Constraints on dark energy from the Ly{\alpha} forest baryon acoustic oscillations measurement of the redshift 2.3 Hubble parameter

We use the Busca et al. (2012) measurement of the Hubble parameter at redshift z = 2.3 in conjunction with 21 lower z measurements, from Simon et al. (2005), Gaztanaga et al. (2009), Stern et al. (2010), and Moresco et al. (2012), to place constraints on model parameters of constant and time-evolving dark energy cosmological models. The inclusion of the new Busca et al. (2012) measurement results in H(z) constraints significantly more restrictive than those derived by Farooq et al. (2012). These H(z) constraints are now more restrictive than those that follow from current Type Ia supernova (SNIa) apparent magnitude measurements (Suzuki et al. 2012). The H(z) constraints by themselves require an accelerating cosmological expansion at about 2-sigma confidence level, depending on cosmological model and Hubble constant prior used in the analysis. A joint analysis of H(z), baryon acoustic oscillation peak length scale, and SNIa data favors a spatially-flat cosmological model currently dominated by a time-independent cosmological constant but does not exclude slowly-evolving dark energy density.


Introduction
Observations indicate the cosmological expansion is accelerating now and that the Universe is spatially flat, provided the dark energy density responsible for the acceleration is close to or time independent. For reviews of dark energy see Jimenez (2011), Li et al. (2011a), Bolotin et al. (2011), Wang (2012), and references therein. 1 In the "standard" spatially-flat ΛCDM cosmological model (Peebles 1984) dark energy -Einstein's cosmological constant Λ -contributes a little more than 70% of the current energy budget. Non-relativistic cold dark matter (CDM) is the next largest contributer (at a little more than 20%), followed by non-relativistic baryonic matter (around 5%). See Ratra & Vogeley (2008) and references therein for reviews of the standard model. It has been known for a while that the standard ΛCDM model is reasonably compatible with most observational constraints (see, e.g., Jassal et al. 2010;Wilson et al. 2006;Davis et al. 2007;Allen et al. 2008). 2 In the ΛCDM model the dark energy density is constant in time and does not vary in space.
It is well known that the standard ΛCDM model has some puzzling features that are easier to accept in a model in which the dark energy density is a slowly decreasing function of time . For recent discussions of time-varying dark energy models see Sheykhi et al. (2012), Brax & Davis (2012), Hollenstein et al. (2012), Cai et al. (2012), Ponce de Leon (2012), Costa et al. (2012), Gu et al. (2012), Basilakos et al. (2012), Xu et al. (2012), and references therein. In this paper we study two dark energy models (with dark energy being either a cosmological constant or a slowly-evolving scalar field φ) as well as a dark energy parameterization.
In the ΛCDM model, time-independent dark energy density (the cosmological constant Λ) is modeled as a spatially homogeneous fluid with equation of state p Λ = −ρ Λ that relates the fluid pressure and energy density. The XCDM parameterization has often been used to describe slowly-decreasing dark energy density. In this case dark energy is modeled as a spatially homogeneous X-fluid with equation of state p X = w X ρ X . Here p X and ρ X are the pressure and energy density of the X-fluid and the equation of state parameter w X < −1/3 is independent of time. When w X = −1 the XCDM parameterization reduces to the complete, consistent ΛCDM model. For all other values of w X < −1/3 the XCDM parameterization is incomplete as it does not describe spatial inhomogeneities (see, e.g. Ratra 1991;Podariu & Ratra 2000). For computational simplicity, in the XCDM case we consider only a spatially-flat cosmological model.
The φCDM model is the simplest, complete and consistent model of slowly-decreasing length scales. See Bolotin et al. (2011), Capozziello & De Laurentis (2011), Starkman (2011, and references therein, for reviews of modified gravity. Here we assume that general relativity is an adequate description of cosmological gravitation. dark energy density . Here dark energy is modeled as the gradually decreasing (in φ and time) potential energy density V (φ) of the scalar field. In this paper we assume an inverse power-law potential energy density V (φ) ∝ φ −α , where α is a nonnegative constant . When α = 0 the φCDM model reduces to the corresponding ΛCDM model. For computational simplicity, we assume a spatially-flat cosmology for φCDM.
