Spatial Curvature and Thermodynamics

Homogeneous and isotropic cosmological models are most aptly described by the Friedmann-Lemaître-Robertson-Walker (FLRW) space-time metric,

The spatial curvature index, $k\in\{-1,0,1\}$, indicates whether the spatial part of the metric is open (negatively curved, i.e., hyperbolic), flat, or closed (positively curved).

This constant index, like the scale factor $a(t)$, is not a directly observable quantity. In principle, however, it can be determined through the knowledge of the spatial curvature density parameter, $\Omega_{k}\equiv-k/(a^{2}H^{2})$, which is accessible to observation, albeit indirectly. Recent estimates of the latter, assuming the universe correctly described by the $\Lambda$CDM model only suggests that its present absolute value is small ($\mid\Omega_{k0}\mid\sim 10^{-2}$ or less (Komatsu et al., 2011; Ade et al., 2014; Aghanim et al., 2020; Vagnozzi et al., 2021a; Vagnozzi et al., 2021b; Dhawan et al., 2021; Park & Ratra, 2019; Handley, 2021; Mukherjee & Banerjee, 2022; Bel et al., 2022; Akarsu et al., 2023)). More recent measurements, not based in the aforesaid model, hint that the universe may not be flat (i.e., $k\neq 0$) (Park & Ratra, 2019; Handley, 2021; Mukherjee & Banerjee, 2022; Bel et al., 2022). So, the sign of $k$ is rather uncertain.

Based on the history of the Hubble factor $H(z)$, (where $H=\dot{a}/a$ ) it has been recently argued (Gonzalez-Espinoza & Pavón, 2019) that the universe is a thermodynamic system in the sense that it satisfies the second law of thermodynamics (the first law is guaranteed by Einstein equations while the third law may not apply). As we shall see soon, the possibilities $k=+1$ and $k=0$ are consistent with the second law while this is not so obvious for the third possibility; in principle $k=-1$ may or may not be compatible with the second law. The aim of our study is to resolve this uncertainty. To this end we shall resort to the 60 $H(z)$ data currently available alongside their $1\sigma$ confidence intervals, the parametrized graph of this history after smoothing the data set by a Gaussian Process (Rasmussen & Williams, 2005), the FLRW metric and the expression of the area of the apparent horizon (Eq. (2) below). As it turns out, the $k=-1$ possibility (open spatial sections) appears also compatible with the second law of thermodynamics. The paper is organized as follows: In section 2 we write the second law at cosmic scales in terms of the deceleration parameter and $\Omega_{k}$. In section 3 we present the $H(z)$ data set and the smoothing process. In section 4 we introduce four parametrizations of the Hubble factor and study whether they are compatible with the second law. In section 5 we consider the impact of the Hubble constant priors on our results. Finally, in section 6 we summarize our findings.

As usual, a subindex zero attached at any given quantity indicates that the quantity is to be evaluated at the present epoch. Likewise, we recall, for future convenience, that after fixing $a_{0}=1$, the redshift, $z$, of any given luminous source is related to the corresponding scale factor by $\,1+z=a^{-1}$. In our units $\,c=G=\hbar=1$.

Given the strong connection between gravity and thermodynamics (Bekenstein, 1974, 1975; Hawking, 1974; Jacobson, 1995; Padmanabhan, 2005), it is natural to expect that the universe behaves as a normal thermodynamic system (Gonzalez-Espinoza & Pavón, 2019) whereby it must tend to a state of maximum entropy in the long run (Radicella & Pavon, 2012; Pavon & Radicella, 2013).

A basic standpoint is that the entropy of the universe is dominated by entropy of the cosmic horizon. In fact its entropy ($\sim 10^{132}\rm{k_{B}}$) exceeds by far the combined entropies of super-massive black holes, the cosmic microwave background radiation, the neutrino sea, etc (Egan & Lineweaver, 2010). As cosmic horizon, we take the apparent horizon rather than other possible choices, since the laws of thermodynamics are fulfilled on it (Wang et al., 2006). Its entropy (proportional to its area) is given by $\,S_{\cal A}=\rm{k_{B}}\pi\,\tilde{r}^{2}_{\cal A}/{\ell_{p}}^{2}\;$ (Bak & Rey, 2000; Cai, 2008), with $\tilde{r}_{\cal A}=(H^{2}+ka^{-2})^{-1/2}\,$ as the radius of the horizon.

Obviously, the area of the apparent horizon,

depends on the Hubble factor and spatial curvature.

By the second law of thermodynamics $S_{\cal A}^{\prime}\geq 0$, thus

where the prime indicates $d/da$.

