Aspects of non-singular bounce in modified gravity theories

This section is organized as follows: in Sec.[II.1], we briefly describe the non-flat warped braneworld model and its four dimensional effective theory. Having set the stage, Sec.[II.2] is dedicated for studying the background cosmological evolution while the evolution of the perturbations and confrontation of the theoretical predictions with the latest Planck observations is discussed in Sec.[II.3]. A summary of our results are explained.
In the context of Randall-Sundrum (RS) 5 dimensional warped braneworld scenario with two 3-branes embedded within the full spacetime, the bulk metric is given by,

$\displaystyle{ds}^{2}=e^{-2A(r_{c},\phi)}g_{\mu\nu}(x){dx}^{\mu}{dx}^{\nu}+r_{c}^{2}{d\phi}^{2}~{}~{}.$

(1)

The extra dimension coordinated by $\phi$ is $S^{1}/Z_{2}$ compactified, $r_{c}$ is the interbrane separation, $\{x^{\mu}\}$ are the brane coordinates and $A(r_{c},\phi)$ is known as the warp factor. In the original RS scenario, the branes are considered to be flat, i.e $g_{\mu\nu}$ is replaced by the Minkowski metric $\eta_{\mu\nu}$, and the warp factor is obtained by solving the 5 dimensional Einstein equations as $A(r_{c},\phi)=k_{0}r_{c}\phi$. Here $k_{0}=\sqrt{-\Lambda/24M^{3}}$ such that $\Lambda$ and $M$ denote the five dimensional cosmological constant and Planck mass respectively. In the case of flat branes, the bulk cosmological constant and the brane tensions are finely adjusted so that the brane curvature gets identically zero. In tis regard, one of our authors proposed a more general solution by relaxing the assumption of $flat$ branes and considered the branes to be either de-Sitter or anti de-Sitter characterized by a positive or negative brane cosmological constant respectively. In the case of non-flat braneworld scenario, induced on-brane metric is the solution of $G_{\mu\nu}=-\Omega g_{\mu\nu}$ where $G_{\mu\nu}$ is the Einstein tensor formed by $g_{\mu\nu}$ and $\Omega$ is the brane cosmological constant ($\Omega>0$ for dS branes and $\Omega<0$ for AdS branes). The warp factor is given by,

	$\displaystyle e^{-A}=\omega\sinh\left(\ln\frac{c_{2}}{\omega}-k_{0}r_{c}|\phi|\right)~{}~{}~{}~{}~{}\mathrm{for~{}dS~{}brane}$
	$\displaystyle e^{-A}=\omega\sinh\left(\ln\frac{\omega}{c_{1}}+k_{0}r_{c}|\phi|\right)~{}~{}~{}~{}~{}\mathrm{for~{}AdS~{}brane}$		(2)

where, $\omega=(\pm\Omega/3k_{0}^{2})$ for dS and AdS case respectively, and $c_{2}=1+\sqrt{1+\omega^{2}}$, $c_{1}=1+\sqrt{1-\omega^{2}}$. Since the observed accelerated expansion of the universe can be explained by a positive brane cosmological constant, we will concentrate mainly on the warped braneworld scenario with de-Sitter branes.

The presence of such non vanishing brane cosmological constant not only generalizes the RS scenario, but also self stabilizes the model by generating a stable radion potential in the four dimensional effective theory. This is unlike to the original RS scenario where the branes are flat, and thus, an adhoc bulk scalar field has to be considered by hand in order to stabilize the braneworld model. Considering the dynamical radion field as $T(x)$, the four dimensional effective action is given by Banerjee:2020uil ,

$\displaystyle\mathcal{A}=\int d^{4}x\sqrt{-g}\Bigg{[}\frac{R}{2\kappa^{2}}-\frac{1}{2}G(\xi)\partial^{\mu}\Phi\partial_{\mu}\Phi-8M^{3}\kappa^{2}V(\xi)\Bigg{]}~{}~{},$

(3)

where $\kappa^{2}=8\pi G$ ($G$ is the four dimensional Newton’s constant), $\Phi=fe^{-k_{0}T(x)\pi}$ (with $f=\sqrt{6M^{3}c_{2}^{2}/k_{0}}$) is the radion field having mass dimension [+1] and $\xi=\frac{\Phi}{f}$ is the dimensionless radion field. The potential and the non-canonical kinetic term for the radion field come with the following expressions Banerjee:2020uil :

