The recent observations of gravitational waves and supermassive black holes can be considered as the main probes of General Relativity (GR) in its fundamental aspects which are: 1) the propagation of space-time perturbations, 2) the existence of singularities. Despite of these undeniable successes, several shortcomings affect GR because the whole phenomenology cannot be addressed in the framework of the Einstein picture. The theory is missing at ultraviolet scales because of the lack of a self-consistent theory of Quantum Gravity, and at infrared scales because it is not capable of encompassing clustering phenomena related to large-scale structure and the observed accelerated expansion of the cosmic fluid. These are generically dubbed as dark matter and dark energy but, up to now, no particle counterpart has been discovered to address them at fundamental level.

In this perspective, extensions and modifications of GR are considered as a reliable way out of the above problems assuming that gravitational field has not been completely explored.

These extensions come from effective theories on curved spacetimes or as alternative formulations like teleparallel gravity and its related models.

A main role to test theories is played by cosmology because phenomena connected to the so called dark side can substantially affect structure formation and cosmic dynamics. Their equivalent geometric explanations could be a major step towards a comprehensive theory of gravity at all scales.

In general, dynamical characteristics of gravitational waves can be the features probing a given theory of gravity. Specifically, speed, damping, dispersion, and oscillations of gravitational waves could be used to fix and reconstruct interactions into gravitational Lagrangian and then be a sort of roadmap inside the wide forest of competing theories of gravity. For example, further gravitational polarization modes, besides the two standard ones of GR, emerge when further degrees of freedom are considered into the theory. In general, as soon as modifications or extensions of GR are taken into account, scalar modes are present into dynamics.

Motivated by these considerations, it is possible to investigate the propagation of gravitational waves in various gravitational models. For example, in F(T) extended teleparallel gravity, in domain wall models, in scalar tensor and F(R) gravity theories, in Chern-Simons Axion Einstein gravity and in several media as in strong magnetic fields or in viscous fluids.

Furthermore, the behavior of gravitational waves can be used to test past and future singularities and then contributes in their classification. Another important issue is connected to thermal effects emerging during the cosmic evolution.

“We point out that the contribution of thermal radiation can heavily affect the dynamics of gravitational waves giving enhancement or dissipation effects both at quantum and classical level. These effects are considered both in General Relativity and in modified theories like F(R) gravity.”— told Odintsov, lead author of the study

Now, Odintsov and colleagues investigated how thermal effects on various cosmological backgrounds affect the propagation of gravitational waves. In particular, they want to take into account such effects in GR, in modified theories of gravity and in presence of future singularities. Lets have a closer look on their findings:

### (A) Thermal effects in Cosmology

At first, they considered thermal effects in cosmology and showed that, when hubble parameter (H) is large, the temperature of the universe becomes large and they may expect the generation of thermal radiation as in the case of the Hawking radiation. The Hawking temperature T is proportional to the inverse of the radius rH of the apparent horizon and the radius rH is proportional to the inverse of the Hubble rate H. Therefore, the temperature T is proportional to the Hubble rate H. In simple terms, at cosmological scales, thermal effects emerge with respect to the Hubble radius and then they can strongly affect the cosmic evolution, in particular, at early epochs or nearby singularities. In other words, thermal effects can dynamically affect the cosmological background and then the evolution of phenomena on it.

### (B) Thermal effects in future cosmological singularities

It is well known that in the cosmic future several kinds of space-time singularity can happen. Such singularities have been classified as follows:

**Type I,**which is also called as “Big rip”, in this type of singularity, scale factor a(t), the total effective pressure peff and the total effective energy density ρeff diverges strongly.**Type II**, which is also called as “sudden/pressure singularity” and it is milder than the Big Rip scenario. Here, only the total effective pressure diverges, and the total effective energy density & the scale factor remain finite.**Type III,**in which both the total effective pressure and the total effective energy density diverges, but the scale factor remains finite. So, this type of singularity is milder than Type I (Big Rip) but stronger than Type II (sudden).**Type IV**, which is the mildest from a phenomenological point of view.

According to authors, the thermal radiation usually makes the singularities less singular, that is, the Big Rip (Type I) singularity or the Type III singularity transit to the Type II singularity.

### (C) Thermal effects in Scalar-field cosmologies

In this sub-section, they showed that the thermal radiation assumes a key role in determining the evolution of the scalar field and then of the universe.

### Gravitational waves in a dynamical background

In this section, they considered the propagation of gravitational waves in a dynamical cosmological background where thermal contributions are present. They showed mathematically, how these terms affect the evolution of gravitational waves. First, they reviewed the propagation of gravitational waves in a general medium. Gravitational waves are derived as perturbations of Einstein field equations. In the Einstein equations, not only the curvature but also the energy-momentum tensor depends on the metric and therefore the variation of the energy-momentum tensor gives a non-trivial contribution to the propagation of gravitational waves.

