−1.
The assumed value for H0 in the simulation was =H 700
km s−1 Mpc−1.Since the number N of data points in the sample is now much
greater than one, the most appropriate information criterion touse is the BIC. The logarithmic penalty in this model selectiontool strongly suppresses overfitting if N is large (the situation
Figure 5. Same as Figure 4, except now for the top-hat distribution tP ( ) given in Equation (7). In this case, both H0 and tc are effectively independent of the assumeddistribution width st .
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we have here, which is deep in the asymptotic regime). WithN = 350, our analysis of the simulated sample shows that theBIC would favor the =R cth universe over ΛCDM by anoverwhelming likelihood of 99.7% versus only 0.3% (i.e., theprescribed s3 confidence limit).
6.2. Assuming ΛCDM as the Background Cosmology
In this case, we assume that the background cosmology isΛCDM, and seek the minimum sample size to rule out =R cthat the s3 confidence level. We have found that a minimum of45 galaxy ages are required to achieve this goal. To allow forthe greatest flexibility in the ΛCDM fit, here too we relax theassumption of flatness, and allow Wde to be a free parameter,along with Wm. In Figure 10, we show the 1D probabilitydistribution for each parameter (Wm, Wde, H0), and 2D plots ofthe s1 and s2 confidence regions for two-parameter combina-tions. The best-fit values for ΛCDM using this simulatedsample with 45 galaxy ages are W = -
+0.28m 0.110.12 s(1 ),
W = 0.30de , and = -+H 60.10 7.6
9.2 s(1 ) km s−1 Mpc−1. Note thatthe simulated ages now give a good constraint on Wm, but onlya weak one on Wde; only an upper limit of ∼0.83 can be set atthe s1 confidence level.The corresponding 1D probability distribution of H0 for the=R cth universe is shown in Figure 11. The best-fit value for
the simulated sample is = -+H 85.00 1.5
1.6 s(1 ) km s−1 Mpc−1. Thisis similar to that in the standard model, but not exactly thesame, reaffirming the importance of reducing the dataseparately for each model being tested. With N = 45, ouranalysis of the simulated sample shows that in this case the BICwould favor ΛCDM over =R cth by an overwhelminglikelihood of 99.7% versus only 0.3% (i.e., the prescribed s3confidence limit).These results were obtained assuming a uniform incubation
time á ñt throughout the mock sample. Of course, if theincubation time is distributed, the corresponding uncertainty inits distribution function will contribute to the variance of the
Figure 6. Same as Figure 4, except now for the =R cth universe.
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“observed” age of the universe. Thus, adding the scatter in theuncertain incubation time tc to the simulations would changethe constructed sample size required to achieve the s3 resultsdiscussed above. We have therefore carried out additionalsimulations using a Gaussian tP ( ) distribution of theincubation time, and an assumed dispersion s =t 0.3. Thecorresponding uncertainty in τ contributes to the variance of
the “observed” age of the universe. From these results, weestimate that a sample of about 55 galaxy ages would beneeded to rule out =R cth at a~99.7% confidence level if thereal cosmology were ΛCDM, while a sample of at least 500ages would be needed to similarly rule out ΛCDM if thebackground cosmology were instead =R cth .
7. DISCUSSION AND CONCLUSIONS
In this paper, we have used the sample of high-redshiftgalaxies with measured ages to compare the predictions ofseveral cosmological models. We have individually optimizedthe parameters in each case by minimizing the c2 statistic.Using a sample of 32 passively evolving galaxies distributedover the redshift interval ⩽ ⩽z0.117 1.845, we have demon-strated how these age–redshift data can constrain parameters,such as H0 andWm. For ΛCDM, these data are not good enoughto improve upon the concordance values yet, but areapproaching the probative levels seen with currently available
Figure 7. Same as Figure 6, except now for the top-hat distribution tP ( ) given in Equation (7).
