The Ricci scalar corresponding to the RW metric (19) is, The point Lagrangian corresponding to the action (6) is given by [10] $L_m$ being the matter Lagrangian, comprised of a barotropic fluid associated with the thermodynamic pressure (p) and energy density ($\rho$), along with the cold darl matter (CDM). The field equations are, Adding the above pair of equations, one obtains, Clearly, the radiation era ($p=\frac{1}{3}\rho$) does not evolve like standard (FLRW) model ($a\propto \sqrt{t}$), for the spatially flat ($k=0$) universe, and as such a viable form of $F\left(R\right)$ remains obscure. For $k\ne 0$, on the other hand, the differential equation cannot be solved, either in the radiation or in the vacuum dominated ($\rho =0=p$) era, other than the fact that, for $k=0$ vacuum era admits a de-Sitter solution. On the contrary, it may be mentioned that different forms of $F\left(R\right)$ emerge in different eras, in view of Noether symmetry [11].

Nonetheless, interesting result emerge from a generalized four-dimensional string effective action associated with higher order curvature invariant terms, being expressed in the following form [12], where B is the coupling constant, $f\left(\varphi \right)$ is the coupling parameter, $\omega \left(\varphi \right)$ is the variable Brans-Dicke parameter and ${\mathcal{L}}_m$ is the matter Lagrangian density. It had been revealed [12] that the above action admits a conserved current $J^{\mu }={(3f^{\prime 2}+2f\frac{\omega }{\varphi })}^{\frac{1}{2}}$ under the condition, where $\lambda$ is a constant. To perceive the very important role of such a conserved current, a particular case was also studied [12]. For example, considering $F\left(R\right)=R^2$, the condition for the existence of conserved current (25) reads as, Now if the scalar field ($\varphi$) is completely used up in the process of driving inflation and reheating under particle creation, then since in the radiation-dominated era ${T^{\mu }}_{\mu }=T=\rho -3p=0$, therefore in view of (26) $\square R=\left(\ddot{R}+3\frac{\dot{a}}{a}R\right)=0$. Further, since in the very early vacuum dominated era sufficient inflation makes the universe spatially flat, i.e., $k=0$, so the above equation (26) admits a Friedmann like solution, $a=a_0\sqrt{t}$ in the radiation dominated era as demonstrated in [12]. This is a unique result, since even in the presence of higher order curvature invariant term Friedmann-like radiation era is admissible.

Finally, we consider pure curvature induced gravity theory, a yet another generalized form of $F\left(R\right)$ action, considered in [13] being expressed as, The above action was found more suitable to explain cosmic evolution right from the very early stage till date, since it satisfies all the strong conditions necessary for a viable $F\left(R\right)$ theory of gravity. It may be mentioned that $R^{\frac{3}{2}}$ term appeared as a consequence of Noether symmetry in R-W metric both in vacuum as well as in the matter dominated eras [14,15,16,17,18]. At the initial stage, $R^2$ term dominates and a de-Sitter solution is realizable. This leads to the inflationary epoch and reheating following the mechanism of particle production via scalaron decay, exploiting gravity only without invoking phase transition [19,20]. After the reheating is over, the universe evolves as Friedmann-like radiation $(a\propto \sqrt{t})$ and early matter $(a\propto t^{\frac{2}{3}})$ dominated eras, and finally accelerated expansion is realized due to the presence of the combination of linear term R and non-linear term $R^{\frac{3}{2}}$[16]. Further, following numerical analysis [13] taking deceleration parameter q as a function of the red-shift parameter z, three distinct cases were analyzed to establish the fact that, within a particular range of $\beta$ and $\gamma$, q versus z depicts that the universe was in pure radiation era at $z>3200$. Thereafter, deceleration parameter falls off from the matter-radiation equality epoch to the decoupling epoch. It falls even sharply afterwards and a Friedmann type $(q\approx 0.5)$ matter dominated era is reached at around $z\approx 200$. The deceleration parameter then starts increasing slowly and it is peaked to $(q=1)$, explaining re-ionization of the inter galactic medium (IGM). Thereafter, late time accelerated expansion is initiated and the phantom divide line is crossed, to make a second transition out of it at $z\approx 0.5$.

