N. Merve Uzun

In this study, we first establish that gravity models incorporating matter-related terms, such as $f({\mathcal{L}}_{\mathrm{m}})$, $f(g_{\mu \nu }{T}^{\mu \nu })$, and $f(T_{\mu \nu }{T}^{\mu \nu })$, into the usual matter Lagrangian density ${\mathcal{L}}_{\mathrm{m}}$, are equivalent to general relativity (GR) with nonminimal matter interactions. Through the redefinition ${\mathcal{L}}_{\mathrm{m}}+f\to {\mathcal{L}}_{\mathrm{m}}^{\mathrm{tot}}$, these models are exactly GR, yet the usual material field $T_{\mu \nu }$ and its accompanying partner, namely, the modification field $T_{\mu \nu }^{\text{mod}}$, engage in nonminimal interactions. Specifically, ${\nabla }^{\mu }{T}_{\mu \nu }=-Q_{\nu }=-{\nabla }^{\mu }{T}_{\mu \nu }^{\text{mod}}$, where $Q_{\nu }$ is the interaction kernel that governs the rate of energy transfer. Our focus narrows on the specific model of $f(T_{\mu \nu }{T}^{\mu \nu })$, known as energy-momentum squared gravity (EMSG), where the usual material field $T_{\mu \nu }$ is accompanied by an energy-momentum squared field (EMSF), $T_{\mu \nu }^{\mathrm{emsf}}$, along with a sui generis nonminimal interaction between them. We demonstrate that a particular $T_{\mu \nu }^{\mathrm{emsf}}$ can be introduced by removing $\frac{{\partial }^2{\mathcal{L}}_{\mathrm{m}}}{\partial g^{\mu \nu }\partial g^{\sigma \epsilon }}$ (the new term emerging in models that incorporate scalars formed from $T_{\mu \nu }$), thanks to the freedom in determining the interaction kernel, but this approach compromises the Lagrangian formulation of EMSG. Additionally, we address the ambiguities regarding the perfect fluid stemming from this new term. We show the proper way of calculating this term for a perfect fluid, revealing that it is indeed nonzero, contrary to common assumption in the literature. Finally, we reexamine cosmological models within the realm of EMSG, offering new insights into the applicability and interpretation of our findings in EMSG and similar theoretical frameworks.

In this study, we first establish that gravity models incorporating matter-related terms, such as $f(\mathcal{L}_{\rm m})$, $f(g_{μν} T^{μν})$, and $f(T_{μν} T^{μν})$, into the usual matter Lagrangian density $\mathcal{L}_{\rm m}$, are equivalent to general relativity with nonminimal matter interactions. Through the redefinition $\mathcal{L}_{\rm m}+f \rightarrow \mathcal{L}_{\rm m}^{\rm tot}$, these models are exactly GR, yet the usual material field $T_{μν}$ and its accompanying partner, the modification field $T_{μν}^{\rm mod}$, engage in nonminimal interactions. Specifically, $\nabla^μT_{μν}=-Q_ν=-\nabla^μT_{μν}^{\rm mod}$, where $Q_ν$ is the interaction kernel that governs the rate of energy transfer. Our focus narrows on the specific model of $f(T_{μν} T^{μν})$, known as Energy-Momentum Squared Gravity, where the usual material field $T_{μν}$ is accompanied by an \textit{energy-momentum squared field} (EMSF), $T_{μν}^{\rm emsf}$, along with a sui generis nonminimal interaction between them. We demonstrate that a particular $T_{μν}^{\rm emsf}$ can be introduced by \textit{removing} $\frac{\partial^2 \mathcal{L}_{\rm m}}{\partial g^{μν} \partial g^{σε}}$ (the new term emerging in models that incorporate scalars formed from $T_{μν}$), thanks to the freedom in determining the interaction kernel, but this approach compromises the Lagrangian formulation of EMSG. Additionally, we address the ambiguities regarding the perfect fluid stemming from this new term. We show the proper way of calculating this term for a perfect fluid, revealing that it is indeed non-zero, contrary to common assumption in the literature. Finally, we re-examine cosmological models within the realm of EMSG, offering new insights into the applicability and interpretation of our findings in EMSG and similar theoretical frameworks.

