We investigate the observational viability of non-minimally coupled scalar-Einstein-Gauss-Bonnet (GB) gravity, during inflation and post-inflationary reheating dynamics, from the perspective of the latest ACT-DR6 combined with the Planck 2018 and BAO data. It turns out that the ACT result considerably affects the inflationary e-fold number compared to the case where only Planck 2018 data is taken into account. The viable parameter spaces corresponding to the inflationary ACT-DR6+Planck18+BAO substantially influence the reheating phenomenology via the reheating equation of state ( <math altimg='si9.svg'><msub><mi>w</mi><mtext>eff</mtext></msub></math>) and the reheating temperature. In particular, the ACT-DR6+Planck18+BAO data seems to disfavor <math altimg='si18.svg'><mrow><msub><mi>w</mi><mtext>eff</mtext></msub><mo linebreak='goodbreak'>&lt;</mo><mn>1</mn><mo linebreak='goodbreak'>/</mo><mn>3</mn></mrow></math> during the reheating stage, which is unlike to that of only Planck 2018 case. These reveal how the ACT-DR6 data hits the early universe phenomenology from inflation to reheating in the context of higher curvature like scalar-Einstein-GB theory of gravity.

Spain

Sergei D. Odintsov

India

Tanmoy Paul

ACT inflation and its influence on reheating era in Einstein-Gauss-Bonnet gravity

Generalized entropy, that has been recently proposed, puts all the known and apparently different entropies like The Tsallis, the Rényi, the Barrow, the Kaniadakis, the Sharma-Mittal and the loop quantum gravity entropy within a single umbrella. However, the microscopic origin of such generalized entropy as well as its relation to thermodynamic system(s) is not clear. In the present work, we will provide a microscopic thermodynamic explanation of generalized entropy(ies) from canonical and grand-canonical ensembles. It turns out that in both the canonical and grand-canonical descriptions, the generalized entropies can be interpreted as the statistical ensemble average of a series of microscopic quantity(ies) given by various powers of <math><msup><mrow><mo>(</mo><mo>−</mo><mi>k</mi><mi>ln</mi><mo>⁡</mo><mi>ρ</mi><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup></math> (with n being a positive integer and ρ symbolizes the phase space density of the respective ensemble), along with a term representing the fluctuation of Hamiltonian and number of particles of the system under consideration (in case of canonical ensemble, the fluctuation on the particle number vanishes).

Japan

Shin'ichi Nojiri

Microscopic interpretation of generalized entropy

In the formalism of generalized holographic dark energy, the infrared cut-off <math><msub><mrow><mi>L</mi></mrow><mrow><mi>IR</mi></mrow></msub></math> is generalized to the form, <math><msub><mrow><mi>L</mi></mrow><mrow><mi>IR</mi></mrow></msub><mo>=</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>IR</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>,</mo><msub><mrow><mover><mrow><mi>L</mi></mrow><mrow><mo>˙</mo></mrow></mover></mrow><mrow><mi>p</mi></mrow></msub><mo>,</mo><msub><mrow><mover><mrow><mi>L</mi></mrow><mrow><mo>¨</mo></mrow></mover></mrow><mrow><mi>p</mi></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>,</mo><msub><mrow><mover><mrow><mi>L</mi></mrow><mrow><mo>˙</mo></mrow></mover></mrow><mrow><mi>f</mi></mrow></msub><mo>,</mo><msub><mrow><mover><mrow><mi>L</mi></mrow><mrow><mo>¨</mo></mrow></mover></mrow><mrow><mi>f</mi></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>a</mi><mo>,</mo><mi>H</mi><mo>,</mo><mover><mrow><mi>H</mi></mrow><mrow><mo>˙</mo></mrow></mover><mo>,</mo><mover><mrow><mi>H</mi></mrow><mrow><mo>¨</mo></mrow></mover><mo>,</mo><mo>⋯</mo><mo>)</mo></mrow></math>, where <math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math> and <math><msub><mrow><mi>L</mi></mrow><mrow><mi>f</mi></mrow></msub></math> are the particle horizon and the future horizon, respectively (moreover, a is the scale factor and H is the Hubble parameter of the universe). Based on such formalism, we establish a holographic realization of constant roll inflation during the early universe, where the corresponding cut-off depends on the Hubble parameter and its derivatives (up to the second order). The viability of this holographic constant roll inflation with respect to the Planck data in turn puts a certain bound on the infrared cut-off at the time of horizon crossing. Such holographic correspondence of constant roll inflation is extended to the scenario where the infrared cut-off is corrected by the ultraviolet one, which may originate due to quantum effects. Besides the mere inflation, we further propose the holographic realization of an <math><mi>u</mi><mi>n</mi><mi>i</mi><mi>f</mi><mi>i</mi><mi>e</mi><mi>d</mi></math> cosmic scenario from constant roll inflation (at the early time) to the dark energy era (at the late time) with an intermediate radiation dominated era followed by a Kamionkowski like reheating stage. In such a unified holographic scenario, the inflationary quantities (like the scalar spectral index and the tensor-to-scalar ratio) and the dark energy quantities (like the dark energy EoS parameter and the present Hubble rate) prove to be simultaneously compatible with observable constraints for suitable ranges of the infrared cut-off and the other model parameters. Moreover the curvature perturbations at super-Hubble scale prove to be a constant (with time) during the entire cosmic era, which in turn ensures the stability of the model under consideration.

