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Abstract

We compile 41 H(z) data from literature and use them to constrain OΛCDM and flat ΛCDM parameters. We show that the available H(z) suffers from uncertainties overestimation and propose a Bayesian method to reduce them. As a result of this method, using H(z) only, we find, in the context of OΛCDM, H0 = 69.5 ± 2.5 km s−1 Mpc−1, Ωm = 0.242 ± 0.036, and |$\Omega _\Lambda =0.68\pm 0.14$|⁠. In the context of flat ΛCDM model, we have found H0 = 70.4 ± 1.2 km s−1 Mpc−1 and Ωm = 0.256 ± 0.014. This corresponds to an uncertainty reduction of up to |${\approx }30\hbox{ per cent}$| when compared to the uncorrected analysis in both cases.

cosmological parameters, dark energy, dark matter, cosmology: observations

1 INTRODUCTION

Measurements of the expansion of the Universe are a central subject in the modern cosmology. In 1998, observations of type Ia supernovae (Riess et al. 1997; Perlmutter et al. 1999) gave strong evidences of a transition epoch between decelerated and accelerated expansion. Those evidences are also consistent with data from Baryon Acoustic Oscillations (BAO) measurements and the Cosmic Microwave Background Anisotropies (CMB).

Among the many viable candidates to explain the cosmic acceleration, the cosmological constant Λ explains very well great part of the current observations and it is also the simplest candidate. It gave to the model formed by cosmological constant plus cold dark matter, the ΛCDM model, the status of standard model in cosmology. On the other hand, the Λ term presents important conceptual problems in its core, e.g. the huge inconsistency of the quantum derived and the cosmological observed values of energy density, the so-called cosmological constant problem (Weinberg 1989). Hence, despite of its observational success, the composition and the history of the Universe are still a question that needs further investigation.

Precise measurements of the cosmic expansion may be obtained through the SNe observations. Although they furnish stringent cosmological constraints, they are not directly measuring the expansion rate H(z) but its integral in the line of sight. Today, three distinct methods are producing direct measurements of H(z) namely, through differential dating of the cosmic chronometers (Simon et al. 2005; Stern et al. 2010; Moresco et al. 2012; Zhang et al. 2012; Moresco 2015; Moresco et al. 2016), BAO techniques (Gaztañaga et al. 2009; Blake et al. 2012; Busca et al. 2012; Anderson et al. 2013; Font-Ribeira et al. 2013; Delubac et al. 2014), and correlation function of luminous red galaxies (LRGs) (Chuang & Wang 2013; Oka et al. 2014), which does not rely on the nature of space–time geometry between the observed object and us.

In this work, we treat the ΛCDM model expansion history as a generative model for the H(z) data (Hogg, Bovy & Lang 2008). However, considering a goodness-of-fit criterion, we discuss a possible overestimation in the uncertainty in the current H(z) data and we propose a new generative model to H(z) data, in order to take into account this overestimation.

This article is structured as follows. In Section 2, we discuss the basic features of the ΛCDM model. In Section 3, we review the H(z) data available on the literature and compile a sample with 41 data.

In Section 4, we discuss the goodness of fit of ΛCDM with H(z) data and in Section 5 we discuss a method to treat H(z) uncertainties and apply it to the ΛCDM with spatial curvature. In Subsection 5.1, we apply the same method to the flat ΛCDM. In Section 6, we compare corrected and uncorrected models by using a Bayesian criterion and in Section 7 we compare our results with other H(z) analyses. Finally, in Section 8, we summarize the results.

2 COSMIC DYNAMICS OF ΛCDM Model

We start by considering the homogeneous and isotropic FRW line element (with c = 1):
(1)
where a is the scale factor, (r, θ, ϕ) are comoving coordinates, and the spatial curvature parameter k can assume values −1, +1 or 0.
In this background, the Einstein Field Equations (EFEs) with a cosmological constant are given by
(2)
(3)
where ρ and p are total density and pressure of the cosmological fluid and Λ is cosmological constant. We may write the Friedmann equation (2) in terms of the observable redshift z, which relates to scale factor as |$a=\frac{a_0}{1+z}$|⁠:
(4)
where |$\rho _\Lambda =\frac{\Lambda }{8\pi G}$| and |$H\equiv \frac{\dot{a}}{a}$| is the expansion rate. The EFEs include energy conservation, so we may deduce the continuity equation from equations (2) and (3):
(5)
where (ρi, pi) stand for each fluid, be it dark matter, baryons, radiation, neutrinos, cosmological constant or anything else that does not exchange energy. For dark matter and baryons, we have pi ∼ 0, so they evolve with ρi ∝ a−3, the cosmological constant has a constant |$\rho _\Lambda$| and radiation and neutrinos follow ρi ∝ a−4, so they may be neglected in our work, as we are interested in low redshifts (up to z ∼ 2). So, we may write for our components of interest:
(6)
(7)
where ρm stands for dark matter+baryons. So, the Friedmann equation can be written as
(8)
and by defining the density parameters |$\Omega _i\equiv \frac{\rho _{i0}}{\rho _{c0}}$|⁠, where |$\rho _{c0}\equiv \frac{3H_0^2}{8\pi G}$| and |$\Omega _k\equiv -\frac{k}{a_0^2H_0^2}$|⁠, we may write
(9)
from which we deduce the normalization condition |$\Omega _{\rm m} + \Omega _\Lambda + \Omega _k = 1$|⁠, or |$\Omega _k = 1 - \Omega _{\rm m} - \Omega _\Lambda$|⁠, so we actually have three free parameters on this equation (⁠|$\Omega _{\rm m}, \Omega _\Lambda ,H_0$|⁠). Finally, we may write for H(z)
(10)
As usual, we will call this model, where we allow for spatial curvature, OΛCDM. The standard, concordance flat ΛCDM model has Ωk = 0, thus:
(11)

3 H(z) DATA

Hubble parameter data as a function of redshift yields one of the most straightforward cosmological tests because it is inferred from astrophysical observations alone, not depending on any background cosmological models.