We explore the weak field and slow motion limits, Newtonian and Post-Newtonian limits, of the energy-momentum powered gravity (EMPG), viz., the energy-momentum squared gravity (EMSG) of the form $f(T_{\mu\nu}T^{\mu\nu})=\alpha (T_{\mu\nu}T^{\mu\nu})^{\eta}$ with $\alpha$ and $\eta$ being constants. We have shown that EMPG with $\eta\geq0$ and general relativity (GR) are not distinguishable by local tests, say, the Solar System tests; as they lead to the same gravitational potential form, PPN parameters, and geodesics for the test particles. However, within the EMPG framework, $M_{\rm ast}$, the mass of an astrophysical object inferred from astronomical observations such as planetary orbits and deflection of light, corresponds to the effective mass $M_{\rm eff}(\alpha,\eta,M)=M+M_{\rm empg}(\alpha,\eta,M)$, $M$ being the actual physical mass and $M_{\rm empg}$ being the modification due to EMPG. Accordingly, while in GR we simply have the relation $M_{\rm ast}=M$, in EMPG we have $M_{\rm ast}=M+M_{\rm empg}$. Within the framework of EMPG, if there is information about the values of $\{\alpha,\eta\}$ pair or $M$ from other independent phenomena (from cosmological observations, structure of the astrophysical object, etc.), then in principle it is possible to infer not only $M_{\rm ast}$ alone from astronomical observations, but $M$ and $M_{\rm empg}$ separately. For a proper analysis within EMPG framework, it is necessary to describe the slow motion condition (also related to the Newtonian limit approximation) by $|p_{\rm eff}/\rho_{\rm eff}|\ll1$ (where $p_{\rm eff}=p+p_{\rm empg}$ and $\rho_{\rm eff}=\rho+\rho_{\rm empg}$), whereas this condition leads to $|p/\rho|\ll1$ in GR.

Scale-independent EMSG is a particular model of energy-momentum squared gravity (EMSG) in which the new terms in the Einstein field equations arising from the EMSG theory enter with the same power as the usual terms from Einstein-Hilbert part of the action. However, the model violates the local energy-momentum conservation and matter-current conservation in general and hence, permits a process of matter creation/annihilation in an expanding universe. Consequently, the scale factor dependencies of the energy densities are modified by the dimensionless model parameter $α$. We revisit some nostalgias such as static universes and de Sitter/steady state universes. We reproduce the original ones, moreover, present some novelties, e.g., a spatially flat static universe, de Sitter expansion by negative vacuum energy, steady state universes in the presence of arbitrary fluids with constant equation of state (EoS) parameter other than dust, etc. We also investigate the possible dynamics of dust dominated and radiation dominated universes. Depending on the value of $α$, dust/radiation dominated universe exhibits power-law accelerated/decelerated expansion, corresponds to a steady state model or may end in a big rip. In the framework of anisotropic cosmology, we reproduce Barrow's quiescent universe in the presence of stiff fluid and extend it to fluids with arbitrary constant EoS parameter. We also relax the condition for isotropic initial singularity (big bang) owing to that EMSG effectively allows ultra-stiff EoS parameters.

We construct a generalization of the standard $\Lambda$CDM model, wherein we simultaneously replace the spatially flat Robertson-Walker metric with its simplest anisotropic generalization (LRS Bianchi I metric), and couple the cold dark matter to the gravity in accordance with the energy-momentum squared gravity (EMSG) of the form $f(T_{\mu\nu}T^{\mu\nu})\propto T_{\mu\nu}T^{\mu\nu}$. These two modifications -- namely, two new stiff fluid-like terms of different nature -- can mutually cancel out, i.e., the shear scalar can be screened completely, and reproduce mathematically exactly the same Friedmann equation of the standard $\Lambda$CDM model. This evades the BBN limits on the anisotropy, and thereby provides an opportunity to manipulate the cosmic microwave background quadrupole temperature fluctuation at the desired amount. We further discuss the consequences of the model on the very early times and far future of the Universe. This study presents also an example of that the EMSG of the form $f(T_{\mu\nu}T^{\mu\nu})\propto T_{\mu\nu}T^{\mu\nu}$, as well as similar type other constructions, is not necessarily relevant only to very early Universe but may even be considered in the context of a major problem of the current cosmology related to the present-day Universe, the so-called $H_0$ problem.

We construct a generalization of the standard $\mathrm{\Lambda }\mathrm{CDM}$ model, wherein we simultaneously replace the spatially flat Robertson-Walker metric with its simplest anisotropic generalization (LRS Bianchi I metric), and couple the cold dark matter to the gravity in accordance with the energy-momentum squared gravity (EMSG) of the form $f(T_{\mu \nu }{T}^{\mu \nu })\propto T_{\mu \nu }{T}^{\mu \nu }$. These two modifications—namely, two new stiff fluid-like terms of different nature—can mutually cancel out, i.e., the shear scalar can be screened completely, and reproduce mathematically exactly the same Friedmann equation of the standard $\mathrm{\Lambda }\mathrm{CDM}$ model. This evades the BBN limits on the anisotropy, and thereby provides an opportunity to manipulate the cosmic microwave background quadrupole temperature fluctuation at the desired amount. We further discuss the consequences of the model on the very early times and far future of the Universe. This study presents also an example of that the EMSG of the form $f(T_{\mu \nu }{T}^{\mu \nu })\propto T_{\mu \nu }{T}^{\mu \nu }$, as well as similar type other constructions, is not necessarily relevant only to very early Universe but may even be considered in the context of a major problem of the current cosmology related to the present-day Universe, the so-called $H_0$ problem.