HUBBLE PARAMETER MEASUREMENT CONSTRAINTS ON THE REDSHIFT OF THE DECELERATION–ACCELERATION TRANSITION, DYNAMICAL DARK ENERGY, AND SPACE CURVATURE

In the standard scenario the currently accelerating cosmological expansion is a consequence of dark energy dominating the current cosmological energy budget; at earlier times nonrelativistic (cold dark and baryonic) matter dominated the energy budget and powered the decelerating cosmological expansion.³ Initial quantitative observational support for this picture came from "lower"-redshift Type Ia supernova (SN Ia) apparent magnitude observations and "higher"-redshift cosmic microwave background (CMB) anisotropy measurements.

More recently, cosmic chronometric and baryon acoustic oscillation (BAO) techniques (see, e.g., Simon et al. 2005; Moresco et al. 2012; Busca et al. 2013) have resulted in the measurement of the cosmological expansion rate or Hubble parameter, H(z), from the present epoch back to a redshift z exceeding 2, higher than currently probed by SN Ia observations. This has resulted in the first mapping out of the cosmological deceleration–acceleration transition, the epoch when dark energy took over from nonrelativistic matter, and the first measurement of the redshift of this transition (see, e.g., Farooq & Ratra 2013a; Farooq et al. 2013a; Moresco et al. 2016).⁴

H(z) measurements have also been used to constrain some more conventional cosmological parameters, such as the density of dark energy and the density of nonrelativistic matter (see, e.g., Samushia & Ratra 2006; Chen & Ratra 2011b; Chimento & Richarte 2013; Farooq & Ratra 2013b; Ferreira et al. 2013; Akarsu et al. 2014; Bamba et al. 2014; Capozziello et al. 2014; Dankiewicz et al. 2014; Forte 2014; Gruber & Luongo 2014; Chen et al. 2015; Meng et al. 2015; Alam et al. 2016; Guo & Zhang 2016; Mukherjee & Banerjee 2016), typically providing constraints comparable to or better than those provided by SN Ia data, but not as good as those from BAO or CMB anisotropy measurements. More recently, H(z) data have been used to measure the Hubble constant H₀ (Verde et al. 2014; Chen et al. 2016a), with the resulting H₀ value being more consistent with recent lower values determined from a median statistics analysis of Huchra's H₀ compilation (Chen & Ratra 2011a), from CMB anisotropy data (Hinshaw et al. 2013; Sievers et al. 2013; Ade et al. 2015), from BAO measurements (Aubourg et al. 2015; Ross et al. 2015; L'Huillier & Shafieloo 2016), and from current cosmological data and the standard model of particle physics with only three light neutrino species (see, e.g., Calabrese et al. 2012).

In this paper, we put together an updated list of H(z) measurements, compared to that of Farooq & Ratra (2013a), and use this compilation to constrain the redshift of the cosmological deceleration–acceleration transition, z_da, as well as other cosmological parameters. In the z_da analysis here we study more models than used by Farooq & Ratra (2013a) and Farooq et al. (2013a), now also allowing for nonzero spatial curvature in the XCDM parameterization of the dynamical dark energy case and in the dynamical dark energy ϕCDM model (Pavlov et al. 2013). The cosmological parameter constraints derived here are based on more, as well as more recent, H(z) data than were used by Farooq et al. (2015), and we also explore a much larger range of parameter space in the nonflat ϕCDM model than they did.

We find, from the likelihood analyses, that the ${z}_{\mathrm{da}}$ values measured from the H(z) data agree within the error bars in all five models. They, however, depend more sensitively on the value of H₀ assumed in the analysis. These results are consistent with those found in Farooq & Ratra (2013a) and Farooq et al. (2013a). In addition, the binned H(z) data in redshift space show qualitative visual evidence for the deceleration–acceleration transition, independent of how they are binned provided that the bins are narrow enough, in agreement with that originally found by Farooq et al. (2013a). Given that the measured z_da are relatively model independent, it is not unreasonable to average the measured values to determine a reasonable summary estimate. We find, for a weighted mean estimate, z_da = 0.72 ± 0.05 (0.84 ± 0.03) if we assume H₀ = 68 ± 2.8 (73.24 ± 1.74) km s⁻¹ Mpc⁻¹.