$\displaystyle V(\xi)=\frac{6\omega^{2}}{h(\xi)}~{}~{}~{}~{}\mathrm{and}~{}~{}~{}~{}G(\xi)=\frac{\mathcal{G}(\xi)}{h(\xi)}+\frac{1}{c_{2}^{2}}\bigg{[}\frac{h^{\prime}(\xi)}{h(\xi)}\bigg{]}^{2}$

(4)

respectively, where

	$\displaystyle h\left(\xi\right)$	$\displaystyle=$	$\displaystyle\left\{\frac{c_{2}^{2}}{4}+\omega^{2}\ln\xi+\frac{\omega^{4}}{4c_{2}^{2}}\left(\frac{1}{\xi^{2}}\right)-\frac{\omega^{4}}{4c_{2}^{2}}-\frac{c_{2}^{2}}{4}\xi^{2}\right\}$
	$\displaystyle\mathcal{G}\left(\xi\right)$	$\displaystyle=$	$\displaystyle 1+\frac{4}{3}\frac{\omega^{2}}{c_{2}^{2}}\left(\frac{1}{\xi^{2}}\right)\ln\xi-\frac{\omega^{4}}{c_{2}^{4}}\left(\frac{1}{\xi^{4}}\right)~{}~{}.$		(5)

Note that $V(\xi)$ is proportional to $\omega^{2}$, which indicates that the potential term for the radion field is solely generated due to the presence of a non-zero brane cosmological constant. The $V(\xi)$ has an inflection point at $\xi=\omega/c_{2}$ and the $G(\xi)$ experiences a zero crossing from positive to negative values. In Fig.[5a] and Fig.[5b] we plot the variation of $V$ and $G$ with the radion field $\xi$ for $\omega=10^{-3}$. Fig.[5b] reveals that the non-canonical coupling to the kinetic term $G(\xi)$ exhibits a transition from a normal to a phantom regime (i.e from $G(\xi)>0$ to $G(\xi)<0$), where the phantom like behavior remains when $\xi$ lies in the range $\xi_{i}\leq\xi\leq\xi_{f}$, with $\xi_{f}$ denoting the zero crossing of $G(\xi)$. During the phantom regime the kinetic energy of the radion field gets negative, which in turn may violate the null energy condition of the radion field. Such violation of null energy condition is the essence for triggering a non-singular bounce during the early stage of the universe. This raises the question whether the radion field can be instrumental in giving rise to a bouncing universe which we address in the next section.

The following metric ansatz will fulfill our purpose,

$\displaystyle ds^{2}=-dt^{2}+a(t)^{2}\bigg{[}dx^{2}+dy^{2}+dz^{2}\bigg{]}$

(6)

with $a(t)$ is known as the scale factor of the universe. Considering $\xi=\xi(t)$, the Friedmann equations for the action Eq.(3) are Banerjee:2020uil ,

	$\displaystyle H^{2}$	$\displaystyle=$	$\displaystyle\frac{\kappa^{2}}{3}\rho(t)=\frac{c_{2}^{2}}{4}G(\xi)\dot{\xi}^{2}+\frac{k_{0}^{2}}{6}V(\xi)$
	$\displaystyle\dot{H}$	$\displaystyle=$	$\displaystyle-\frac{\kappa^{2}}{2}(\rho+p)=-\frac{3}{4}c_{2}^{2}G(\xi)\dot{\xi}^{2}$		(7)

where $H=\dot{a}/a$ denotes the Hubble parameter. The equation of motion for the radion field is given by,

$\displaystyle\ddot{\xi}+3H\dot{\xi}+\frac{G^{\prime}(\xi)}{2G(\xi)}\dot{\xi}^{2}+\frac{k_{0}^{2}}{3c_{2}^{2}}\frac{V^{\prime}(\xi)}{G(\xi)}=0$

(8)

Eq.(7) reveals that the model has a possibility to show a bounce phenomena when the non-canonical function $G(\xi)$ becomes negative i.e when the radion field is in the phantom regime. $G(\xi)$ becomes negative in the regime $\xi\sim\omega$. Thus, at first we analytically solve the background equations near $\xi\sim\omega$ to investigate the bounce and then we numerically determine the background evolution for a wide range of $\xi$ (or equivalently for a wide range of cosmic time), where the boundary conditions of the numerical calculation are provided from the analytic solutions.
In particular we consider,