At the present time, the most important methods for obtaining H(z) data are1 (i) through ‘cosmic chronometers’, for example, the differential age of galaxies (Simon et al. 2005; Stern et al. 2010; Moresco et al. 2012; Zhang et al. 2012; Moresco 2015; Moresco et al. 2016), (ii) measurements of peaks of acoustic oscillations of baryons (BAO) (Gaztañaga et al. 2009; Blake et al. 2012; Busca et al. 2012; Anderson et al. 2013; Font-Ribeira et al. 2013; Delubac et al. 2014), and (iii) through a correlation function of LRGs (Chuang & Wang 2013; Oka et al. 2014).

The data we work here are a combination of the compilations from Sharov & Vorontsova (2014) and Moresco et al. (2016). Sharov & Vorontsova (2014) add six H(z) data in comparison to Farooq & Ratra (2013) compilation, which had 28 measurements. Moresco et al. (2016), on their turn, have added seven new H(z) measurements in comparison to Sharov & Vorontsova (2014). By combining both data sets, we arrive at 41 H(z) data, as can be seen in Table 1 and Fig. 1.

41 H(z) data and corresponding best-fitting ΛCDM model.
Figure 1.

41 H(z) data and corresponding best-fitting ΛCDM model.

Open in new tabDownload slide
Table 1.
Open in new tab

41 Hubble parameter versus redshift data.

zH(z)σHReference
0.0706919.6Zhang et al. (2012)
0.0906912Simon et al. (2005)
0.12068.626.2Zhang et al. (2012)
0.170838Simon et al. (2005)
0.179754Moresco et al. (2012)
0.199755Moresco et al. (2012)
0.20072.929.6Zhang et al. (2012)
0.24079.696.65Gaztañaga et al. (2009)
0.2707714Simon et al. (2005)
0.28088.836.6Zhang et al. (2012)
0.30081.76.22Oka et al. (2014)
0.35082.78.4Chuang & Wang (2013)
0.3528314Moresco et al. (2012)
0.38028313.5Moresco et al. (2016)
0.4009517Simon et al. (2005)
0.40047710.02Moresco et al. (2016)
0.424787.111.2Moresco et al. (2016)
0.43086.453.68Gaztañaga et al. (2009)
0.44082.67.8Blake et al. (2012)
0.449792.812.9Moresco et al. (2016)
0.478380.99Moresco et al. (2016)
0.4809762Stern et al. (2010)
0.57092.9007.855Anderson et al. (2013)
0.59310413Moresco et al. (2012)
0.687.96.1Blake et al. (2012)
0.68928Moresco et al. (2012)
0.7397.37.0Blake et al. (2012)
0.78110512Moresco et al. (2012)
0.87512517Moresco et al. (2012)
0.889040Stern et al. (2010)
0.911723Simon et al. (2005)
1.03715420Moresco et al. (2012)
1.30016817Simon et al. (2005)
1.36316022.6Moresco (2015)
1.4317718Simon et al. (2005)
1.5314014Simon et al. (2005)
1.7520240Simon et al. (2005)
1.965186.550.4Moresco (2015)
2.3002248Busca et al. (2012)
2.342227Delubac et al. (2014)
2.362268Font-Ribeira et al. (2013)
zH(z)σHReference
0.0706919.6Zhang et al. (2012)
0.0906912Simon et al. (2005)
0.12068.626.2Zhang et al. (2012)
0.170838Simon et al. (2005)
0.179754Moresco et al. (2012)
0.199755Moresco et al. (2012)
0.20072.929.6Zhang et al. (2012)
0.24079.696.65Gaztañaga et al. (2009)
0.2707714Simon et al. (2005)
0.28088.836.6Zhang et al. (2012)
0.30081.76.22Oka et al. (2014)
0.35082.78.4Chuang & Wang (2013)
0.3528314Moresco et al. (2012)
0.38028313.5Moresco et al. (2016)
0.4009517Simon et al. (2005)
0.40047710.02Moresco et al. (2016)
0.424787.111.2Moresco et al. (2016)
0.43086.453.68Gaztañaga et al. (2009)
0.44082.67.8Blake et al. (2012)
0.449792.812.9Moresco et al. (2016)
0.478380.99Moresco et al. (2016)
0.4809762Stern et al. (2010)
0.57092.9007.855Anderson et al. (2013)
0.59310413Moresco et al. (2012)
0.687.96.1Blake et al. (2012)
0.68928Moresco et al. (2012)
0.7397.37.0Blake et al. (2012)
0.78110512Moresco et al. (2012)
0.87512517Moresco et al. (2012)
0.889040Stern et al. (2010)
0.911723Simon et al. (2005)
1.03715420Moresco et al. (2012)
1.30016817Simon et al. (2005)
1.36316022.6Moresco (2015)
1.4317718Simon et al. (2005)
1.5314014Simon et al. (2005)
1.7520240Simon et al. (2005)
1.965186.550.4Moresco (2015)
2.3002248Busca et al. (2012)
2.342227Delubac et al. (2014)
2.362268Font-Ribeira et al. (2013)
Table 1.
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41 Hubble parameter versus redshift data.

