We present a complete analytic and semi-analytic study of gravitational collapse and primordial black hole (PBH) formation in the quadratic $f(R)$ model $f(R)=R+αR^2$. We first derive the perturbative expansion around General Relativity (GR), working to first order in the small parameter $α$. For a collapsing flat FLRW dust interior we compute the explicit first-order corrections to the scale factor, the stellar radius, and the horizon formation time. { We then use these results to infer the expected trend in the PBH formation threshold $δ_c$, rather than a direct quantitative determination.
Within this perturbative framework, the quadratic correction modifies the dust collapse dynamics at first order, while the flat radiation-dominated background does not receive a nontrivial correction at the same order. As result, any modification of PBH formation in the radiation era must arise from nonlinear or non-perturbative effects.
The perturbative analysis therefore provides qualitative insight into how curvature corrections influence collapse, particularly in high-curvature regimes.} To access this regime we reformulate the theory in the Einstein frame, where the model becomes GR plus the scalaron field $ϕ$ with the Starobinsky potential. We provide the complete ODE system governing both the cosmological background and the evolution of an overdense closed FLRW patch. This system can be numerically integrated to obtain the critical overdensity $δ_c(k)$ for PBH formation near the end of inflation.
We consider the primordial gravity wave background produced by inflation. We compute the small anisotropy produced by the primordial scalar fluctuations.
We study slow-roll inflation and the generation of primordial fluctuations in $F(T)$ gravity's rainbow. We obtain the second order action for scalar and tensor perturbations and then calculate the primordial power spectrum for them. Thus, after calculating the inflationary observables up to first order in slow-roll approximation, namely the scalar spectral index $n_{s}$ and the tensor-to scalar ratio $r$, we confront the predictions of the model with the current PLANCK and BICEP/Keck data.
In this work we consider a combined theoretical framework comprised by $F(R)$ gravity and a kinetic scalar field. The kinetic energy of the scalar field dominates over its potential for all cosmic times, and the kinetic scalar potential is chosen to be small and non-trivial. In this case, we show that the primordial gravitational wave energy spectrum of vacuum $F(R)$ gravity is significantly enhanced and can be detectable in future interferometers. The kinetic scalar thus affects significantly the inflationary era, since it extends its duration, but also has an overall amplifying effect on the…
We study primordial tensor power-spectra generated during inflation in bimetric gravity. More precisely, we examine a homogeneous expanding spacetime in a minimal bimetric model with an inflaton and calculate tensor perturbations on the homogeneous background under slow-roll approximation. In terms of the mass eigenstates, only the power-spectrum of the massless state remains constant and both the power-spectrum of the massive state and the cross power-spectrum rapidly decay during inflation. The amplitude of the physical power-spectrum is suppressed due to the flavor mixing. All power-spectra…
We investigate primordial gravitational waves and curvature perturbations in de Rham-Gabadadze-Tolley (dRGT) bimetric gravity. We evaluate the power-spectra in the leading order in slow roll. Taking into account the decay of massive graviton, we find that the action up to the second order reduces to the Einstein theory with a non-minimally coupled scalar field, which is simplified to a minimally coupled model by conformal transformation. We also find that the tensor to scalar ratio for large field inflation with power law potential is larger than the general relativity counterpart for any choi…
The observability of primordial cosmic fluctuations does not require a geometric horizon $H^{-1}$, which is exceeded temporarily by the wavelength of fluctuations. The primordial information can be protected against later thermal washout even if all relevant wavelengths remain smaller than $H^{-1}$. This is demonstrated by formulating the equations governing the cosmic fluctuations in a form that is manifestly invariant under conformal field transformations of the metric. Beyond the field equations this holds for the defining equation for the correlation function, as expressed by the inverse o…
We investigate primordial perturbations and non-gaussianities in the Hořava-Lifshitz theory of gravitation. In the UV limit, the scalar perturbation in the Hořava theory is naturally scale-invariant, ignoring the details of the expansion of the Universe. One may thus relax the exponential inflation and the slow-roll conditions for the inflaton field. As a result, it is possible that the primordial non-gaussianities, which are ``slow-roll suppressed'' in the standard scenarios, become large. We calculate the non-gaussianities from the bispectrum of the perturbation and find that the equilateral…
The gravitational potential of a gas of initially randomly distributed primordial black holes (PBH) can induce a stochastic gravitational-wave (GW) background through second-order gravitational effects. This GW background can be abundantly generated in a cosmic era dominated by ultralight primordial black holes, with masses $m_\mathrm{PBH}<10^{9}\mathrm{g}$. In this work, we consider $f(R)$ gravity as the underlying gravitational theory and we study its effect at the level of the gravitational potential of Poisson distributed primordial black holes. After a general analysis, we focus on the $R…
In this work we shall consider the effects of a geometrically generated post-inflationary era on the energy spectrum of the primordial gravitational waves. Specifically, we shall consider a post-inflationary constant equation of state era, generated by the synergistic effect of $f(R)$ gravity and of radiation and matter perfect fluids. Two cases of interest shall be studied, one with equation of state parameter $w=-1/3$, in which case the Universe neither accelerates nor decelerates, and one with $w=0$ so an early matter domination era. For the evaluation of the inflationary observational indi…
We calculate tensor bispectrum in a theory of massive tensor gravitons which predicts blue-tilted and largely amplified primordial gravitational waves. We find that a new 3-point interaction can produce a larger tensor bispectrum than the conventional one from general relativity. The bispectrum peaks at the squeezed configuration and its slope towards the squeezed limit is determined by the graviton mass. This squeezed tensor bispectrum may be observed as the quadrupolar modulation of the tensor power spectrum by interferometers.
