HUBBLE PARAMETER MEASUREMENT CONSTRAINTS ON THE COSMOLOGICAL DECELERATION–ACCELERATION TRANSITION REDSHIFT

In the standard picture of cosmology, dark energy powers the current accelerating cosmological expansion but played a less significant role in the past when non-relativistic (cold dark and baryonic) matter dominated and powered the then-decelerating cosmological expansion.¹ It is of some interest to determine the redshift of the deceleration–acceleration transition predicted to exist in dark energy cosmological models. There have been a number of attempts to do so; see, e.g., Lu et al. (2011a), Giostri et al. (2012), Lima et al. (2012), and references therein. However, until very recently, this has not been possible because there has not been much high-quality data at high enough redshift (i.e., for z above the transition redshift in standard dark energy cosmological models).

The recent Busca et al. (2012) detection of the baryon acoustic oscillation (BAO) peak at z = 2.3 in the Lyα forest has dramatically changed the situation by allowing for a high-precision measurement of the Hubble parameter H(z) at z = 2.3, well in the matter-dominated epoch of the standard dark energy cosmological model. Busca et al. (2012) use this and 10 other H(z) measurements, largely based on BAO-like data, and the Riess et al. (2011) Hubble Space Telescope determination of the Hubble constant, in the context of the standard ΛCDM cosmological model, to estimate a deceleration–acceleration transition redshift of z_da = 0.82 ± 0.08.

In this Letter, we extend the analysis of Busca et al. (2012). We first compile a list of 28 independent H(z) measurements.² We then use these 28 measurements to constrain cosmological parameters in three different dark energy models and establish that the models are a good fit to the data and that the data provide tight constraints on the model parameters. Finally, we use the models to estimate the redshift of the deceleration–acceleration transition. Busca et al. (2012) have 1 measurement (of 11) above their estimated z_da = 0.82, while we have 9 of 28 above this (and 10 of 28 above our estimated redshift z_da = 0.74). Granted, the Busca et al. (2012) z = 2.3 measurement carries great weight because of the small, 3.6%, uncertainty, but 9 of our 10 high-redshift measurements from Simon et al. (2005), Stern et al. (2010), and Moresco et al. (2012) include three 11%, 13%, and 14% measurements from Moresco et al. (2012) and three 10% measurements from Simon et al. (2005), all six of which carry significant weight.

Dark energy, most simply thought of as a negative pressure substance, dominates the current cosmological energy budget. In this Letter, we consider three dark energy models.

The first one is the "standard" spatially flat ΛCDM cosmological model (Peebles 1984). In this model, a little over 70% of the current energy budget is dark energy (Einstein's cosmological constant Λ), non-relativistic cold dark matter (CDM) being the next largest contributor (a little over 20%), followed by non-relativistic baryonic matter (about 5%). In the ΛCDM model, the dark energy density is constant in time and does not vary in space. ΛCDM has a number of well-known puzzling features (see, e.g., Peebles & Ratra 2003).

These puzzles could be eased if the dark energy density is a slowly decreasing function of time (Ratra & Peebles 1988).³ In this Letter, we consider a slowly evolving dark energy scalar field model as well as a time-varying dark energy parameterization.

In ΛCDM, time-independent dark energy density is modeled as a spatially homogeneous fluid with equation of state p_Λ = −ρ_Λ, where p_Λ and ρ_Λ are the fluid pressure and energy density. Much use has been made of a parameterization of slowly decreasing dark energy density known as XCDM where dark energy is modeled as a spatially homogeneous fluid with equation of state p_X = w_Xρ_X. The equation of state parameter w_X < −1/3 is independent of time and p_X and ρ_X are the pressure and energy density of the X-fluid. When w_X = −1 the XCDM parameterization reduces to the complete and consistent ΛCDM model. For any other value of w_X < −1/3 the XCDM parameterization is incomplete as it cannot describe spatial inhomogeneities (see, e.g., Ratra 1991; Podariu & Ratra 2000). For computational simplicity, in the XCDM case we assume a spatially flat cosmological model. The ϕCDM model is the simplest, consistent, and complete model of slowly decreasing dark energy density (Ratra & Peebles 1988). Here dark energy is modeled as a scalar field, ϕ, with a gradually decreasing (in ϕ) potential energy density V(ϕ). In this Letter, we assume an inverse power-law potential energy density V(ϕ)∝ϕ^−α, where α is a nonnegative constant (Peebles & Ratra 1988). When α = 0 the ϕCDM model reduces to the corresponding ΛCDM case. For computational simplicity, we again only consider the spatially flat cosmological case for ϕCDM.

Many different data sets have been used to derive constraints on the three cosmological models we consider here.⁴ Of interest to us here are 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; Duan et al. 2011; Aviles et al. 2012; Seikel et al. 2012). Table 1 lists 28 H(z) measurements.⁵ We only include independent measurements of H(z), listing only the most recent result from analyses of a given data set. The values in Table 1 have been determined using a number of different techniques; for details, see the papers listed in the table caption. Table 1 is the largest set of independent H(z) measurements considered to date.