zH(z)σHReference
0.0706919.6Zhang et al. (2012)
0.0906912Simon et al. (2005)
0.12068.626.2Zhang et al. (2012)
0.170838Simon et al. (2005)
0.179754Moresco et al. (2012)
0.199755Moresco et al. (2012)
0.20072.929.6Zhang et al. (2012)
0.24079.696.65Gaztañaga et al. (2009)
0.2707714Simon et al. (2005)
0.28088.836.6Zhang et al. (2012)
0.30081.76.22Oka et al. (2014)
0.35082.78.4Chuang & Wang (2013)
0.3528314Moresco et al. (2012)
0.38028313.5Moresco et al. (2016)
0.4009517Simon et al. (2005)
0.40047710.02Moresco et al. (2016)
0.424787.111.2Moresco et al. (2016)
0.43086.453.68Gaztañaga et al. (2009)
0.44082.67.8Blake et al. (2012)
0.449792.812.9Moresco et al. (2016)
0.478380.99Moresco et al. (2016)
0.4809762Stern et al. (2010)
0.57092.9007.855Anderson et al. (2013)
0.59310413Moresco et al. (2012)
0.687.96.1Blake et al. (2012)
0.68928Moresco et al. (2012)
0.7397.37.0Blake et al. (2012)
0.78110512Moresco et al. (2012)
0.87512517Moresco et al. (2012)
0.889040Stern et al. (2010)
0.911723Simon et al. (2005)
1.03715420Moresco et al. (2012)
1.30016817Simon et al. (2005)
1.36316022.6Moresco (2015)
1.4317718Simon et al. (2005)
1.5314014Simon et al. (2005)
1.7520240Simon et al. (2005)
1.965186.550.4Moresco (2015)
2.3002248Busca et al. (2012)
2.342227Delubac et al. (2014)
2.362268Font-Ribeira et al. (2013)
zH(z)σHReference
0.0706919.6Zhang et al. (2012)
0.0906912Simon et al. (2005)
0.12068.626.2Zhang et al. (2012)
0.170838Simon et al. (2005)
0.179754Moresco et al. (2012)
0.199755Moresco et al. (2012)
0.20072.929.6Zhang et al. (2012)
0.24079.696.65Gaztañaga et al. (2009)
0.2707714Simon et al. (2005)
0.28088.836.6Zhang et al. (2012)
0.30081.76.22Oka et al. (2014)
0.35082.78.4Chuang & Wang (2013)
0.3528314Moresco et al. (2012)
0.38028313.5Moresco et al. (2016)
0.4009517Simon et al. (2005)
0.40047710.02Moresco et al. (2016)
0.424787.111.2Moresco et al. (2016)
0.43086.453.68Gaztañaga et al. (2009)
0.44082.67.8Blake et al. (2012)
0.449792.812.9Moresco et al. (2016)
0.478380.99Moresco et al. (2016)
0.4809762Stern et al. (2010)
0.57092.9007.855Anderson et al. (2013)
0.59310413Moresco et al. (2012)
0.687.96.1Blake et al. (2012)
0.68928Moresco et al. (2012)
0.7397.37.0Blake et al. (2012)
0.78110512Moresco et al. (2012)
0.87512517Moresco et al. (2012)
0.889040Stern et al. (2010)
0.911723Simon et al. (2005)
1.03715420Moresco et al. (2012)
1.30016817Simon et al. (2005)
1.36316022.6Moresco (2015)
1.4317718Simon et al. (2005)
1.5314014Simon et al. (2005)
1.7520240Simon et al. (2005)
1.965186.550.4Moresco (2015)
2.3002248Busca et al. (2012)
2.342227Delubac et al. (2014)
2.362268Font-Ribeira et al. (2013)
From these data, we perform a χ2-statistics, generating the χ2 function of free parameters:
(12)
where |$E(z)\equiv \frac{H(z)}{H_0}$| and H(z) is given by equation (10).

4 DATA ANALYSIS AND GOODNESS OF FIT

In order to minimize the χ2 function (12) and find the constraints over the free parameters |$(H_0, \Omega _{\rm m}, \Omega _\Lambda )$|⁠, we have sampled the likelihood |$\mathcal {L}\propto e^{-\chi ^2/2}$| through Monte Carlo Markov Chain (MCMC) analysis. A simple and powerful MCMC method is the so-called Affine Invariant MCMC Ensemble Sampler by Goodman & Weare (2010), which was implemented in Python language with the emcee software by Foreman et al. (2013). This MCMC method has the advantage over simple Metropolis-Hasting (MH) methods of depending on only one scale parameter of the proposal distribution and on the number of walkers, while MH methods in general depend on the parameter covariance matrix, that is, it depends on n(n + 1)/2 tuning parameters, where n is dimension of parameter space. The main idea of the Goodman–Weare affine-invariant sampler is the so-called ‘stretch move’, where the position (parameter vector in parameter space) of a walker (chain) is determined by the position of the other walkers. Foreman-Mackey et al. (2013) modified this method, in order to make it suitable for parallelization, by splitting the walkers in two groups, then the position of a walker in one group is determined by only the position of walkers of the other group.2

We used the freely available software emcee to sample from our likelihood in our three-dimensional parameter space. We have used flat priors over the parameters. In order to plot all the constraints in the same figure, we have used the freely available software getdist,3 in its Python version. The results of our statistical analyses from equation (12) correspond to the red lines in Fig. 3 and Table 2. From this analysis, we have obtained |$\chi ^2_\nu =\frac{\chi ^2_{{\rm min}}}{\nu }=18.551/38=0.488\,19$|⁠, where ν = n − p is the number of degrees of freedom.

Table 2.
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Mean values of parameters of OΛCDM model from H(z) data, without uncertainties correction and with uncertainties correction factor f. Uncertainties correspond to 68 per cent c.l.

H(z) onlyH(z) + H0
ParameterUncorrectedCorrectedUncorrectedCorrected
H069.1 ± 3.569.5 ± 2.572.4 ± 1.572.5 ± 1.1
Ωm0.237 ± 0.0510.242 ± 0.0360.267 ± 0.0380.268 ± 0.028
|$\Omega _\Lambda$|0.66 ± 0.200.68 ± 0.14|$0.825^{+0.11}_{-0.095}$|0.831 ± 0.073
f–|$0.723^{+0.084}_{-0.085}$|–|$0.728^{+0.067}_{-0.098}$|
H(z) onlyH(z) + H0
ParameterUncorrectedCorrectedUncorrectedCorrected
H069.1 ± 3.569.5 ± 2.572.4 ± 1.572.5 ± 1.1
Ωm0.237 ± 0.0510.242 ± 0.0360.267 ± 0.0380.268 ± 0.028
|$\Omega _\Lambda$|0.66 ± 0.200.68 ± 0.14|$0.825^{+0.11}_{-0.095}$|0.831 ± 0.073
f–|$0.723^{+0.084}_{-0.085}$|–|$0.728^{+0.067}_{-0.098}$|
Table 2.
Open in new tab

Mean values of parameters of OΛCDM model from H(z) data, without uncertainties correction and with uncertainties correction factor f. Uncertainties correspond to 68 per cent c.l.

