MNRAS 477, 2867–2873 (2018) doi:10.1093/mnras/sty813 Advance Access publication 2018 March 28 Bayesian correction of H(z) data uncertainties J. F. Jesus,1,2‹ T. M. Gregório,1 F. Andrade-Oliveira,3,4 R. Valentim5 and C. A. O. Matos1 1Universidade Estadual Paulista (UNESP), Câmpus Experimental de Itapeva, Rua Geraldo Alckmin 519, 18409-010 Vila N. Sra. de Fátima, Itapeva, SP, Brazil 2Universidade Estadual Paulista (UNESP), Faculdade de Engenharia, Guaratinguetá, Departamento de Fı́sica e Quı́mica, Av. Dr. Ariberto Pereira da Cunha 333, 12516-410 Guaratinguetá - SP, Brazil 3Universidade Estadual Paulista (UNESP), Instituto de Fı́sica Teórica, São Paulo SP - 01140-070, Brazil 4Laboratório Interinstitucional de e-Astronomia - LIneA, Rua General José Cristino, 77, Rio de Janeiro, RJ 20921-400, Brazil 5Departamento de Fı́sica, Instituto de Ciências Ambientais, Quı́micas e Farmacêuticas - ICAQF, Universidade Federal de São Paulo (UNIFESP), Unidade José Alencar, Rua São Nicolau No. 210, 09913-030 Diadema, SP, Brazil Accepted 2018 March 26. Received 2018 March 16; in original form 2017 November 24 ABSTRACT We compile 41 H(z) data from literature and use them to constrain O�CDM and flat �CDM 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 O�CDM, H0 = 69.5 ± 2.5 km s−1 Mpc−1, �m = 0.242 ± 0.036, and �� = 0.68 ± 0.14. In the context of flat �CDM model, we have found H0 = 70.4 ± 1.2 km s−1 Mpc−1 and �m = 0.256 ± 0.014. This corresponds to an uncertainty reduction of up to ≈30 per cent when compared to the uncorrected analysis in both cases. Key words: cosmological parameters – dark energy – dark matter – cosmology: observations. 1 IN T RO D U C T I O N Measurements of the expansion of the Universe are a central sub- ject 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 acceler- ated 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 accel- eration, the cosmological constant � 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 �CDM model, the status of standard model in cosmol- ogy. On the other hand, the � term presents important conceptual problems in its core, e.g. the huge inconsistency of the quantum de- rived 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 ob- tained through the SNe observations. Although they furnish strin- gent cosmological constraints, they are not directly measuring the � E-mail: jfjesus@itapeva.unesp.br 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; An- derson 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 na- ture of space–time geometry between the observed object and us. In this work, we treat the �CDM 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 pos- sible 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 �CDM 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 �CDM with H(z) data and in Section 5 we discuss a method to treat H(z) uncertainties and apply it to the �CDM with spatial curvature. In Subsection 5.1, we apply the same method to the flat �CDM. In Section 6, we compare corrected and uncorrected models by using a Bayesian C© 2018 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society D ow nloaded from https://academ ic.oup.com /m nras/article-abstract/477/3/2867/4956051 by U niversidade Estadual Paulista J� lio de M esquita Filho user on 23 July 2019 mailto:jfjesus@itapeva.unesp.br 2868 J. F. Jesus et al. criterion and in Section 7 we compare our results with other H(z) analyses. Finally, in Section 8, we summarize the results. 