Table 1AIC Likelihood Estimation
Distributed τ st ΛCDM =R cth
Gaussian 0.1 50% 50%0.3 47% 53%0.5 41% 59%
Top-hat 0.3 51% 49%0.5 50% 50%0.8 52% 48%
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GRB luminosity data (Wei et al. 2013), strong gravitational-lensing measurements (see, e.g., Suyu et al. 2013), andmeasurements of the Hubble parameter as a function of redshift(Melia & Maier 2013).
Based solely on these 32 passively evolving galaxies, acomparison of the cdof
2 for the =R cth universe and theconcordance ΛCDM model shows that the age–redshift data donot yet favor either model. The =R cth universe fits the datawith c = 0.435dof
2 for a Hubble constant = -+H 67.20 4.0
4.5
km s−1 Mpc−1 and a average delay time á ñt = 2.72 Gyr. Bycomparison, the concordance model fits these same data with a
reduced c = 0.523dof2 , with a delay timeá ñt = 1.36 Gyr. Both
are consistent with the view that none of these galaxies shouldhave started forming prior to the transition from Population IIIto Population II stars at ∼300Myr. However, if we relax someof the priors, and allow both Wm and H0 to be optimized inΛCDM, we obtain best-fit values W = -
+0.12m 0.110.54 and
= -+H 94.30 35.8
32.7 km s−1 Mpc−1. The delay factor correspondingto this best fit is á ñt = 1.62 Gyr, with a c = 0.428dof
2 . Thecurrent sample favors =R cth over the standard model with alikelihood of » -66.5% 80.5% versus » -19.5% 33.5%.We also analyzed the age–redshift relationship in cases
where the delay factor τ may be different from galaxy togalaxy, and considered two representative distributions: (i) aGaussian; and (ii) a top-hat. We found that the optimizedcosmological parameters change quantitatively, though thequalitative results and conclusions remain the same, indepen-dent of what kind of the distribution one assumes for τ. Thoughone does not in reality expect the delay factor to be uniform, thefact that its distribution does not significantly affect the resultscan be useful for a qualitative assessment of the data. It alsosuggests that the outcome of our analysis is insensitive to theunderlying assumptions we have made.But though galaxy age estimates currently tend to slightly
favor =R cth over ΛCDM, the known sample of suchmeasurements is still too small for us to completely rule outeither model. We have therefore considered two syntheticsamples with characteristics similar to those of the 32 knownage measurements, one based on a ΛCDM backgroundcosmology, the other on =R cth . From the analysis of thesesimulated ages, we have estimated that a sample of about45–55 galaxy ages would be needed to rule out =R cth at a~99.7% confidence level if the real cosmology were in factΛCDM, while a sample of 350–500 ages would be needed to
Figure 8. 1D probability distributions and 2D regions with the s1 and s2 contours corresponding to the parameters Wm, Wde, and H0 in the best-fit ΛCDM model,using the simulated sample with 350 ages, assuming =R cth as the background cosmology.
Figure 9. 1D probability distribution for the parameter H0 in the =R cthuniverse, using a sample of 350 ages, simulated with =R cth as thebackground cosmology. The assumed value for H0 in the simulation was
=H 700 km s−1 Mpc−1.
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similarly rule out ΛCDM if the background cosmology wereinstead =R cth . These ranges allow for the possible contribu-tion of an uncertainty in τ to the variance of the observed age ofthe universe at each redshift. The difference in required samplesize is due to ΛCDMʼs greater flexibility in fitting the data,since it has a larger number of free parameters.