H(z) onlyH(z) + H0
ParameterUncorrectedCorrectedUncorrectedCorrected
H069.1 ± 3.569.5 ± 2.572.4 ± 1.572.5 ± 1.1
Ωm0.237 ± 0.0510.242 ± 0.0360.267 ± 0.0380.268 ± 0.028
|$\Omega _\Lambda$|0.66 ± 0.200.68 ± 0.14|$0.825^{+0.11}_{-0.095}$|0.831 ± 0.073
f–|$0.723^{+0.084}_{-0.085}$|–|$0.728^{+0.067}_{-0.098}$|
H(z) onlyH(z) + H0
ParameterUncorrectedCorrectedUncorrectedCorrected
H069.1 ± 3.569.5 ± 2.572.4 ± 1.572.5 ± 1.1
Ωm0.237 ± 0.0510.242 ± 0.0360.267 ± 0.0380.268 ± 0.028
|$\Omega _\Lambda$|0.66 ± 0.200.68 ± 0.14|$0.825^{+0.11}_{-0.095}$|0.831 ± 0.073
f–|$0.723^{+0.084}_{-0.085}$|–|$0.728^{+0.067}_{-0.098}$|
As it is well known (Vuolo 1996; Bevington & Robinson 2003), when one analyses the probability distribution of |$\chi ^2_\nu$| it has an expected value |$\chi ^2_\nu = 1$|⁠. |$\chi ^2_\nu$| values very far from this are unlikely. High |$\chi ^2_\nu$| values may indicate underestimation of uncertainties or poor fitting of the model, while low values of |$\chi ^2_\nu$| indicate, in general, overestimation of uncertainties. The |$\chi ^2_\nu$| distribution is given by
(13)
where Γ is complete gamma function. It can be shown that the mean |$\chi ^2_\nu$| is given by |$\overline{\chi ^2_\nu }=1$|⁠, while the mode is given by |$\widehat{\chi ^2_\nu }=1-\frac{2}{\nu }$|⁠. In the limit of a large sample and few parameters, both converge to the same value |$\chi ^2_\nu \approx 1$|⁠. From (13), we may also define the cumulative distribution function (cdf) or probability of obtaining a value of |$\chi ^2_\nu$| as low as Q as
(14)
In order to illustrate the untypical small value of the |$\chi ^2_\nu$| value we have obtained, namely, |$\chi ^2_\nu =0.488\,19$|⁠, we have plotted the pdf |$h_\nu (\chi ^2_\nu )$| (13) and the cdf (14) for ν = 38 in Fig. 2.
$h_\nu (\chi ^2_\nu )$ and corresponding cdf for ν = 38.
Figure 2.

|$h_\nu (\chi ^2_\nu )$| and corresponding cdf for ν = 38.

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As one may see in the Fig. 2, the probability of obtaining |$\chi ^2_\nu$| as low as |$\chi ^2_\nu =0.488$| for ν = 38 is quite small. In fact, by calculating the integral (14), we have obtained |$P(\chi ^2_\nu <0.488\,19)=0.3342\hbox{ per cent}$|⁠. Hence, it is a very small and unlikely χ2 value, which, in turn, from equation (12) indicates overestimated H(z) uncertainties.

5 H(z) UNCERTAINTIES CORRECTION

How one may try to correct uncertainties? Ideally, at the moment of data acquisition, a better control of systematic uncertainties is desirable and new methods less prone to errors are to be used. In fact, in general, data coming from BAO and Lyman α have smaller errors than data coming from differential ages. However, not being able to reobtain the measurements or reanalyze them through new methods, we are left with the available data. Then, can nothing be done? From the Bayesian viewpoint, not necessarily. In fact, we may view the data as a collection of (zi, Hi, σHi). Very often, we are interested in a likelihood given by |$\mathcal {L}=Ne^{-\chi ^2/2}$|⁠, where N is only a normalization constant and one is interested in maximizing the likelihood, which is equivalent to maximizing the χ2. Let us recall from where this expression comes from.

As discussed in Hogg et al. (2008), the likelihood may be seen as an objective function, that is, a function that represents monotonically the quality of the fit. Given a scientific problem at hand as fitting a model to the data, one must define some objective function that represents this ‘goodness of fit’, then try to optimize it in order to determine the best set of free parameters of the model that describe the data.

Hogg et al. (2008) argue that the only choice of the objective function that is truly justified, in the sense that it leads to probabilistic inference, is to make a generative model for the data. We may think of the generative model as a parametrized statistical procedure to reasonably generate the given data.

For instance, assuming Gaussian uncertainties in one dimension, we can create the following generative model: Imagine that the data really come from a function y = f(x, θ) given by the model and that the only reason for any data point deviates from this model is that to each of the true y values a small y-direction offset has been added, where that offset was drawn from a Gaussian distribution of zero mean and known variance |$\sigma _y^2$|⁠. In this model, given an independent position xi, an uncertainty σyi, and free parameters θ, the frequency distribution p(yi|xi, σyi, θ) for yi is
(15)
Thus, if the data points are independently drawn, the likelihood |$\mathcal {L}$| is the product of conditional probabilities
(16)
Taking the logarithm,
(17)

In equation above, the second term |$-\frac{1}{2}\sum _i\ln (2\pi \sigma _{yi}^2)$| is in general absorbed in the likelihood normalization constant, because the variances |$\sigma _{yi}^2$| are considered fixed by the data. Here, we consider σi as parameters to be obtained by optimization of the objective function |$\mathcal {L}$|⁠. As discussed in Hogg et al. (2008), it can be considered a correct procedure from the Bayesian point of view, although an involved one, and the obtained σi can be quite prior dependent.