Cosmological observations that provide the strongest evidence for dark energy are: SNIa apparent magnitude versus redshift data (Suzuki et al. 2012;Li et al. 2011b;Astier & Pain 2012;Ruiz et al. 2012, and references therein); cosmic microwave background (CMB) anisotropy measurements (e.g., Podariu et al. 2001b;Komatsu et al. 2011) combined with low estimates of the cosmological mass density , and references therein), provided the dark energy density is close to or time independent; and, baryon acoustic oscillation (BAO) peak length scale data (e.g., Beutler et al. 2011;Blake et al. 2011;Mehta et al. 2012;Anderson et al. 2012). Current error bars associated with these three types of data are still too large to allow for a significant observational discrimination between the ΛCDM model and the two simple time-varying dark energy models discussed above. Additional data are needed for this task, as well as to provide a cross check on the above results.
Other data that have been used for this purpose include lookback time as a function of redshift Dantas et al. 2011;Tonoiu et al. 2011;Thakur et al. 2012, and references therein), gamma-ray burst luminosity distance as a function of redshift (e.g., Samushia & Ratra 2010;Wang & Dai 2011;Busti et al. 2012;Poitras 2012), strong gravitational lensing (Chae et al. 2004;Lee & Ng 2007;Zhang & Wu 2010;Cao et al. 2012, and references therein), HII starburst galaxy apparent magnitude as a function of redshift (e.g., Plionis et al. 2010Plionis et al. , 2011, angular size as a function of redshift (Chen & Ratra 2012;Lima & Cunha 2012;Jackson 2012, and references therein), and galaxy cluster properties (e.g., Campanelli et al. 2012;Devi et al. 2011;De Boni et al. 2012;Gonzalez et al. 2012). The constraints from these data are, at present, significantly weaker than those from SNIa, BAO, and CMB anisotropy measurements, but it is anticipated that future data of these kinds will provide significant constraints. 3 Two other current data sets provide interesting constraints on cosmological parameters, somewhat comparable to those from SNIa, BAO, and CMB anisotropy data. These are galaxy cluster gas mass fraction as a function of redshift measurements (e.g., Allen et al. 2008;Samushia & Ratra 2008;Tong & Noh 2011;Lu et al. 2011;Solano & Nucamendi 2012) and measurements of the Hubble parameter as a function of redshift (e.g., Jimenez et al. 2003;Samushia & Ratra 2006;Samushia et al. 2007;Sen & Scherrer 2008;Chen & Ratra 2011b;Kumar 2012;Wang & Zhang 2011;Duan et al. 2011;Aviles et al. 2012;Seikel et al. 2012). Interestingly, most measurements now provide largely compatible constraints on cosmological parameters that are consistent with a currently accelerating cosmological expansion. This provides confidence that the broad outlines of a standard cosmological model are now in place.
In this paper we use the 21 H(z) measurements of Simon et al. (2005), Gaztañaga et al. (2009), Stern et al. (2010, and Moresco et al. (2012) [listed in Table 1 of Farooq et al. (2012)], in conjunction with the H(z = 2.3) measurement of Busca et al. (2012), determined from BAO in the Lyα forest (in combination with WMAP CMB anisotropy data, Komatsu et al. 2011), to constrain the ΛCDM and φCDM models and the XCDM parametrization. The inclusion of the new Busca et al. (2012) measurement results in tighter constraints than those derived by Farooq et al. (2012) from the 21 H(z) measurements alone. The new H(z) constraints derived here are more restrictive than those derived from the recent SNIa data compilation of Suzuki et al. (2012), which more carefully accounts for the systematic errors in SNIa measurements. 4 In addition to deriving H(z)-data only constraints, we also use these H(z) measurements in combination with recent SNIa and BAO data to jointly constrain cosmological parameters. Adding the H(z) data tightens the constraints, quite significantly in some parts of parameter space. More precisely, the H(z) measurements more significantly tighten constraints on the nonrelativistic matter density parameter than on the parameter that more closely controls the time evolution of the dark energy density Our paper is organized as follows. In Sec. 2 we present the basic equations of the three dark energy models we consider. Constraints from the data are derived in Sec. 3. We conclude in Sec. 4.