Both the Gaussian and Top-hat distributions that we haveincorporated into this study have assumed that the mean andscatter of the incubation time are constant with redshift.However, it would not be unreasonable to suppose that thesequantities could have a systematic dependence on z. Toexamine how the results might change in this case, we have
therefore also analyzed the real data using a Gaussiandistribution
tt t
sµ -
éëê - + ù
ûúéëê + ù
ûú
a
ta
( )
( )P
z
z( ) exp
· 1
2 · 1, (10)
c i
i
2
2
with s =t 0.3 and, for simplicity, a = 1. Figure 12(a) showsthe corresponding distributions of tc and H0 for theconcordance ΛCDM model, with best-fit values
t =H( , ) (82.5, 0.26)c0 . The analogous distributions for the=R cth universe are shown in Figure 12(b). In this case, the
best fit corresponds to t =H( , ) (80.0, 0.73)c0 . The addedredshift dependence has not changed the result that bothmodels fit the passive galaxy age–redshift relationshipcomparably well, based solely on their reduced c2ʼs. Notethat in this case, both the concordance ΛCDM and =R cth
models have the same free parameters (i.e., H( 0 and tc), so theinformation criteria should all provide the same results.Therefore, we show only the AIC results here. We find thatthe AIC does not favor either =R cth or the concordanceΛCDM model, with relatively likelihoods of »54% versus»46%. Note, however, that the best-fit value of tc in ΛCDMdoes not appear to be consistent with the supposition that all ofthe galaxies should have formed after the transition fromPopulation III to Population II stars at ~t 300 Myr.An additional limitation of this type of work is the degree of
uncertainty in the galaxy-age measurement itself. It is difficultto precisely constrain stellar ages for systems that are spatiallyresolved using stellar evolution models. We may be grosslyunderestimating how uncertain the age measurements of distant
Figure 10. Same as Figure 8, except now with a flat ΛCDM as the (assumed) background cosmology. The simulated model parameters wereW = 0.27m and =H 700km s−1 Mpc−1.
Figure 11. Same as Figure 9, except now with ΛCDM as the (assumed)background cosmology.
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galaxies are. One ought to acknowledge this possibility andconsider its impact on cosmological inferences. For example,redoing our analysis using a Gaussian tP ( ) and an assumeddispersion s =t 0.3, but now with an additional 24%uncertainty on the age measurements, (i.e., twice as big asthe value quoted in Dantas et al. 2009, 2011, and Samushiaet al. 2010), produces the results shown in Figure 13. Panel (a)in this plot shows the corresponding distributions of tc and H0
in the concordance ΛCDM model, with best-fit valuest =H( , ) (77.5, 0.86)c0 . Not surprisingly, a comparison of
Figure 13(a) with the left-middle panel in Figure 4 showsthat, as the uncertainty in the age measurement increases, theconstraints on model parameters weaken; nonetheless, the best-fit values of H0 and tc are more or less the same.
The comparison between ΛCDM (Figure 13(a)) and=R cth (Figure 13(b)) may be summarized as follows: the
best-fit results are more or less the same for both the 12% and24% uncertainties. Based solely on their minimum c2 values,both models fit the passive galaxy age–redshift relationshipcomparably well. The AIC does not favor either =R cth or theconcordance ΛCDM model, regardless of how uncertain thegalaxy-age measurements are, with relative likelihoods of»51% versus »49%.
We are very grateful to the anonymous referee for providinga thoughtful and helpful review, and for making severalimportant suggestions to improve the presentation in themanuscript. This work is partially supported by the NationalBasic Research Program (“973” Program) of China (Grants
Figure 12. (a): ΛCDM with a Gaussian distribution for τ and a redshift dependent dispersion and mean value (see text). (b): Same as (a), except now for the =R cthuniverse.
Figure 13. (a): ΛCDM with a Gaussian distribution of τ and an (assumed) uncertainty of 24% in the age measurement. (b): Same as (a), except now for the =R cthUniverse.
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2014CB845800 and2013CB834900), the National NaturalScience Foundation of China (grants Nos. 11322328 and11373068), the One-hundred-talents Program, the YouthInnovation Promotion Association, and the Strategic PriorityResearch Program “The Emergence of Cosmological Struc-tures” (Grant No. XDB09000000) of the Chinese Academy ofSciences, and the Natural Science Foundation of JiangsuProvince (Grant No. BK2012890). F. M. is grateful to AmherstCollege for its support through a John Woodruff SimpsonLectureship, and to Purple Mountain Observatory in Nanjing,China, for its hospitality while part of this work was beingcarried out. This work was partially supported by grant2012T1J0011 from The Chinese Academy of Sciences VisitingProfessorships for Senior International Scientists, and grantGDJ20120491013 from the Chinese State Administration ofForeign Experts Affairs.
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