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The cosmological model executing hybrid expansion law (HEL) of scale factor in Randall–Sundrum type II (RS) braneworld gravity with Gauss–Bonnet terms in the presence of bulk viscosity described by Eckart theory, Truncated Israel–Stewart (TIS) theory, Full Israel–Stewart (FIS) theory and nonlinear Israel–Stewart (nIS) theory is studied. In the hybrid expansion law, the scale factor of the universe is described by the product of power-law and exponential expansions. In the HEL model, the early inflation and its transition from deceleration phase to present accelerated phase of expansion can be explored. The constraints of hybrid expansion law model parameters are determined using the recent observational data. Thereafter, the estimated parameters are considered to explore the present value of deceleration parameter, jerk parameter and transition epoch from early deceleration to the present accelerating phase.

In this paper, we examine the stability condition of $f(T)$ gravity theory where T is the Torsion scalar using interacting and non-interacting models with the help of dynamical system. We let the $f(T)$ function be $f(T)=\alpha T+\beta$, where T is Torsion scalar and $\alpha$ and $\beta$ are the arbitrary constants. We calculated the critical points and study the stability behavior for this model. We analyze the system’s phase graphs and examined the physical interpretation. We demonstrate all of the cosmological parameters including ${\mathrm{\Omega }}_m$, ${\mathrm{\Omega }}_{\varphi }$, q, and ${\omega }_{\mathrm{\text{Tot}}}$ at each critical point and contrast them with values from observations. Then, we assume hybrid scale factor and transform it into the equation of redshift and time. Using this equation, we convert all the parameters into terms of redshift and analyze their behavior. Our model illustrates the Universe’s accelerating expansion. The energy conditions are examined in terms of redshift for the model. As a result, we conclude that our $f(T)$ model is stable and all the observed values are in agreement with observations.

In this work, we study bulk viscous Friedmann–Robertson–Walker cosmologies with hybrid expansion law. The bulk viscous theory of dissipative effects described by Eckart theory, truncated Israel–Stewart theory and full Israel–Stewart theory are implemented here. The hybrid expansion law model of scale factor is a general analytic type of evolution from which one can recover power-law and exponential expansion as a special cases. Hybrid expansion law model are applied to describe the present accelerating phase and early phases of evolution. We have determined the cosmological parameters such as Hubble parameter, deceleration parameter, jerk parameter, energy density, bulk viscous pressure and coefficient of bulk viscosity of the universe to construct physically acceptable cosmological model. We have considered both flat and closed space–time of Friedmann–Robertson–Walker cosmology to implement hybrid expansion law with dissipative effect. The variations of the coefficient of bulk viscosity with cosmic evolution are studied here in Eckart, truncated and full Israel–Stewart theory for flat and closed space–time to obtain physically acceptable hybrid expansion models driven by viscosity. We have also estimated observational constraints of the parameters of hybrid expansion law model by considering recent observational data set. We further reveal possible connections of our models with $H_0$ tension problem.

We investigated the stability condition of $f(Q)$ gravity theory with interacting and noninteracting models by using dynamical system. We assume the $f(Q)$ function as $f(Q)=\left(\right.\sqrt{Q}+\frac{M}{2}{\left)\right.}^2$, where $M$ is the free parameter. We evaluated the critical points for this model and examined the stability behavior. We found two stable critical points for interacting model. The phase plots for this system are examined and the physical interpretation is discussed. We illustrate all the cosmological parameters such as ${\mathrm{\Omega }}_m$, ${\mathrm{\Omega }}_{\varphi }$, $q$ and ${\omega }_{\mathrm{\text{Tot}}}$ at each fixed point and compare the parameters with observational values. Further, we assume hybrid scale factor and the equation of redshift and time is $t(z)=\frac{\mu }{\nu }W[\frac{\nu {(1+z)}^{-\frac{m}{\mu }}}{\mu }]$. We transform all the parameters in terms of redshift by using this equation and examine the behavior of these parameters. Our model represents the accelerated expansion of the universe. The energy conditions are examined in terms of redshift and strong energy conditions are not satisfied for the model. We also find the statefinder parameters $\{r,s\}$ in terms of z and discuss the nature of r–s and r–q plane. For both pairs $\{r,s\}$ and $\{r,q\}$ our model represents the $\mathrm{\Lambda }$CDM model. Hence, we determine that our $f(Q)$ model is stable and it satisfies all the observational values.

In this paper, evolution of a Friedmann–Robertson–Walker universe is studied in a higher derivative theory of gravity. The relativistic solutions admitting hybrid expansion law of the universe are explored here. Hybrid expansion law is a general form of scale factor from which one can recover both the power-law expansion and exponential expansion as a special case. The hybrid expansion law is interesting as it addresses the early deceleration phase and presents accelerating phase satisfactorily. It is found that an inflationary scenario with hybrid expansion law is permitted in the $R^2$ gravity fairly well. We consider universe filled with cosmic fluid that describes by an equation of state (EoS) parameter which varies with time. Consequently, we analyze the time variation of energy density parameter, cosmic pressure, equation of state parameter, deceleration parameter and jerk parameter in the cosmological model. The constraints of the model parameters imposed by the cosmological observational data set are determined. The present value of the deceleration parameter $(q)$, EoS parameter and the epoch at which the transition of decelerated phase to accelerated phase are estimated. In the higher derivative theory, we obtain some new and interesting cosmological solutions relevant for building cosmological models.