Holographic realization of constant roll inflation and dark energy: An unified scenario

In the realm of the Bekenstein-Hawking entropy, the thermodynamics of apparent horizon bridges with the usual FLRW (Friedmann-Lemaître-Robertson-Walker) equation only for a special case where the matter field is given by a perfect fluid having equation of state (EoS) parameter =−1, i.e. <math><mi>p</mi><mo>=</mo><mo>−</mo><mi>ρ</mi></math> with ρ and p representing the energy density and the pressure of the fluid, respectively. To include the case <math><mi>p</mi><mo>≠</mo><mo>−</mo><mi>ρ</mi></math>, we consider the modification of the Bekenstein-Hawking entropy in the present work. In particular, we develop an entropy function that leads to the usual FLRW equations, for a <math><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>l</mi></math> EoS of the matter fluid given by <math><mi>p</mi><mo>=</mo><mi>w</mi><mi>ρ</mi></math>, directly from the thermodynamics of the apparent horizon. The newly developed entropy acquires a correction over the Bekenstein-Hawking entropy and differs from the known entropies like the Tsallis, Rényi, Barrow, Sharma-Mittal, Kaniadakis, and Loop Quantum Gravity entropies proposed so far. Based on this finding, we examine how the Friedmann equations of the apparent horizon cosmology are accordingly modified if one starts with a general entropy depending on the Bekenstein-Hawking entropy. This results in some interesting cosmological consequences during the early and late stages of the universe.

Modified cosmology from the thermodynamics of apparent horizon

We propose a new four-parameter entropy function that generalizes the Tsallis, Rényi, Barrow, Sharma-Mittal, Kaniadakis and Loop Quantum Gravity entropies for suitable limits of the parameters. Consequently, we address the early and late universe cosmology corresponding to the proposed four-parameter entropy function. As a result, it turns out that the entropic cosmology from the generalized entropy function can unify the early inflation to the late dark energy era of the universe. In such a unified scenario, we find that – (1) the inflation era is described by a quasi de-Sitter evolution of the Hubble parameter, which has an exit at around 58 e-folding number, (2) the inflationary observable quantities like the spectral index for primordial scalar perturbation and the tensor-to-scalar ratio are simultaneously compatible with the recent Planck data, and (3) regarding the late time cosmology, the dark energy EoS parameter is found to be consistent with the Planck result for the same values of the entropy parameters that lead to the viable inflation during the early universe. Furthermore, we show that the entropic cosmology from the proposed entropy function is equivalent to holographic cosmology, where the respective holographic cut-offs are determined in terms of either particle horizon and its derivative or future horizon and its derivative.

