1. Introduction
Currently, the estimates of the value of
, obtained from the
CDM fit to the cosmic microwave background (CMB) differ between 2–3
with respect to the value of
obtained with the analysis of galaxy clustering using two-point correlation functions (2PCFs) [
1]. This discrepancy is the so-called
tension [
2,
3]. Additionally, the most recent estimate of the value of the Hubble constant,
, obtained with the calibration of the cosmic distance ladder scale through Cepheid stars and supernovae type Ia is
different [
4] from the value
CDM obtained with CMB observations.
Specifically, the primary anisotropies of the CMB exhibit a tension in the matter clustering strength at the level of 2–3
when compared to lower
z probes such as weak gravitational lensing and galaxy clustering (e.g., [
1,
5,
6,
7,
8,
9,
10,
11,
12,
13]). The lower
z probes (see Figure 4 from [
14]) select a lower value of
compared to the high
z CMB estimates. The measured
value is model-dependent and in the majority of the scenarios it is considered a standard flat
CDM model. Of course, this latter model provides a good fit to the data from all probes but predicts a lower value of structure formation compared to what we expect from the CMB [
6]. In [
14], it is reported the latter parameter estimates and constraints, where we notice that there might be slight differences in the selection that enters each study. For example, on one hand, a statistical property of the
distribution might be selected, such as its mean or mode together with the asymmetric
C.L around this value or the standard deviation of the data points. On the other hand, the statistics to the full posterior distribution can be adopted, such as the maximum a posteriori point or the best-fitting values and their errors. In any case, these considerations can affect the estimated values of the parameters, in particular when the posterior distributions are significantly non-Gaussian.
Many models beyond
CDM models have been proposed as tentative explanations to this observed
tension; however, a nonspecific model has been proven much better than the standard one. In such a case, we should look for a reason why the tension between the CMB and cosmic shear in the inferred value of
can arise from unaccounted baryonic physics, other unknown systematic errors or a statistical fluke. Furthermore, we require that
be independent of the CMB and cosmic shear. In this matter, the redshift-space distortion (RSD) data also prefer a small value of
, that is 2–3
lower than the Planck result [
15]. However, the RSD values on
are sensitive to the cosmological model.
In this paper, we perform the reconstructions of the
observations using Gaussian processes (GP) to analyze if the reconstructions suggest possible deviations with respect to the standard cosmological model. Unlike previous studies [
16,
17,
18], we take into account the Planck 2018 confidence contours when comparing the predictions of
CDM with those of the reconstructions. This allows us to control the statistical uncertainties. The possibility of using GP to distinguish between modified gravity (MG) and general relativity (GR) has been analyzed [
19], e.g., by treating the perturbations of a disformally scalar field model whose background mimics the
CDM [
20].
Previous articles [
17] have used the reconstructions of the Hubble parameter to reconstruct
and to show how different values of
change the value of the
tension. However, the large number of free parameters associated with that approach leads to large uncertainties on the reconstructions, which could explain why a ∼4
change in the value of
only causes a ∼1
change in the value of the
tension.
Unlike the standard way to reconstruct
[
17], in this work, we reconstruct
(z) by directly using the estimations of
, therefore it is possible to remove
and
as free parameters and reduce the uncertainty in the confidence contours. However, we should mention that our approach does not allow us to obtain a direct estimate of the aforementioned parameters.
In this line of thought, several numerical methods have been developed over the years, showing a great advance in the precision cosmology road, e.g., methods that involves artificial neural network (ANN) to reconstruct late–time cosmology data [
21], the creation of mock datasets through machine learning (ML) based on the LSST survey and using a fiducial cosmology [
22] and Bayesian analyses in order to reassess the
discrepancy between the CMB and weak lensing data [
15].
Furthermore, we analyze two different approaches to obtain the value of the confidence contours of the reconstructions. The first approach consists in the maximization of the likelihood associated with the GP in order to obtain the value of the free parameters from the reconstruction. This approach has been frequently used in the literature to perform GP in the context of cosmology [
17,
19,
23,
24,
25]; however, in some works [
26,
27], it has been shown that this process could lead to an underestimation of the confidence contours of the reconstructions. The second approach consists in an exploration of the parameter space of the free parameters through Markov chain Monte Carlo (MCMC) methods [
28]. This method allows us to obtain an estimate of the uncertainties associated with each parameter, which contributes to solve the underestimation problem.
1.1. Treatment of Data Samples and Methodology
To use the GP method, we need to assume that our set of observations “
” given in the set of redshifts “
” is Gaussian distributed around the underlying function that we seek to reconstruct,
g(
z). This allows us to associate a probability distribution with the data:
where
is the mean value of the observations,
C is its covariance matrix and
K is the covariance matrix associated with the Gaussian processes also known as the kernel function. It can be shown that the mean value and covariance matrix of the reconstructed function in the set of redshifts
are given by [
26].
The measurements of the clustering pattern of matter in the redshift space allow us to infer parameters such as the linear growth of matter perturbations and the variance of matter fluctuations on a given scale R. The variance is usually reported on a scale Mpc, that is, .
However, an exact determination of the value of
turns out to be complex, since the galaxy redshift surveys provide an estimate of the perturbations in terms of galaxy densities
instead of directly in terms of matter density
. Unfortunately, the exact value of the parameter
b remains uncertain [
29]; for that reason, the measurements are reported in terms of
, since it is independent of the parameter
b [
17].
To perform the reconstructions we used the compilation of measurements of
shown in Table 1 of [
30]. This table contains 30 measurements of
, together with their uncertainties and their covariance matrices.