The constraints on the more conventional cosmological parameters, such as the density of dark energy, derived from the likelihood analysis of the H(z) data here, indicate that these data are quite consistent with the spatially flat ΛCDM model, the standard model of cosmology where the cosmological constant Λ is the dark energy. These H(z) data, however, do not rule out the possibility of dynamical dark energy or space curvature, especially when included simultaneously, in agreement with the conclusions of Farooq et al. (2015). Currently available SN Ia, BAO, growth factor, CMB anisotropy, and other data can tighten the constraints on these parameters, and it will be interesting to study these data sets in conjunction with the H(z) data we have compiled here, but this is beyond the scope of our paper. Near-future data will also result in interesting limits (see, e.g., Podariu et al. 2001a; Pavlov et al. 2012; Santos et al. 2013; Basse et al. 2014).

The outline of our paper is as follows. In the next section, we discuss and tabulate our new H(z) data compilation. In Section 3 we summarize how we bin the H(z) data in redshift space and list binned H(z) data. Section 4 summarizes the cosmological models we consider. In Section 5 we discuss how we compute and measure the deceleration–acceleration transition redshift and tabulate numerical values of z_da determined from the H(z) measurements. Section 6 presents the constraints on cosmological parameters, and we conclude in the last section.

In Table 1 we collect 38 Hubble parameter H(z) measurements from Simon et al. (2005), Stern et al. (2010), Moresco et al. (2012), Blake et al. (2012), Zhang et al. (2012), Font-Ribera et al. (2014), Delubac et al. (2015), Moresco (2015), Alam et al. (2016), and Moresco et al. (2016). These data are plotted in the top panel of Figure 1.

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These 38 H(z) measurements are not completely independent. The three measurements taken from Blake et al. (2012) are correlated with each other, and the three measurements of Alam et al. (2016) also are correlated. Also, in these and other cases, when BAO observations are used to measure H(z), one has to apply a prior on the radius of the sound horizon, ${r}_{d}={\int }_{{z}_{d}}^{\infty }{c}_{s}(z){dz}/H(z)$, evaluated at the drag epoch z_d, shortly after recombination, when photons and baryons decouple. This prior value of r_d is generally derived from CMB observations.

Table 1 here is based on Table 1 of Farooq & Ratra (2013a) with the following modifications. We drop older Sloan Digital Sky Survey galaxy clustering H(z) determinations from Chuang & Wang (2013) in favor of the more recent measurements from Alam et al. (2016). We have added the new Moresco et al. (2016) measurements. We have dropped the older Busca et al. (2013) Lyα forest measurement in favor of the newer Font-Ribera et al. (2014) and Delubac et al. (2015) ones. We have also added two new measurements from Moresco (2015).

There are many other compilations of H(z) data available in the literature (see, e.g., Cai et al. 2015; Meng et al. 2015; Duan et al. 2016; Nunes et al. 2016; Qi et al. 2016; Solà et al. 2016; Yu & Wang 2016; Zhang & Xia 2016). We emphasize that our compilation here does not include older, less reliable data, a few with a lot of weight because of anomalously small error bars.

There are two reasons to compute "average" H(z) values for bins in redshift space. First, the weighted mean technique of binning data can indicate whether the original unbinned data have error bars inconsistent with Gaussianity, an important consistency check. Second, data binned in redshift space can more clearly visually illustrate trends as a function of redshift, with the additional advantage of not having to assume a particular cosmological model.

The 38 Hubble parameter measurements in Table 1 are binned to ensure as many measurements as possible per bin, while also retaining as many (narrow) redshift bins as possible. The ideal case is $\sqrt{38}$ measurements in each of $\sqrt{38}$ bins. Here we consider about 3–4, 4–5, 4–5–6, and 5–7 measurements per bin. The last four measurements are binned by twos in all but the 4–5–6 measurement per bin case. In all cases, data points in a given bin are not correlated with each other.

After binning the data, we use weighted mean statistics⁵ to find a representative central estimate for each bin. Following Podariu et al. (2001b), the weighted mean is given by

$\begin{eqnarray}&&\overline{H}(z)=\displaystyle \frac{{\displaystyle \sum }_{i=1}^{N}H({z}_{i})/{\sigma }_{i}^{2}}{{\displaystyle \sum }_{i=1}^{N}1/{\sigma }_{i}^{2}},\end{eqnarray} \tag{ 1 }$

where H(z_i) and σ_i are the Hubble parameter and one standard deviation of i = 1, 2, 3,..., N measurements in the bin. We also compute the weighted bin redshift using

$\begin{eqnarray}&&\overline{z}=\displaystyle \frac{{\displaystyle \sum }_{i=1}^{N}{z}_{i}/{\sigma }_{i}^{2}}{{\displaystyle \sum }_{i=1}^{N}1/{\sigma }_{i}^{2}}.\end{eqnarray} \tag{ 2 }$