Dark energy models
In this section we list relevant characteristics of the two models (ΛCDM and φCDM) and the one parametrization (XCDM) we use in our analyses of the data.
In the ΛCDM model with spatial curvature the Hubble parameter evolves as where H 0 is the current value of Hubble parameter (the Hubble constant), the current value of the spatial curvature density parameter is Ω K0 = 1 − Ω m0 − Ω Λ , and the model parameter set we want to constrain is p = (Ω m0 , Ω Λ ).
Here Ω m0 is the nonrelativistic (baryonic and cold dark) matter density parameter and Ω Λ is the time-independent cosmological constant density parameter. Below we shall have need for the dimensionless Hubble parameter E(z) = H(z)/H 0 .

The XCDM parameterization Friedmann equation is
where for computational simplicity we consider only flat spatial hypersurfaces and the model parameters p = (Ω m0 , ω X ). The XCDM parametrization is incomplete, as it cannot describe the evolution of energy density inhomogeneities.
The φCDM model ) is the simplest, complete and consistent dynamical dark energy model. In this model dark energy is a slowly-rolling scalar field φ with an, e.g., inverse-power-law potential energy density V (φ) = κm 2 p φ −α where m p = 1/ √ G is the Planck mass, G is the Newtonian gravitational constant, and α is a non-negative free parameter that determines κ. The scalar field part of the φCDM model action is where g µν is the metric tensor and α and κ are related as with corresponding scalar field equation of motion where the overdot denotes a derivative with respect to time. In the spatially-flat case the Friedmann equation for the φCDM model is where Ω φ (z, α) is determined by the φ field energy density Equations (5)-(7) constitute a system of differential equations which can be solved numerically for the φCDM model Hubble parameter H(z), using the initial conditions described in . In this case the model parameter set is p = (Ω m0 , α).
To constrain cosmological parameters p of the models of interest we follow the procedure of Farooq et al. (2012). We again marginalize over the nuisance parameter H 0 using two different Gaussian priors withH 0 ± σ H 0 = 68 ± 2.8 km s −1 Mpc −1 and withH 0 ± σ H 0 = 73.8 ± 2.4 km s −1 Mpc −1 . As discussed there, the Hubble constant measurement uncertainty can significantly affect cosmological parameter estimation (for a recent example see, e.g., Calabrese et al. 2012). The lower of the two values we use is from a median statistics analysis (Gott et al. 2001) of 553 measurements of H 0 (Chen & Ratra 2011a); this estimate has been stable for over a decade now (Gott et al. 2001;. The other value is a recent, HST based one (Riess et al. 2011). Other recent estimates are compatible with at least one of the two values we use (see, e.g., Freedman et al. 2012;Sorce et al. 2012;Colless et al. 2012;Tammann & Reindl 2012).
We maximize the likelihood L H (p) with respect to the parameters p to find the best- fit parameter values p 0 . In the models we consider χ 2 H = −2ln[L H (p)] depends on two parameters. We define 1, 2, and 3 σ confidence intervals as two-dimensional parameter sets bounded by χ 2 H (p) = χ 2 H (p 0 ) + 2.3, χ 2 H (p) = χ 2 H (p 0 ) + 6.17, and χ 2 H (p) = χ 2 H (p 0 ) + 11.8, respectively. Figures 1-3 show the constraints from the H(z) data derived here, as well as those derived by Farooq et al. (2012), for the three dark energy models, and for the two different H 0 priors. Clearly, the H(z = 2.3) measurement of Busca et al. (2012) significantly tightens the constrains. Given that the nonrelativistic matter density is larger relative to the dark energy density at z = 2.3, it is perhaps not unexpected that the Busca et al. (2012) measurement tightens the constraints on Ω m0 much more significantly than it does for the constraints on the other parameter which more strongly affects the evolution of the dark energy density, Comparing the H(z) constraints derived here, and shown in Figs. 1-3 here, to the SNIa constraints shown in Fig. 4 of Farooq et al. (2012), we see that the new H(z) data constraints are significantly more restrictive than those that follow on using the SNIa data. This is a remarkable result. Qualitatively, because of the dependence on the H 0 prior and on the model used in the analysis, Figs. 1-3 show that the H(z) data alone require accelerated cosmological expansion at approximately the two standard deviation confidence level.