The above inequality tells us: $(i)$ if $H^{\prime}$ is or has been positive at any stage of cosmic expansion (excluding, possibly, the pre-Planckian era), then $k=+1$, $(ii)$ $H^{\prime}<0$ may, in principle, be compatible with any sign of $k$, and $(iii)$ the possibilities $k=+1$ and $k=0$ are consistent with the second law. In particular the data analysis carried out by the 2018 Planck mission produced the 99$\%$ probability region $-0.095<\Omega_{k0}<-0.007$ on the spatial curvature parameter (Aghanim et al., 2020). Note that this bound on $\Omega_{k0}$ is fully compatible with the second law because it corresponds to the possibility $k=+1$. However, the third possibility, $k=-1$, may or may not be compatible.

The inequality $HH^{\prime}\leq\frac{k}{a^{3}}$ can be recast as

where $q=-\ddot{a}/(aH^{2})=-(1+\dot{H}H^{-2})$ is the deceleration parameter.

Equation (4) expresses the second law of thermodynamics at cosmic scales (Gonzalez-Espinoza & Pavón, 2019).

To see that it is perfectly reasonable, let us assume the universe dominated by a perfect fluid of energy density $\rho$ and hydrostatic pressure $P$. The corresponding Einstein equations can be written as,

If the fluid is dust ($P=0$), one has $\,q>0$, and these equations lead to $\,\Omega_{k}=1-2q$.

If the second law holds, i.e., $1+q\geq\Omega_{k}$, it follows that $\,q\geq 0$ which is consistent with the fact that a pressureless matter dominated universe decelerates ($q>0$).

By contrast, should the second law not hold, $1+q<\Omega_{k}$, the absurd result $0<-q$ (namely, zero less than a negative quantity), would follow.

Likewise, if the equation of state of the fluid filling the universe were $P=w\rho$ with $\,w\geq-1$ the expression $1+q\geq\Omega_{k}$ also proves to be consistent with the corresponding sign of the deceleration parameter and not so, if $1+q<\Omega_{k}$. Values of $w$ lower that $-1$ correspond to phantom fields. As is well known these fields are unstable both classically (Dabrowski, 2015) and quantum mechanically¹¹1Should $H^{\prime}$ be positive, the FLRW universe could neither be flat nor open, just closed ($k=+1$) which would be puzzling in view of the current data that clearly allow for $k=0$.   (Cline et al., 2004).

Further, by combining the Friedmann equation in the case of a universe dominated by pressureless matter, the cosmological constant and spatial curvature with the acceleration equation (5.b), we obtain

The latter can be rewritten as

which makes apparent the reasonableness of the expression $\,1+q\geq\Omega_{k}$. Moreover, the above shows the compatibility of the said expression with general relativity.

Our data set consists of: $(i)$ recent 32 cosmic chronometer (CC) $H(z)$ measurements, presented in Table 1, which do not assume any particular cosmological model, and $(ii)$ the 28 $H(z)$ measurements from baryon acoustic oscillations (BAO) from different galaxy surveys, listed in Table 2. In both cases, the datasets are given with their $1\sigma$ confidence interval. Because the BAO measurements are not entirely model-independent, particularly due to the assumption of a fiducial radius of the comoving sound horizon, ${r_{d}}^{\text{fid}}$, we adopted a full marginalization over the GP hyperparameter space (see below) with the comoving sound horizon at drag epoch $r_{d}$ for the BAO data set as free parameter. This results in a model-independent Hubble data set from CC and the calibrated BAO.

As is apparent, a handful of $H(z)$ data are afflicted by a rather large $1\sigma$ confidence intervals whence some statistically founded smoothing process must be applied to the full data set to arrive to sensible results. This is why we resort to the machine-learning model Gaussian Process (GP) which infers a function from labelled training data (Rasmussen & Williams, 2005; Seikel et al., 2012), and reconstruct the Hubble diagram as a function of the scale factor. The said process is able to reproduce an ample range of behaviors with just a few parameters and allows a Bayesian interpretation (Zhao et al., 2008). Then, after smoothing the Hubble data set, we constrain the free parameters that enter the proposed parametric expressions (Eqs. (10), (13) and (15), below) of the history of the Hubble factor and check whether the corresponding curves, $H(a)$, are consistent with the second law. That is to say, whether they comply with the inequality $\,1+q\geq\Omega_{k}$.

To undertake the GP, we consider the squared exponential (SqExp hereafter), Matérn 7/2 (Mat72 hereafter), Cauchy and rational quadratic (R.Quad hereafter) covariance functions,

(also called “kernels") where $\sigma_{f}$, $l$ and $\alpha$ are the hyperparameters. Throughout this work, we assume a zero mean function to characterize the GP. We investigate if the different covariance functions lead to significant differences in the results²²2For details on GP, visit http://www.gaussianprocess.org.

We adopt a python implementation of the ensemble sampler for MCMC, the emcee³³3https://github.com/dfm/emcee, introduced by Foreman-Mackey et al. (2013). The two dimensional confidence contours showing the uncertainties along with the one dimensional marginalized posterior probability distributions, are shown in Fig. 1 with the help of the GetDist⁴⁴4https://github.com/cmbant/getdist module of python, developed by Lewis (2019).

