In the formalism of generalized holographic dark energy (HDE), the holographic cut-off is generalized to depend upon $L_{\mathrm{IR}}=L_{\mathrm{IR}}\left(L_{\mathrm{p}},,,{\dot{L}}_{\mathrm{p}},,,{\ddot{L}}_{\mathrm{p}},,,\cdots ,,,L_{\mathrm{f}},,,{\dot{L}}_{\mathrm{f}},,,\cdots ,,,a\right)$ with $L_{\mathrm{p}}$ and $L_{\mathrm{f}}$ being the particle horizon and the future horizon, respectively (moreover, a is the scale factor of the Universe). Based on such formalism, in the present paper, we show that a wide class of dark energy (DE) models can be regarded as different candidates for the generalized HDE family, with respective cut-offs. This can be thought as a symmetry between the generalized HDE and different DE models. In this regard, we considered several entropic dark energy models—such as the Tsallis entropic DE, the Rényi entropic DE, and the Sharma–Mittal entropic DE—and found that they are indeed equivalent with the generalized HDE. Such equivalence between the entropic DE and the generalized HDE is extended to the scenario where the respective exponents of the entropy functions are allowed to vary with the expansion of the Universe. Besides the entropic DE models, the correspondence with the generalized HDE was also established for the quintessence and for the Ricci DE model. In all the above cases, the effective equation of state (EoS) parameter corresponding to the holographic energy density was determined, by which the equivalence of various DE models with the respective generalized HDE models was further confirmed. The equivalent holographic cut-offs were determined by two ways: (1) in terms of the particle horizon and its derivatives, (2) in terms of the future horizon horizon and its derivatives.

In the present paper, we use the holographic approach to describe the early-time acceleration and the late-time acceleration eras of our Universe in a unified manner. Such “holographic unification” is found to have a correspondence with various higher curvature cosmological models with or without matter fields. The corresponding holographic cutoffs are determined in terms of the particle horizon and its derivatives, or the future horizon and its derivatives. As a result, the holographic energy density we propose is able to merge various cosmological epochs of the Universe from a holographic point of view. We find the holographic correspondence of several $F(R)$ gravity models, including axion-$F(R)$ gravity models, of several Gauss-Bonnet $F(G)$ models and finally of $F(T)$ models, and in each case we demonstrate that it is possible to describe in a unified way inflation and late-time acceleration in the context of the same holographic model.

We have explicitly demonstrated that scalar coupled Gauss-Bonnet gravity in four dimensions can have nontrivial effects on the early inflationary stage of our Universe. In particular, we have shown that the scalar coupled Gauss-Bonnet term alone is capable of driving the inflationary stages of the Universe without incorporating slow roll approximation, while remaining compatible with the current observations. Subsequently, to avoid the instability of the tensor perturbation modes we have introduced a self-interacting potential for the inflaton field and have shown that, in this context as well, it is possible to have an inflationary scenario. Moreover, it turns out that presence of the Gauss-Bonnet term is incompatible with the slow roll approximation and hence one must work with the field equations in the most general context. Finally, we have shown that the scalar coupled Gauss-Bonnet term attains smaller and smaller values as the Universe exits from inflation. Thus, at the end of the inflation, the Universe makes a smooth transition to Einstein gravity.

Primordial gravitational waves (GWs) carry the imprints of the dynamics of the universe during its earliest stages. With a variety of GW detectors being proposed to operate over a wide range of frequencies, there is great expectation that observations of primordial GWs can provide us with an unprecedented window to the physics operating during inflation and reheating. In this work, we closely examine the effects of the regime of reheating on the spectrum of primordial GWs observed today. We consider a scenario wherein the phase of reheating is described by an averaged equation of state (EoS) parameter with an abrupt transition to radiation domination as well as a scenario wherein there is a gradual change in the effective EoS parameter to that of radiation due to the perturbative decay of the inflaton. We show that the perturbative decay of the inflaton leads to oscillations in the spectrum of GWs, which, if observed, can possibly help us decipher finer aspects of the reheating mechanism. We also examine the effects of a secondary phase of reheating arising due to a brief epoch driven possibly by an exotic, noncanonical, scalar field. Interestingly, we find that, for suitable values of the EoS parameter governing the secondary phase of reheating, the GWs can be of the strength as suggested by the recent NANOGrav observations. We conclude with a discussion of the wider implications of our analysis.