Abstract We study static, spherically symmetric stellar configurations in the power-law class of energy-momentum squared gravity defined by F ( R , T ) = R + η T n using the covariant 1 + 1 + 2 semi-tetrad formalism. For perfect physical fluids, we show that the nonlinear matter corrections can be reinterpreted as an effective perfect fluid, so that the stellar equilibrium equations retain the standard Tolman–Oppenheimer–Volkoff (TOV) form when written in terms of effective variables. The resulting covariant structure equations are formulated in both metric and dimensionless variables and, whenever an effective closure relation exists, reduce to an autonomous planar dynamical system. This provides a global qualitative description of the stellar phase space in terms of finite and asymptotic critical points. Specializing to linear physical equations of state, we recover the general relativistic benchmark and identify sectors that are exactly, asymptotically, or piecewise equivalent to general relativity, as well as sectors for which the planar reduction breaks down and the full three-dimensional covariant flow must be considered. We further recover the standard metric TOV equation in terms of effective variables and show that, although the exterior spacetime remains Schwarzschild, the natural matching condition at the stellar surface is p eff ( R ) = 0 , which need not coincide with p ( R ) = 0 for self-bound matter. Finally, in the appendix, we show how our procedure can be applied to a realistic neutron-star equation of state.

Modified gravity theories are interesting alternatives to describe the accelerated expansion in the early universe, without the need of an extra scalar field. In this work, we discuss the post-inflationary preheating phase in f(R) theory of gravity. We investigate the correspondance between the Einstein and Jordan frame Lagrangian, in order to explain the parametric resonance effect needed for preheating. We consider different forms of potentials in the Einstein Frame and a conformal-like coupling ∼ R ψ ^2 in the Jordan Frame, from which we obtain a non-trivial f(R) and a non-trivial coupling between the fields ϕ̃ and ψ̃ that yield parametric resonance in the Einstein Frame. This study is a first step towards building a mathematical relationship between the two frames that will allow further investigations on the production of primordial gravitational waves and black holes in f(R) theories.

Modified (R) theories of gravity have been investigated for quite a long time in the literature as a possible explanation for the inflationary period of the universe. The correspondence to General Relativity with an extra scalar field in the so-called Einstein Frame via a conformal transformation is a major tool in this class of theories. Since the trivial forms of f(R) theory of gravity have already been ruled out from observations, the quest for the "correct" f(R) remains unfinished. We argue that the function (R) can be more soundly justified if one starts from its corresponding potential (4) in the Einstein Frame. To shed some light on the subject, we focus on the phase of preheating in this work. We assume three suitably chosen potentials in the Einstein Frame and a conformal-like coupling Ry in the Jordan Frame, from which we then obtain both a non-trivial (R) and a non-trivial coupling between the fields and that yield Parametric Resonance in the Einstein Frame. We propose that a corresponding so-called vacuum awakening mechanism in the Jordan frame explains the exponential amplification of the field. This study is a first step towards building an important relationship between the two frames that will allow further investigations on the production of primordial gravitational waves and black holes during preheating.

Two different dynamical system formulations are presented for the generic f(R,ϕ,X) family of gravity theories. As illustrative examples, the first and the second formulation is applied to study the phase space of a toy model of the Non-Minimal Derivative Coupling (NMDC) without a potential, and the mixed R^2-Higgs inflation model, respectively. The first dynamical system formulation applied to the toy NMDC model, although able to identify several invariant submanifolds, fails to fully investigate the fixed point structure, as all the fixed points turn out to be non-hyperbolic. We, however, discover an interesting feature that the qualitative dynamics are independent of the coupling strength between the Ricci scalar and the scalar field derivative. The second dynamical system formulation applied to the mixed R^2-Higgs inflation model performs much better, being able to correctly reduce to the individual phase spaces of the R^2 and Higgs inflation separately in special cases, as well as correctly delivering the expected invariant submanifolds and fixed points. For the mixed R^2-Higgs case, illustrative phase portraits are provided for a somewhat better understanding of the dynamics.

Warm inflation is a well-motivated and generalized framework of inflation, describing a coupled inflaton-radiation bath. In this work, we investigate a warm inflation model with a quartic potential and a composite dissipation coefficient Υ(ϕ, T) = C_1 T^3/M_Pl^2 + C_2 T^3/ϕ^2. The two terms in Υ dominate at different scales: the first term governs the early inflationary dynamics at large (CMB) scales, while the second term becomes significant at smaller scales. The model features two distinct stages of inflation: an initial phase where strong dissipation (Q ≫ 1) generates a red-tilted primordial spectrum consistent with CMB observations (from ACT), followed by a second phase producing a blue-tilted spectrum with a significant amplification of power at small scales, leading to primordial black hole formation. We analyze the effects of key parameters – like the duration of each inflationary phase, the slow-roll parameter at the end of the first phase, the dissipation strength at the pivot scale, and the choice of the growth function – on the primordial power spectrum and its spectral index. Additionally, we examine the consistency of the model with the swampland distance conjecture and trans-Planckian conjecture, needed for embedding these models with some UV complete theories. This work highlights the potential of warm inflation with a composite dissipation coefficient to reconcile large-scale CMB measurements with small-scale structure formation.