Early and late universe holographic cosmology from a new generalized entropy

The holographic cut-off, in the formalism of generalized holographic dark energy (HDE), is generalized to depend on <math><msub><mrow><mi>L</mi></mrow><mrow><mi>IR</mi></mrow></msub><mo>=</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>IR</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>,</mo><msub><mrow><mover><mrow><mi>L</mi></mrow><mrow><mo>˙</mo></mrow></mover></mrow><mrow><mi>p</mi></mrow></msub><mo>,</mo><msub><mrow><mover><mrow><mi>L</mi></mrow><mrow><mo>¨</mo></mrow></mover></mrow><mrow><mi>p</mi></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>,</mo><msub><mrow><mover><mrow><mi>L</mi></mrow><mrow><mo>˙</mo></mrow></mover></mrow><mrow><mi>f</mi></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>a</mi><mo>)</mo></mrow></math>, where <math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math> and <math><msub><mrow><mi>L</mi></mrow><mrow><mi>f</mi></mrow></msub></math> are the particle horizon and future horizon respectively, and a is the scale factor of the universe. Based on such formalism, we showed that the Barrow entropic dark energy (DE) model is equivalent to the generalized HDE where the respective holographic cut-off is determined by two ways – (1) in terms of particle horizon and its derivative and (2) in terms of future horizon and its derivative. Interestingly, such cut-off turns out to depend up-to first order derivative of <math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math> or <math><msub><mrow><mi>L</mi></mrow><mrow><mi>f</mi></mrow></msub></math> respectively. Such equivalence between the Barrow entropic dark energy and the generalized HDE is extended to the scenario where the exponent of the Barrow entropy allows to vary with the cosmological expansion of the universe. In both the cases (whether the Barrow exponent is a constant or varies with the cosmological evolution), we determine effective equation of state (EoS) parameter from the generalized holographic point of view, which, by comparing with the Barrow DE EoS parameter, further ensures the equivalence between the Barrow entropic dark energy and the generalized HDE.

Russia

Barrow entropic dark energy: A member of generalized holographic dark energy family

We apply the bottom-up reconstruction technique in the context of bouncing cosmology in F(R) gravity, where the starting point is a suitable ansatz of observable quantity (like spectral index or tensor to scalar ratio) rather than a priori form of Hubble parameter. In inflationary scenario, the slow roll conditions are assumed to hold true, and thus the observational indices have general expressions in terms of the slow-roll parameters, as for example the tensor to scalar ratio in F(R) inflation can be expressed as <math><mi>r</mi><mo>=</mo><mn>48</mn><msubsup><mrow><mi>ϵ</mi></mrow><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msubsup></math> with <math><msub><mrow><mi>ϵ</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>=</mo><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msubsup><mrow><mi>H</mi></mrow><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac><mfrac><mrow><mi>d</mi><msub><mrow><mi>H</mi></mrow><mrow><mi>F</mi></mrow></msub></mrow><mrow><mi>d</mi><msub><mrow><mi>t</mi></mrow><mrow><mi>F</mi></mrow></msub></mrow></mfrac></math> and <math><msub><mrow><mi>H</mi></mrow><mrow><mi>F</mi></mrow></msub></math>, <math><msub><mrow><mi>t</mi></mrow><mrow><mi>F</mi></mrow></msub></math> are the Hubble parameter, cosmic time respectively. However, in the bouncing cosmology (say in F(R) gravity theory), the slow-roll conditions are not satisfied, in general, and thus the observable quantities do not have any general expressions that will hold true irrespective of the form of F(R). Thus, in order to apply the bottom-up reconstruction procedure in F(R) bouncing model, we use the conformal correspondence between F(R) and scalar-tensor model where the conformal factor in the present context is chosen in a way such that it leads to an inflationary scenario in the scalar-tensor frame. Due to the reason that the scalar and tensor perturbations remain invariant under conformal transformation, the observable viability of the scalar-tensor inflationary model confirms the viability of the conformally connected F(R) bouncing model. Motivated by these arguments, here we construct a viable non-singular bounce in F(R) gravity directly from the observable indices of the corresponding scalar-tensor inflationary model.

S.D. Odintsov

Greece

V.K. Oikonomou

Bottom-up reconstruction of non-singular bounce in F(R) gravity from observational indices

