Barrow Entropy Cosmology: an observational approach with a hint of stability analysis

For a flat Universe ($k=0$) filled by matter, the Friedmann and Raychaudhuri equations [20] are

where $\rho_{m}$ denotes the energy density of matter (baryons plus dark matter) and we assume the equation of state $p_{m}=0$ corresponding to dust (pressureless) matter. The energy density and pressure of this effective dark energy are written in the form

where $\Lambda\equiv 4{C}G(4\pi)^{\Delta/2}$, $C$ is an appropriate integration constant and $\beta\equiv 4(4\pi)^{\Delta/2}G/A_{0}^{1+\Delta/2}$. Firstly, we keep $\beta$ as a free parameter. Next, by setting $A_{0}=4G$ we have $\beta\equiv(\pi/G)^{\Delta/2}$.

Although Eqs. (2.4)-(2.5) does not has a dark energy component, we have dubbed effective dark energy the extra-terms introduced by the Barrow entropy.

Furthermore, the flatness constraint $E(0)=1$, gives the equation

In addition, the deceleration parameter is defined as

By substituting Eq. (2.13) and replacing Eqs. (2.6), (2.7) into (2.5), the $q(z)$ results

and the DE EoS in terms of Eqs. (2.13)-(2.14) is written as

where

It is worthy to note that for $\bar{\beta}=1$, $\Delta=0$ and $z\to 1$ the EoS for the cosmological constant is recovered, i.e. $w_{DE}=-1$.

By considering two fluids, matter and radiation, for a flat Universe, the Friedmann and Raychaudhuri equations in Barrow cosmology result as follows

where $\rho_{r}$ indicates the radiation energy density and the radiation pressure is $p_{r}=\rho_{r}/3$. We define

Using the previous variable definitions (2.11), the dimensionless Friedmann equation becomes

Finally, from Eq. (2.23) it follows

The dimensionless Friedmann equation can be written as

where the radiation component evolves in the traditional way $\rho_{r}=\rho_{r0}(z+1)^{4}$, and $\Omega_{r0}$ is obtained from (2.22) evaluated at $a=1$ ($z=0$). This can be calculated as $\Omega_{r0}=2.469\times 10^{-5}h^{-2}(1+0.2271N_{eff})$, with $N_{eff}=3.04$ as the number of relativistic species [26] and $h=H_{0}/100\mathrm{kms}^{-1}\mathrm{Mpc}^{-1}$ as the current Hubble dimensionless parameter.

The flatness constriction $E(0)=1$, gives the equation

In addition, the deceleration parameter reads

The DE EoS can be calculated by substituting Eq. (2.25) into Eq. (2.18) and $E^{\prime}(z)$ as

Notice that for $\bar{\beta}=1$, $\Delta=0$ and $z\to 1$ the EoS for the cosmological constant is recovered.

A cosmological-independent measurement of the Hubble parameter is acquired through the differential age (DA) technique [32] in cosmic chronometers (i.e passive elliptic galaxies). In this paper, we consider the OHD compilation provided by [27] containing $31$ data points in the range $0.07<z<1.965$. Hence, the chi square function for OHD can be constructed through the following equation

where $H_{th}({\bf\Theta},z_{i})$ and $H_{obs}(z_{i})$, are the theoretical and observational Hubble parameters respectively at the redshift $z_{i}$, and $\sigma^{i}_{obs}$ is the observational error. Notice that ${\bf\Theta}$ is a vector related to the number of free parameters of the studied cosmological model.

The Pantheon sample [33] contains $1048$ SNIa data points in the redshift range $0.001<z<2.3$. The observational distance modulus $\mu_{\mathrm{PAN}}$ for Pantheon SNIa can be measured as

HIIG are galaxies with large HII regions, product of young and hot stars (O and/or B type stars) ionizing the medium. For these galaxies there is a correlation between the measured luminosity, $L$, and the inferred velocity dispersion, $\sigma$, of the ionized gas. Several authors have shown that the correlation $L-\sigma$ could be used as a cosmological tracer [34, 35, 36, 37, 38, and references therein]. A HIIG data sample was compiled by [38] containing $107$ low redshift ($0.0088$ $\leq$ $z$ $\leq$ $0.16417$) galaxies, and $46$ high redshift ($0.636427$ $\leq$ $z$ $\leq$ $2.42935$) galaxies. [29] used such HIIG sample to constrain the cosmological parameters for six different cosmological models. Recently, [39] presented a new sample which contains $181$ local and high-z HIIG data points in the redshift range $0.01<z<2.6$. In this paper, we use such HIIG sample, and follow their methodology [see 39].

The correlation between $L$ and $\sigma$ can be written as

where $\gamma$ and $\beta_{II}$ are the intercept and slope functions, respectively. Following [29, 38, 39], we set $\beta_{II}$ = 5.022, and $\gamma$ = 33.268. Therefore, the distance modulus takes the form

where $f$ is the flux emitted by the HIIG. Moreover, the theoretical distance modulus is

being $d_{L}({\bf\Theta},z)$ the luminosity distance (in Mpc).