$\displaystyle\xi(t)=\frac{\omega}{c_{2}}[1+\delta(t)]$

(9)

with $\delta(t)\ll 1$. In this regime of $\xi$, $V(\xi)$ and $G(\xi)$ are approximated by,

$\displaystyle V(\xi)\simeq\frac{24\omega^{2}}{c_{2}^{2}}~{}~{}~{}~{}~{}\mathrm{and}~{}~{}~{}~{}~{}G(\xi)\simeq-\frac{16}{3c_{2}^{2}}\bigg{[}ln\bigg{(}\frac{c_{2}}{\omega}\bigg{)}-\bigg{\{}4+2ln\bigg{(}\frac{c_{2}}{\omega}\bigg{)}\bigg{\}}\delta\bigg{]}~{}~{}.$

Using the above expressions of $V(\xi)$ and $G(\xi)$, Eq.(7) and Eq.(8) become

$\displaystyle\dot{H}+3H^{2}-\frac{12k_{0}^{2}\omega^{2}}{c_{2}^{2}}=0~{}~{}~{}~{}~{}\mathrm{and}~{}~{}~{}~{}~{}\dot{\delta}^{2}=\frac{c_{2}^{2}}{\omega^{2}}\frac{\dot{H}}{4ln\big{(}\frac{c_{2}}{\omega}\big{)}}\bigg{[}1+\delta\bigg{\{}\frac{4+2ln\big{(}\frac{c_{2}}{\omega}\big{)}}{ln\big{(}\frac{c_{2}}{\omega}\big{)}}\bigg{\}}\bigg{]}$

(10)

respectively. Solving which, we get the background solutions near $\xi\sim\omega$ as Banerjee:2020uil ,

$\displaystyle\xi(t)=\frac{\omega}{c_{2}}(1+\delta(t)),\begin{cases}H(t)=2k_{0}\frac{\omega}{c_{2}}tanh\bigg{[}6\frac{\omega}{c_{2}}k_{0}t\bigg{]}&\\ \delta(t)=\frac{2}{A}\bigg{[}exp\bigg{\{}-\frac{A}{6}\frac{\omega}{c_{2}}\sqrt{\frac{3}{ln\big{(}\frac{c_{2}}{\omega}\big{)}}}\bigg{(}tan^{-1}tanh\bigg{(}\frac{3\omega}{c_{2}}k_{0}t\bigg{)}-\frac{\pi}{4}\bigg{)}\bigg{\}}-1\bigg{]},\end{cases}$

(11)

where, $A=\frac{4+2ln\big{(}\frac{c_{2}}{\omega}\big{)}}{ln\big{(}\frac{c_{2}}{\omega}\big{)}}$. To derive the above solutions, we consider $\lim_{t\rightarrow\infty}\xi(t)=\frac{\omega}{c_{2}}$. Recall that $\xi=\frac{\omega}{c_{2}}$ is the inflection point of $V(\xi)$, and thus $\lim_{t\rightarrow\infty}\xi(t)=\frac{\omega}{c_{2}}$implies that the radion asymptotically reaches to its stable value.

We now solve the coupled equations for $H(t)$ and $\xi(t)$ (i.e Eq.(7)) for a wide range of cosmic time numerically. In regard to the numerical calculation, the boundary conditions are provided from the analytic solutions as determined in 11, in particular, $H(0)=0$ and $\xi(0)=6.0041\times 10^{-4}$, where we consider $\omega=10^{-3}$ (later, during the perturbation calculation, we show that such a value of $\omega$ is consistent with the Planck 2018 constraints). The time evolution of the Hubble parameter and the radion field are shown in Fig.[6a] and Fig.[6b] respectively (more descriptions about the figure are given in the caption of the figure).

Fig.[2] reveals – (1) the Hubble parameter becomes zero and increases with respect to cosmic time at $t=0$, which confirms a non-singular bounce at $t=0$; (2) the radion field starts its journey from the normal regime and dynamically moves to the phantom era with time by monotonically decreasing in magnitude and asymptotically stabilizes to the value $\langle\xi\rangle\to\omega/c_{2}$ which for $\omega=10^{-3}\Rightarrow\langle\xi\rangle=5\times 10^{-4}$.