In order to avoid having more free parameters than data, here we consider the σi to be all overestimated by a constant factor f, thus, σi, true = fσi. This can be seen just as a simplifying hypothesis. More elaborated methods could be gather the data in some groups, then correct the σi for each group. However, as discussed in Hogg et al. (2008), it is not an easy task to separate good data from bad data, and not necessarily the bad data are the ones with bigger uncertainties. So, we limit ourselves here with just one overall correction factor and then we investigate if this is a good approximation. Taking f as a free parameter, we constrain it in a joint analysis with the cosmological parameters, similar to what is made in SNe Ia analyses (Amanullah et al. 2010; Suzuki et al. 2012; Betoule et al. 2014). For ΛCDM model, our set of free parameters now is |$\theta =(H_0, \Omega _{\rm m}, \Omega _\Lambda ,f)$|⁠. A simpler but less justified hypothesis would be to simply find the value for f which provides |$\chi ^2_\nu \equiv 1$|⁠. However, as we expect |$\chi ^2_\nu$| to have some variance, such a procedure is not much trustworthy. With f as a free parameter, it may include some uncertainty into the analysis, when compared to the standard, uncorrected analysis, but at the same time, it may also reduce the cosmological parameter uncertainties.

Instead of equation (17), we must work here with the following objective function:
(18)

By maximizing the above likelihood, we find not only the best-fitting cosmological parameters, but also the best correction factor f which will furnish the best model to describe the data. By doing the same procedure of last section, now with the additional parameter f, we find the constraints shown by the black lines in Fig. 3.

The results of statistical analysis for OΛCDM model. H0 is in km s−1 Mpc−1. Diagonal: Marginalized constraints from H(z) data for each parameter. Below diagonal: Marginalized contour constraints for each indicated combination of parameters, with contours for 68.3 and 95.4 per cent confidence levels.
Figure 3.

The results of statistical analysis for OΛCDM model. H0 is in km s−1 Mpc−1. Diagonal: Marginalized constraints from H(z) data for each parameter. Below diagonal: Marginalized contour constraints for each indicated combination of parameters, with contours for 68.3 and 95.4 per cent confidence levels.

Open in new tabDownload slide

From Fig. 3, we may already see the difference in the parameter space if we introduce the f parameter. The corrected contours (black lines) are narrower than the uncorrected contours (red lines). This can be quantified by the parameter constraints shown in Table 2.

As can be seen in Table 2, σH0 has been reduced from 3.5 to 2.5, |$\sigma _{\Omega _{\rm m}}$| has been reduced from 0.051 to 0.036, and |$\sigma _{\Omega _\Lambda }$| has been reduced from 0.20 to 0.14. The mean value for f was |$f=0.723^{+0.084}_{-0.085}$|⁠. An interesting feature we may see from Fig. 3 is that the f parameter is much uncorrelated to cosmological parameters (confidence contours quite aligned with parameter axes). As we show in the next section, the best fit for the cosmological parameters (⁠|$H_0, \Omega _{\rm m}, \Omega _\Lambda$|⁠) is independent from the best fit for f. On the other hand, this is not true for the likelihoods, that is, |$\mathcal {L}\ne \mathcal {L}_1(H_0, \Omega _{\rm m}, \Omega _\Lambda )\mathcal {L}_2(f)$|⁠, as one may see from equation (18). This small unequality explains the small shift on mean values of cosmological parameters from Table 2. Saying in another way, the central values of cosmological parameters are weakly dependent on overall shifts on Hi uncertainties, but their variances are directly affected by f.

5.1 Flat ΛCDM

For completeness, as flat ΛCDM model is favoured from many observations, in this section we analyse this model similarly to OΛCDM. Equation (10) now reads
(19)
The results of this analysis may be seen in Fig. 4 and Table 3.
The results of statistical analysis for flat ΛCDM model. H0 is in km s−1 Mpc−1. Diagonal: Marginalized constraints from H(z) data for each parameter. Below diagonal: Marginalized contour constraints for each indicated combination of parameters, with contours for 68.3 and 95.4 per cent confidence levels.
Figure 4.

The results of statistical analysis for flat ΛCDM model. H0 is in km s−1 Mpc−1. Diagonal: Marginalized constraints from H(z) data for each parameter. Below diagonal: Marginalized contour constraints for each indicated combination of parameters, with contours for 68.3 and 95.4 per cent confidence levels.

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Table 3.
Open in new tab

Mean values of parameters of flat ΛCDM model from H(z) data, without uncertainties correction and with uncertainties correction factor f. Uncertainties correspond to 68 per cent c.l.

H(z) onlyH(z) + H0
ParameterUncorrectedCorrectedUncorrectedCorrected
H070.3 ± 1.770.4 ± 1.271.8 ± 1.271.80 ± 0.89
Ωm0.257 ± 0.0200.256 ± 0.014|$0.243^{+0.014}_{-0.015}$|0.242 ± 0.011
f–0.714 ± 0.082–|$0.728^{+0.066}_{-0.096}$|
H(z) onlyH(z) + H0
ParameterUncorrectedCorrectedUncorrectedCorrected
H070.3 ± 1.770.4 ± 1.271.8 ± 1.271.80 ± 0.89
Ωm0.257 ± 0.0200.256 ± 0.014|$0.243^{+0.014}_{-0.015}$|0.242 ± 0.011
f–0.714 ± 0.082–|$0.728^{+0.066}_{-0.096}$|
Table 3.
Open in new tab

Mean values of parameters of flat ΛCDM model from H(z) data, without uncertainties correction and with uncertainties correction factor f. Uncertainties correspond to 68 per cent c.l.

H(z) onlyH(z) + H0
ParameterUncorrectedCorrectedUncorrectedCorrected
H070.3 ± 1.770.4 ± 1.271.8 ± 1.271.80 ± 0.89
Ωm0.257 ± 0.0200.256 ± 0.014|$0.243^{+0.014}_{-0.015}$|0.242 ± 0.011
f–0.714 ± 0.082–|$0.728^{+0.066}_{-0.096}$|
H(z) onlyH(z) + H0
ParameterUncorrectedCorrectedUncorrectedCorrected
H070.3 ± 1.770.4 ± 1.271.8 ± 1.271.80 ± 0.89
Ωm0.257 ± 0.0200.256 ± 0.014|$0.243^{+0.014}_{-0.015}$|0.242 ± 0.011
f–0.714 ± 0.082–|$0.728^{+0.066}_{-0.096}$|

As one may see from Fig. 4, f is again uncorrelated to cosmological parameters, so it does not change their central values.

As one may see in Table 3, the H0 uncertainty, for instance, is reduced from 1.7 to 1.2, which now corresponds to 1.7 per cent relative uncertainty. Ωm uncertainty has reduced from 0.020 to 0.014.