The figure-of-merit is given by the following equation

where $\epsilon_{i}$ is the uncertainty of the $i_{th}$ measurement and it can be calculated propagating the errors of Eq. 3.6 [see further details in 39].

Several authors have shown that strong lensing systems can be used as cosmological tool to constrain cosmological parameters [30]. The method consists in comparing a theoretical distance ratio of angular diameter distances in the lens geometry with its observational counterpart. It can be obtained from the Einstein radius of a lens (modeled with a singular isothermal sphere) given by

where $\sigma_{l}$ is the observed velocity dispersion of the lens galaxy, $D_{s}$ is the angular diameter distance to the source at redshift $z_{s}$, and $D_{ls}$ is the angular diameter distance from the lens (at redshift $z_{l}$) to the source. Then, the observational distance ratio of angular diameter distances is defined as

To measure the theoretical distance ratio $D^{th}({\bf\Theta},z_{l},z_{s})\equiv D_{ls}/D_{s}$, we calculate $D_{s}$ using the definition of angular diameter distance of a source at redshift $z$

and $D_{ls}$ through the definition of the angular diameter distance between two objects at redshift $z_{1}$ and $z_{2}$

The most recent compilation of Strong-Lensing Systems (SLS) given by [30] consists of 204 SLS spanning the redshift region $0.0625<z_{l}<0.958$ for the lens and $0.196<z_{s}<3.595$ for the source. To avoid convergence problems and discarding (unphysical) systems with $D_{ls}>D_{s}$, the authors provided a fiducial sample with an observational lens equation ($D^{obs}$) within the region $0.5\leq D^{obs}\leq 1$.

In this work, we use such a fiducial sample consisting of 143 SLS, and the chi-square function takes the form

where

being $\delta\theta_{E}$ and $\delta\sigma$ the uncertainties of the Einstein radius and velocity dispersion, respectively.

BAO are considered as standard rulers, being primordial signatures of the interaction of baryons and photons in a hot plasma on the matter power spectrum in the pre-recombination epoch. Authors in [40] collected 6 correlated data points measured by [41, 42, 43]. To confront cosmological models to these data, it is useful to build the $\chi^{2}$-function in the form

where $\Delta\vec{X}$ is the difference between the theoretical and observational values of $d_{A}(z_{drag})/D_{V}(z_{i})$ where $z_{drag}$ is defined by the sound horizon at baryon drag epoch measured at the redshift $z_{i}$, and $\rm{Cov}^{-1}$ is the inverse of covariance matrix, the dilation scale ($D_{V}$) is defined as [44]

where $d_{A}({\bf\Theta},z)=(1+z)D_{A}({\bf\Theta},z)$ is the comoving angular-diameter distance and $D_{A}$ is the angular diameter distance at $z$ presented in Eq. (3.11). Additionally, $r_{drag}$ is the sound horizon at baryon drag epoch. We use $r_{drag}=147.21\pm 0.23$ reported in [2].

The inference of the cosmological parameters under Barrow cosmology, for both model I (Eq. 2.13) and II (Eq. 2.25), is performed using a Bayesian Markov Chain Monte Carlo (MCMC) approach using the emcee Python module [45]. We set $3000$ chains with $250$ steps each one. The burn-in phase is stopped up to obtain convergence according to the auto-correlation time criteria. We build a Gaussian log-likelihood as the merit-of-function to minimize through the equation $-2\log(\mathcal{L}_{\rm data})\varpropto\chi^{2}_{\rm data}$ for each dataset, and consider Gaussian priors on $h$ and $\Omega_{m0}$ centered at $0.6766\pm 0.0042$ and $0.3111\pm 0.0056$ [2], respectively, and a flat prior for $\Delta$ in the ranges: $\Delta:\,[0,2]$. The parameter $\bar{\beta}$ is calculated using Eq. (2.14), where we have set $A_{0}=4G$. Additionally, a joint analysis can be constructed through the sum of their function-of-merits, i.e.,

where subscripts indicate the observational measurements under consideration.

Figure 1 shows the 2D confidence contours at $68\%$ ($1\sigma$) and $99.7\%$ ($3\sigma$) confidence level (CL) respectively, and 1D posterior distribution of the parameters in Barrow cosmology with a matter component (top panel) and matter plus radiation. In the case of the $\Delta$ parameter, although the contours for most of the samples are consistent with each other, the ones obtained using SLS data are in tension with those estimated with the other samples. However, this is not surprising inasmuch as reported by [30], the use of their fiduciary sample of 143 strong lensing systems while performs better constraining the cosmological models tested in such work (compared with other lensing samples), the parameters are not tightly constraint. Indeed, they reported that the range on the studied cosmological parameters were in agreement with those expected from other astrophysical observations, but they also discussed that the method needs improvement, in particular to take into account systematic biases (e.g. not fully confirmed lenses, multiple arcs, uncertain redshifts, complex lens substructure).

