Forecasting cosmological constraints from age of highz galaxies
^{1}
Departamento de Astronomia, Observatório Nacional, 20921400 Rio de Janeiro  RJ, Brasil
email: carlosap@on.br; alcaniz@on.br
^{2}
Departamento de Física, Universidade Estadual do Rio Grande
do Norte, 59625620 Mossoró  RN, Brasil
email:
aldinezdantas@uern.br
^{3}
Departamento de Física, Universidade Federal do Rio Grande do
Norte, 59072970 Natal  RN, Brasil
email:
carvalho@dfte.ufrn.br
Received:
12
August
2013
Accepted:
13
September
2013
We perform Monte Carlo simulations based on current age estimates of highz objects to forecast constraints on the equation of state (EoS) of the dark energy. In our analysis, we use two different EoS parameterizations, namely, the socalled ChevallierPolarskiLinder (CPL) and its uncorrelated form, and calculate the improvements on the figure of merit (FoM) for both cases. Although there is a clear dependence of the FoM with the size and accuracy of the synthetic age samples, we find that the most substantial gain in FoM comes from a joint analysis involving age and baryon acoustic oscillation data.
Key words: cosmology: theory / cosmology: observations / cosmological parameters / dark energy
© ESO, 2013
1. Introduction
Over the last decade, a significant amount of evidence has been accumulated for the existence of a dark energy component that fuels current cosmic acceleration. This evidence comes mostly from distance measurements of type Ia supernovae (SN Ia; Riess et al. 1998; Permultter et al. 1999), the baryon acoustic oscillation (BAO) feature in the largescale distribution traced by the galaxy distribution (Peebles & Yu 1970; Blake & Glazebrook 2003; Eisenstein et al. 2005), and measurements of the cosmic microwave background (CMB) anisotropies (Spergel et al. 2005; Planck Collaboration 2013). Together, these results provide strong support for the standard cosmological scenario and an interesting link connecting the inflationary flatness prediction with current astronomical observations.
Another important class of evidence comes from estimates of the age of the Universe. In reality, since the days of predark energy, this kind of observation has been one of the most pressing pieces of data supporting the idea of a latetime cosmic acceleration (see, e.g., Krauss & Turner 1995; Bolte & Hogan 1995; Dunlop et al. 1996; Alcaniz & Lima 1999; Jimenez & Loeb 2002). In this regard, age estimates of highz objects provide effective constraints on cosmological parameters since the evolution of the age of the Universe differs from scenario to scenario, which means that models that are able to explain the total expanding age may not be compatible with age estimates of highz objects (see Friaça et al. 2005; Dantas et al. 2007; 2011). This kind of analysis, therefore, is particularly interesting and complementary to those mentioned ealier, which are essentially based on distance measurements to a particular class of objects or physical rulers (see Jimenez & Loeb 2002, for discussion on a cosmological test based on relative galaxy ages).
Our goal in this paper is to investigate the constraining power of future age data on the parameters of the dark energy equation of state (EoS). To this end, we assume the observational error distribution of a sample of 32 passively evolving galaxies studied by Simon et al. (2005) and run Monte Carlo simulations to generate synthetic samples of galaxy ages with different sizes and characteristics. To perform our analysis we assume the socalled ChevalierPolarskiLinder (CPL; Chevalier & Polarski 2001; Linder 2003) dark energy EoS parameterization and its uncorrelated form (Wang 2008). We discuss the improvement in the figure of merit (FoM) for the EoS parameters of both parameterizations with the size and precision of age samples as well as with the combination of age data and current baryonic acoustic oscillation (BAO) measurements.
Fig. 1 Δt(z)/t(z) versus Δp_{i}/p_{i} assuming the model parameters discussed in the text for three values of z. 

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Fig. 2 Δt(z)/t(z) versus z for some values of w_{0} and w_{a} and w_{0.5}. 

