Constraining the curvature density parameter in cosmology

The universe on a large scale is described by the spatially homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) metric,

The scale factor $a(t)$ is the only unknown function to be determined by the field equations. The isotropy and homogeneity of the space section demand the spatial curvature to be a constant, which can thus be scaled to pick up values from $-1,+1,0$. This constant spatial curvature is termed the curvature index and is denoted as $k$. This index is not determined by the field equations but is rather fixed by hand, essentially from observational requirements.

The effect of the spatial curvature $k$ in the evolution of the universe is estimated through the curvature density parameter, defined as,

where $H=\frac{\dot{a}}{a}$ is the Hubble parameter. $\Omega_{k}$ is positive, negative or zero corresponding to $k=-1,+1,0$, which in turn correspond to open, closed and flat space sections respectively.

For the standard cosmological model to correctly describe the present state of the evolution, the initial value of $\Omega_{k}$ has to be tantalizingly close to zero, indicating that the universe essentially starts with a zero spatial curvature. This is known as the flatness or fine-tuning problem for the standard cosmology which is believed to be taken care of by an early accelerated expansion called inflation. For a brief but systematic description, we refer to the monograph by Liddle and Lythliddle . Indeed inflation can wash out an early effect of spatial curvature, in comparison with the inflaton energy and the Hubble expansion. However, if $\Omega_{k}$ is negligible but $k$ itself is non-zero, it may reappear in course of evolution and make its presence felt as the universe evolves. Reconstruction of some dark energy parameters indicate that a non-flat space section may not be easily ruled out. The use of $\Omega_{k}$ as a free parameter is found to affect the reconstruction of dark energy equation of state parameter $w(z)$, as shown by Clarkson, Cortes, and Bassettclarkson2007 . A reconstruction of the deceleration parameter $q(z)$ by Gong and Wanggong2007 shows that although a flat universe is still consistent, $|\Omega_{k0}|$ is less than only 0.05 for a one-parameter dark energy model and lies between -0.064 and 0.028 for a $\Lambda$CDM model with spatial curvature, where a subscript 0 indicates the present value of the quantity. The recent Planckplanck data also indicates that a universe with a non-zero spatial curvature may not be completely ruled out.

The motivation of the present work is to constrain the curvature density parameter $\Omega_{k0}$ hence attempt to ascertain the signature of the curvature index $k$, directly from observational data without assuming any background cosmological model. We do not start from any theory of gravity or use any form of matter distribution in the universe. The only a priori assumption is that the universe is homogeneous and isotropic, and thus described by the FLRW metric. There is quite a lot of interest in this direction, which is normally pursued along with constraining other cosmological parameters pertaining to the alleged accelerated expansion of the universe. Most of these investigations depend on some chosen parametric form of cosmological quantities related to the late-time expansion behaviour of the universeleo2016 ; witz2018 ; deni2018 ; cao2019 ; li2020 ; gratton2020 ; park2020 ; nunes2020 ; benisty2021 ; handly2021 . This approach is indeed biased by the parametrization, as the functional form of the quantity is already chosen.

Another way of reconstruction involves a verification of the FLRW metric from datasets, and ascertaining the value of $\Omega_{k0}$ as a by-product by combining the dimensionless reduced Hubble parameter $E(z)$ and the normalised comoving distance $D(z)$clarkson2007 ; clarkson2008 ; cai2016 ; rana2017 ; liu2020 ; arjona2021 ; ref_new2014 .

The present work does not assume any functional form of $\Omega_{k}$, but rather resorts to a non-parametric reconstruction of $\Omega_{k0}$, the present value of the curvature density parameter. The idea is to obtain constraints on the geometrical quantity $\Omega_{k0}$ using recent observational data provided by the high precision cosmological probes, namely, the Supernova (SN) distance modulus data, the Cosmic Chronometer (CC) and the radial Baryon Acoustic Oscillations ($r$BAO) measurements of the Hubble parameter. We also combine these data from background measurements with the data from redshift space distortions (RSD) due to the growth of large scale structures. The reconstruction is performed adopting the non-parametric Gaussian Processes (GP). The resulting marginalized constraints on $\Omega_{k0}$ are obtained via a Markov Chain Monte Carlo (MCMC) analysis, independent of any parametric model of the expansion history.