In a nutshell, higher order gravity theory is a viable option to explain cosmological evolution, from the early vacuum dominated era till date via a Friedmann-like radiation and early matter dominated eras. Additionally, re-ionization of the IGM may also be explained in the process. Inflation in the Starobinski model [19,20] with $R^2$ term is widely explored in the literature, and therefore we leave it.

Although, $F\left(R\right)$ seems to produce a viable theory of gravity, the fact that it sometimes suffer from Ostrogradsky’s instability led to consider alternative telleparallel theories of gravity, as already discussed. Here, we shall consider generalized metric telleparallel gravity theory.

In the recent years, a generalized version of the ‘teleparallel gravity’ with torsion, namely the $F\left(\mathrm{T}\right)$ theory of gravity (where, $\mathrm{T}$ stands for the torsion scalar), also dubbed as generalized ‘gravity with torsion’ has been proposed as an alternative to both the dark energy theories and the modified theories of gravity. Primarily $F\left(\mathrm{T}\right)$ theory of gravity was proposed to drive inflation. Later, it was applied to drive the current accelerated expansion of the present universe without considering dark energy [21,22]. It is worth mentioning that ‘teleparallel equivalent of general relativity’ (TEGR) is established for $F\left(\mathrm{T}\right)\propto \mathrm{T}$, since dynamically it leads to GTR. Further as demonstrated in [13] the trace of electro-magnetic radiation field tensor $(T=\rho -3p)$ being zero, the contracted GTR equation $R\propto T$, enforces Ricci scalar to vanish $(R=0)$ and the Friedmann-like decelerated expansion $[a\left(t\right)\propto \sqrt{t}]$ results in automatically, in the radiation dominated era. This is true also in the case of modified theory of gravity, as demonstrated above. However, in the case of torsion, although $F\left(\mathrm{T}\right)\propto \mathrm{T}$ leads to GTR, the trace of the energy-momentum tensor $T=0$, does not lead to the static solution $\mathrm{T}=0$. Therefore, even though all the results of GTR hold, the pathology of discontinuous evolution of the Ricci scalar (large initially, vanishing in the middle and small at present) is averted. However unlike GTR and $F\left(R\right)$ theories, gravity with torsion is not generally covariant by default. Nonetheless, it has been argued that it may be made so by introducing a new variable, viz, a spin-connection [8,23]. The reason being, the spin-connection enters the teleparallel action only as a surface term, and does not contribute to the field equations.

Now, the components of vierbein field for the RW metric (19) are expressed in terms of the cosmological scale factor $a\left(t\right)$ as, and the torsion scalar reads as [24], where, $H=\frac{\dot{a}}{a}$ is the Hubble parameter. Thus the field equations (12) for RW metric are, where $\rho$ and p are the energy density and thermodynamic pressure of a barotropic fluid (inclusive of dark matter component) respectively and $F_{,\mathrm{T}}$ stands for derivative of $F\left(\mathrm{T}\right)$ with respect to $\mathrm{T}$. Bianchi identity does not hold naturally in teleparallel gravity theories. However, in the RW metric under consideration, it holds even for extended models such as $F\left(\mathrm{T}\right)$ [24]. For example, taking the time derivative of the first equation (30), and also adding the the two (30) and (31), one gets, Combining the above two one finds which is the Bianchi identity. Thus, one may also write for the barotropic equation of state $p=\omega \rho$, as usual.

Let us now proceed to find the form of $F\left(\mathrm{T}\right)$ for flat space $(k=0)$, already obtained earlier [25], in view of cosmological evolution. First, combining equations (30) and (52), for flat space $k=0$, one finds for the vacuum era ($p=0=\rho$),

Clearly, two possibilities emerge from the equation (35): i) $F=F_0\sqrt{\mathrm{T}}$, where $F_0$ is a constant of integration. In view of this form of $F\left(\mathrm{T}\right)$ either of the field equations (30) or (52) yields simply the definition, and hence, no dynamics results. Indeed it is expected, since the above form of $F\left(\mathrm{T}\right)$, only results in a divergent term in the action. ii) $H=\lambda$, (where $\lambda$ is a constant) leading to the de-Sitter solution, $a=a_{0}exp\left(\lambda t\right)$, while $F\left(\mathrm{T}\right)=F_{0}exp(-\frac{t}{12{\Lambda }^2})$ emerges as an exponentially decaying function, which is not much promising.

