Narrow Results

In this paper, we show how the Gambier equation arises in connection to Friedmann–Lemaître–Robertson–Walker (FLRW) cosmology and a Dark Matter equation of state. Moreover, we provide a correspondence between the Friedmann equations and the Gambier equations that possess the Painlevé property in (2 + 1) dimensions. We also consider special cases of the Gambier G27 equation such as the generalized Pinney equation. For an extended FLRW model with dynamic scalar field as matter model, the Einstein equations correspond to the Milne–Pinney equation which in turn can be mapped to the parametric Gambier equation of second order.

This study explores the interaction between non-extensive entropic FLRW cosmology and the power-law inflationary model, with a focus on the overlap between the scalar spectral index “$n_s$” and the tensor-to-scalar ratio “r”. Based on a conjecture that non-extensive entropy alters the energy–momentum content of the cosmic fluid, the analysis examines how these overlaps shift with different model parameters and compares the findings to those from Bekenstein–Hawking (BH) entropic cosmology. The study highlights the impact of Tsallis, Rényi, and Sharma–Mittal entropies, uncovering a significant correlation between “$n_s$” and “r” that suggests a deeper connection in power-law inflationary dynamics. The results demonstrate that non-extensive entropies not only enable viable inflation with a graceful exit but also address limitations inherent in the standard BH entropic framework, emphasizing the importance of precise parameter estimation. Specifically, Tsallis entropy allows for power-law inflation with $n=1$ to $n=2$ in alignment with Planck 2018 data. Moreover, the $\alpha$ parameter in Rényi and Sharma–Mittal entropy models must be extremely small ($\alpha \le 10^{-8}$ in Planck mass units) to achieve successful power-law inflation with an e-folding number around 55–65, suggesting a unified thermodynamic perspective in cosmological studies.

In this paper, we construct the covariant or model independent induced Einstein–Yang–Mills field equations on a four-dimensional brane embedded isometrically in an D-dimensional bulk space, assuming the matter fields are confined to the brane. Applying this formalism to cosmology, we derive the generalized Friedmann equations. We derive the density parameter of dark energy in terms of width of the brane, normal curvature radii and the number of extra large dimensions. We show that dark energy could actually be the manifestation of the local extrinsic shape of the brane. It is shown that the predictions of this model are in good agreement with observation if we consider an 11-dimensional bulk space.

We come back on the dynamical properties of $k$-essential cosmological models and show how the interesting phenomenological features of those models are related to the existence of boundaries in the phase surface. We focus our attention to the branching curves where the energy density has an extremum and the effective speed of sound diverges. We discuss the behaviour of solutions of a general class of cosmological models exhibiting such curves and give two possible interpretations; the most interesting possibility regards the arrow of time that is reversed in trespassing the branching curve. This study teaches to us something new about general FLRW cosmologies where the fluids driving the cosmic evolution have equations of state that are multivalued functions of the energy density and other thermodynamical quantities.

So-called “Buchert averaging” is actually a coarse-graining procedure, where fine detail is “smeared out” due to limited spatio-temporal resolution. For technical reasons, (to be explained herein), “averaging” is not really an appropriate term, and I shall consistently describe the process as a “coarse-graining”. Because Einstein gravity is nonlinear the coarse-grained Einstein tensor is typically not equal to the Einstein tensor of the coarse-grained spacetime geometry. The discrepancy can be viewed as an “effective” stress-energy, and this “effective” stress-energy often violates the classical energy conditions. To keep otherwise messy technical issues firmly under control, I shall work with conformal-FLRW (CFLRW) cosmologies. These CFLRW-based models are particularly tractable, and are also particularly attractive observationally: the CMB is not distorted. In this CFLRW context one can prove some rigorous theorems regarding the interplay between Buchert coarse-graining, tracelessness of the effective stress-energy, and the classical energy conditions.