Attempts towards obtaining constraints on $\Omega_{k0}$ using the non-parametric approach started to gain momentum in the recent past. Li et al.li2016 , Wei and Wuwei2017 proposed to constrain the cosmic curvature in a model-independent way by combining the CC-$H(z)$ with Union 2.1union2.1 , and Joint Light-curve Analysis (JLA)jla SN-Ia data respectively. Model-independent constraints on cosmic curvature and opacity was carried out by Wang et. al.wang2017 using the CC-H(z) and JLA SN-Ia data. Liaoliao2019 studied constraints on cosmic curvature with lensing time delays and gravitational waves (GWs). Model-independent distance calibration and $\Omega_{k0}$ measurement using Quasi-Stellar Objects (QSOs) and CCs was done by Wei and Meliawei2019 . Ruan et al.ruan2019 obtained constraints on $\Omega_{k0}$ using the CC-$H(z)$ data and HII galaxy Hubble diagram. Model-independent estimation for $\Omega_{k0}$ from the latest strong gravitational lens systems (SGLs) was performed by Zhou and Lizhou2019 . Wang et al.wang2019 constrained $\Omega_{k0}$ from SGL and Pantheonpan1 SN-Ia observations. Wang, Ma and Xiawang2020 employed a machine learning algorithm called Artificial Neural Network (ANN) to constrain $\Omega_{k0}$ using data from CC, SN-Ia and GWs. Recently, Yang and Gongyang2021 constrained the $\Omega_{k0}h^{2}$ using CC-$H(z)$, Pantheon SN-Ia and RSD data where $h=\frac{H_{0}}{100~{}\mbox{\small km}~{}\mbox{\small Mpc}^{-1}~{}\mbox{\small s}^{-1}}$ is the dimensionless Hubble parameter at the present epoch. Non-parametric spatial curvature inference using CC and Pantheon data was performed by Dhawan, Alsing and Vagnozzidhawan2021 . A majority of these investigations use GP as their numerical tool.

We use observational data more recent than most of these investigations, but the major difference is that we include a wider variety of data in combination, measuring different features of the evolution. We also include a section where the RSD dataset which has mostly eluded the attention so far, except the work of Yang and Gongyang2021 in the reconstruction of $\Omega_{k0}h^{2}$ despite its utmost relevance in this connection, as the growth of perturbations has to be consistent with the spatial curvature.

The other crucial addition in the present work is that we also check the consistency of the constraints on spatial curvature with thermodynamic requirements. Very recently, Ferreira and Pavónpavon imposed a relation using the generalized second law of thermodynamics, which reads as $1+q\geq\Omega_{k}$, where $q$ is the deceleration parameter. It is quite reassuring to see that constraints on $\Omega_{k0}$ quite comfortably satisfies the requirement.

The results obtained indicate that a spatial curvature may indeed exist at the present epoch. But the estimated sign of the curvature depends on the strategies for measuring $H_{0}$ to some extent. But the results are statistically not too significant, as a zero curvature is mostly included in 1$\sigma$ and always at least in 2$\sigma$.

The paper is organized as follows. Section 2 contains the details on the reconstruction method. In section 3, the observational data used in the present work have been briefly reviewed. The methodology is discussed in section 4. Reconstruction using background data is performed in section 5. Section 6 shows the consistency of $\Omega_{k0}$ constraints with the second law of thermodynamics. Reconstruction using the perturbation data are presented in section 7. The final section 8 contains an overall discussion on the results.

We shall employ the well-known Gaussian processes (GP)william ; mackay ; rw for the reconstruction of ${\Omega}_{k0}$. Assuming that the observational data obey a Gaussian distribution with mean and variance, the posterior distribution of the reconstructed function (say $f$) and its derivatives can be expressed as a joint Gaussian distribution. In this method, the covariance function $\kappa(z,\tilde{z})$ plays a key role. It correlates the values of $f(z)$ at two redshift points $z$ and $\tilde{z}$. This covariance function depends on a set of hyperparameters which are optimised by maximizing the log marginal likelihood. With the optimised covariance function, the data can be extended to any redshift point. The GP method has been widely applied in cosmology gp0 ; gp1 ; gp2 ; gp3 ; gp4 ; gp5 ; gp6 ; gp7 ; gp8 ; gp9 ; gp10 ; keeley2020 ; xia_q ; lin_q ; jesus_q ; purba_q ; purba_cddr ; purba_j ; purba_int ; benisty ; kamal .