5.2 Alternative analysis

In order to test the consistency of the above results, we have made an alternative analysis, considering only the data with lower redshifts and larger errors on H(z). Namely, we have ignored the data with z ≥ 2.3, which, although being distant, are reported with small uncertainties |$(3.15{\rm -}3.57\hbox{ per cent})$|⁠, when compared with lower redshift data with bigger uncertainties. Thus, here we use a new sample with 38 H(z) data and z < 2.3. In the present analysis, we do not consider H0 constraints, for simplicity.

As can be seen in Fig. 5 and Table 4, the result is that, without this ‘anchor’ at high redshift, the OΛCDM model is quite poorly constrained, mainly if we do not correct uncertainties. The result for Ωm, for example, is compatible with the absence of dark matter at a 2.6σ c.l. (Ωm ∼ 0.04 ∼ Ωb in its 2.6σ c.l. inferior limit). The constraints are slightly improved when we introduce the f correction (Ωm ≥ 0.04 at a 3.7σ c.l.). Concerning the flat ΛCDM model (Fig. 6 and Table 4), the result already is good with no correction (σH0 = 2.3) but is improved with the f correction (σH0 = 1.7).

The results of statistical analysis for OΛCDM model with 38 H(z) data with z < 2.3. H0 is in km s−1 Mpc−1. Diagonal: Marginalized constraints from H(z) data for each parameter. Below diagonal: Marginalized contour constraints for each indicated combination of parameters, with contours for 68.3 and 95.4 per cent confidence levels.
Figure 5.

The results of statistical analysis for OΛCDM model with 38 H(z) data with z < 2.3. H0 is in km s−1 Mpc−1. Diagonal: Marginalized constraints from H(z) data for each parameter. Below diagonal: Marginalized contour constraints for each indicated combination of parameters, with contours for 68.3 and 95.4 per cent confidence levels.

Open in new tabDownload slide
The results of statistical analysis for flat ΛCDM model with 38 H(z) data with z < 2.3. H0 is in km s−1 Mpc−1. Diagonal: Marginalized constraints from H(z) data for each parameter. Below diagonal: Marginalized contour constraints for each indicated combination of parameters, with contours for 68.3 and 95.4 per cent confidence levels.
Figure 6.

The results of statistical analysis for flat ΛCDM model with 38 H(z) data with z < 2.3. H0 is in km s−1 Mpc−1. Diagonal: Marginalized constraints from H(z) data for each parameter. Below diagonal: Marginalized contour constraints for each indicated combination of parameters, with contours for 68.3 and 95.4 per cent confidence levels.

Open in new tabDownload slide
Table 4.
Open in new tab

Mean values of parameters of OΛCDM and flat ΛCDM models from H(z) data, without uncertainties correction and with uncertainties correction factor f. Uncertainties correspond to 68 per cent c.l.

OΛCDMFlat ΛCDM
ParameterUncorrectedCorrectedUncorrectedCorrected
H071.7 ± 4.272.2 ± 3.069.2 ± 2.369.3 ± 1.7
Ωm|$0.40^{+0.18}_{-0.14}$||$0.41^{+0.12}_{-0.10}$||$0.290^{+0.041}_{-0.053}$||$0.286^{+0.030}_{-0.037}$|
|$\Omega _\Lambda$||$0.92^{+0.34}_{-0.23}$||$0.96^{+0.23}_{-0.17}$|––
f–|$0.72^{+0.069}_{-0.10}$|–|$0.730^{+0.069}_{-0.10}$|
OΛCDMFlat ΛCDM
ParameterUncorrectedCorrectedUncorrectedCorrected
H071.7 ± 4.272.2 ± 3.069.2 ± 2.369.3 ± 1.7
Ωm|$0.40^{+0.18}_{-0.14}$||$0.41^{+0.12}_{-0.10}$||$0.290^{+0.041}_{-0.053}$||$0.286^{+0.030}_{-0.037}$|
|$\Omega _\Lambda$||$0.92^{+0.34}_{-0.23}$||$0.96^{+0.23}_{-0.17}$|––
f–|$0.72^{+0.069}_{-0.10}$|–|$0.730^{+0.069}_{-0.10}$|
Table 4.
Open in new tab

Mean values of parameters of OΛCDM and flat ΛCDM models from H(z) data, without uncertainties correction and with uncertainties correction factor f. Uncertainties correspond to 68 per cent c.l.

OΛCDMFlat ΛCDM
ParameterUncorrectedCorrectedUncorrectedCorrected
H071.7 ± 4.272.2 ± 3.069.2 ± 2.369.3 ± 1.7
Ωm|$0.40^{+0.18}_{-0.14}$||$0.41^{+0.12}_{-0.10}$||$0.290^{+0.041}_{-0.053}$||$0.286^{+0.030}_{-0.037}$|
|$\Omega _\Lambda$||$0.92^{+0.34}_{-0.23}$||$0.96^{+0.23}_{-0.17}$|––
f–|$0.72^{+0.069}_{-0.10}$|–|$0.730^{+0.069}_{-0.10}$|
OΛCDMFlat ΛCDM
ParameterUncorrectedCorrectedUncorrectedCorrected
H071.7 ± 4.272.2 ± 3.069.2 ± 2.369.3 ± 1.7
Ωm|$0.40^{+0.18}_{-0.14}$||$0.41^{+0.12}_{-0.10}$||$0.290^{+0.041}_{-0.053}$||$0.286^{+0.030}_{-0.037}$|
|$\Omega _\Lambda$||$0.92^{+0.34}_{-0.23}$||$0.96^{+0.23}_{-0.17}$|––
f–|$0.72^{+0.069}_{-0.10}$|–|$0.730^{+0.069}_{-0.10}$|

Furthermore, the results for f are consistent with the ones we have obtained in the full 41 H(z) sample data, which indicates some robustness of the method.