We study a theory of massive tensor gravitons which predicts blue-tilted and largely amplified primordial gravitational waves. After inflation, while their mass is significant until it diminishes to a small value, gravitons are diluted as non-relativistic matter and hence their amplitude can be substantially amplified compared to the massless gravitons which decay as radiation. We show that such gravitational waves can be detected by interferometer experiments, even if their signal is not observed on the CMB scales.
One-loop renormalised quantum effective action for gravity contains quadratic in curvature terms. We have found an approximate analytic black hole solution in quadratic gravity by keeping only the radial spherically symmetric fluctuations and dimensionally reducing the 4-dimensional (4D) theory down to the 2-dimensional (2D) dilaton gravity with a potential. The solution reduces to the Schwarzschild black hole in the limit of Einstein's gravity, but otherwise admits non-negative Arnowitt-Deser-Misner (ADM) and positive quasi-local Misner-Sharp masses that can differ significantly. We then stud…
In the context of modified teleparallel gravity, we study the generation of primordial density fluctuations in a general scalar-torsion theory whose Lagrangian density is an arbitrary function $f(T,ϕ)$ of the torsion scalar $T$ and a scalar field $ϕ$, plus the kinetic term of this latter. It is well known that generic modifications of teleparallel gravity are not invariant under six-parameter local Lorentz transformations. In order to restore the local Lorentz symmetry, we have incorporated six additional degrees of freedom in the form of Goldstone modes of the symmetry breaking through a Lore…
In this work we examine the effects of a pre-inflationary de Sitter bounce on the energy spectrum of the primordial gravitational waves. Specifically we assume that the Universe is described by several evolution patches, starting with a de Sitter pre-inflationary bounce which is followed by an quasi-de Sitter slow-roll inflationary era, followed by a constant equation of state parameter abnormal reheating era, which is followed by the radiation and matter domination eras and the late-time acceleration eras. The bounce and the inflationary era can be realized by vacuum $f(R)$ gravity and the ab…
We examine the effect of the stress tensor of a quantum matter field, such as the electromagnetic field, on the spectrum of primordial gravity waves expected in inflationary cosmology. We find that the net effect is a small reduction in the power spectrum, especially at higher frequencies, but which has a different form from that described by the usual spectral index. Thus this effect has a characteristic signature, and is in principle observable. The net effect is a sum of two contributions, one of which is due to quantum fluctuations of the matter field stress tensor. The other is a quantum …
The Eddington-inspired-Born-Infeld (EiBI) model is reformulated within the mimetic approach. In the presence of a mimetic field, the model contains non-trivial vacuum solutions which could be free of spacetime singularity because of the Born-Infeld nature of the theory. We study a realistic primordial vacuum universe and prove the existence of regular solutions, such as primordial inflationary solutions of de Sitter type or bouncing solutions. Besides, the linear instabilities present in the EiBI model are found to be avoidable for some interesting bouncing solutions in which the physical metr…
For primordial perturbations, deviations from Gaussian statistics on the tail of the probability distribution can be associated with non-perturbative effects of inflation. In this paper, we present some particular examples in which the tail of the distribution becomes highly non-Gaussian although the statistics remains almost Gaussian in the perturbative regime. We begin with an extension of the ultra-slow-roll inflation that incorporates a transition process, where the inflaton climbs up a tiny potential step at the end of the non-attractor stage before it converges to the slow-roll attractor…
In this study, we investigate the consequence of the constant-roll condition and examine the role of $f(Q,T)$ gravity in the cosmological inflation process. We analyze the inflationary scenario by calculating modified Friedmann equations, and giving an alternative technique that enables relating modified slow-roll parameters to the constant roll parameter $β$. Considering both chaotic and hilltop models, we calculate the spectral index and the tensor-to-scalar ratio and compare their compatibility with Planck's data for different choices of the constant roll parameter $β$. We examine the evolu…
In this work we shall exhaustively study the effects of modified gravity on the energy spectrum of the primordial gravitational waves background. S. Weinberg has also produced significant works related to the primordial gravitational waves with the most important one being the effects of neutrinos on primordial gravitational waves. With this sort review, our main aim is to gather all the necessary information for studying the effects of modified gravity on primordial gravitational waves in a concrete and quantitative way and in a single paper. After reviewing all the necessary techniques for e…