Here we consider the spacetime perturbations over the background FRW metric and consequently determine the primordial observable quantities like the scalar spectral index ($n_{s}$), tensor to scalar ratio ($r$) and the amplitude of scalar perturbations ($A_{s}$). For the present bounce scenario, the comoving Hubble radius asymptotically goes to zero at both sides of the bounce, which in turn depicts that the perturbation modes generate near the bounce when the Hubble horizon is infinite in size to contain all the relevant modes within it. Therefore following we solve the perturbation equations near the bouncing point $t=0$. In this regard we further mention that due to the reason that the comoving Hubble radius decreases (with time) at both sides of the bounce, the effective EoS parameter during the contracting stage is less than unity. In effect, the anisotropic energy density during the contracting phase grows faster than that of the bouncer, and leads to the BKL instability.

The scalar metric perturbation ($\Psi$) over the background FRW metric obeys the following equation in the longitudinal gauge,

$\displaystyle\ddot{\Psi}-\frac{1}{a^{2}}\nabla^{2}\Psi+\bigg{[}7H+\frac{2k_{0}^{2}~{}V^{\prime}(\xi_{0})}{3c_{2}^{2}G(\xi_{0})\dot{\xi}_{0}}\bigg{]}\dot{\Psi}+\bigg{[}2\dot{H}+6H^{2}+\frac{2k_{0}^{2}H~{}V^{\prime}(\xi_{0})}{3c_{2}^{2}G(\xi_{0})\dot{\xi}_{0}}\bigg{]}\Psi=0$

(12)

where $H=\frac{\dot{a}}{a}$ is the Hubble parameter in cosmic time and recall, $\xi_{0}$ is the dimensionless background radion field. Using the background solutions, we determine $\frac{V^{\prime}(\xi_{0})}{G(\xi_{0})\dot{\xi}_{0}}$ (present in the above equation) near the bounce where we retain the term up-to the leading order in $t$:

	$\displaystyle\frac{V^{\prime}(\xi_{0})}{G(\xi_{0})\dot{\xi}_{0}}$	$\displaystyle=$	$\displaystyle\frac{36\omega^{2}\delta^{2}(t)}{\dot{\delta}\big{[}\ln{\frac{\omega}{c_{2}}}+2\big{(}2-\ln{\frac{\omega}{c_{2}}}\big{)}\delta(t)\big{]}}$		(13)
		$\displaystyle=$	$\displaystyle-\frac{48B\omega c_{2}\sinh^{2}(B\pi/8)}{k_{0}\big{(}3-2e^{B\pi/4}\big{)}\big{(}2-\ln{\frac{\omega}{c_{2}}}\big{)}}+\frac{72\omega^{2}\big{(}2-2\cosh(B\pi/4)+\sinh(B\pi/4)\big{)}}{\big{(}3-2e^{B\pi/4}\big{)}^{2}\big{(}2-\ln{\frac{\omega}{c_{2}}}\big{)}}~{}t$

with $B=\frac{A}{6}\frac{\omega}{c_{2}}\sqrt{\frac{3}{\ln{\big{(}\frac{c_{2}}{\omega}}\big{)}}}$. Consequently Eq.(78) in Fourier space becomes,

$\displaystyle\ddot{\Psi}_{k}+\big{[}-\sqrt{\alpha}p+(q+14)\alpha t\big{]}\dot{\Psi}_{k}+\big{[}k^{2}+4\alpha-2\alpha\sqrt{\alpha}p~{}t\big{]}\Psi_{k}(t)=0~{}~{},$

(14)

where $\alpha=\frac{6k_{0}^{2}\omega^{2}}{c_{2}^{2}}$ and $p$ and $q$ have the following expressions,

$\displaystyle p=16\sqrt{\frac{2}{3}}\bigg{(}\frac{B\sinh^{2}(B\pi/8)}{\big{(}3-2e^{B\pi/4}\big{)}\big{(}2-\ln{\frac{\omega}{c_{2}}}\big{)}}\bigg{)}~{}~{}~{}~{}~{}\mathrm{and}~{}~{}~{}~{}~{}q=\frac{8\big{(}2-2\cosh(B\pi/4)+\sinh(B\pi/4)\big{)}}{\big{(}3-2e^{B\pi/4}\big{)}^{2}\big{(}2-\ln{\frac{\omega}{c_{2}}}\big{)}}$

respectively. Considering the Bunch-Davies initial condition of scalar Mukhanov-Sasaki variable $t=0$, Eq.(14) is solved to get,