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2. The ageredshift relation
In our analyses we consider a flat universe dominated by nonrelativistic matter (baryonic and dark) and a dark energy component. In this background, the theoretical agez relation t(z_{i}) of an object at redshift z_{i} can be written as (Sandage 1988; Peebles 1993) (1)where θ stands for the parameters of the cosmological model under consideration and h(z,θ) is the normalized Hubble parameter, given by (2)with (3)For the dark energy EoS, w_{DE}, we consider the CPL parametrization (4)Wang (2008) derived an uncorrelated form for the above parameterization by rewriting it at the value a_{c} (or, equivalently, z_{c}) at which the parameters w_{0} and w_{a} are uncorrelated, i.e., (5)In our analyses, we follow Wang (2008) and consider a_{c} = 2/3 (z_{c} = 0.5), so that the above equation can be written in terms of z as (6)where w_{0.5} ≡ w(z = 0.5). As mentioned earlier, Eq. (6) is a rearrangement of parametrization (4) that minimizes the correlation between the parameters w_{0} and w_{a} and allows us to obtain tighter constraints on the parametric space. The parameters w_{0.5}, w_{0}, and w_{a} are directly related by (see Wang 2008; and Sendra & Lazkoz 2012, for more details).
From the above equations, we calculate the relative error in the expansion age as a function of the relative error in the EoS parameters from , where w_{i} stands for w_{0}, w_{a} and w_{0.5}. Neglecting errors on z and fixing Ω_{m} = 0.27, w_{0} = −1.0, w_{a} = −0.25 (Panels a and b) and, equivalently, w_{0.5} = −1.08 (panels c and d), Fig. 1 shows Δt(z)/t(z) versus Δw_{i}/w_{i} for z = 0.3,0.6, and 1.0. Panels a and b refer to the CPL parameterization, where we note only a slight dependence of Δt(z)/t(z) with redshift. In order to constrain w_{0} at a 10% level, there must be an accuracy for Δt(z)/t(z) of 1.65% at z = 0.3 and of 1% at z = 1.0. We also note that much better measurements are required to constrain w_{a} at a level of 20%. In this case, we estimate Δt(z)/t(z) ≃ 0.20% for z = 0.3 and 0.6, and Δt(z)/t(z) ≃ 0.18% for z = 1.0, which are beyond the accuracy expected in current planned observations (see, e.g., Simon et al. 2005; Crawford et al. 2010).
We also performed the same analysis for parametrization (6), displayed in panels c and d. Compared to the previous case, we note that the accuracy required to measure w_{0} at a 10% level should be improved by a factor of 5, whereas to obtain a 20% measurement of w_{0.5} the accuracy of Δt(z)/t(z) could be reduced by a factor of 25. This clearly shows the effectiveness of t(z) data in measuring the parameter w_{0.5}, which is also directly related to the timedependent part of the dark energy EoS. For completeness, we also show the dependence of Δt(z)/t(z) with redshift for some selected values of w_{i} (Fig. 2). For these values, the curve Δt(z)/t(z) presents a maximum at lowz, which indicates that age data at this redshift interval must impose more restrictive bounds on w_{i} than those at highz.
Fig. 3 Figure of merit as a function of the number of data points N for parametrization (4) (panel a)), parametrization (6) (panel b)), and parametrization (5) (panel c)) assuming σ_{t} = 10%. Solid squares correspond to the age simulated data only whereas solid circles stand for a joint analysis with the BAO data described in the text. Solid (dashed) red lines represent the FoM for the current observational t(z) (t(z) + BAO) data while solid (dashed) blue lines represent the FoM obtained from the Union2.1 compilation (SNe Ia + BAO). 

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Fig. 4 Same as Fig. 3 for σ_{t} = 5%. 