The associated weighted error is given by

$\begin{eqnarray}&&\overline{\sigma }={\left(\displaystyle \sum _{i=1}^{N}1/{\sigma }_{i}^{2}\right)}^{-1/2}.\end{eqnarray} \tag{ 3 }$

A goodness of fit, χ², can be found for each bin where the reduced χ² is

$\begin{eqnarray}&&{\chi }_{\nu }^{2}=\displaystyle \frac{1}{N-1}\displaystyle \sum _{i=1}^{N}\,\displaystyle \frac{{[H({z}_{i})-\overline{H}(z)]}^{2}}{{\sigma }_{i}^{2}}.\end{eqnarray} \tag{ 4 }$

The number of standard deviations that χ_ν deviates from unity (the expected value) is given by

$\begin{eqnarray}&&{N}_{\sigma }=| {\chi }_{\nu }-1| \sqrt{2(N-1)}.\end{eqnarray} \tag{ 5 }$

A large N_σ can be the result of non-Gaussian measurements, the presence of unaccounted-for systematic errors, or correlations between measurements. Table 2 lists the weighted mean results for the binned H(z) measurements.

The last column of Table 2 shows reasonably small N_σ for all binnings, thus suggesting that the error bars of the H(z) data of Table 1 are not inconsistent with Gaussianity. As in Farooq et al. (2013a), we find that the cosmological constraints that follow from the weighted mean binned data are almost identical to those derived using the unbinned data, while the median statistics binned data typically result in somewhat weaker constraints. A possible reason for this could be that some of the unbinned H(z) data error bars might be a bit larger than they really should be. This would be consistent with the low reduced χ² shown in the last line of Table 1 in Chen et al. (2016a).

The binned data are plotted in the four lower panels of Figure 1. It is reassuring that, independent of the binning used, all the binned data sets show clear visual qualitative evidence for the cosmological deceleration–acceleration transition, as in Farooq et al. (2013a). This is model-independent qualitative evidence for the existence of the cosmological deceleration–acceleration transition. We shall see, in Section 5, that all cosmological models we use in the analysis of the H(z) data to measure z_da result in z_da values that overlap within the error bars (for a given H₀ prior). This is additional model-independent evidence for the presence of the deceleration–acceleration transition.

In this section we briefly describe the five models we use to analyze the H(z) data. These are the ΛCDM model, which allows for spatial curvature and where dark energy is the cosmological constant Λ (Peebles 1984), and the ϕCDM model, in which dynamical dark energy is represented by a slowly evolving scalar field ϕ (Peebles & Ratra 1988; Ratra & Peebles 1988). We also consider an incomplete, but popular, parameterization of dynamical dark energy, XCDM, where dynamical dark energy is represented by an X-fluid. In the ϕCDM and XCDM cases, we consider both spatially flat and nonflat models (Pavlov et al. 2013).

In the ΛCDM model with spatial curvature the Hubble parameter is

$\begin{eqnarray}H(z;{H}_{0},{\boldsymbol{p}}) & = & {H}_{0}[{{\rm{\Omega }}}_{m0}{(1+z)}^{3}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}\\ & & +{(1-{{\rm{\Omega }}}_{m0}-{{\rm{\Omega }}}_{{\rm{\Lambda }}}){(1+z)}^{2}]}^{1/2},\end{eqnarray} \tag{ 6 }$

where we have made use of ${{\rm{\Omega }}}_{K0}=1-{{\rm{\Omega }}}_{m0}-{{\rm{\Omega }}}_{{\rm{\Lambda }}}$ to eliminate the current value of the space curvature energy density parameter in favor of the current value of the nonrelativistic matter energy density parameter, Ω_m0, and the cosmological constant energy density parameter, Ω_Λ. Here ${\boldsymbol{p}}=({{\rm{\Omega }}}_{m0},{{\rm{\Omega }}}_{{\rm{\Lambda }}})$ are the two cosmological parameters that conventionally characterize ΛCDM, and H₀ is the value of the Hubble parameter at the present time and is called the Hubble constant.