Since vacuum dominated era does not yield a reasonably viable form of $F\left(\mathrm{T}\right)$, so let us now advance further to study the radiation-dominated era. Starting from the action (8), if a solution in the form $a=a_0{t}^n$ is sought in the radiation dominated era ($p=\frac{1}{3}\rho$), for which the Bianchi identity yields $\rho a^4={\rho }_{r0}$, where $a_0$, n and ${\rho }_{r0}$ are constants, then the following form of $F\left(\mathrm{T}\right)$ (suffix r stands for radiation dominated era) is found [11], where $f_1$ is a constant. Note that the first term of the above equation is essentially a divergent term in the RW metric under consideration. Thus, only the second term is left and hence, It is quite apparent that under the choice $n=\frac{1}{2}$, the radiation era evolves exactly like the standard (FLRW) model, however the action too reduces to that of GTR since $\left[F\right(T)\propto \mathrm{T}]$ (apart from a total derivative term), which is TEGR, as already mentioned. Of-course, $n=\frac{3}{4}$ also leads to decelerated expansion, but such a slow deceleration tells upon the formation of CMBR at latter times than observed. Let us next, focus on the matter dominated era ($p=0$), for which Bianchi identity yields $\rho a^3={\rho }_{m0}$, (where ${\rho }_{m0}$ is a constant, and suffix m stands for the matter dominated era). Proceeding in the similar manner as before, one finds, where the suffix m stands for the matter dominated era. Thus, for $n=\frac{2}{3}$, the matter dominated era evolves ($a\propto t^{\frac{2}{3}}$) in the same manner as the standard (FLRW) model of cosmology, and GTR is recovered through TEGR. Whatsoever, it was also discussed in [11] that when torsion is attributed to gravity, usually a form such as $F\left(\mathrm{T}\right)=f_0\mathrm{T}+f_1{\mathrm{T}}^2+\cdots$ is chosen to combat early deceleration in the Friedmann form $[a\left(t\right)\propto t^{\frac{2}{3}}]$ followed by late-time cosmic acceleration in the matter (pressure-less dust) dominated era. In view of (36) and (38), it is clear that $F\left(\mathrm{T}\right)\propto \mathrm{T}$ gives exactly Friedmann-like radiation dominated era $[a\left(t\right)\propto t^{\frac{1}{2}}]$, and early pressure-less dust dominated era $[a\left(t\right)\propto t^{\frac{2}{3}}]$ respectively. Thus, following generalized form of $F\left(\mathrm{T}\right)$, might be useful to study cosmic evolution. On the contrary, Noether symmetry analysis [11] demands that instead of ${\mathrm{T}}^2$, one should associate ${\mathrm{T}}^3$ and higher odd integral powers in the action. That is, a viable form that might explain the cosmic evolutionary history may be in the form, $F\left(\mathrm{T}\right)=f_0\mathrm{T}+f_1{\mathrm{T}}^3+\cdots$. It may be mentioned that the current analysis also reveals the fact that pure $F(\mathrm{T}$) gravity in vacuum ($\rho =p=0$) does not give rise to any dynamics. Therefore to drive inflation, and also to avert the pathological behaviour of pure $F\left(\mathrm{T}\right)$ gravity in the very early vacuum-dominated era, either unimodular $F\left(\mathrm{T}\right)$ gravity has to be considered [26], or a scalar field should be associated with $F\left(\mathrm{T}\right)$ gravity theory [25] and the action may be proposed as, where, the scalar field drives inflation at the very early stage and decayed to an insignificant value in the process of particle creation. Therefore, once the inflation is over, the universe enters the radiation and thereafter the matter dominated eras, whence $\mathrm{T}$ dominates to envisage the standard model. Now since $\mathrm{T}=-6H^2<0$ by definition, therefore as the Hubble parameter decreases further, the odd-integral higher degree terms start dominating, and become responsible for late-time cosmic acceleration. The great conceptual advantage over modified theories of gravity is that, unlike the Ricci scalar, $\mathrm{T}\ne 0$, at any stage of evolution in the middle.