2 C O S M I C DY NA M I C S O F �C D M M O D E L We start by considering the homogeneous and isotropic FRW line element (with c = 1): ds2 = dt2 − a2(t) ( dr2 1 − kr2 + r2dθ2 + r2sin2θdφ2 ) , (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 8πGρ = 3 ȧ2 a2 + 3 k a2 − � (2) − 8πGp = 2 ä a + ȧ2 a2 + k a2 − �, (3) where ρ and p are total density and pressure of the cosmological fluid and � is cosmological constant. We may write the Friedmann equation (2) in terms of the observable redshift z, which relates to scale factor as a = a0 1+z : H 2 = 8πG(ρ + ρ�) 3 − k(1 + z)2, (4) where ρ� = � 8πG and H ≡ ȧ a is the expansion rate. The EFEs in- clude energy conservation, so we may deduce the continuity equa- tion from equations (2) and (3): ρ̇i + 3H (ρi + pi) = 0, (5) where (ρ i, pi) stand for each fluid, be it dark matter, baryons, radia- tion, neutrinos, cosmological constant or anything else that does not exchange energy. For dark matter and baryons, we have pi ∼ 0, so they evolve with ρ i ∝ a−3, the cosmological constant has a constant ρ� and radiation and neutrinos follow ρ i ∝ 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: ρm = ρm0(1 + z)3 (6) ρ� = ρ�0, (7) where ρm stands for dark matter+baryons. So, the Friedmann equa- tion can be written as( H H0 )2 = 8πGρm0(1 + z)3 3H 2 0 + 8πGρ�0 3H 2 0 − k(1 + z)2 H 2 0 (8) and by defining the density parameters �i ≡ ρi0 ρc0 , where ρc0 ≡ 3H 2 0 8πG and �k ≡ − k a2 0H 2 0 , we may write( H H0 )2 = �m(1 + z)3 + �k(1 + z)2 + ��, (9) from which we deduce the normalization condition �m + �� + �k = 1, or �k = 1 − �m − ��, so we actually have three free parameters on this equation (�m, ��, H0). Finally, we may write for H(z) H (z) = H0 [ �m(1 + z)3 + (1 − �m − ��)(1 + z)2 + �� ] 1 2 . (10) As usual, we will call this model, where we allow for spatial curvature, O�CDM. The standard, concordance flat �CDM model has �k = 0, thus: H (z) = H0 [ �m(1 + z)3 + 1 − �m ] 1 2 . (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. From these data, we perform a χ2-statistics, generating the χ2 function of free parameters: χ2 = 41∑ i=1 [ H0E(zi, �m, ��) − Hi σHi ]2 , (12) where E(z) ≡ H (z) H0 and H(z) is given by equation (10). 4 DATA A NA LY S I S A N D G O O D N E S S O F F I T In order to minimize the χ2 function (12) and find the constraints over the free parameters (H0, �m, ��), we have sampled the like- lihood L ∝ e−χ2/2 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 em- cee 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 1 See Lima et al. (2012) for a review. 2 See Allison & Dunkley (2014) for a comparison among various MCMC sampling techniques. MNRAS 477, 2867–2873 (2018) D ow nloaded from https://academ ic.oup.com /m nras/article-abstract/477/3/2867/4956051 by U niversidade Estadual Paulista J� lio de M esquita Filho user on 23 July 2019 Bayesian correction of H(z) uncertainties 2869 Table 1. 41 Hubble parameter versus redshift data. z H(z) σH Reference 0.070 69 19.6 Zhang et al. (2012) 0.090 69 12 Simon et al. (2005) 0.120 68.6 26.2 Zhang et al. (2012) 0.170 83 8 Simon et al. (2005) 0.179 75 4 Moresco et al. (2012) 0.199 75 5 Moresco et al. (2012) 0.200 72.9 29.6 Zhang et al. (2012) 0.240 79.69 6.65 Gaztañaga et al. (2009) 0.270 77 14 Simon et al. (2005) 0.280 88.8 36.6 Zhang et al. (2012) 0.300 81.7 6.22 Oka et al. (2014) 0.350 82.7 8.4 Chuang & Wang (2013) 0.352 83 14 Moresco et al. (2012) 0.3802 83 13.5 Moresco et al. (2016) 0.400 95 17 Simon et al. (2005) 0.4004 77 10.02 Moresco et al. (2016) 0.4247 87.1 11.2 Moresco et al. (2016) 0.430 86.45 3.68 Gaztañaga et al. (2009) 0.440 82.6 7.8 Blake et al. (2012) 0.4497 92.8 12.9 Moresco et al. (2016) 0.4783 80.9 9 Moresco et al. (2016) 0.480 97 62 Stern et al. (2010) 0.570 92.900 7.855 Anderson et al. (2013) 0.593 104 13 Moresco et al. (2012) 0.6 87.9 6.1 Blake et al. (2012) 0.68 92 8 Moresco et al. (2012) 0.73 97.3 7.0 Blake et al. (2012) 0.781 105 12 Moresco et al. (2012) 0.875 125 17 Moresco et al. (2012) 0.88 90 40 Stern et al. (2010) 0.9 117 23 Simon et al. (2005) 1.037 154 20 Moresco et al. (2012) 1.300 168 17 Simon et al. (2005) 1.363 160 22.6 Moresco (2015) 1.43 177 18 Simon et al. (2005) 1.53 140 14 Simon et al. (2005) 1.75 202 40 Simon et al. (2005) 1.965 186.5 50.4 Moresco (2015) 2.300 224 8 Busca et al. (2012) 2.34 222 7 Delubac et al. (2014) 2.36 226 8 Font-Ribeira et al. (2013) Figure 1. 41 H(z) data and corresponding best-fitting �CDM model. 