In this section we study whether hyperbolic, i.e., open, spatial sections (which correspond to $k=-1$) are compatible with the second law of thermodynamics as expressed by Eq, (4). To this end we consider three parametrizations (equations (10), (13) and (15), below) of the history of the Hubble factor and use the corresponding parameter values obtained after the smoothing of the Hubble data (tables 3, 4 and 5 for the first, second and third parametrizations, respectively⁵⁵5In these tables $a_{t}$ is the scale factor at which the transition from deceleration to acceleration occurs.). Figure 2 shows the contour plots of the different parameters occurring the three parametrizations. Then we check whether the corresponding curves, $H(a)$, comply with the inequality (4). These curves are shown in Fig. 3.

We express $\Omega_{k}$ as

with $\Omega_{k0}\lesssim 10^{-2}$ the upper bound on the likely present value of the spatial curvature parameter assuming $k=-1$, see Komatsu et al. (2011); Ade et al. (2014); Park & Ratra (2019); Handley (2021); Mukherjee & Banerjee (2022); Bel et al. (2022).

In the cases considered below $H_{\star}\equiv H(a\rightarrow\infty)$. The analysis of the fourth parametrization does not call for a numerical study.

Before going any further, it is worthwhile to note that since $0<\Omega_{k=-1}<1$ and that $q>0$ ($<0$) when the universe is decelerating (accelerating), the said inequality cannot be violated when the universe is decelerating.

We further examine to what extent (if at all) the rising tension between the local measurements of the Hubble constant (Riess et al., 2022; Freedman, 2021), and its inferred values via an extrapolation of data on the early universe (Aghanim et al., 2020) affects our results.

To this end we consider $(i)$ the local measurements by the SH0ES team [$H_{0}^{\text{R22}}$ = $73.3\pm 1.04$ km Mpc^-1 s^-1 (R hereafter)], $(ii)$ the updated TRGB calibration from the Carnegie Supernova Project [$H_{0}^{\text{F21}}$ = $69.8\pm 1.7$ km Mpc^-1 s^-1 (F hereafter)], and $(iii)$ the inferred value via an extrapolation of data on the early universe by the Planck survey [$H_{0}^{\text{P20}}$ = $67.4\pm 0.5$ km Mpc^-1 s^-1 (P hereafter)], respectively. We assume Gaussian prior distributions with the mean and variances corresponding to the best-fit and $1\sigma$ confidence interval reported about $H_{0}$ values.

We can conclude this section by saying that our overall result, that the constrain $1+q\geq\Omega_{k}$ is satisfied even if $k=-1$, is not affected whether we take the $H_{0}$ value from the local measurements or the one obtained from extrapolating the precise data drawn from the cosmic microwave background radiation.

The second law of thermodynamics applied to our expanding FLRW universe implies that the area of its apparent horizon cannot decrease, i.e., ${\cal A}^{\prime}\geq 0$. This, in its turn, dictates $HH^{\prime}\leq k/a^{3}$. Since in absence of phantom fields, which are unstable, $H^{\prime}<0$ leads to two possibilities, namely, $k=0$ and $k=+1$, result immediately compatible with the second law. The third possibility, $k=-1$, may or may not be compatible with it. Here we resorted to the set of observational data regarding the evolution of the Hubble factor to discern whether the said law is satisfied when $k=-1$. So, with the help of the non-parametric Gaussian Process we smoothed the set of $60$ currently available data of the Hubble factor and introduced four different parametrizations of the ensuing curve, $H(a)$. After using the latter to constrain the free parameters entering the first three parametrizations (the fourth one was not in need of a numerical study), we examined whether the resulting expressions, with the curvature index $k$ fixed to $-1$, satisfy Eq. (4). As it turned out, all of them did (Fig. 4 and Eq. (18)). Further, a quick inspection of the said expressions shows that they also comply with Eq. (4) in the limit $a\rightarrow\infty$. Therefore, we are led to conclude that the evolution of FLRW universes with open spatial sections does not appear to conflict with the second law of thermodynamics. Regrettably, a final verdict cannot be attained at this stage since currently we have no cosmological model that deserves our unreserved confidence. In fact, the most reliable model, the so-called “concordance model", is afflicted by some drawbacks whose relevance may not be small (Perivolaropoulos & Skara, 2022; Di Valentino, 2022). This is why we did resort to parametrize the Hubble factor rather than taking it from any given model.

One may argue that this conclusion could be reached from Eq.(7) straightaway. However this is not so because that equation only concerns the adiabatic evolution of the fluid that sources the gravitational field. Further, the second law does not enter its derivation whereby it cannot be ascertained whether it is respected or violated.