While the H(z) data provide tight constraints on a linear combination of cosmological parameters, the banana-like constraint contours of Figs. 1-3 imply that these data alone cannot significantly discriminate between cosmological models. To tighten the constraints we must add other data to the mix. Following Farooq et al. (2012), we derive constraints on cosmological parameters of the three models from a joint analysis of the H(z) data with the 6 BAO peak length scale measurements of Percival et al. (2010), Beutler et al. (2011), andBlake et al. (2011), and the Union2.1 compilation of 580 SNIa apparent magnitude measurements (covering a redshift range 0.015 < z < 1.4) from Suzuki et al. (2012). Figures 4-6 show the constraints on cosmological parameters for the ΛCDM and φCDM models and the XCDM parametrization, from a joint analysis of the BAO and SNIa data, as well as from a joint analysis of the BAO, SNIa, and H(z) data. Including the H(z) data in the analysis tightens the constraints, somewhat significantly (sometimes by more than -Thick solid (thin dot-dashed) lines are 1, 2, and 3 σ constraint contours for the φCDM model from a joint analysis of the BAO and SNIa (with systematic errors) data, with (without) the H(z) data. The full (empty) circle marks the best-fit point determined from the joint analysis with (without) the H(z) data. The α = 0 horizontal axes correspond to spatially-flat ΛCDM models. In the left panel we use the H 0 = 68 ± 2.8 km s −1 Mpc −1 prior. Here the empty circle corresponds to best-fit pair (Ω m0 , α) = (0.30, 0) with χ 2 min = 551 while the full circle indicates best-fit pair (Ω m0 , α) = (0.29, 0) with χ 2 min = 567. In the right panel we use the H 0 = 73.8 ± 2.4 km s −1 Mpc −1 prior. Here the empty circle corresponds to best-fit pair (Ω m0 , α) = (0.30, 0) with χ 2 min = 551 while the full circle demarcates best-fit pair (Ω m0 , α) = (0.27, 0) with χ 2 min = 569.
two standard deviations), in parts of the parameter spaces. Figure 7 shows the H(z) data and the two best-fit ΛCDM models. The H(z) data do support the idea of a deceleration to acceleration transition somewhere in the range 0.5 < z < 1. Table 1 lists the two standard deviation bounds on the individual cosmological parameters, determined from their one-dimensional posterior probability distributions functions (which are derived by marginalizing the two-dimensional likelihood over the other cosmological parameter).

Conclusion
Adding the Busca et al. (2012) z = 2.3 measurement of the Hubble parameter, from BAO in the Lyα forest, to the 21 H(z) data points tabulated in Farooq et al. (2012), results in an H(z) data set that provides quite restrictive constraints on cosmological parameters. These constraints are tighter than those that follow from the SNIa data of Suzuki et al. (2012), which carefully accounts for all known systematic uncertainties. The H(z) field is much less mature than the SNIa one, and there might be some as yet undetected H(z) systematic errors that could broaden the H(z) error bars, as has happened in the SNIa case. However, we emphasize that the observers have done a careful analysis and the error bars we have used in our analysis have been carefully estimated. In addition to providing more restrictive constraints, the H(z) data alone requires accelerated cosmological expansion at the current epoch at approximately 2 σ confidence level, depending on model and H 0 prior used in the analysis.
In summary, the results of the joint analysis of the H(z), BAO, and SNIa data are quite consistent with the predictions of the standard spatially-flat ΛCDM cosmological model, with current energy budget dominated by a time-independent cosmological constant. However, currently-available data cannot rule out slowly-evolving dark energy density. We anticipate that soon to be available better quality data will more clearly discriminate between constant and slowly-evolving dark energy density.