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 con- straints in the same figure, we have used the freely available soft- ware getdist,3 in its Python version. The results of our sta- tistical analyses from equation (12) correspond to the red lines in Fig. 3 and Table 2. From this analysis, we have obtained χ2 ν = χ2 min ν = 18.551/38 = 0.488 19, where ν = n − p is the number of degrees of freedom. As it is well known (Vuolo 1996; Bevington & Robinson 2003), when one analyses the probability distribution of χ2 ν it has an ex- pected value χ2 ν = 1. χ2 ν values very far from this are unlikely. High χ2 ν values may indicate underestimation of uncertainties or poor fitting of the model, while low values of χ2 ν indicate, in general, overestimation of uncertainties. The χ2 ν distribution is given by hν(χ2 ν ) = ν ν 2 (χ2 ν ) 1 2 (ν−2)e− ν 2 χ2 ν 2ν/2�(ν/2) , (13) where � is complete gamma function. It can be shown that the mean χ2 ν is given by χ2 ν = 1, while the mode is given by χ̂2 ν = 1 − 2 ν . In the limit of a large sample and few parameters, both converge to the same value χ2 ν ≈ 1. From (13), we may also define the cumulative distribution function (cdf) or probability of obtaining a value of χ2 ν as low as Q as P (χ2 ν < Q) ≡ ∫ Q 0 hν(Q′)dQ′. (14) In order to illustrate the untypical small value of the χ2 ν value we have obtained, namely, χ2 ν = 0.488 19, we have plotted the pdf hν(χ2 ν ) (13) and the cdf (14) for ν = 38 in Fig. 2. As one may see in the Fig. 2, the probability of obtaining χ2 ν as low as χ2 ν = 0.488 for ν = 38 is quite small. In fact, by calculating the in- tegral (14), we have obtained P (χ2 ν < 0.488 19) = 0.3342 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) U N C E RTA I N T I E S C O R R E C T I O N 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, σ Hi). Very often, we are interested in a likelihood given by L = Ne−χ2/2, 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 mono- tonically the quality of the fit. Given a scientific problem at hand as fitting a model to the data, one must define some objective func- tion 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. 3 getdist is part of the great MCMC sampler and CMB power spectrum solver COSMOMC, by Lewis & Bridle (2002). MNRAS 477, 2867–2873 (2018) D ow nloaded from https://academ ic.oup.com /m nras/article-abstract/477/3/2867/4956051 by U niversidade Estadual Paulista J� lio de M esquita Filho user on 23 July 2019 2870 J. F. Jesus et al. Table 2. Mean values of parameters of O�CDM model from H(z) data, without uncertainties correction and with uncertainties correction factor f. Uncertainties correspond to 68 per cent c.l. H(z) only H(z) + H0 Parameter Uncorrected Corrected Uncorrected Corrected H0 69.1 ± 3.5 69.5 ± 2.5 72.4 ± 1.5 72.5 ± 1.1 �m 0.237 ± 0.051 0.242 ± 0.036 0.267 ± 0.038 0.268 ± 0.028 �� 0.66 ± 0.20 0.68 ± 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 Figure 2. hν (χ2 ν ) and corresponding cdf for ν = 38. Hogg et al. (2008) argue that the only choice of the objective function that is truly justified, in the sense that it leads to probabilis- tic 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 σ 2 y . In this model, given an independent position xi, an uncertainty σ yi, and free parameters θ , the frequency distribution p(yi|xi, σ yi, θ ) for yi is p(yi |xi, σyi , θ ) = 1 (2π )1/2σyi exp [ − (yi − f (xi, θ ))2 2 σ 2 yi ] , (15) Thus, if the data points are independently drawn, the likelihood L is the product of conditional probabilities L = n∏ i=1 p(yi |xi, σyi , θ ). (16) Taking the logarithm, lnL = −1 2 n∑ i=1 [ (yi − f (xi, θ ))2 σ 2 yi + ln(2πσ 2 yi) ] . (17) In equation above, the second term − 1 2 ∑ i ln(2πσ 2 yi) is in gen- eral absorbed in the likelihood normalization constant, because the variances σ 2 yi are considered fixed by the data. Here, we consider σ i as parameters to be obtained by optimization of the objective function 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 σ i can be quite prior dependent. In order to avoid having more free parameters than data, here we consider the σ i to be all overestimated by a constant factor f, thus, σ i, true = fσ i. This can be seen just as a simplifying hypoth- esis. More elaborated methods could be gather the data in some groups, then correct the σ i 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 big- ger uncertainties. So, we limit ourselves here with just one overall correction factor and then we investigate if this is a good approx- imation. 