$\displaystyle\Psi_{k}(t)=\frac{\sqrt{3}}{2k^{3/2}}\bigg{(}\frac{\omega}{c_{2}}\bigg{)}\bigg{(}\frac{k_{0}}{M}\bigg{)}^{3/2}e^{B\pi/4}\big{(}3-2e^{B\pi/4}\big{)}^{1/2}e^{[p\sqrt{\alpha}~{}t~{}-7\alpha t^{2}-\frac{q}{2}\alpha t^{2}]}~{}\Bigg{\{}\frac{H\big{[}-1+\frac{k^{2}+4\alpha}{\alpha(q+14)},\frac{-p+(q+14)\sqrt{\alpha}~{}t}{\sqrt{2(q+14)}}\big{]}}{H\big{[}-1+\frac{k^{2}+4\alpha}{\alpha(q+14)},\frac{-p}{\sqrt{2(q+14)}}\big{]}}\Bigg{\}}~{}~{}.$

(15)

The solution of $\Psi_{k}(t)$ immediately leads to the scalar power spectrum for $k$-th modes as,

$\displaystyle P_{\Psi}(k,t)=\frac{3}{8\pi^{2}}\bigg{(}\frac{\omega}{c_{2}}\bigg{)}^{2}\bigg{(}\frac{k_{0}}{M}\bigg{)}^{3}e^{B\pi/2}\big{(}3-2e^{B\pi/4}\big{)}e^{[2p\sqrt{\alpha}~{}t~{}-14\alpha t^{2}-q\alpha t^{2}]}~{}\Bigg{\{}\frac{H\big{[}-1+\frac{k^{2}+4\alpha}{\alpha(q+14)},\frac{-p+(q+14)\sqrt{\alpha}~{}t}{\sqrt{2(q+14)}}\big{]}}{H\big{[}-1+\frac{k^{2}+4\alpha}{\alpha(q+14)},\frac{-p}{\sqrt{2(q+14)}}\big{]}}\Bigg{\}}^{2}~{}.$

(16)

The tensor perturbation near the bounce follows the equation like,

$\displaystyle\ddot{h}_{k}+6\alpha\dot{h}_{k}~{}t+k^{2}h_{k}(t)=0$

(17)

with recall, $\alpha=\frac{6k_{0}^{2}\omega^{2}}{c_{2}^{2}}$. Considering the Bunch-Davies state for tensor perturbation at $t=0$, we solve Eq.(17) to get,

$\displaystyle h_{k}(t)=\bigg{(}\frac{2\kappa~{}\Gamma\big{(}1-\frac{k^{2}}{12\alpha}\big{)}}{\sqrt{2\pi k}~{}2^{\frac{k^{2}}{6\alpha}}}\bigg{)}~{}e^{-3\alpha t^{2}}~{}H\bigg{[}-1+\frac{k^{2}}{6\alpha},\sqrt{3\alpha}~{}t\bigg{]}~{}~{},$

(18)

which immediately leads to the tensor power spectrum as,

$\displaystyle P_{h}(k,t)=\frac{2k^{2}}{\pi^{3}}~{}\frac{\bigg{(}\kappa~{}\Gamma\big{(}1-\frac{k^{2}}{12\alpha}\big{)}\bigg{)}^{2}}{~{}2^{\frac{k^{2}}{3\alpha}}}e^{-6\alpha t^{2}}~{}\bigg{\{}H\bigg{[}-1+\frac{k^{2}}{6\alpha},\sqrt{3\alpha}~{}t\bigg{]}\bigg{\}}^{2}$

(19)

Now one can explicitly confront the model at hand with the latest Planck observational data Akrami:2018odb , so we calculate the spectral index of the primordial curvature perturbations $n_{s}$ and the tensor-to-scalar ratio $r$, which are defined and have the respective constraints from the Planck observation as follows,

	$\displaystyle n_{s}-1$	$\displaystyle=$	$\displaystyle\frac{\partial\ln{P_{\Psi}}}{\partial\ln{k}}\bigg{|}_{h.c}~{}~{},~{}~{}~{}~{}\mathrm{Constraint}:~{}~{}n_{s}=0.9649\pm 0.0042$
	$\displaystyle r$	$\displaystyle=$	$\displaystyle\frac{P_{h}(k,t)}{P_{\Psi}(k,t)}\bigg{|}_{h.c}~{}~{},~{}~{}~{}~{}\mathrm{Constraint}:~{}~{}r<0.064$		(20)