We address the question of how one can modify the inflationary tensor spectrum without changing at all the successful predictions on the curvature perturbation. We show that this is indeed possible, and determine the two quadratic curvature corrections that are free from instabilities and affect only the tensor sector at the level of linear cosmological perturbations. Both of the two corrections can reduce the tensor amplitude, though one of them generates large non-Gaussianity of the curvature perturbation. It turns out that the other one corresponds to so-called Lorentz-violating Weyl gravit…
We investigate the scalar and the tensor perturbations of the $φ^2$ inflation model in the strong-gravity limit of Eddington-inspired Born-Infeld (EiBI) theory. In order to consider the strong EiBI-gravity effect, we take the value of $κ$ large, where $κ$ is the EiBI theory parameter. The energy density of the Universe at the early stage is very high, and the Universe is in a strong-gravity regime. Therefore, the perturbation feature is not altered from what was investigated earlier. At the attractor inflationary stage, however, the feature is changed in the strong EiBI-gravity limit. The corr…
In order to describe inflation in general relativity, scalar fields must inevitably be used, with all the setbacks of that description. On the other hand, $f(R)$ gravity and other modified gravity theories seem to provide a unified description of early and late-time dynamics without resorting to scalar or phantom theories. The question is, can modified gravity affect directly the mysterious radiation domination era? Addressing this question is the focus in this work, and we shall consider the case for which in the early stages of the radiation domination era, namely during the reheating era, t…
In this work we shall develop a quantitative approach for extracting predictions on the primordial gravitational waves energy spectrum for $f(R)$ gravity. We shall consider two distinct models which yield different phenomenology, one pure $f(R)$ gravity model and one Chern-Simons corrected potential-less $k$-essence $f(R)$ gravity model in the presence of radiation and non-relativistic perfect matter fluids. The two $f(R)$ gravity models were carefully chosen in order for them to describe in a unified way inflation and the dark energy era, in both cases viable and compatible with the latest Pl…
We investigate the co-evolution of dark matter (DM) density perturbation and metric perturbation in the WIMP paradigm. Instead of adopting the conventional assumption that DM starts out in thermal equilibrium, we propose a simple phase of DM production for the WIMP paradigm and extend our analysis to this phase. Being free from the envelop of thermal equilibrium, an amplified perturbative resonance between DM density perturbation and scalar modes of metric perturbation takes place during the DM production phase, and consequently results in a suppression of the tensor-to-scalar ratio of metric …
In this work we study non-singular{ bounce cosmology} in the context of the Lagrange multiplier generalized $F(R)$ gravity theory of gravity. We specify our study by using a specific variant form of the well known matter{ bounce cosmology}, with scale factor $a(t) = \left(a_0t^2 + 1 \right)^n$, and we demonstrate that for $n < 1/2$, the primordial curvature perturbations are generated deeply in the contraction era. Particularly, we show explicitly that the perturbation modes exit the horizon at a large negative time during the contraction era, which in turn makes the ``low-curvature'' regime, …
Recent measurements of the cosmic microwave background (CMB) radiation have shown an apparent tension with the present value of the Hubble parameter inferred from local observations of supernovae, which look closer, i.e. brighter, than what is expected in a homogeneous model with a value of $H_0$ equal to the one estimated from CMB observations. We examine the possibility that such a discrepancy is the consequence of the presence of a local inhomogeneity seeded by primordial curvature perturbations, finding that a negative peak of the order of less than two standard deviations could allow to f…
The tests of parity symmetry in the gravitational interaction is an attractive issue in the gravitational-wave astronomy. In the general theories of gravity with parity violation, one of the fundamental results is that the primordial gravitational waves (PGWs) produced during the slow-roll inflation is circularly polarized. In this article, we investigate the polarization of PGWs in the recently proposed ghost-free parity-violating gravity, which generalizes the Chern-Simons gravity by including the higher derivatives of the coupling scalar field. For this purpose, we first construct the appro…