It has become fashionable to parameterize dynamical dark energy as a spatially homogeneous X-fluid, with a constant equation-of-state parameter, ${\omega }_{X}={p}_{X}/{\rho }_{X}\lt -1/3$ (here p_X and ρ_X are the pressure and energy density of the X-fluid, respectively). For the spatially flat XCDM parameterization, using ${{\rm{\Omega }}}_{X0}=1-{{\rm{\Omega }}}_{m0}$ (where Ω_X0 is the current value of the X-fluid energy density parameter), we have

$\begin{eqnarray}H(z;{H}_{0},{\boldsymbol{p}}) & = & {H}_{0}[{{\rm{\Omega }}}_{m0}{(1+z)}^{3}\\ & & +{(1-{{\rm{\Omega }}}_{m0}){(1+z)}^{3(1+{\omega }_{X})}]}^{1/2}.\end{eqnarray} \tag{ 7 }$

In this spatially flat case the two cosmological parameters are ${\boldsymbol{p}}=({{\rm{\Omega }}}_{m0},{\omega }_{X})$. The XCDM parameterization is incomplete as it cannot describe the evolution of energy density inhomogeneities. In the nonflat XCDM parameterization case, ${{\rm{\Omega }}}_{K0}$ is the third free parameter and

$\begin{eqnarray}H(z;{H}_{0},{\boldsymbol{p}}) & = & {H}_{0}[{{\rm{\Omega }}}_{m0}{(1+z)}^{3}+(1-{{\rm{\Omega }}}_{m0}-{{\rm{\Omega }}}_{K0})\\ & & \times {{(1+z)}^{3(1+{\omega }_{X})}+{{\rm{\Omega }}}_{K0}{(1+z)}^{2}]}^{1/2},\end{eqnarray} \tag{ 8 }$

where the three cosmological parameters are ${\boldsymbol{p}}\ =({{\rm{\Omega }}}_{m0},{\omega }_{X},{{\rm{\Omega }}}_{K0})$. ϕCDM is the simplest, most complete, and most consistent dynamical dark energy model. Here dark energy is modeled as a slowly rolling scalar field ϕ with, e.g., an inverse-power-law potential energy density $V(\phi )=\kappa {m}_{p}^{2}{\phi }^{-\alpha }/2$, where m_p is the Planck mass and α is a non-negative parameter that determines the coefficient κ(m_p, α) (Peebles & Ratra 1988). The equation of motion of the scalar field is

$\begin{eqnarray}&&\ddot{\phi }+3\displaystyle \frac{\dot{a}}{a}\dot{\phi }-\displaystyle \frac{\kappa }{2}\alpha {m}_{p}^{2}{\phi }^{-(\alpha +1)}=0,\end{eqnarray} \tag{ 9 }$

where an overdot represents a time derivative and a is the scale factor. For the spatially flat ϕCDM model

$\begin{eqnarray}&&H(z;{H}_{0},{\boldsymbol{p}})={H}_{0}{[{{\rm{\Omega }}}_{m0}{(1+z)}^{3}+{{\rm{\Omega }}}_{\phi }(z,\alpha )]}^{1/2},\end{eqnarray} \tag{ 10 }$

where the time-dependent scalar field energy density parameter is

$\begin{eqnarray}&&{{\rm{\Omega }}}_{\phi }(z,\alpha )=\displaystyle \frac{1}{12{H}_{0}^{2}}({\dot{\phi }}^{2}+\kappa {m}_{p}^{2}{\phi }^{-\alpha }).\end{eqnarray} \tag{ 11 }$

In this case the two cosmological parameters are ${\boldsymbol{p}}=({{\rm{\Omega }}}_{m0},\alpha )$. In the nonflat ϕCDM model

$\begin{eqnarray}H(z;{H}_{0},{\boldsymbol{p}}) & = & {H}_{0}[{{\rm{\Omega }}}_{m0}{(1+z)}^{3}+{{\rm{\Omega }}}_{\phi }(z,\alpha )\\ & & +{{{\rm{\Omega }}}_{K0}{(1+z)}^{2}]}^{1/2},\end{eqnarray} \tag{ 12 }$

and the three cosmological parameters are ${\boldsymbol{p}}=({{\rm{\Omega }}}_{m0},\alpha ,{{\rm{\Omega }}}_{K0})$.

Solving the coupled differential equations of motion allows for a numerical computation of the Hubble parameter $H(z;{H}_{0},{\boldsymbol{p}})$ (Peebles & Ratra 1988; Samushia 2009; Farooq 2013; Pavlov et al. 2013).⁶

In Section 6 we use these expressions for the Hubble parameter in conjunction with the H(z) measurements in Table 1 to constrain the cosmological parameters of these models. In our analyses here we study the following parameter ranges: 0 ≤ Ω_m0 ≤ 1, 0 ≤ Ω_Λ ≤ 1.4, −2 ≤ ω_X ≤ 0, 0 ≤ α ≤ 5, and −0.7 ≤ Ω_K0 ≤ 0.7 for nonflat XCDM and −0.4 ≤ Ω_K0 ≤ 0.4 for nonflat ϕCDM (which is double the Ω_K0 range used in Farooq et al. 2015).