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 �CDM model, our set of free parameters now is θ = (H0, �m, ��, f ). A simpler but less justified hypothe- sis would be to simply find the value for f which provides χ2 ν ≡ 1. However, as we expect χ2 ν 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: lnL = −1 2 n∑ i=1 { [Hi − H (zi, H0, �m, ��)]2 f 2σ 2 Hi + ln(2πf 2σ 2 Hi) } . (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. 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, σ H0 has been reduced from 3.5 to 2.5, σ�m has been reduced from 0.051 to 0.036, and σ�� has been re- duced 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 con- tours quite aligned with parameter axes). As we show in the next section, the best fit for the cosmological parameters (H0, �m, ��) is independent from the best fit for f. On the other hand, this is not true for the likelihoods, that is, L �= L1(H0,�m, ��)L2(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 parame- ters are weakly dependent on overall shifts on Hi uncertainties, but their variances are directly affected by f. MNRAS 477, 2867–2873 (2018) D ow nloaded from https://academ ic.oup.com /m nras/article-abstract/477/3/2867/4956051 by U niversidade Estadual Paulista J� lio de M esquita Filho user on 23 July 2019 Bayesian correction of H(z) uncertainties 2871 Figure 3. The results of statistical analysis for O�CDM 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 in- dicated combination of parameters, with contours for 68.3 and 95.4 per cent confidence levels. 5.1 Flat �CDM For completeness, as flat �CDM model is favoured from many observations, in this section we analyse this model similarly to O�CDM. Equation (10) now reads H (z) = H0 [ �m(1 + z)3 + 1 − �m ] 1 2 . (19) The results of this analysis may be seen in Fig. 4 and Table 3. As one may see from Fig. 4, f is again uncorrelated to cosmolog- ical 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 rel- ative uncertainty. �m 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 red- shifts 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 (3.15−3.57 per cent), when compared with lower red- shift 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 O�CDM model is quite poorly constrained, mainly if we do not correct uncertainties. The result for �m, for example, is compatible with the absence of dark matter at a 2.6σ c.l. (�m ∼ 0.04 ∼ �b in its 2.6σ c.l. inferior limit). The constraints are slightly improved when we introduce the f correction (�m ≥ 0.04 at a 3.7σ c.l.). Concerning the flat �CDM model (Fig. 6 and Table 4), the result already is good with no correction (σ H0 = 2.3) but is improved with the f correction (σ H0 = 1.7). Figure 4. The results of statistical analysis for flat �CDM 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 in- dicated combination of parameters, with contours for 68.3 and 95.4 per cent confidence levels. Table 3. Mean values of parameters of flat �CDM model from H(z) data, without uncertainties correction and with uncertainties correction factor f. Uncertainties correspond to 68 per cent c.l. H(z) only H(z) + H0 Parameter Uncorrected Corrected Uncorrected Corrected H0 70.3 ± 1.7 70.4 ± 1.2 71.8 ± 1.2 71.80 ± 0.89 �m 0.257 ± 0.020 0.256 ± 0.014 0.243+0.014 −0.015 0.242 ± 0.011 f – 0.714 ± 0.082 – 0.728+0.066 −0.096 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 BAY E S I A N C R I T E R I O N C O M PA R I S O N Here, we use the Bayesian Information Criterion (BIC) (Schwarz 1978; Jesus, Valentim & Andrade-Oliveira 2017) in order to com- pare the models with uncertainties correction and without uncer- tainties 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 com- parison tools, because they incorporate the Ockham’s razor princi- ple, which