We first use these data to derive constraints on cosmological parameters of the three models described above. The constraints derived here are compatible with cosmological parameter constraints determined by other techniques. These constraints are more restrictive than those derived by Farooq & Ratra (2012) using the previous largest set of H(z) measurements as well as those derived from the recent SNIa data compilation of Suzuki et al. (2012). The H(z) data considered here require accelerated cosmological expansion at the current epoch at about or more than 3σ confidence.

Our Letter is organized as follows. In the next section we present constraints from the H(z) data on cosmological parameters of the three models we consider, establish that the three models are very consistent with the H(z) data, and use the models to estimate the redshift of the cosmological deceleration–acceleration transition. We conclude in Section 3.

Following Farooq et al. (2013), we use the 28 independent H(z) data points listed in Table 1 to constrain cosmological model parameters. The observational data consist of measurements of the Hubble parameter H_obs(z_i) at redshifts z_i, with the corresponding one standard deviation uncertainties σ_i. To constrain cosmological parameters p of the models of interest, we build the posterior likelihood function $\mathcal {L}_{H}(\textbf {p})$ that depends only on the p by integrating the product of exp $(-\chi _H^2 /2)$ and the H₀ prior likelihood function exp$[-(H_0-\bar{H}_0)^2/(2\sigma ^2_{H_0})]$, as in Equation (18) of Farooq et al. (2013). We marginalize over the nuisance parameter H₀ using two different Gaussian priors with $\bar{H_0}\pm \sigma _{H_0}$ = 68 ± 2.8 km s⁻¹ Mpc⁻¹ (Chen et al. 2003; Chen & Ratra 2011a) and with $\bar{H_0}\pm \sigma _{H_0}$ = 73.8 ± 2.4 km s⁻¹ Mpc⁻¹ (Riess et al. 2011). 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). We determine the parameter values that maximize the likelihood function and find 1σ, 2σ, and 3σ constraint contours by integrating the likelihood function, starting from the maximum and including 68.27%, 95.45%, and 99.73% of the probability.

Figures 1–3 show the constraints from the H(z) data for the three dark energy models we consider and for the two different H₀ priors. In all six cases the H(z) data of Table 1 require accelerated cosmological expansion at the current epoch, at or better than 3σ confidence. The previous largest H(z) data set used, that in Farooq & Ratra (2012), required this accelerated expansion at or better than 2σ confidence. Comparing Figures 1–3 here to Figures 1– 3 of Farooq & Ratra (2012), we see that in the XCDM and ϕCDM cases the H(z) data we use in this Letter significantly tighten the constraints on w_X and α, but do not much affect the Ω_m0 constraints. However, in the ΛCDM case the H(z) data used here tighten constraints on both Ω_Λ and Ω_m0. We found that as we increase the value of the nuisance parameter H₀ the best-fit point for ΛCDM moves from the spatially flat case to the closed case, and for XCDM the best-fit point moves almost orthogonally to the flat ΛCDM line, toward more negative values of ω_X.

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As indicated by the $\chi ^2_{\rm min}$ values listed in the captions of Figures 1–3, all six best-fit models are very consistent with the H(z) data listed in Table 1. It is straightforward to compute the cosmological deceleration–acceleration transition redshift in these cases. They are 0.706 [0.785], 0.695 [0.718], and 0.698 [0.817] for the ΛCDM, XCDM, and ϕCDM models with prior $\bar{H}_0 \pm \sigma _{H_0} = 68 \pm 2.8$ km s⁻¹ Mpc⁻¹ [$\bar{H}_0 \pm \sigma _{H_0} = 73.8 \pm 2.4$ km s⁻¹ Mpc⁻¹]. The mean and standard deviation give z_da = 0.74 ± 0.05, in good agreement with the recent Busca et al. (2012) determination of z_da = 0.82 ± 0.08 based on less data, possibly not all independent. Figure 4 shows H(z)/(1 + z) data from Table 1 and the six best-fit model predictions as a function of redshift. The deceleration–acceleration transition is not impossible to discern in the data.

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From Figure 4 one sees that there are only six data points for z > 1 but 22 data points for z < 1. The larger errors of some of the z < 1 data, as compared to those of the z > 1 measurements, are likely responsible for the excellent reduced χ² values of the best-fit models.

In summary, we have extended the analysis of Busca et al. (2012) to a larger independent set of 28 H(z) measurements and determined the cosmological deceleration–acceleration transition redshift z_da = 0.74 ± 0.05. These H(z) data are well described by all six best-fit models and provide tight constraints on the model parameters. The H(z) data require accelerated cosmological expansion at the current epoch, and are consistent with the decelerated cosmological expansion at earlier times predicted and required in standard dark energy models. While the standard spatially flat ΛCDM model is very consistent with the H(z) data, current H(z) data are not able to rule out slowly evolving dark energy. More, and better quality, data are needed to better discriminate between constant and slowly evolving dark energy density; these data are likely to soon be in hand.

We thank Data Mania and Mikhail Makouski for useful discussions and helpful advice. We thank the referee for the prompt and helpful report. This work was supported in part by DOE grant DEFG03-99EP41093 and NSF grant AST-1109275.