We consider a homogeneous and isotropic Universe, described by the minisuperspace Lagrangian with the scale factor as a generalized coordinate. We show that the energy of a closed Universe is zero. We apply the uncertainty principle to this Lagrangian and propose that the quantum uncertainty of the scale factor causes the primordial fluctuations of the matter density. We use the dynamics of the early Universe in the Einstein$-$Cartan theory of gravity with spin and torsion, which eliminates the big-bang singularity and replaces it with a nonsingular bounce. Quantum particle production in highl…
We investigate the evolution of cosmological perturbations during de Sitter inflation in the Einstein-Chern-Simons-Weyl gravity. Primordial massive gravitational waves are composed of one scalar, two vector and four tensor circularly polarized modes. We show that the vector power spectrum decays quickly like a transversely massive vector in the superhorizon limit $z\to 0$. In this limit, the power spectrum coming from massive tensor modes decays quickly, leading to the conventional tensor power spectrum. Also, we find that in the limit of $m^2 \to 0$ (keeping the Weyl-squared term only), the v…
We investigate the production of primordial black holes (PBHs) and scalar-induced gravitational waves (GWs) for cosmological models in the Horndeski theory of gravity. The cosmological models of our interest incorporate the derivative self-interaction of the scalar field and the kinetic coupling between the scalar field and gravity. We show that the scalar power spectrum of the primordial fluctuations can be enhanced on small scales due to these additional interactions. Thus, the formation of PBHs and the production of induced GWs are feasible for our model. Parameterizing the scalar power spe…
Nonlocal gravity has been shown to provide a phenomenologically viable infrared modification of GR. A natural question is whether the required nonlocality can emerge from perturbative quantum loop corrections due to light particles. We show that this is not the case. For the value of the mass scale of the non-local models required by cosmology, the perturbative form factors obtained from the loop corrections, in the present cosmological epoch, are in the regime where they are local. The mechanism behind the generation of the required nonlocality must be more complex, possibly related to strong…
We study the superposition of primordial massive particles and compute the associated decoherence time scale in the radiation dominated universe. We observe that for lighter primordial particles with masses up to $10^7\,\rm{kg}$, the corresponding decoherence time scale is significantly larger than the age of the observable universe, demonstrating that a primordial particle would persist in a pure quantum state, with its wavefunction spreading freely. For heavier particles, they can still be in a quantum state while their position uncertainties are limited by the wavelength of background photo…
We develop an exact formalism for the computation of the abundance of primordial black holes (PBHs) in the presence of local non-gaussianity (NG) in the curvature perturbation field. For the first time, we include NG going beyond the widely used quadratic and cubic approximations, and consider a completely generic functional form. Adopting threshold statistics of the compaction function, we address the computation of the abundance both for narrow and broad power spectra. While our formulas are generic, we discuss explicit examples of phenomenological relevance considering the physics case of t…
It is believed that primordial black holes (PBHs), if they exist, can serve as a powerful tool to probe the early stage of the cosmic history. Essentially, in the radiation dominated universe, PBHs could form by the gravitational collapse of overdense primordial perturbations produced during inflation. In this picture, one important ingredient is the threshold of density contrast, which defines the onset of PBH formation. In the literature, most of the estimations of threshold, no matter numerically or analytically, are implemented in the framework of general relativity (GR). In this paper, by…
We study fluctuations and correlations between spacial regions, generated by the primordial quantum gravitational phase of the universe. We do so by a numerical evaluation of Lorentzian amplitudes in Loop Quantum Gravity, in a non-semiclassical regime. We find that the expectation value of the quantum state of the geometry emerging from the early quantum phase of the universe is a homogeneous space but fluctuations are very large and correlations are strong, although not maximal. In particular, this suggests that early quantum gravitational effects could be sufficient to solve the cosmological…
It has been recently suggested that small mass black holes (BHs) may become unstable due to quantum-gravitational effects and eventually decay, producing radiation, on a timescale shorter than the Hawking evaporation time. We argue that the existence of a population of low-mass Primordial Black Holes (PBHs) acting as a fraction of the Universe dark matter component can be used to test proposed models of quantum decay of BHs via their effect on galaxy number counts. We study what constraints future galaxy clustering measurements can set on quantum-gravity parameters governing the BH lifetime an…