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3. Numerical simulations
We perform Monte Carlo (MC) simulations to generate t(z) samples with different sizes and accuracy and study the expected improvement on the FoM for parameterizations (4)–(6). Our simulations assume the current observational error distribution (σ_{t} = 10%) of the t(z) data given by Simon et al. (2005), which consist of 32 old passively evolving galaxies distributed over the redshift interval 0.11 ≤ z ≤ 1.84. We then use a normal distribution centered at the t(z_{i}) prediction of the chosen fiducial model, namely, a spatially flat Lambda Cold Dark Matter (ΛCDM) model with Ω_{m} = 0.27 and H_{0} = 74.3 ± 3.6 km s^{1} Mpc^{1}, which is consistent with current data from CMB (Komatsu et al. 2011) and differential measurements of Cepheids variable observations (Riess et al. 2009).
According to some authors (see, e.g., Simon et al. 2005; Crawford et al. 2010), future observations of passively evolving galaxies will be able to provide age estimates with σ_{t} ≤ 10%. In our simulations, therefore, we adopt two values of σ_{t}, i.e., σ_{t} = 5% and σ_{t} = 10%, and divide our samples into groups of 100, 500 and 1000 data points evenly spaced in the redshift range 0.1 ≤ z ≤ 1.5. This makes it possible to study the expected improvement of the FoM not only as a function of the number of objects N, but also as a function of the precision of future cosmological observations.
4. Results
In order to calculate the FoM, we follow Wang (2008) and define , where C(θ) is the covariance matrix of the set of parameters θ. Using the prescription of the previous section, we perform 30 realizations of t(z) for each group of N = 100, 500, and 1000 data points, with σ_{t} = 5% and 10%. The central values of the FoM and the corresponding error bars are obtained using a bootstrap method on the original 30 Monte Carlo realizations.
Constraints on (w_{0} − w_{a}) and (w_{0} − w_{0.5}).
Figures 3 and 4 show the main results of our analysis. The expected FoM is shown as a function of the number of data points for σ_{t} = 10% (Fig. 3) and σ_{t} = 5% (Fig. 4). In all panels, solid red lines represent the FoM for the current observational t(z) sample (Simon et al. 2009) whereas solid blue lines represent the FoM obtained from a magnituderedshift test using 580 type Ia supernovae (SNe Ia) of the Union2.1 compilation (Suzuki et al. 2012). We also performed joint analyses involving t(z), SNe Ia, and six measurements of the BAO peak length scale taken from Percival et al. (2007), Blake et al. (2011), and Beutler et al. (2011). Dashed red and blue lines stand for the joint analyses involving t(z) + BAO and SNe Ia + BAO, respectively. The results for these combinations of age, SNe Ia, and BAO data are shown in Table 1.
Panels 3a–4a and 3b–4b correspond, respectively, to the results obtained for parameterizations (4) and (6). Solid squares represent the FoM for each group of age simulated data, whereas solid circles correspond to the joint analysis with the six BAO data points. There is a slight dependence of the FoM with the increase of N and a clear and substantial gain with the combination with BAO data. For instance, considering age data only and σ_{t} = 10% we find that increasing the number of data points improves the FoM by a factor of 1.3–2.3, whereas the ratio between FoM_{Age + BAO} and FoM_{Age} increases by a factor of 4–9. In Table 2 we summarize the main results of our analysis and put into numbers the results displayed in Figs. 3 and 4.
Another aspect that is worth emphasizing concerns the use of parameterization (6). Clearly, there is a significant improvement in the FoM for the w_{0} − w_{0.5} plane relative to the one for the w_{0} − w_{a}. This is an expected result since the former set of parameters is less correlated than the latter one, which is in full agreement with the results discussed by Wang (2008) using SNe Ia + CMB + BAO data. In our analysis we find that the FoM_{Age} and FoM_{Age + BAO} increase, respectively, by a factor of 4 and 6 relative to the same quantities for parameterization (4). We also observe that a similar result can also be obtained from the current observational data (Table 1). In this case, regardless of whether we consider the age only, or the age + BAO analysis, we obtain an improvement factor for the figure of merit of ≃3.
For the sake of completeness, we also performed our analysis assuming z_{c} in Eq. (5) to be a free parameter of the model so that the plane w_{0} − w_{a} becomes completely uncorrelated. The results for this analysis are shown in Figs. 3c and 4c. Compared to the analysis for parameterization (6), we find an improvement in the FoM_{Age + BAO} and FoM_{Age} that varies, respectively, by a factor of 1.5–3.4 and 2.0–4.6 (see Table 3).
Figure of merit obtained from each group of simulated dataset for the parametric spaces (w_{0} − w_{a}) and (w_{0} − w_{0.5}).
5. Conclusions
Age estimates of highz objects constitute a complementary probe to distancebased observations such as SNe Ia and BAO measurements. In this paper, we have explored t(z) constraints on the dark energy EoS from two different routes, namely: calculating the relative error in the expansion age as
a function of the relative error in the EoS parameters, and performing MC simulations from current observational data.
Using synthetic samples of t(z) with different sizes and accuracy and their combinations with BAO data, we have found a significant improvement in the FoM for the planes w_{0} − w_{0.5} and w_{0} − w_{zc} relative to the one for the w_{0} − w_{a} space. We have also studied the dependence of the figure of merit with the number of data points N, with σ_{t}, as well as with the combination of t(z) and BAO observations, and found that the latter two provide the more substantial gains. One of the results of our analysis is that t(z) data may become competitive with SNe Ia observations only for σ_{t} < 5%. This result certainly reinforces the importance of a better understanding of the systematic errors in the age determination of highz objects as important probes to the late stages of the Universe.
Acknowledgments
The authors thank CAPES, CNPq, and FAPERJ for the grants under which this work was carried out.
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All Tables
Figure of merit obtained from each group of simulated dataset for the parametric spaces (w_{0} − w_{a}) and (w_{0} − w_{0.5}).
All Figures
Fig. 1 Δt(z)/t(z) versus Δp_{i}/p_{i} assuming the model parameters discussed in the text for three values of z. 

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In the text 
Fig. 2 Δt(z)/t(z) versus z for some values of w_{0} and w_{a} and w_{0.5}. 

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In the text 
Fig. 3 Figure of merit as a function of the number of data points N for parametrization (4) (panel a)), parametrization (6) (panel b)), and parametrization (5) (panel c)) assuming σ_{t} = 10%. Solid squares correspond to the age simulated data only whereas solid circles stand for a joint analysis with the BAO data described in the text. Solid (dashed) red lines represent the FoM for the current observational t(z) (t(z) + BAO) data while solid (dashed) blue lines represent the FoM obtained from the Union2.1 compilation (SNe Ia + BAO). 

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In the text 
Fig. 4 Same as Fig. 3 for σ_{t} = 5%. 

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In the text 