where the suffix ’h.c’ indicates the horizon crossing instance of the large scale modes. As mentioned earlier that the relevant perturbation modes cross the horizon near the bounce, and thus the horizon crossing condition becomes $k=aH=2\alpha t_{h}$ (where $t_{h}$ is the horizon crossing time). With this relation along with Eq.(20), the parametric plot of $n_{s}$ vs. $r$ is shown in Fig.[3] which clearly demonstrates the simultaneously compatibility of $n_{s}$ and $r$ with the Planck data. If one combines the scalar perturbation amplitude ($A_{s}$) with $n_{s}$ and $r$, then $n_{s}$, $r$ and $A_{s}$ in the present bounce scenario become simultaneously compatible with the Planck constraints for the parameter ranges: $\omega=10^{-3}$, $14\lesssim\frac{R_{h}}{\alpha}\lesssim 19$, $\frac{k_{0}}{M}=[0.601,0.607]$; where $R_{h}$ is the Ricci scalar at the horizon crossing instant and recall that $k_{0}=\sqrt{-\Lambda/24M^{3}}$ such that $\Lambda$ and $M$ denote the five dimensional cosmological constant and Planck mass respectively.

To determine the scalar and tensor power spectra, we use the Hubble parameter as

$\displaystyle H(t)=12k_{0}^{2}\left(\frac{\omega}{c_{2}}\right)^{2}t~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathrm{or~{}equivalently}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}a(t)=1+6k_{0}^{2}\left(\frac{\omega}{c_{2}}\right)^{2}t^{2}~{}~{},$

(21)

where we keep up to the linear order in t from Eq.(11), and consequently, the evolution equations for scalar and tensor perturbations are considered up to the linear order in t. Therefore the scalar and tensor power spectra obtained in Eq.(16) and Eq.(19) respectively, are valid for

$\displaystyle t<\frac{1}{k_{0}}\left(\frac{c_{2}}{6\omega}\right)=t_{v}(\mathrm{say})~{}~{}.$

(22)

As we have shown that the model stands to be a viable one in regard to the Planck constraints for the parameter ranges : $\omega=10^{-3}$ and $\frac{k_{0}}{M}=[0.601,0.607]$ respectively. Such parametric regime results to $t_{v}\sim 10^{-16}\mathrm{GeV}^{-1}$. On other hand, the horizon crossing condition for $k$-th mode is given by $k=aH$. Here we would like to mention that the scale of interest in the present context is around the CMB scale given by $k_{CMB}\approx 0.05\mathrm{Mpc}^{-1}\approx 10^{-40}\mathrm{GeV}$, as we are interested to investigate whether the theoretical predictions of $n_{s}$, $\mathcal{A}_{s}$ and $r$ match with the Planck 2018 results which put a constraint on these observable quantities around the CMB scale. With Eq.(21), we determine the expression of the time when $k_{CMB}$ crosses the horizon by using the horizon crossing relation $k=aH$, and is given by,

$\displaystyle t_{h}=\frac{k_{CMB}}{12k_{0}^{2}}\bigg{(}\frac{c_{2}^{2}}{\omega^{2}}\bigg{)}~{}~{},$

(23)

where, $t_{h}$ is the horizon crossing time of the CMB scale and recall, $k_{0}$ being the bulk curvature scale. Using the parametric regime mentioned above, we get the horizon crossing instance of $k_{CMB}$ as $t_{h}\sim 10^{-68}\mathrm{GeV}^{-1}$. This leads to $t_{h}\ll t_{v}$, which in turn justifies that the power spectra are evaluated when the large scale modes are on super-Hubble scales.

Thus as a whole, the presence of a non-vanishing brane cosmological constant results to a phantom phase of the radion field during its evolution. The existence of such phantom phase leads to a violation of null energy condition and triggers a non-singular bounce which predicts a low tensor-to scalar ratio and gets well consistent with the Planck data for suitable regime of the model parameters.