Table 1 presents the chi-square and mean values of the parameters obtained from the different data set and their uncertainties at $1\sigma$ for both Barrow cosmologies. We obtain $\Delta=(5.912^{+3.353}_{-3.112})\times 10^{-4}$, $\bar{\beta}=0.920^{+0.042}_{-0.042}$, and $\Delta=(6.245^{+3.377}_{-3.164})\times 10^{-4}$, $\bar{\beta}=0.915^{+0.043}_{-0.043}$ for model I and II, respectively. Both models are consistent at $2\sigma$ with the standard cosmological model, i.e. $\Delta=0$ and $\bar{\beta}=1$. Moreover, the $\Delta$ bounds suggest the entropy-area relation is consistent with the Bekenstein entropy. From now on, we focus our discussion on the model II ( matter and radiation components) since it is a more realistic model.

The top panel of the Figure 2 shows that the expansion rate estimated from the mean values of the Barrow cosmology parameters are consistent with the OHD. In addition, the reconstruction of the deceleration parameter as function of redshift is shown in the middle panel of Fig. 2. The $q(z)$ behavior is similar to the standard one, i.e. there is a transition at $z_{t}\simeq 0.711^{+0.035}_{-0.034}$ from a decelerated stage to an accelerated stage with $q_{0}=-0.573^{+0.019}_{-0.019}$, suggesting a de Sitter solution. However, in the Barrow scenario the late cosmic acceleration is driven by the new terms in the dynamical equations. Finally, the bottom panel of Fig. 2 illustrates the reconstruction of the equation of state of the effective dark energy as function of redshift. It is worth noting that it has a transition at $z_{wde}\simeq 1.070^{+0.114}_{-0.104}$ from a quintessence-like regime to a phantom-like one, yielding $w_{DE}(0)\simeq-1.000134^{+0.000069}_{-0.000068}$ at current times, which is consistent with the cosmological constant at $1\sigma$. This behavior of $w_{DE}(z)$ has been also discussed by [20], the effective dark energy can undergo the phantom-divide crossing but it tends asymptotically to a de Sitter solution at late times.

Furthermore, the age of the Universe can be estimated by solving the integral $t_{0}=\int_{0}^{1}{\text{d}a}/aH(a)$. Considering the constraints from the joint analysis, we obtain $t_{0}\simeq 14.062^{+0.179}_{-0.170}$ ($14.045^{+0.179}_{-0.167}$) Gyr for matter+radiation (matter) model, consistent with $2\sigma$ confidence level with the measurements of Planck [2]. Thus, the constraints obtained from several data at different scales indicate that, by modifying the entropy-area relation, Barrow cosmology is a plausible scenario to explain the late cosmic acceleration without the need to include an exotic component.

In the following sections, we perform a dynamical system analysis of the Barrow cosmology.

To start the dynamical analysis for the Barrow cosmology (§2), we use the dynamical variables $(\Omega_{m},\Omega_{\Lambda})$ defined in (2.11), say,

As we commented before, the normalized Friedmann equation (2.9) is then transformed to

Notice that by substituting $\beta=1,\Delta=0$, in Eq. (2.9), we obtain the usual relation in FRW cosmology. For $\beta\neq 1,\Delta\neq 0$, by substituting Eqs. (2.6) and (2.7) into Eq. (2.5), we obtain

Then, for $H\neq 0$, $\beta\neq 1$, and $\Delta\neq 0$, the deceleration parameter (Eq. (2.16)) results

for $\Lambda$CDM ($\beta=1,\Delta=0$) we obtain the usual relation $q=-1+\frac{3}{2}\Omega_{m}$.

In general, $H(t)$ satisfies the differential equation

Furthermore, we have

Finally, we obtain the dynamical system

where the prime means derivative with respect $\tau=\ln(a)$, and $q$ is defined by (4.6). The main difference with the $\Lambda$CDM model is that the term $\frac{\beta(\Delta+2)H^{-\Delta}}{2-\Delta}$ in Eq. (4.2) is unbounded as $H\rightarrow 0$, resulting in unbounded $\Omega_{\Lambda},\Omega_{m}$. The equilibrium points in the finite part of the phase space are

1. 1.

   the line $A(\Omega_{\Lambda}):$ $\Omega_{m}=0$, $\Omega_{\Lambda}=\text{arbitrary}$, for $\Delta=\text{arbitrary}$, with eigensystem $\left(\begin{array}[]{cc}0&-3\\ \{1,0\}&\left\{\frac{2}{\Delta-2},1\right\}\\ \end{array}\right)$; and

2. 2.

   the line $B(\Omega_{m}):$ $\Omega_{m}=\text{arbitrary},\Omega_{\Lambda}=0$, for $\Delta=0$, with eigensystem $\left(\begin{array}[]{cc}3&0\\ \{-1,1\}&\{0,1\}\\ \end{array}\right)$.

The line of points $B(\Omega_{m})$ exists only for $\Delta=0$. All these lines of equilibrium points are normally hyperbolic because the tangent vector at a given point of each line is parallel to the corresponding eigenvector associated to the zero eigenvalue. This implies that the stability conditions can be inferred from the eigenvalues with non-zero real parts [55]. Therefore, the line $A(\Omega_{\Lambda})$ is the attractor of the system, representing de Sitter solutions. For $\Delta=0$, the line $B(\Omega_{m})$ contains the past attractors, which represents matter dominated solutions.