“We showed thermal radiation contribution play a key role in the evolution of the Gravitational waves”

### (D) Thermal corrections in quantum matter

In this sub-section, they considered a real scalar field φ as the source of matter. They deal with the scalar field as a quantum field at finite temperature. In the case of high temperature or in the massless case, the scalar field plays the role of radiation. On the other hand, in the limit where the temperature is vanishing but the density is finite, they obtained the dust, which can be considered as cold dark matter.

### Enhancement and dissipation of gravitational waves with thermal effects

In this section, they investigated the propagation of gravitational massless spin-two modes, and showed that thermal contribution affects the evolution of the gravitational wave amplitude.

### (A) The behavior of gravitational waves near the singularities

In this sub-section, they studied the behavior of gravitational waves near the Type II singularity and the Big Rip (the Type I) singularity. They showed that the enhancement of the gravitational wave occurs near the Type II singularity but it could not occur near the Big Rip (Type I) singularity.

### (B) Gravitational waves in the early universe

In this subsection, they showed that if there is no thermal effect, that is α = 0, there is no enhancement or dissipation of the gravitational wave, but if we include the thermal effect, enhancement or dissipation occur.

### The propagation of scalar modes with thermal effects

In this section, they developed similar discussion in a generalized context where scalar modes are included. In case of the Type II singularity, just before the singularity, the scalar field oscillates very rapidly. In the case of Big Rip (the Type I) singularity, the amplitude of scalar field increases or decreases very rapidly. While, in the case of bouncing universe, they found that, scalar field (ω(η)) vanishes at t = ±t0 and therefore mass, m² diverges. If m² >, which may depend on the parameters near t = ±t0, the scalar field oscillates very rapidly and if m² < 0, the amplitude of the scalar field increases or decreases very rapidly.

They also mentioned that expanding universe can be realized by the perfect fluid. The perfect fluid also generates scalar waves, whose propagating velocity is known as the sound speed Cs, which is given by:

Therefore if we find the equation of state (EoS), we can find the speed. Thus, in case of the Type II singularity, they found,

Therefore C²s diverges to negative infinity. Because c²s is negative, the amplitude of the perfect fluid wave rapidly decreases or increases without oscillation. But, this behavior is much different from that in the scalar field, where the scalar field oscillates very rapidly.

In case of the Big Rip (the Type I) singularity, they found

which is finite but because C²s is negative, the amplitude of the perfect fluid wave decreases or increases exponentially without oscillation. Even in case of the scalar field, the amplitude of the scalar field increases or decreases very rapidly but m² diverges in the case of scalar field, the increase or decrease is much more rapid.

In case of the bouncing universe, the sound speed is given by,

Which diverges at t=t0 as the m² of the scalar field. Then, the propagation of the perfect fluid wave might be similar to that in the scalar field.

In case of the inflation, H ∼ H0, they found

which is finite but could be negative as long as ηH is small enough. Therefore the perfect scalar wave does not propagate although the scalar wave can propagate.

All the results given above, tell us that the propagation of scalar modes depends on the mechanism which generates the expansion of the universe and, since thermal effects affect the effective mass, they have to be considered in the evolution. And last,

### (A) Scalar waves in modified gravity vs compressional waves of cosmic fluid

In this sub-section they showed how we can distinguish scalar modes with respect to the compressional waves of a perfect fluid? They showed that a perfect fluid, where the EoS parameter is w < 0, does not generate a compressional wave. Therefore if we find massive scalar waves, it could be an evidence for modified gravity. However, even in modified gravity, there are various models. In the case of F(R) gravity, however, the coupling of massive scalar with matter is universal, that is, it does not depend on the kind of matter because the coupling appears by the rescaling of metric. This structure is rather characteristic of F(R) gravity and it may give some clue to observationally discriminate F(R) gravity with respect to other modified gravity models.

“In the case of F(R) gravity, the effects of scalar modes or compressional fluids strictly depend on the “representation” of the theory in the Einstein or the Jordan frames. This could constitute an important feature in order to distinguish the true physical frame by the observations”— wrote authors of the study

Finally, they concluded that dynamics related to the above discussion could be observationally tested by interferometers. In fact, at suitable sensitivities, it seems realistic to disentangle GR contributions with respect to other contributions in the stochastic background of gravitational waves. In a future study, they will develop this topic in detail.

**Reference***: Salvatore Capozziello, Shin’ichi Nojiri, Sergei D. Odintsov, “Thermal effects and scalar modes in cosmological gravitational waves”, pp. 1-20, Arxiv, 2021. https://arxiv.org/abs/2104.10936*

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