It deserves mention that the choice of $\kappa(z,\tilde{z})$, affects the reconstruction to some extent. The more commonly used covariance function is the squared exponential covariance, which is infinitely differentiable,

In this particular work we consider the squared exponential, Matérn 9/2, Cauchy and rational quadratic covariance functions. The Matérn 9/2 covariance function is given by,

The Cauchy covariance function is

and the rational quadratic covariance function is

where $\sigma_{f}$ , $l$ and $\alpha$ are the kernel hyperparameters. Throughout this work, we assume a zero mean function a priori to characterize the GP.

For more details on the GP method, one can refer to the Gaussian Process website¹¹1http://www.gaussianprocess.org. The publicly available GaPP²²2https://github.com/carlosandrepaes/GaPP (Gaussian Processes in python) code by Seikel et al.gp1 has been used in this work.

In this work we use both the background data and the perturbation data for the reconstruction of ${\Omega}_{k0}$. The background level includes different combinations of datasets involving the Cosmic Chronometer data (CC), the Supernova distance modulus data (SN), the Baryon Acoustic Oscillation data (BAO). For the perturbation level data, the growth rate of structure $f\sigma_{8}$ from the redshift-space distortions (RSD) are utilized. A brief summary of the datasets is given below.

In an FLRW universe, the proper distance from the observer to a celestial object at redshift $z$ along the line of sight is given by,

and the transverse comoving distance can be expressed as,

in which the $\sin n$ function is a shorthand for,

We define the reduced Hubble parameter as,

Here, a suffix 0 indicates the value of the relevant quantity at the present epoch and $z$ is the redshift, defined as $1+z\equiv\frac{a}{a_{0}}$. The dimensionless parameter $\Omega_{k0}$, namely the cosmic curvature density parameter, defined as

is positive, negative or zero corresponding to the spatial curvature $k=-1,+1,0$ which signifies an open, closed, or flat universe, respectively.

For convenience, we can define the normalized proper distance,

and the normalized transverse comoving distance,

as dimensionless cosmological distance measures which will be used later in our work.

In the very beginning we use the GP method to reconstruct the Hubble parameter $H(z)$ from the CC data and CC+$r$BAO data. We then normalize the datasets with the reconstructed value of $H_{0}$ i.e., $H(z=0)$ to obtain the dimensionless or reduced Hubble parameter $E(z)$. Considering the error associated with the Hubble data as $\sigma_{H}$, we calculate the uncertainty in $E(z)$ as,

where $\sigma_{H_{0}}$ is the error associated with $H_{0}$.

With the function $E(z)$ reconstructed from the Hubble data, as described in equation (9), the normalised proper distance $D_{p}$ is calculated via a numerical integration using the composite trapezoidaltrapez rule.

Thus we get $D_{p}$ without assuming any prior fiducial cosmological model. The error associated with $D_{p}$, say $\sigma_{D_{p}}$, is obtained from the reconstructed function $E(z)$ along with its associated error uncertainties $\sigma_{E}(z)$ described in equation (13), and is given by,

From this reconstructed $D_{p}$, we can calculate the normalised transverse comoving distance $D$ from the Hubble data as,

The error $\sigma_{D}$ of the reconstructed $D$ from the Hubble data is,

Steinhardt et al.pan_correct lists the corrected distance modulus $\mu$ corresponding to different redshift $z$, along with their respective error uncertainties, from supernovae observations following the BEAMS with Bias Corrections (BBC)beams framework.

The total uncertainty matrix of observed distance modulus given by,

where both the statistical covariance matrix $\mathbf{C}_{\mbox{\tiny stat}}$ and the systematic errors $\mathbf{C}_{\mbox{\tiny sys}}$ are included in our calculation.

With another Gaussian Process on the observed distance modulus of the SN-Ia data, we reconstruct $\mu_{\mbox{\tiny SN}}$ and the associated error uncertainties ${\sigma}_{\mu_{\mbox{\tiny SN}}}$, at the same redshift as that of the Hubble data. The subscript SN stands for supernova.

The distance modulus is theoretically given by,

Here, $d_{L}$ is the luminosity distance. This $d_{L}$ is related to the normalised transverse comoving distance $D$ as,

Substituting equation (16) in equation (19), we estimate the reconstructed distance modulus from the Hubble data, say $\mu_{\mbox{\tiny H}}$ along with its 1$\sigma$ error uncertainty $\sigma_{\mu_{\mbox{\tiny H}}}$ as,

Equations (16), (17), (21) and (22) will finally be utilized for obtaining the contour plots between $\Omega_{k0}$ and $H_{0}$ at different confidence levels.