6 BAYESIAN CRITERION COMPARISON

Here, we use the Bayesian Information Criterion (BIC) (Schwarz 1978; Jesus, Valentim & Andrade-Oliveira 2017) in order to compare the models with uncertainties correction and without uncertainties correction. As an approximation for the Bayesian Evidence (BE) (Trotta 2008), BIC is useful because it is, in general, easier to calculate. As explained in, e.g. Kass & Raftery (1995), Trotta (2008), and Jesus et al. (2017), BE and BIC are great model comparison tools, because they incorporate the Ockham's razor principle, which penalizes models with excess of parameters due to their unnecessary complexity. They are different from other model selection tools, like Akaike Information Criterion (Akaike 1974), for instance, which does not take into account the excess of parameters. Let us discuss now for our case, if the introduction of the f parameter is necessary to better describe the H(z) data. BIC is given by
(20)
where |$\mathcal {L}_{{\rm max}}$| is the likelihood maximum and p is the number of free parameters. The two models we want to compare are: M1: f = 1, that is, ΛCDM model without uncertainties correction is enough to describe the data; and M2: f ≠ 1 such that some correction f to uncertainties is necessary in order for the ΛCDM model to explain the H(z) data. We may write the log-likelihood as
(21)
where χ2 is the uncorrected |$\chi ^2\equiv \sum _{i=1}^n\frac{\left[H_i - H(z_i,H_0, \Omega _{\rm m}, \Omega _\Lambda )\right]^2}{\sigma _{Hi}^2}$|⁠. In order to calculate BIC, we must find the maximum of |$\ln \mathcal {L}$|⁠. By deriving (21) with respect to f:
(22)
When it vanishes, we find the best fit:
(23)
From (20) and (23), we find:
(24)
(25)
where pj is the number of free parameters in Mj. So,
(26)
For p1 = 3 and p2 = 4, it simplifies to
(27)
For n = 41 and |$\chi ^2_{{\rm min}}=18.551$|⁠, it yields: ΔBIC = 6.352. As discussed in Jesus et al. (2017), for example, values of ΔBIC > 5 correspond to a decisive or strong statistical difference. That is, by this criterion, the model M1 (no correction) may be discarded against model M2 (with correction).

So, according to the BIC, the inclusion of the f parameter is necessary and, in the context of ΛCDM model, it leads to a more appropriate analysis of H(z) data.

7 COMPARISON WITH OTHER H(z) DATA ANALYSES

Farooq & Ratra (2013) have constrained OΛCDM model with 28 H(z) data and two possible priors over H0. With the most stringent prior, namely, the one from Riess et al. (2011), they have found, at 2σ, 0.20 ≤ Ωm ≤ 0.44 and |$0.62\le \Omega _\Lambda \le 1.14$|⁠. We have found 0.13 ≤ Ωm ≤ 0.34 and |$0.23\le \Omega _\Lambda \le 1.04$| for 41 H(z) data without correction and 0.162 ≤ Ωm ≤ 0.31 and |$0.38\le \Omega _\Lambda \le 0.96$| with the f correction. By considering the prior from Riess et al. (2011), namely, H0 = 73.8 ± 2.4 km s−1 Mpc−1, we have found 0.18 ≤ Ωm ≤ 0.34 and |$0.57\le \Omega _\Lambda \le 1.04$| without correction and 0.21 ≤ Ωm ≤ 0.32 and |$0.65\le \Omega _\Lambda \le 0.99$| with the f correction.

With 34 H(z) data, Sharov & Vorontsova (2014) find a more stringent result, namely, H0 = 70.26 ± 0.32, |$\Omega _{\rm m}=0.276^{+0.009}_{-0.008}$|⁠, and |$\Omega _\Lambda =0.769\pm 0.029$|⁠. However, they have combined H(z) data with SNe Ia and BAO data, which is beyond the scope of our present work. However, by comparing their result with our Table 2, we may see that both constraints are compatible at a 1σ c.l.

Moresco et al. (2016) have used their compilation of 30 H(z) data combined with H0 from Riess et al. (2011) to constrain the transition redshift from deceleration to acceleration, in the context of OΛCDM (Lima et al. 2012):
(28)

They have found |$z_t=0.64^{+0.11}_{-0.07}$|⁠. By using the present 41 H(z) data, we find zt = 0.77 ± 0.22 without correction and zt = 0.78 ± 0.15 with the f correction. The results are in full agreement without the correction and are compatible at a 2σ c.l. with the f correction. We have mentioned the mean value for zt, while Moresco et al. (2016) refer to the best-fitting value.

The constraints over H0 are quite stringent today from many observations (Planck Collaboration XIII et al. 2016; Riess et al. 2016). However, there is some tension among H0 values estimated from different observations (Bernal, Verde & Riess 2016), so we choose not to use H0 in our main results here, Figs 3 and 4. We combine H(z) + H0 only in Tables 2 and 3 and in the present section, using Riess et al. (2011) result, in order to compare with other earlier analyses.

8 CONCLUSION

In this work, we have compiled 41 H(z) data and proposed a new method to better constrain models using H(z) data alone, namely, by reducing overestimated uncertainties through a Bayesian approach. The BIC was used to show the need for correcting H(z) data uncertainties. The uncertainties in the parameters were quite reduced when compared with methods of parameter estimation without correction and we have obtained an estimate of an overall correction factor in the context of OΛCDM and flat ΛCDM models.

Further investigations may include constraining other cosmological models or trying to optimally group H(z) data and then correcting uncertainties.

ACKNOWLEDGEMENTS

JFJ is supported by Fundação de Amparo à Pesquisa do Estado de São Paulo – FAPESP (Processes no. 2013/26258-4 and 2017/05859-0). FAO is supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – CAPES. TMG is supported by Unesp (Pró Talentos grant), and RV is supported by Fundação de Amparo à Pesquisa do Estado de São Paulo – FAPESP (Processes no. 2013/26258-4 and 2016/09831-0). We thank the support of the Instituto Nacional de Ciência e Tecnologia (INCT) e-Universe (CNPq grant 465376/2014-2).

Footnotes

1

See Lima et al. (2012) for a review.

2

See Allison & Dunkley (2014) for a comparison among various MCMC sampling techniques.

3

getdist is part of the great MCMC sampler and CMB power spectrum solver COSMOMC, by Lewis & Bridle (2002).