This series of three lectures was presented at “Escola de Cosmologia e Gravitação" and webcast at – in Portuguese, but slides in English. We will go through a brief review on f(R) theories (in the metric approach) and the usual requirements for successful modifications of General Relativity. Then we will open a large parenthesis to talk about a non-standard approach to Phase Transitions: the Catastrophe Theory. Finally, we will connect the previous lectures to introduce a new Thermodynamic interpretation of f(R) theories.

The macroscopic properties of compact stars in modified gravity theories can be significantly different from the general relativistic (GR) predictions. Within the gravitational context of scalar–tensor theories, with a scalar field ϕ and coupling function Φ(ϕ)=exp[2ϕ/3], we investigate the hydrostatic equilibrium structure of neutron stars for the simple potential V(ϕ)=ωϕ2/2 defined in the Einstein frame (EF). From the scalar field in the EF, we also interpret such theories as f(R) gravity in the corresponding Jordan frame (JF). The mass–radius relations, proper mass, and binding energy are obtained for a polytropic equation of state (EoS) in the JF. Our results reveal that the maximum-mass values increase substantially as ω gets smaller, while the radius and mass decrease in the low-central-density region as we move further away from the pure GR scenario. Furthermore, a cusp is formed when the binding energy is plotted as a function of the proper mass, which indicates the appearance of instability. Specifically, we find that the central-density value where the binding energy is a minimum corresponds precisely to dM/dρcJ=0 on the M(ρcJ)-curve.

In this work we further extend the analysis of $f(R)$ theories of gravity in the metric formalism under the approach of a Thermodynamics analogy, proposed in arXiv:1911.04830v3. Here we assume a double-well inflationary potential in the Einstein frame and obtain a parametric form of $f(R)$ in the corresponding Jordan frame. The whole Thermodynamics picture then follows: an equation of state, binodal and spinodal curves, phase transition, critical quantities (pressure, volume and temperature), entropy jumps, specific-heat divergence (and the corresponding critical exponent) and a butterfly catastrophe.

We investigate the structure of quark stars in the framework of f(R) = R + alpha R-2 gravity using an equation of state for cold quark matter obtained from perturbative QCD, parametrized only by the renormalization scale. We show that a considerably large range of the free parameter alpha, within and even beyond the constraints previously reported in the literature, yield non-negligible modifications in the mass and radius of stars with large central mass densities. Their stability against baryon evaporation is analyzed through the behavior of the associated total binding energies which are slightly affected by the modified gravity term in the regime of high proper (baryon) masses.

We provide the modified TOV equations for the hydrostatic equilibrium of charged compact stars within the metric f ( R ) gravitational background. We adopt the MIT bag model EoS for the dense matter and assume a charge distribution where the electric charge density ρ ch is proportional to the standard energy density ρ . Using the Starobinsky model, we explore the role of the αR 2 term, where α is a free constant and R the Ricci scalar, on the global properties of charged stars such as radius, mass and total charge. We present the dependence of the structure of the star for several values of α and for different values of the constant parameter β ≡ ρ ch / ρ . Remarkably, we find that the radius decreases with respect to its GR value for low central densities, while the opposite occurs in the high-central-density region. The mass measured at the surface always decreases and the maximum-total charge undergoes a substantial increase as the parameter α increases. We also illustrate the variations of the asymptotic mass as a consequence of the electric charge and the extra quadratic term.

We investigate the equilibrium and radial stability of spherically symmetric relativistic stars, considering a polytropic equation of state (EoS), within the framework of f(R,T) gravity with a conservative energy-momentum tensor. Both modified stellar structure equations and Chandrasekhar's pulsation equations are derived for the f(R,T)= R+ h(T) gravity model, where the function h(T) assumes a specific form in order to safeguard the conservation equation for the energy-momentum tensor. The neutron star properties, such as radius, mass, binding energy and oscillation spectrum are studied in detail. Our results show that a cusp — which signals the appearance of instability — is formed when the binding energy is plotted as a function of the compact star proper mass. We find that the squared frequency of the fundamental vibration mode passes through zero at the central-density value corresponding to such a cusp where the binding energy is a minimum.

We examine the static structure configurations and radial stability of compact stars within the context of f(R, T) gravity, with R and T standing for the Ricci scalar and trace of the energy-momentum tensor, respectively. Considering the f(R, T)=R+2β T functional form, with β being a constant, we derive the corresponding hydrostatic equilibrium equation and the modified Chandrasekhar's pulsation equation. The mass-radius relations and radial mode frequencies are obtained for some realistic equations of state. Our results show that the traditional stellar stability criteria, namely, the necessary condition d M/dρc >0 and sufficient condition ω2 >0, still hold in this theory of gravity.

