A new covariant generalization of Einstein’s general relativity is developed which allows the existence of a term proportional to $T_{\alpha \beta }{T}^{\alpha \beta }$ in the action functional of the theory ($T_{\alpha \beta }$ is the energy-momentum tensor). Consequently, the relevant field equations are different from general relativity only in the presence of matter sources. In the case of a charged black hole, we find exact solutions for the field equations. Applying this theory to a homogeneous and isotropic spacetime, we find that there is a maximum energy density ${\rho }_{\max }$, and correspondingly a minimum length $a_{\min }$, at the early Universe. This means that there is a bounce at early times, and this theory avoids the existence of an early-time singularity. Moreover, we show that this theory possesses a true sequence of cosmological eras. We also argue that, although in the context of the standard cosmological model the cosmological constant $\mathrm{\Lambda }$ does not play any important role in the early times and becomes important only after the matter-dominated era, in this theory the “repulsive” nature of the cosmological constant plays a crucial role at early times in resolving the singularity.

A simple generalization to Einstein’s general relativity (GR) was recently proposed which allows a correction term $T_{\alpha \beta }{T}^{\alpha \beta }$ in the action functional of the theory. This theory is called energy-momentum squared gravity (EMSG) and introduces a new coupling parameter $\eta$. EMSG resolves the big bang singularity and has a viable sequence of cosmological epochs in its thermal history. Interestingly, in the vacuum EMSG is equivalent to GR, and its effects appear only inside the matter-energy distribution. More specifically, its consequences appear in high curvature regime. Therefore it is natural to expect deviations form GR inside compact stars. In order to study spherically symmetric compact stars in EMSG, we find the relativistic governing equations. More specifically, we find the generalized version of the Tolman-Oppenheimer-Volkov equation in EMSG. Finally we present two analytical solutions, and two numerical solutions for the field equations. For obtaining the numerical solutions we use polytropic equation of state which is widely used to understand the internal structure of neutron stars in the literature. Eventually we find a mass-radius relation for neutron stars. Also, we found that EMSG, depending on the central pressure of the star and the magnitude of free parameter $\eta$, can lead to larger or smaller masses for neutron stars compared with GR. Existence of high-mass neutron stars with ordinary polytropic equation of state in EMSG is important in the sense that these stars exist in GR when equation of state is more complicated.

Many observed disc galaxies harbour a central bar. In the standard cosmological paradigm, galactic bars should be slowed down by dynamical friction from the dark matter halo. This friction depends on the galaxy’s physical properties in a complex way, making it impossible to formulate analytically. Fortunately, cosmological hydrodynamical simulations provide an excellent statistical population of galaxies, letting us quantify how simulated galactic bars evolve within dark matter haloes. We measure bar strengths, lengths, and pattern speeds in barred galaxies in state-of-the-art cosmological hydrodynamical simulations of the IllustrisTNG and EAGLE projects, using techniques similar to those used observationally. We then compare our results with the largest available observational sample at redshift z = 0. We show that the tension between these simulations and observations in the ratio of corotation radius to bar length is 12.62σ (TNG50), 13.56σ (TNG100), 2.94σ (EAGLE50), and 9.69σ (EAGLE100), revealing for the first time that the significant tension reported previously persists in the recently released TNG50. The lower statistical tension in EAGLE50 is actually caused by it only having five galaxies suitable for our analysis, but all four simulations give similar statistics for the bar pattern speed distribution. In addition, the fraction of disc galaxies with bars is similar between TNG50 and TNG100, though somewhat above EAGLE100. The simulated bar fraction and its trend with stellar mass both differ greatly from observations. These dramatic disagreements cast serious doubt on whether galaxies actually have massive cold dark matter haloes, with their associated dynamical friction acting on galactic bars.

MOdified Gravity (MOG) is a covariant modification of Einstein’s general relativity. This theory is one of the current alternatives to dark matter models. We describe the dynamics of collisionless self-gravitating systems in the context of MOG. By studying the weak field approximation of this theory, we derive the equations governing the dynamics of the self-gravitating systems. More specifically, we consider the Jeans instability for self-gravitating fluid and stellar systems and derive new Jeans mass limit ${\tilde{M}}_J$ and wave-number ${\tilde{k}}_J$. Furthermore, considering the gravitational instability in star-forming regions, we show that MOG has not a significant difference with general relativity on this astrophysical scale. However, at larger scales, such as intergalactic space, MOG may lead to different galaxy- and structure-formation processes.

We study the generalized version of energy-momentum squared gravity (EMSG) in the Palatini formalism. This theory allows the existence of a scalar constructed with energy-momentum tensor as $T_{\alpha \beta }{T}^{\alpha \beta }$ in the generic action of the theory. We study the most general form of this theory in the Palatini framework and present the underlying field equations. The equations of motion of a massive test particle have been derived. The weak field limit of the theory is explored and the generalized version of the Poisson equation is obtained. Moreover, we explore the cosmological behavior of the theory with emphasis on bouncing solutions. Some new bouncing solutions for the specific Palatini EMSG model given by the Lagrangian density $\mathcal{L}=R+\beta R^2+\eta T_{\mu \nu }{T}^{\mu \nu }$ are introduced. We show that only the case $\eta >0$ can lead to viable cosmic bounce.

When bars form within galaxy formation simulations in the standard cosmological context, dynamical friction with dark matter (DM) causes them to rotate rather slowly. However, almost all observed galactic bars are fast in terms of the ratio between corotation radius and bar length. Here, we explicitly display an 8σ tension between the observed distribution of this ratio and that in the EAGLE simulation at redshift 0. We also compare the evolution of Newtonian galactic discs embedded in DM haloes to their evolution in three extended gravity theories: Milgromian Dynamics (MOND), a model of non-local gravity, and a scalar–tensor–vector gravity theory (MOG). Although our models start with the same initial baryonic distribution and rotation curve, the long-term evolution is different. The bar instability happens more violently in MOND compared to the other models. There are some common features between the extended gravity models, in particular the negligible role played by dynamical friction − which plays a key role in the DM model. Partly for this reason, all extended gravity models predict weaker bars and faster bar pattern speeds compared to the DM case. Although the absence of strong bars in our idealized, isolated extended gravity simulations is in tension with observations, they reproduce the strong observational preference for ‘fast’ bar pattern speeds, which we could not do with DM. We confirm previous findings that apparently ‘ultrafast’ bars can be due to bar-spiral arm alignment leading to an overestimated bar length, especially in extended gravity scenarios where the bar is already fast.