Finally we constrain the curvature density parameter $\Omega_{k0}$ and the Hubble parameter $H_{0}$ simultaneously by minimizing the $\chi^{2}$ statistics. The $\chi^{2}$ function is given by,

$\Delta\mu=\mu_{{\mbox{\tiny SN}}}-\mu_{{\mbox{\tiny H}}}$ is the difference between the distance moduli of Pantheon SN-Ia and that of the $H(z)$ data. $\mathbf{\Sigma}={\sigma}^{2}_{\mu_{\mbox{\tiny SN}}}+\sigma_{\mu_{\mbox{\tiny H}}}^{2}$ is the total uncertainty matrix from combined Pantheon and Hubble data.

We attempt to reconstruct $\Omega_{k0}$ directly for the following combination of data sets,

• •

  Set I

  1. 1.

     N1 - CC+SN

  2. 2.

     P1 - CC+SN+P18

  3. 3.

     R1 - CC+SN+R19

• •

  Set II

  1. 1.

     N2 - CC+$r$BAO+SN

  2. 2.

     P2 - CC+$r$BAO+SN+P18

  3. 3.

     R2 - CC+$r$BAO+SN+R19

We get the constraints on $\Omega_{k0}$ and $H_{0}$ along with their respective error uncertainties by a Markov Chain Monte Carlo (MCMC) analysis with the assumption of a uniform prior distribution for $\Omega_{k0}\in[-1,1]$ and $H_{0}\in[50,100]$ in case of the N1 and N2 combinations respectively. For the P1 and P2 combinations, we consider the P18 Gaussian $H_{0}$ prior whereas, for R1 and R2 combinations, the R19 Gaussian $H_{0}$ prior has been used.

In this work, we adopt a python implementation of the ensemble sampler for MCMC, the publicly available emcee³³3https://github.com/dfm/emcee, introduced by Foreman-Mackey et al.emcee . The best fit results along with their respective 1$\sigma$, 2$\sigma$ and 3$\sigma$ uncertainties is given in Table 1. We plot the results using the GetDist⁴⁴4https://github.com/cmbant/getdist module of python, developed by Lewisgetdist . The plots for the marginalized distributions with 1$\sigma$ and 2$\sigma$ confidence contours for $\Omega_{k0}$ and $H_{0}$ are shown in Figures 1, 2, 3 and 4 considering the squared exponential, Matérn $9/2$, Cauchy and rational quadratic covariance respectively.

The reconstructed $\Omega_{k0}$ for the N1 combination are consistent with spatial flatness within 2$\sigma$ confidence level (CL) for the squared exponential, Matérn 9/2 and Cauchy covariance functions, and within 3$\sigma$ CL for the rational quadratic covariance. With the addition of $r$BAO data in Set II, the constraints on $\Omega_{k0}$ become tighter. From the combined N2 data-set, we find that $\Omega_{k0}$ is consistent with a spatially flat universe at 1$\sigma$ CL for the squared exponential, Matérn 9/2 and Cauchy covariance, whereas in 2$\sigma$ for the rational quadratic kernel. The best-fit values shows an inclination towards a closed universe for N1 and N2 data-sets. The degeneracy between $H_{0}$ and $\Omega_{k0}$ along with their correlation has also been shown.

We also examine if the two different strategies for determining value of $H_{0}$, with conflicting results, affect the reconstruction significantly. We plot the marginalized distributions with 1$\sigma$ and 2$\sigma$ confidence contours for $\Omega_{k0}$ and $H_{0}$ using the P1 and P2 combinations considering the P18 prior on $H_{0}$, and for the R1 and R2 combinations considering the R19 prior on $H_{0}$ in figures 1-4. With the inclusion of the P18 data prior we see that the best-fit values of $\Omega_{k0}$ favour a spatially open universe, whereas in case of the choice of R19 as a prior, the best-fit values of the constrained $\Omega_{k0}$ shows that the combined data favours a spatially closed universe. However, a spatially flat universe is mostly included at 2$\sigma$ CL for both cases.