According to the latter, the inferred values of
depend on two things: (i) the anisotropies in the power spectrum of the peculiar velocities of galaxies and (ii) the fiducial cosmology chosen to estimate the values of
. If the cosmology chosen to perform the data analysis does not adequately describe the geometry of the universe, then nontrivial anisotropies are introduced in the 2PCFs, which are directly correlated with the estimated value of
; this effect is known as Alcock–Paczynski (AP) effect. It is important to note that the
values sometimes are reported assuming [
31] different fiducial cosmologies, e.g.,
When considering a fiducial cosmology with
,
km/s/Mpc,
and
,
is obtained [
32].
With a fiducial cosmology with
,
km/s/Mpc,
and
, we can obtain
[
33].
1.2. Alcock-Paczynski Corrections
If the
reconstructions are performed without including corrections to the AP effect, then all the information from different fiducial cosmologies are mixed, i.e., it is not possible to properly estimate the tension between the reconstructions and the
CDM model. We applied the correction to the AP effect given in [
31], whose corrections state that if a measurement of
has been obtained assuming a fiducial cosmology with a Hubble parameter
and angular diameter
, then the corresponding value of
assuming a different fiducial cosmology with
and
can be approximated as:
To correct the AP effect, we considered a vanilla
CDM cosmology with the parameters inferred from the Planck 2018 Collaboration [
34], such that the high redshift data from Planck TT, TE, EE + lowE are
. Combining these data with secondary CMB anisotropies, in the form of CMB lensing, serves to tighten the constraint to
.
To proceed with this calculation, we set the following steps:
Suppose that
and
are two measurements of
that were obtained assuming the fiducial cosmology
with
,
the mean value of each measurement,
,
their 1
uncertainties and also suppose that the measurements are correlated through a covariance matrix
CIn order to compute the value of
and
in a fiducial cosmology
, we have to perform the AP correction. To do so, we multiply
A and
B with constants
and
given by Equation (
4) and then the covariance matrix for
and
is given by [
35].
1.3. Kernel Metrics
It has been shown that the kernel chosen to perform the reconstructions can affect the uncertainties of the parameters derived from the reconstructions [
26,
27,
36]. In particular, the Gaussian kernel can lead to uncertainties up to three times smaller than the value of the uncertainties obtained with the Matérn covariance functions [
37]. Since this could be associated with the problem of underestimation of uncertainties, we chose to use only Matérn kernels, specifically the Matérn 3/2 and Matérn 5/2 kernels. These covariance functions have been used, e.g., in studies that analyze how the choice between different kernels affect the value of
that is obtained from the reconstructions of the cosmic late expansion [
36] and in an analysis on how different kernels can affect the constraints on modified theories of gravity [
23]. The Matérn 3/2 and Matérn 5/2 kernels are, respectively, defined as:
where
and
l are free parameters that measure the width of the reconstructed function and the correlation between the function evaluated at two given points
and
. With these kernels at hand, we are ready to proceed with two different methods to obtain the value of the hyperparameters of the kernel:
Method (i) consists in the maximization of the likelihood associated with the observations Equation (
1).
Method (ii) consists in a full exploration of the parameter space for the hyperparameters through MCMC methods. This approach is convenient when the parameter space is multidimensional, and it could help us to find the true maximum likelihood estimate in cases where the algorithm for maximization gets stuck in a local minima.
Since
is a squared quantity in both covariance Functions (
9) and (
10), if (
,
) are the values of the hyperparameters that maximize Equation (
1), then (
,
) also maximize Equation (
1), and this leads to a bimodal posterior distribution for the hyperparameter
. Since with one mode, we can obtain the full information to carry out the reconstructions, it is convenient to establish priors that only take into account the positive (or negative) branch of the parameter space for
. With this method, it is possible to reduce the computational time required to calculate the posterior distribution of the hyperparameters and it also allows us to avoid convergence problems within the numerical code.
Notice that covariance Functions (
9) and (
10) are not symmetric on
l. However, we expected positive values of
l, otherwise the correlation between the points
,
would grow proportionally to
Since hyperparameters were restricted to be positive, we considered for them gamma probability distributions as priors. The probability density function
for a random variable
X that follows a gamma distribution with parameters
and
is given by
with
the gamma function. The mean value of
is
, the mode is
and the variance is Var
[
35]. In order to find the value of the hyperparameters of the covariance functions that maximizes Equation (
1), we performed a maximum likelihood estimation (MLS) with the code [
26], and afterwards, we proceeded with a standard MCMC analysis using the publicly available code PyMC3 [
38]. For both hyperparameters, we chose gamma priors with
and
equal to their MLS. We estimated the convergence of the chains using a Gelman–Rubin convergence criterion [
39] with
. The reconstructions were estimated with two different approaches: (1) we used the mean value of the hyperparameters to estimate the mean value of the reconstructions and (2) we used the mode. Our results are detailed in
Table 1 and
Table 2 and in
Figure 1 and
Figure 2.
In order to distinguish between reconstructions, we defined the tension metric as follows:
where
and
are the mean values of the reconstructions and
, respectively.
and
are the
statistical uncertainties. Notice that Equation (
13) quantifies the difference in standard deviations between the reconstructions and
.
From
Table 3 and
Table 4, we can notice that for both kernels the tension for all methods is of the order of 2
in the observable redshift region. The mean value of the reconstructions obtained through the different methods represent
of its total value.
Additionally, to test the performance of each model describing by the observations, we estimated the chi-square statistics as follows
with
the inverse of the covariance matrix of the data. The results are shown in
Table 5 and
Table 6. With these results and with those obtained using the
function, we conclude that there is no statistically significant difference between
CDM and the reconstructions presented here (see
Table 5 and
Table 6).