Cosmological interacting models via energy-momentum squared gravity

Abstract

It was recently shown in the literature that gravity models that modify the material part of the standard Einstein-Hilbert action with $f(\mathcal{L}_{\rm m})$, $f(T)$, and $f(T_{\mu\nu}T^{\mu\nu})$ terms are equivalent to general relativity, encompassing non-minimal matter interactions between the material field and its accompanying partner, uniquely formed by the function $f$. In Energy-momentum squared gravity (EMSG), the ``squared" terminology arises from the self contraction of EMT $f(T_{\mu\nu}T^{\mu\nu})$ added to Einstein Hilbert action, nontrivial interaction kernels have been obtained and these models diverge from phenomenological interacting models (constructed in ad hoc way); this is due to the function $f$ and its variations with respect to both its argument and the metric, which intricately intertwine the interaction kernel $\mathcal{Q}(f,\delta f/\delta\mathbf{T^2},\delta, \mathbf{T^2}/\delta g^{\mu\nu})$. This makes the interaction kernel as Equation of State (EoS) parameter dependent as well. Bianchi identity $\nabla^{\mu}G_{\mu\nu}=0$ implies the conservation of total energy momentum tensor (EMT of the standard source plus its EMSF partner's), $\nabla^{\mu}(T_{\mu\nu}+T_{\mu\nu}^{{\rm EMSF} })=0$, leading cosmological models having an interaction between these sectors $\nabla^{\mu}T_{\mu\nu}=\mathcal{Q}_{\nu}$ and $\nabla^{\mu}T_{\mu\nu}^{\rm EMSF}=-\mathcal{Q}_{\nu}$ where $\mathcal{Q}_{\nu}\neq0$. In this thesis, different than the literature, still consistent with the Bianchi Identity, we focus on a scenario where the sector comprising conventional fluids (standard material fields) overall interacts minimally with the sector associated with their EMSF partners, i.e., satisfying $\nabla^{\mu}T_{\mu\nu}=0=\nabla^{\mu} T_{\mu\nu}^{{\rm EMSF}}$. Specifically, we consider the case characterized by $\mathcal{Q}=0$. Accordingly, we will consider a two-fluid model (perfect fluids described by constant EoS parameters) leading to the following conservation equations, $\nabla^{\mu}\left(T_{\mu\nu,1}+T_{\mu\nu,2}\right)=0$, and $\nabla^{\mu}\left(T_{\mu\nu,1}^{\rm EMSF}+T_{\mu\nu,2}^{\rm EMSF}\right)=0$ where we name the partner arisen from EMSG corrections as ``Energy Momentum Squared Field" (EMSF). We will explore this choice in detail within the framework of scale-independent EMSG which introduces a simple interaction kernel: a kernel linear in energy density. Then, we examine alternative cosmologies wherein the sector comprising conventional fluids minimally interacts with the sector associated with their EMSF partners, represented by $\nabla^{\mu}\left(T_{\mu\nu}^1+T_{\mu\nu}^{\rm 2}\right)=-\nabla^{\mu}\left(T_{\mu\nu}^{\rm EMSF1}+T_{\mu\nu}^{\rm EMSF2}\right)=\mathcal{Q}_{\nu}$ with $\mathcal{Q}_{\nu}=0$, diverging from the more commonly studied scenarios in literature where $\mathcal{Q}_{\nu}\neq0$. We also show that this model is reminiscent of the cosmological model with energy exchange studied by Barrow and Clifton in [Phys. Rev. D 73, 103520 (2006)] where the interaction term is taken ad hoc to be proportional to energy density, $\mathcal{Q}(H\rho)$. Unlike their model, the coefficients in our work are not arbitrary constants but are dependent on the species. Moreover, with an additional sector associated with the EMSF partners of the conventional fluids in the Friedmann equation, it is possible to negate one of the fluid's contributions in the Friedmann equation via its EMSF partner for a specific choice of $\alpha$ and two sources may superpose in their energy densities in the Friedmann equation, resulting in a joint (degenerate) scale factor dependence even if $w_1 \neq w_2$ reproducing interesting cosmologies such as power-law universes where the scale factor of the universe grows as a EoS parameter dependent power of time in the presence of a perfect fluid and vacuum energy density/stiff fluid, de Sitter universe in the presence of a perfect fluid and vacuum energy density. In this thesis, we show a simple mathematical description of the exchange of energy between two standard fluids from matter modified theories within GR choosing the simplest case study, yet some non-trivial functions/behaviors are favored by observations to alleviate tensions, non-linear interactions and non-linear energy density contributions from matter-type modified theories which may work for the change of direction of energy transfer in dark sector are prospects for future research.