We address the issue of the existence of inequivalent definitions of gravitational mass in $R^{2}$-gravity. We present several definitions of gravitational mass, and discuss the formal relations between them. We then consider the concrete case of a static and spherically symmetric neutron star, and solve numerically the equations of motion for several values of the free parameter of the model. We compare the features of the mass-radius relations obtained for each definition of gravitational mass, and we comment on their dependence on the free parameter. We then argue that $R^{2}$-gravity is a valuable proxy to discuss the existence of inequivalent definitions of gravitational mass in a generic modified gravity theory, and present some comments on the general case.

In the metric approach of f(R) theories of gravity, the fourth-order field equations are often recast as effective Einstein equations in the presence of standard matter and a curvature fluid (which gathers all the extra terms), always in the Jordan frame. In this picture, we investigate the strong gravity regime of the f (R) = R + alpha R-2 model. In particular, we focus on the stability of a compact star composed by a mixture of ordinary matter - described by a polytropic equation of state - and an effective curvature fluid in an otherwise standard Einstein gravity, so that we are able to apply the usual equations that govern the radial adiabatic oscillations of relativistic stars. Our new restriction on the free parameter is alpha less than or similar to 2.4 x 10(8) cm(2) in order to guarantee stellar stability, about 100 times more restrictive than previous results (based on mass-radius relations alone) in the literature.

This paper starts from a toy model for inflation in a class of modified theories of gravity in the metric formalism. Instead of the standard procedure - assuming a non-linear Lagrangian f(R) in the Jordan frame - we start from a simple phi(2) potential in the Einstein frame and investigate the corresponding f (R) in the former picture. The addition of an adhoc Cosmological Constant in the Einstein frame leads to a Thermodynamical interpretation of this physical system, which allows further insight on its (meta)stability and evolution.

We study the tunneling probability of a massive (m(w)) uncharged scalar packet out from a near-extremal, static charged black hole (with mass M and charge Q less than or similar to M). We show that there is indeed a net probability that a massive uncharged particle tunnels out from the black hole so that the final state (with new mass M' equivalent to M - m(w) < Q) does violate the cosmic censorship conjecture. Nevertheless, the typical time for such a black hole to discharge (i.e., to absorb charge-Q from its surroundings and then become neutral) is much smaller than the tunneling time; therefore, the violation is never attained in practice. Even for a completely isolated black hole (should it exist), the standard time dilation near the horizon stretches the typical violation time scale to unobservable values.

We investigate the linear regime of f(R) = R + alpha R-2 gravity for static, spherically symmetric and asymptotically flat configurations of matter. We show that, in vacuum and deep inside the range of the extra scalar degree of freedom, the post-Newtonian parameter gamma is not equal to 1/2, as established in the literature, but it assumes larger values depending on the pressure of the star. We provide an explicit expression for gamma in terms of the mass, of the integrated pressure of the star and of the ratio between the star's radius and the range of the extra degree of freedom. We corroborate our results by providing numerical solutions for the case of a neutron star.

Understanding the influence of dark energy on the formation of structures is currently a major challenge in Cosmology, since it can distinguish otherwise degenerated viable models. In this work we consider the Top-Hat Spherical-Collapse (SC) model with dark energy, which can partially (or totally) cluster, according to a free parameter γ. The lack of energy conservation has to be taken into account accordingly, as we will show. We determine characteristic quantities for the SC model, such as the critical contrast density and radius evolution, with particular emphasis on their dependence on the clustering parameter γ.

Critical overdensity dc is a key concept in estimating the number count of halos for different redshift and halo-mass bins, and therefore, it is a powerful tool to compare cosmological models to observations. There are currently two different prescriptions in the literature for its calculation, namely, the differential-radius and the constant-infinity methods. In this work we show that the latter yields precise results only if we are careful in the definition of the so-called numerical infinities. Although the subtleties we point out are crucial ingredients for an accurate determination of dc both in general relativity and in any other gravity theory, we focus on f(R)-modified gravity models in the metric approach; in particular, we use the so-called large (F = 1/3) and small-field (F = 0) limits. For both of them, we calculate the relative errors (between our method and the others) in the critical density dc, in the comoving number density of halos per logarithmic mass interval n(lnM), and in the number of clusters at a given redshift in a given mass bin N-bin, as functions of the redshift. We have also derived an analytical expression for the density contrast in the linear regime as a function of the collapse redshift z(c) and Omega(m0) for any F.

We construct a state in the Schwarzschild-de Sitter spacetime which is invariant under the action of its group of symmetries. Our state is not defined in the whole Kruskal extension of this spacetime, but rather in a subset of the maximally extended conformal diagram. The construction is based on a careful use of the bulk-to-boundary technique. We will show that our state is Hadamard and that it is not a KMS state, differently from the case of states constructed in spacetimes containing only one event horizon.