At the current epoch, dark energy dominates the cosmological energy budget and accelerates the cosmological expansion. At earlier times nonrelativistic (baryonic and cold dark) matter dominated the energy budget and the cosmological expansion decelerated. The cosmological deceleration–acceleration transition redshift, z_da, is defined as the redshift at which $\ddot{a}=0$, in the cosmological model under consideration. $\ddot{a}$ is proportional to the active gravitational mass density, the sum of the energy densities and three times the pressure of the constituents.

For ΛCDM, setting $\ddot{a}=0$, we find

$\begin{eqnarray}&&{z}_{\mathrm{da}}={\left(\displaystyle \frac{2{{\rm{\Omega }}}_{{\rm{\Lambda }}}}{{{\rm{\Omega }}}_{m0}}\right)}^{1/3}-1.\end{eqnarray} \tag{ 13 }$

For the case of the spatially flat XCDM parameterization

$\begin{eqnarray}&&{z}_{\mathrm{da}}={\left(\displaystyle \frac{{{\rm{\Omega }}}_{m0}}{({{\rm{\Omega }}}_{m0}-1)(1+3{\omega }_{X})}\right)}^{1/3{\omega }_{X}}-1,\end{eqnarray} \tag{ 14 }$

while for nonflat XCDM

$\begin{eqnarray}&&{z}_{\mathrm{da}}={\left(\displaystyle \frac{{{\rm{\Omega }}}_{m0}}{({{\rm{\Omega }}}_{m0}+{{\rm{\Omega }}}_{K0}-1)(1+3{\omega }_{X})}\right)}^{1/3{\omega }_{X}}-1.\end{eqnarray} \tag{ 15 }$

For the spatially flat ϕCDM model, defining the time-dependent equation-of-state parameter for the scalar field

$\begin{eqnarray}&&{\omega }_{\phi }(z)=\displaystyle \frac{\tfrac{1}{2}{\dot{\phi }}^{2}-V(\phi )}{\tfrac{1}{2}{\dot{\phi }}^{2}+V(\phi )},\end{eqnarray} \tag{ 16 }$

the redshift z_da (Ω_m0, α) is determined by numerically solving

$\begin{eqnarray}&&{{\rm{\Omega }}}_{m0}{(1+{z}_{\mathrm{da}})}^{3}+{{\rm{\Omega }}}_{\phi }({z}_{\mathrm{da}},\alpha )[1+3\,{\omega }_{\phi }({z}_{\mathrm{da}})]=0,\end{eqnarray} \tag{ 17 }$

where ${{\rm{\Omega }}}_{\phi 0}=1-{{\rm{\Omega }}}_{m0}$. In the nonflat ϕCDM model z_da (Ω_m0, α, Ω_K0) is determined by numerically solving the same equation, but now setting ${{\rm{\Omega }}}_{\phi 0}=1-{{\rm{\Omega }}}_{m0}-{{\rm{\Omega }}}_{K0}$.

To compute the expected values $\langle {z}_{\mathrm{da}}\rangle$ and $\langle {z}_{\mathrm{da}}^{2}\rangle$ for the two-parameter models, we use

$\begin{eqnarray}\langle {z}_{\mathrm{da}}\rangle & = & \displaystyle \frac{\displaystyle \int \displaystyle \int {z}_{\mathrm{da}}({\boldsymbol{p}}){ \mathcal L }({\boldsymbol{p}})d{\boldsymbol{p}}}{\displaystyle \int \displaystyle \int { \mathcal L }({\boldsymbol{p}})d{\boldsymbol{p}}},\\ \langle {z}_{\mathrm{da}}^{2}\rangle & = & \displaystyle \frac{\displaystyle \int \displaystyle \int {z}_{\mathrm{da}}^{2}({\boldsymbol{p}}){ \mathcal L }({\boldsymbol{p}})d{\boldsymbol{p}}}{\displaystyle \int \displaystyle \int { \mathcal L }({\boldsymbol{p}})d{\boldsymbol{p}}}.\end{eqnarray} \tag{ 18 }$

Here ${ \mathcal L }({\boldsymbol{p}})$ is the H(z) data likelihood function after marginalization over the Gaussian H₀ prior in the two-parameter model under consideration, as explained in Farooq et al. (2013b, 2015), but this time accounting for the nondiagonal correlation matrices of the Blake et al. (2012) and Alam et al. (2016) measurements, which have a small effect. ${ \mathcal L }({\boldsymbol{p}})$ depends only on the model parameters $({{\rm{\Omega }}}_{m0},{{\rm{\Omega }}}_{{\rm{\Lambda }}})$ for ΛCDM, $({{\rm{\Omega }}}_{m0},{\omega }_{X})$ for flat XCDM, and $({{\rm{\Omega }}}_{m0},\alpha )$ for flat ϕCDM. The generalization for the three-parameter models is straightforward. The standard deviation in z_da is computed from the standard formula ${\sigma }_{{z}_{\mathrm{da}}}=\sqrt{\langle {z}_{\mathrm{da}}^{2}\rangle -\langle {z}_{\mathrm{da}}{\rangle }^{2}}$. The results of this computation are summarized in Table 3.