penalizes models with excess of parameters due to their unnecessary complexity. They are different from other model se- lection 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 BIC = −2 lnLmax + p ln n, (20) where Lmax is the likelihood maximum and p is the number of free parameters. The two models we want to compare are: M1: f = 1, MNRAS 477, 2867–2873 (2018) D ow nloaded from https://academ ic.oup.com /m nras/article-abstract/477/3/2867/4956051 by U niversidade Estadual Paulista J� lio de M esquita Filho user on 23 July 2019 2872 J. F. Jesus et al. Figure 5. The results of statistical analysis for O�CDM 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: Marginal- ized contour constraints for each indicated combination of parameters, with contours for 68.3 and 95.4 per cent confidence levels. Table 4. Mean values of parameters of O�CDM and flat �CDM mod- els from H(z) data, without uncertainties correction and with uncertainties correction factor f. Uncertainties correspond to 68 per cent c.l. O�CDM Flat �CDM Parameter Uncorrected Corrected Uncorrected Corrected H0 71.7 ± 4.2 72.2 ± 3.0 69.2 ± 2.3 69.3 ± 1.7 �m 0.40+0.18 −0.14 0.41+0.12 −0.10 0.290+0.041 −0.053 0.286+0.030 −0.037 �� 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 that is, �CDM 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 �CDM model to explain the H(z) data. We may write the log-likelihood as lnL = −1 2 [ χ2 f 2 + n∑ i=1 ln(2πf 2σ 2 i ) ] , (21) where χ2 is the uncorrected χ2 ≡ ∑n i=1 [Hi−H (zi ,H0,�m,��)]2 σ 2 Hi . In or- der to calculate BIC, we must find the maximum of lnL. By deriving (21) with respect to f: ∂ lnL ∂f = − 1 f [ n − χ2 f 2 ] . (22) When it vanishes, we find the best fit: f̂ = √ χ2 min n . (23) From (20) and (23), we find: BIC1 = χ2 min + n∑ i=1 ln(2πσ 2 i ) + p1 ln n (24) Figure 6. The results of statistical analysis for flat �CDM 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: Marginal- ized contour constraints for each indicated combination of parameters, with contours for 68.3 and 95.4 per cent confidence levels. BIC2 = n + n ln ( 2πχ2 min n ) + n∑ i=1 ln(σ 2 i ) + p2 ln n, (25) where pj is the number of free parameters in Mj. So, �BIC = BIC1 − BIC2 = χ2 min − n ln ( χ2 min ) + (n − p2 + p1) ln n − n. (26) For p1 = 3 and p2 = 4, it simplifies to �BIC = χ2 min − n ln ( χ2 min ) + (n − 1) ln n − n. (27) For n = 41 and χ2 min = 18.551, it yields: �BIC = 6.352. As dis- cussed in Jesus et al. (2017), for example, values of �BIC > 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 �CDM model, it leads to a more appropriate analysis of H(z) data. 7 C O M PA R I S O N W I T H OT H E R H( z) DATA A NA LY S E S Farooq & Ratra (2013) have constrained O�CDM 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 ≤ �m ≤ 0.44 and 0.62 ≤ �� ≤ 1.14. We have found 0.13 ≤ �m ≤ 0.34 and 0.23 ≤ �� ≤ 1.04 for 41 H(z) data without correction and 0.162 ≤ �m ≤ 0.31 and 0.38 ≤ �� ≤ 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 ≤ �m ≤ 0.34 and 0.57 ≤ �� ≤ 1.04 without correction and 0.21 ≤ �m ≤ 0.32 and 0.65 ≤ �� ≤ 0.99 with the f correction. MNRAS 477, 2867–2873 (2018) D ow nloaded from https://academ ic.oup.com /m nras/article-abstract/477/3/2867/4956051 by U niversidade Estadual Paulista J� lio de M esquita Filho user on 23 July 2019 Bayesian correction of H(z) uncertainties 2873 With 34 H(z) data, Sharov & Vorontsova (2014) find a more stringent result, namely, H0 = 70.26 ± 0.32, �m = 0.276+0.009 −0.008, and �� = 0.769 ± 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 O�CDM (Lima et al. 2012): zt = [ 2�� �m ]1/3 − 1. (28) They have found zt = 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 C O N C L U S I O N 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 un- certainties. The uncertainties in the parameters were quite reduced when compared with methods of parameter estimation without cor- rection and we have obtained an estimate of an overall correction factor in the context of O�CDM and flat �CDM models. 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