In the context of a ghost free <math><mi>f</mi><mo>(</mo><mi>R</mi><mo>,</mo><mi>G</mi><mo>)</mo></math> model, an extended matter bounce scenario is studied where the form of the scale factor is given by <math><mi>a</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><msup><mrow><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow></msub><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup></math>. The ghost free character of the model is ensured by the presence of a Lagrange multiplier, as developed in [1]. The conditions under which, in this model, the speed of gravitational waves becomes equal to the speed of light (equal to one, in natural units), thus becoming compatible with the striking event GW170817, is investigated. It is shown that this happens for a class of Gauss-Bonnet (GB) coupling functions (<math><mi>h</mi><mo>(</mo><mi>t</mi><mo>)</mo></math>) which satisfies a constraint equation of the form <math><mover><mrow><mi>h</mi></mrow><mrow><mo>¨</mo></mrow></mover><mo>=</mo><mover><mrow><mi>h</mi></mrow><mrow><mo>˙</mo></mrow></mover><mi>H</mi></math>, with H the Hubble parameter. This constraint is then imposed on the ghost free <math><mi>f</mi><mo>(</mo><mi>R</mi><mo>,</mo><mi>G</mi><mo>)</mo></math> gravity theory to be consistent with the GW170817 event, subsequently, the corresponding non-singular bouncing cosmology with the aforementioned scale factor is extensively studied. The forms of the coupling function and Lagrange multiplier in the “low curvature limit” of the theory are reconstructed, yielding a viable approximation for <math><mi>n</mi><mo>&lt;</mo><mn>1</mn><mo>/</mo><mn>2</mn></math>. Correspondingly, by solving the cosmological perturbation equation, the main observable quantities, namely the spectral index, tensor to scalar ratio, and the running index are determined and confronted with the latest Planck 2018 data. Consistency with the data is proven for those parametric regimes that which correspond to <math><mi>n</mi><mo>&lt;</mo><mn>1</mn><mo>/</mo><mn>2</mn></math>. This makes the low curvature approximation a viable one for calculating the scalar and tensor power spectra.

E. Elizalde

Extended matter bounce scenario in ghost free gravity compatible with GW170817

Matter and quasi-matter bounce scenarios are studied for an F(R) gravity model with holonomy corrections and a Lagrange multiplier, with a scale factor $$a(t) = \left( a_0t^2 + 1 \right) ^n$$ <math><mrow><mi>a</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><msup><mfenced><msub><mi>a</mi><mn>0</mn></msub><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mfenced><mi>n</mi></msup></mrow></math> , where the Hubble parameter squared has a linear and a quadratic dependence on the effective energy density. Provided $$n &lt; 1/2$$ <math><mrow><mi>n</mi><mo>&lt;</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></math> , it is shown that the primordial curvature perturbations are generated deeply into the contracting era, at large negative time, which makes the low-curvature limit a good approximation for calculating the perturbation power spectrum. Moreover, it is shown that, for n within this range, the obtained cosmological quantities are fully compatible with the Planck constraints, and that the “low curvature limit” comes as a viable approximation to calculate the power spectra of both scalar and tensor perturbations. Using reconstruction techniques for F(R) gravity with the Lagrange multiplier, the precise form of the effective F(R) gravity is found, from which one determines the power spectra of scalar and tensor perturbations in such bouncing scenario. Correspondingly, the spectral index for curvature perturbations and the tensor to scalar ratio are obtained, and these values are successfully confronted with the latest Planck observations. Further, it is shown that both the weak and the null energy conditions are satisfied, thanks to the holonomy corrections performed in the theory–which are then proven to be necessary for achieving this goal. In fact, when approaching the bouncing era, the holonomy corrections become significant and play a crucial role in order to restore the energy conditions. Summing up, a cosmological bouncing scenario with the scale factor above and fulfilling the energy conditions can be adequately described by the F(R) model with a Lagrange multiplier and holonomy corrections, which prove to be very important.

Emilio Elizalde

S. Odintsov

Viable non-singular cosmic bounce in holonomy improved F(R) gravity endowed with a Lagrange multiplier

In this work we investigate the inflationary phenomenological implications of a recently developed ghost-free Gauss–Bonnet theory of gravity. The resulting theory can be viewed as a scalar Einstein–Gauss–Bonnet theory of gravity, so by employing the formalism for cosmological perturbations for the latter theory, we calculate the slow-roll indices and the observational indices, and we compare these with the latest observational data. Due to the presence of a freely chosen function in the model, in principle any cosmological evolution can be realized, so we specify the Hubble rate and the freely chosen function and we examine the phenomenology of the model. Specifically we focus on de Sitter, quasi-de Sitter and a cosmological evolution in which the Hubble rate evolves exponentially, with the last two being more realistic choices for describing inflation. As we demonstrate, the ghost-free model can produce inflationary phenomenology compatible with the observational data. We also briefly address the stability of first order scalar and tensor cosmological perturbations, for the exponential Hubble rate, and as we demonstrate, stability is achieved for the same range of values of the free parameters that guarantee the phenomenological viability of the models.