Table 3.  Deceleration–Acceleration Transition Redshifts^a

Model	h Priorb	BFc	${\chi }_{\min }^{2}$	${z}_{\mathrm{da}}\pm {\sigma }_{{z}_{\mathrm{da}}}$ d	${z}_{\mathrm{da}}\pm {\sigma }_{{z}_{\mathrm{da}}}$ e
	0.68 ± 0.028	Ωm0 = 0.23	22.4	0.723 ± 0.089	0.690 ± 0.096
ΛCDM		ΩΛ = 0.60
	0.7324 ± 0.0174	Ωm0 = 0.25	24.2	0.832 ± 0.055	0.781 ± 0.067
		ΩΛ = 0.78
	0.68 ± 0.028	Ωm0 = 0.26	22.5	0.753 ± 0.091	0.677 ± 0.097
Flat XCDM		ωX = −0.86
	0.7324 ± 0.0174	Ωm0 = 0.24	23.9	0.813 ± 0.062	0.696 ± 0.082
		ωX = −1.06
	0.68 ± 0.028	Ωm0 = 0.27	22.9	0.703 ± 0.104	0.724 ± 0.148
Flat ϕCDM		α = 0.50
	0.7324 ± 0.0174	Ωm0 = 0.25	25.2	0.885 ± 0.056	0.850 ± 0.116
		α = 0
	0.68 ± 0.028	Ωm0 = 0.15	21.9	0.684 ± 0.117	⋯
		ωX = −1.68
Nonflat XCDM		ΩK0 = 0.45
	0.7324 ± 0.0174	Ωm0 = 0.13	20.3	0.709 ± 0.090	⋯
		ωX = −2
		ΩK0 = 0.41
	0.68 ± 0.028	Ωm0 = 0.23	22.6	0.690 ± 0.118	⋯
		α = 0
Nonflat ϕCDM		ΩK0 = 0.18
	0.7324 ± 0.0174	Ωm0 = 0.25	25.0	0.853 ± 0.053	⋯
		α = 0
		ΩK0 = −0.03

Notes.

^aEstimated using the unbinned data of Table 1. ^bHubble constant in units of 100 km s⁻¹ Mpc⁻¹. ^cBest-fit parameter values. ^dComputed using Equations (13)–(18) of this work. ^eThe deceleration–acceleration transition redshift in the model, as computed in Table 1 of Farooq et al. (2013a). Note that the best-fit cosmological parameter values found in Farooq et al. (2013a) differ from those found here and listed in this table.

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Table 3 shows best-fit cosmological parameter values and the corresponding minimum χ² for the five different cosmological models and for the two Gaussian H₀ priors. The second-to-last column in Table 3 shows the average deceleration–acceleration transition redshift with corresponding standard deviation for each model. It is very reassuring that the z_da values we measure in the five different models (for a given H₀ prior) overlap reasonably well. (The main effect on the measured z_da value is the assumed H₀ prior value.) Given that the measured z_da are almost independent of the other model parameters, within the errors, we may conclude that to leading order we have measured a model-independent z_da value. However, it is useful to have a single summary value for this cosmological parameter.

By taking the simple average of the penultimate column z_da values and computing the population standard deviation for the five values in this column, we find z_da = 0.71 ± 0.03 (0.82 ±0.06) for ${H}_{0}\pm {\sigma }_{{H}_{0}}=68\pm 2.8$ (73.24 ± 1.74) km s⁻¹ Mpc⁻¹. Using all 10 z_da values in the penultimate column of Table 3, we find z_da = 0.76 ± 0.07.

A more reliable summary value of the deceleration–acceleration transition redshift is determined from a weighted mean analysis. Using Equations (1)–(3), we find z_da = 0.72 ± 0.05 (0.84 ± 0.03) for ${H}_{0}\pm {\sigma }_{{H}_{0}}=68\pm 2.8$ (73.24 ± 1.74) km s⁻¹ Mpc⁻¹, and using all 10 values in the penultimate column of Table 3, we get z_da = 0.80 ± 0.02. By looking at the fourth and fifth columns of Table 3, it appears that all five models discussed here fit better with the lower value of H₀, while the uncertainty in z_da is more sensitive to ${\sigma }_{{H}_{0}}$.

