Anisotropic Bianchi-I cosmological model in non-conservative unimodular gravity

TL;DR

This work explores anisotropic Bianchi-I cosmologies within non-conservative unimodular gravity (NUG), addressing the theory’s underdetermined equations by imposing a condition on the enthalpy combination $\rho+p$. The authors analyze vacuum Kasner-like solutions and several cases with constant or scale-factor–dependent enthalpy $\rho+p$, revealing how the anisotropy parameter $\Omega_A$ and the integration-constant term $\Lambda_U$ shape cosmic evolution, isotropization, and the universe’s age. They find GR-like behavior for $\rho+p$ with dust or radiation when anisotropy is small, and show that the sign of the constant enthalpy $l$ drives qualitatively different futures (recollapse vs. ghost-like acceleration). The results highlight the role of $\Omega_A$ and $\Lambda_U$ in NUG cosmology and motivate further data-driven constraints and thermodynamic interpretation of enthalpy in this framework.

Abstract

In this article, we propose an anisotropic Bianchi-I type cosmological model in non-conservative Unimodular Gravity ($\mathrm{NUG}$). We show that simply using the Bianchi-I type metric does not resolve a striking characteristic of the field equations in $\mathrm{NUG}$: their underdetermination. This fact led us to implement extra conditions on the combination $\left(ρ+p\right)$ and, consequently, obtain a consistent background cosmological analysis. In the vacuum case, we obtain an analogy between the Kasner solutions and the equations in $\mathrm{NUG}$. We also propose a new analysis of a non-homogeneous equation of state, the combination $\left(ρ+p\right)=l$. We identify that the cosmological dynamics are strictly dependent on the value of the constant $l$. The physically interesting case is at the value $l<0$, which seems to indicate a super-accelerated, ghost-like universe. This case still requires a more detailed analysis, for example, from a thermodynamic point of view, keeping in mind that $\left(ρ+p\right)$ may be interpreted as enthalpy of the system. For the cases $\left(ρ+p\right)\propto a^{-3}$ and $\left(ρ+p\right) \propto a^{-4}$, we obtain a description consistent with the anisotropic cosmological model described by $\mathrm{GR}$. In all cases analyzed, a small value for the anisotropic parameter $Ω_{A}$ (on the order of $10^{-2}$) is required in order to have agreement, for example, with the age of the universe to be approximately $12-14\, \mathrm{Gyr}$, agreeing with the age of globular clusters. As the universe expands an isotropization is verified, with the anistropies going to zero asymptotically, similarly with what happens in an anistropic cosmological model based on $\mathrm{GR}$.

Figures (2)

• Figure 1: Graph showing the time evolution of the universe for three cases proposed in the combination $\left(\rho+p\right)$.
• Figure 2: Graph of the evolution of the deceleration parameter $q\left(z\right)$ analyzed in the anisotropic model in NUG plus $\Lambda \mathrm{CDM}$ model. In both graphs, the horizontal axis represented by $\left(1+z\right)$ is on the logarithmic scale. The dashed black line in the graphs above represents the redshift today $z=0$. For the case $\left(\rho+p\right)=0$, represented by the solid red line. To create the graph we set $\Omega_{A}=0.01$. For the case $\left(\rho+p\right)=l$ we plot two cases: $\Omega_{l}>0$ (represented by the dashed blue line) and $\Omega_{l}<0$ (represented by the solid black line). To create the graph we set $\Omega_{A}=0.001$, $\Omega_{l}=\pm 0.09$. For the case $\left(\rho+p\right)=\bar{\rho}_{0}/a^{-3}$, represented by the solid gray line. To create the graph we set $\Omega_{A}=0.01$, $\bar{\Omega}=0.317$. For the case $\left(\rho+p\right)=\bar{\rho}_{0}/a^{-4}$, represented by the solid green line. To create the graph we set $\Omega_{A}=0.001$, $\bar{\Omega}=0.0009$. For the $\Lambda \mathrm{CDM}$ model, represented by solid orange line, we use $\Omega_{m}=0.317$.