REFERENCES

Akaike
H.
,
1974
,
Trans. Autom. Control
,
19
,
6

 

Allison
R.
,
Dunkley
J.
,
2014
,
MNRAS
,
437
,
3918

Crossref
Search ADS
 

Amanullah
R.
et al. ,
2010
,
ApJ
,
716
,
712
10.1088/0004-637X/716/1/712

Crossref
Search ADS
 

Anderson
L.
et al. ,
2013
,
MNRAS
,
439
,
83

Crossref
Search ADS
 

Bernal
J. L.
,
Verde
L.
,
Riess
A. G.
,
2016
,
J. Cosmology Astropart. Phys.
,
1610
,
019

Crossref
Search ADS
 

Betoule
M.
et al. ,
2014
,
A&A
,
568
,
A22

Crossref
Search ADS
 

Bevington
P. R.
,
Robinson
D. K.
,
2003
,
Data Reduction and Error Analysis for the Physical Sciences
.
McGraw-Hill Book Company
,
New York, NY

Google Scholar

OpenURL Placeholder Text

 

Blake
C.
et al. ,
2012
,
MNRAS
,
425
,
405

Crossref
Search ADS
 

Busca
N. G.
et al. ,
2013
,
A&A
,
552
,
A96

Crossref
Search ADS
 

Chuang
C. H.
,
Wang
Y.
,
2013
,
MNRAS
,
435
,
255

Crossref
Search ADS
 

Delubac
T.
et al. ,
2015
,
A&A
,
574
,
A59

Crossref
Search ADS
 

Farooq
O.
,
Ratra
B.
,
2013
,
ApJ
,
766
,
L7
10.1088/2041-8205/766/1/L7

Crossref
Search ADS
 

Font-Ribera
A.
et al. ,
2013
,
J. Cosmology Astropart. Phys.
,
05
,
027

 

Foreman-Mackey
D.
,
Hogg
D. W.
,
Lang
D.
,
Goodman
J.
,
2013
,
PASP
,
125
,
306
10.1086/670067

Crossref
Search ADS
 

Gaztañaga
E.
,
Cabré
A.
,
Hui
L.
,
2009
,
MNRAS
,
399
,
1663

Crossref
Search ADS
 

Goodman
J.
,
Weare
J.
,
2010
,
Commun. Appl. Math. Comput. Sci.
,
5
,
65

Crossref
Search ADS
 

Hogg
D. W.
,
Bovy
J.
,
Lang
D.
,
2008
,
preprint (arXiv:1008.4686)

Jesus
J. F.
,
Valentim
R.
,
Andrade-Oliveira
F.
,
2017
,
J. Cosmology Astropart. Phys.
,
1709
,
030

Crossref
Search ADS
 

Kass
R. E.
,
Raftery
A. E.
,
1995
,
J. Am. Stat. Assoc.
,
90
,
773

Crossref
Search ADS
 

Lewis
A.
,
Bridle
S.
,
2002
,
Phys. Rev. D
,
66
,
103511
10.1103/PhysRevD.66.103511

Crossref
Search ADS
 

Lima
J. A. S.
,
Jesus
J. F.
,
Santos
R. C.
,
Gill
M. S. S.
,
2012
,
preprint (arXiv:1205.4688)

Moresco
M.
,
2015
,
MNRAS
,
450
,
L16
10.1093/mnrasl/slv037

Crossref
Search ADS
 

Moresco
M.
et al. ,
2012
,
J. Cosmology Astropart. Phys.
,
8
,
006

Crossref
Search ADS
 

Moresco
M.
et al. ,
2016
,
J. Cosmology Astropart. Phys.
,
1605
,
014

Crossref
Search ADS
 

Oka
A.
,
Saito
S.
,
Nishimichi
T.
,
Taruya
A.
,
Yamamoto
K.
,
2014
,
MNRAS
,
439
,
2515

Crossref
Search ADS
 

Perlmutter
S.
et al. ,
1999
,
ApJ
,
517
,
565
10.1086/307221

Crossref
Search ADS
 

Planck Collaboration XIII
et al. ,
2016
,
A&A
,
594
,
A13

Crossref
Search ADS
 

Riess
A. G.
et al. ,
1997
,
AJ
,
116
,
1009
10.1086/300499

Crossref
Search ADS
 

Riess
A. G.
et al. ,
2011
,
ApJ
,
730
,
119
10.1088/0004-637X/730/2/119

Crossref
Search ADS
 

Riess
A. G.
et al. ,
2016
,
ApJ
,
826
,
56

Crossref
Search ADS
 

Schwarz
G.
,
1978
,
Ann. Stat.
,
5
,
461

Crossref
Search ADS
 

Sharov
G. S.
,
Vorontsova
E. G.
,
2014
,
J. Cosmology Astropart. Phys.
,
1410
,
057

Crossref
Search ADS
 

Simon
J.
,
Verde
L.
,
Jimenez
R.
,
2005
,
Phys. Rev. D
,
71
,
123001
10.1103/PhysRevD.71.123001

Crossref
Search ADS
 

Stern
D.
,
Jimenez
R.
,
Verde
L.
,
Kamionkowski
M.
,
Stanford
S. A.
,
2010
,
J. Cosmology Astropart. Phys.
,
02
,
008

Crossref
Search ADS
 

Suzuki
N.
et al. ,
2012
,
ApJ
,
746
,
85
10.1088/0004-637X/746/1/85

Crossref
Search ADS
 

Trotta
R.
,
2008
,
Contemp. Phys.
,
49
,
71
10.1080/00107510802066753

Crossref
Search ADS
 

Vuolo
J. H.
,
1996
,
Fundamentos da teoria de erros
.
Edgard Blücher
,
São Paulo
(in Portuguese)

Google Scholar

OpenURL Placeholder Text

 

Weinberg
S.
,
1989
,
Rev. Mod. Phys.
,
61
,
1
10.1103/RevModPhys.61.1

Crossref
Search ADS
 

Zhang
C.
,
Zhang
H.
,
Yuan
S.
,
Liu
S.
,
Zhang
T.-J.
,
Sun
Y.-C.
,
2012
,
Res. Astron. Astrophys.
,
14
,
1221

Crossref
Search ADS
 
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