These results are listed in Table 4 and compared with the previously computed summary values of Farooq et al. (2013a). Note that only three models (ΛCDM, flat XCDM, and flat ϕCDM) were considered in Farooq et al. (2013a). Here we also consider nonflat XCDM and nonflat ϕCDM. We see that there is good agreement between the old and new weighted mean z_da for h = 0.68, less so for h = 0.7324. From Table 4 we see that for a given H₀ the weighted average values of z_da for all five models and for the two sets of (non-nested) triplets of models agree to within the error bars.

In this section, we use the 38 Hubble parameter measurements (over 0.07 ≤ z ≤ 2.36) listed in Table 1 to determine constraints on the parameters of the five different cosmological models. We use the technique of Farooq et al. (2015) to find constraints on (Ω_m0, Ω_Λ) in the ΛCDM model, (Ω_m0, ω_X) for the spatially flat XCDM parameterization, (Ω_m0, α) in the spatially flat ϕCDM model, $({{\rm{\Omega }}}_{m0},{\omega }_{X},{{\rm{\Omega }}}_{K0})$ for the XCDM parameterization with space curvature, and $({{\rm{\Omega }}}_{m0},\alpha ,{{\rm{\Omega }}}_{K0})$ in the ϕCDM model with space curvature. For the H(z) cosmological test, cosmological parameter constraints depend on the value of the Hubble constant (see, e.g., Samushia et al. 2007). We use two different Gaussian priors for the Hubble constant; the lower value is 68 ± 2.8 km s⁻¹ Mpc⁻¹, and the higher is 73.24 ± 1.74 km s⁻¹ Mpc⁻¹. The lower value is from a median statistics analysis (Gott et al. 2001) of 553 measurements of H₀ tabulated by Huchra (Chen & Ratra 2011a). It agrees with earlier median statistics estimates of H₀ from smaller compilations (Gott et al. 2001; Chen et al. 2003) and is consistent with a number of other recent determinations of H₀ from Wilkinson Microwave Anisotropy Probe, Atacama Cosmology Telescope, and Planck CMB anisotropy data (Hinshaw et al. 2013; Sievers et al. 2013; Ade et al. 2015; Addison et al. 2016), from BAO measurements (Aubourg et al. 2015; Ross et al. 2015; L'Huillier & Shafieloo 2016), and from Hubble parameter data (Chen et al. 2016a), as well as with what is expected in the standard model of particle physics with only three light neutrino species given current cosmological data (see, e.g., Calabrese et al. 2012). The higher value is a relatively local measurement, based on Hubble Space Telescope data (Riess et al. 2016). It is consistent with other recent local measurements of H₀ (Riess et al. 2011; Freedman et al. 2012; Efstathiou 2014).

We compute the likelihood function ${ \mathcal L }({\boldsymbol{p}})$ for the models under discussion using Equation (18) of Farooq et al. (2013b) for the ranges of the cosmological parameters listed at the end of Section 4. We need these likelihood functions for the z_da computation of the previous section, which is the main result of the paper. In this section, we use these likelihood functions to constrain cosmological parameters such as the dark energy density.

For the two-parameter models, maximizing the likelihood function ${ \mathcal L }({\boldsymbol{p}})$ is performed by minimizing the corresponding ${\chi }^{2}({\boldsymbol{p}})\equiv -2\mathrm{ln}[{ \mathcal L }({\boldsymbol{p}})]$ following the procedure of Farooq et al. (2015). The corresponding minimum values of χ² and best-fit parameter values for the two-parameter models are summarized in Table 3. The 1σ, 2σ, and 3σ confidence contours are computed following the procedure of Farooq et al. (2015), and results are shown in Figure 2. The generalization of this procedure for the three-parameter models is straightforward, and best-fit three-dimensional parameter values and minimum χ² are also summarized in Table 3.

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For the three-parameter models we next compute three two-dimensional likelihood functions by marginalizing the three-dimensional likelihood function over each of the three parameters (assuming flat priors) in turn. These three two-dimensional likelihood functions are maximized as above, and the corresponding best-fit parameter values and minimum χ² are listed in Table 5. The confidence contours for these two-dimensional likelihood functions are shown in Figure 3 for the nonflat XCDM parameterization and in Figure 4 for the nonflat ϕCDM model.