In this section, the consistency of constraints obtained on $\Omega_{k0}$ with the second law of thermodynamics is looked at. We assume the universe as a system is bounded by a cosmological horizon, and the matter content of the universe is enclosed within a volume defined by a radius not bigger than the horizon gibbons ; jacob ; padma . In cosmology, the apparent horizon $r_{A}$ serves as the cosmological horizon, which is given by the equation $g^{\mu\nu}R_{,\mu}R_{,\nu}=0$, where $R=a(t)r$ is the proper radius of the 2-sphere and $r$ is the comoving radius. For the FLRW universe with a spatial curvature index $k$, the apparent horizon is thus given by

For $k=0$, the apparent horizon reduces to the Hubble horizon $r_{H}=\frac{1}{H}$.

Now, the entropy of the horizon $S_{A}$ can be written as bak ,

For the second law to be valid, the entropy $S$ should be non-decreasing with respect to the expansion of the universe. If $S_{f}$ and $S_{A}$ stand for the entropy of the fluid describing the observable universe, and that of the horizon containing the fluid, respectively, then the total entropy of the system, i.e., $S=S_{f}+S_{A}$, should satisfy the relation

Recently Ferreira and Pavónpavon gave a prescription to ascertain the signature of $k$ from the second law of thermodynamics. It is a fair assumption that the entropy of the observable universe is dominated by that of the cosmic horizon $S_{A}$ f_p_21 . So, the second law can be safely written as pavon ,

Using equation (25) in (27) one can obtain the following condition,

The inequality (28) can be rewritten, with a bit of simple algebraic exercise, as

Here $q$ is the deceleration parameter which gives a dimensionless measure of the cosmic acceleration and is defined as,

Testing the thermodynamic validity for the obtained constraints on $\Omega_{k0}$ requires a reconstruction of $q_{0}$ from the respective combination of data sets. Quite a lot of work on a non-parametric reconstruction of the cosmic deceleration parameter $q$ is already there in the literature. Some of them can be found in lin_q ; xia_q ; purba_q ; jesus_q ; purba_j ; keeley2020 . The list, however, is far from being exhaustive. We use the same datasets, that were used for the reconstruction of $\Omega_{k0}$, to find the corresponding values of $q_{0}$. A very brief methodology is the following. For a more detailed technical description, we refer to purba_int ; purba_q . The comoving distance $D(z)$, and its derivatives $D^{\prime}(z)$ and $D^{\prime\prime}(z)$ are reconstructed w.r.t $z$ for different combinations of data sets. The uncertainty in $D(z)$ from the corresponding data set is taken into account. For the CC and $r$BAO data, we convert the $H$-$\sigma_{H}$ data to $E$-$\sigma_{E}$ data set using Eq. (9) and (13). $D^{\prime}(z)$ is then connected to $D(z)$ and $E(z)$ via Eq. (12) as,

Thus, we take into account the $E$ data points, the uncertainty associated $\sigma_{E}$ while performing the GP reconstruction. We add two extra points $D(0)=0$ to the $D$ data set, and $E(0)=1$ to the $E$ data before proceeding with the reconstruction. We obtain the reconstructed values of $D(z)$, $D^{\prime}(z)$ and $D^{\prime\prime}(z)$ at the present epoch, along with their error uncertainties. Now, $q$ can be rewritten as a function of $D(z)$ and its derivatives as,

Using the reconstructed $D(0)$, $D^{\prime}(0)$ and $D^{\prime\prime}(0)$, we obtain the values of $1+q_{0}$, shown in the third column of Table 1. Eq. (29) asserts that $\Omega_{k0}\leq 1+q_{0}$ for the second law to be valid. We see that for all combinations, the second law is satisfied by a generous margin, independent of the choice of the kernel.

Redshift-space distortions are an effect in observational cosmology where the spatial distribution of galaxies appears distorted when their positions are looked at as a function of their redshift, rather than as functions of their distances. This effect occurs due to the peculiar velocities of the galaxies causing a Doppler shift in addition to the redshift caused by the cosmological expansion. The growth of large structure can not only probe the background evolution of the universe, but also distinguish between GR and different modified gravity theories ref39 ; ref41 . Recently, non-parametric constraints on the Hubble parameter $H$ and the matter density parameter $\Omega_{m}$ were obtained using the data from cosmic chronometers, type-Ia supernovae, baryon acoustic oscillations and redshift-space distortions, assuming a spatially flat universe ref_new2022 . In this section, we propose a non-parametric method to use the growth rate data measured from RSDs to constrain the spatial curvature.