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Table 5.  Two-dimensional Best-fit Parameters for Three-parameter, Nonflat Models

Model	h Priora	Marginalized Parameter	BFb	${\chi }_{\min }^{2}$
	0.68 ± 0.028	ΩK0	Ωm0 = 0.38	25.3
			ωX = −0.64
		ωX	Ωm0 = 0.16	22.5
			ΩK0 = 0.43
		Ωm0	ωX = −1.80	27.3
Nonflat XCDM			ΩK0 = 0.47
	0.7324 ± 0.0174	ΩK0	Ωm0 = 0.13	25.0
			ωX = −2
		ωX	Ωm0 = 0.15	22.1
			ΩK0 = 0.39
		Ωm0	ωX = −2	26.8
			ΩK0 = 0.41
	0.68 ± 0.028	ΩK0	Ωm0 = 0.28	25.7
			α = 1.33
		α	Ωm0 = 0.26	22.4
			ΩK0 = −0.02
		Ωm0	α = 0.01	28.7
Nonflat ϕCDM			ΩK0 = 0.19
	0.7324 ± 0.0174	ΩK0	Ωm0 = 0.25	29.0
			α = 0.01
		α	Ωm0 = 0.28	26.6
			ΩK0 = −0.19
		Ωm0	α = 0.01	31.6
			ΩK0 = −0.04

Notes.

^aHubble constant in units of 100 km s⁻¹ Mpc⁻¹. ^bBest-fit parameter values.

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To get two one-dimensional likelihood functions from each of the two-dimensional likelihood functions, we marginalize (with a flat prior) over each parameter in turn. We then determine the best-fit parameter values by maximizing each one-dimensional likelihood function and compute 1σ and 2σ intervals for each parameter in each model and for both H₀ priors. The best-fit parameter values and 1σ and 2σ intervals are given in Table 6 for the two-parameter models and in Table 7 for the three-parameter models.

The best-fit (two- and three-dimensional) model predictions are shown in Figure 1, for the five different cosmological models, ΛCDM in red, flat XCDM in blue, flat ϕCDM in green, nonflat XCDM in orange, and nonflat ϕCDM in brown, for the two H₀ priors, with ${H}_{0}\pm {\sigma }_{{H}_{0}}=68\pm 2.8$ km s⁻¹ Mpc⁻¹ in solid lines and ${H}_{0}\pm {\sigma }_{{H}_{0}}=73.24\pm 1.74$ km s⁻¹ Mpc⁻¹ in dot-dashed lines.

While the main purpose of our paper was to improve on the characterization of the deceleration–acceleration transition studied in Farooq & Ratra (2013a) and Farooq et al. (2013a), we see from Figure 2 and the left panels of Figures 3 and 4 that the H(z) data by themselves indicate that the cosmological expansion is currently accelerating.

From these figures, it is clear that the H(z) data of Table 1 are very consistent with the standard spatially flat ΛCDM cosmological model, although even for the two-parameter model constraint contours shown in Figure 2 there are a large range of dynamical dark energy models and spatially curved models that are consistent with the data. In Figures 3 and 4 for the nonflat dynamical dark energy models, it is clear that allowing for nonzero space curvature considerably broadens the dynamical dark energy options and vice versa. It is interesting to note that in the nonflat ϕCDM model Chen et al. (2016b) find that the cosmological data bound on the sum of neutrino masses is considerably weaker than if the model were spatially flat.

While the error bars are large, it is curious that Table 7 entries show that the nonflat XCDM parameterization mildly favors open spatial hypersurfaces while the nonflat ϕCDM model mildly prefers closed ones.

From the new list of H(z) data we have compiled, we find evidence for the cosmological deceleration–acceleration transition to have taken place at a redshift z_da = 0.72 ± 0.05 (0.84 ± 0.03), depending on the value of H₀ = 68 ± 2.8 (73.24 ± 1.74) km s⁻¹ Mpc⁻¹, but otherwise only mildly dependent on other cosmological parameters. In addition, the binned H(z) data in redshift space show qualitative visual evidence for the deceleration–acceleration transition, independent of how they are binned provided that the bins are narrow enough, in agreement with that originally found by Farooq et al. (2013a). These H(z) data are consistent with the standard spatially flat ΛCDM cosmological model but do not rule out nonzero space curvature or dynamical dark energy, especially in models that allow for both. Other data, such as currently available SN Ia, BAO, growth factor, or CMB anisotropy data, can tighten the constraints on these parameters (see, e.g., Farooq et al. 2015), and it is of interest to study how the other data constrain parameters when used in conjunction with the H(z) data we have compiled here.

O.F. and M.F. greatfully acknowledge the partial funding from the Department of Physical Sciences, Embry-Riddle Aeronautical University. S.C. and B.R. were supported in part by DOE grant DE-SC0011840.