We start with a second rank antisymmetric Kalb-Ramond (KR) field in F(R) gravity, and the action is Paul:2022mup ,

$\displaystyle S=\int d^{4}x\sqrt{-g}\left[\left(\frac{1}{2\kappa^{2}}\right)F(R)-\frac{1}{12}H_{\mu\nu\alpha}H_{\rho\beta\delta}g^{\mu\rho}g^{\nu\beta}g^{\alpha\delta}\right]~{},$

(24)

$\frac{1}{2\kappa^{2}}=M_{\mathrm{Pl}}^{2}$ ($M_{\mathrm{Pl}}$ being the Planck mass). $H_{\mu\nu\alpha}$ is the field strength tensor of KR field, defined by $H_{\mu\nu\alpha}=\partial_{[\mu}B_{\nu\alpha]}$. The above action can be mapped into the Einstein frame by using the following conformal transformation: $g_{\mu\nu}\longrightarrow\widetilde{g}_{\mu\nu}=\sqrt{\frac{2}{3}}\kappa\Phi(x^{\mu})~{}g_{\mu\nu}$, with $\Phi$ being the conformal factor and related to the spacetime curvature as $\Phi(R)=\frac{1}{\kappa}\sqrt{\frac{3}{2}}F^{\prime}(R)$. Consequently the action in Eq.(24) can be expressed as a scalar-tensor theory,

$\displaystyle S=\int d^{4}x\sqrt{-\widetilde{g}}\left[\frac{\widetilde{R}}{2\kappa^{2}}-\frac{3}{4\kappa^{2}}\left(\frac{1}{\Phi^{2}}\right)\widetilde{g}^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi-V(\Phi)-\frac{1}{12}\left(\sqrt{\frac{2}{3}}\kappa\Phi\right)H_{\mu\nu\rho}H^{\mu\nu\rho}\right]~{}.$

(25)

where $\widetilde{R}$ is the Ricci scalar formed by $\widetilde{g}_{\mu\nu}$ and $H^{\mu\nu\rho}=H_{\alpha\beta\delta}\widetilde{g}^{\mu\alpha}\widetilde{g}^{\nu\beta}\widetilde{g}^{\rho\delta}$. The scalar field potential depends on the form of F(R), and is given by

$\displaystyle V(\Phi)=\frac{1}{2\kappa^{2}}\left[\frac{RF^{\prime}(R)-F(R)}{F^{\prime}(R)^{2}}\right]~{}.$

(26)

For our present purpose, we consider the isotropic and homogeneous FRW metric:

$\displaystyle ds^{2}=-dt^{2}+a^{2}(t)\delta_{\mathrm{ij}}dx^{\mathrm{i}}dx^{\mathrm{j}}$

(27)

where $t$ and $a(t)$ are the cosmic time and the scale factor of the universe respectively. Here we would like to emphasize that $H_{\mu\nu\lambda}$ has four independent components in four dimensional spacetime due to its antisymmetric nature, and they can be expressed as,

$\displaystyle H_{\mathrm{012}}=h_{\mathrm{1}}~{}~{}~{},~{}~{}~{}H_{\mathrm{013}}=h_{\mathrm{2}}~{}~{}~{},~{}~{}~{}H_{\mathrm{023}}=h_{\mathrm{3}}~{}~{}~{},~{}~{}~{}H_{\mathrm{123}}=h_{\mathrm{4}}~{}.$

(28)

However due to the isotropic and homogeneous spacetime, the off-diagonal Friedmann equations lead to the following solutions:

$\displaystyle h_{\mathrm{1}}=h_{\mathrm{2}}=h_{\mathrm{3}}=0~{}~{}~{}~{}~{}~{}~{}~{}\mathrm{and}~{}~{}~{}~{}~{}~{}~{}h_{\mathrm{4}}\neq 0~{}.$

(29)

Using this solution, one easily obtains the independent field equations as follows,

	$\displaystyle H^{2}$	$\displaystyle=$	$\displaystyle\frac{\kappa^{2}}{3}\left[\frac{3}{4\kappa^{2}}\left(\frac{\dot{\Phi}}{\Phi}\right)^{2}+V(\Phi)+\frac{1}{2}\sqrt{\frac{2}{3}}\kappa\Phi~{}h_{\mathrm{4}}h^{\mathrm{4}}\right]~{},$		(30)
	$\displaystyle\frac{\ddot{\Phi}}{\Phi}$	$\displaystyle-$	$\displaystyle\left(\frac{\dot{\Phi}}{\Phi}\right)^{2}+3H\left(\frac{\dot{\Phi}}{\Phi}\right)+\left(\frac{2\kappa^{2}}{3}\right)\Phi~{}V^{\prime}(\Phi)+\left(\frac{\sqrt{2}}{3\sqrt{3}}\right)\kappa^{3}\Phi~{}h_{\mathrm{4}}h^{\mathrm{4}}=0$		(31)
	$\displaystyle 0$	$\displaystyle=$	$\displaystyle\frac{1}{a^{3}}\partial_{\mu}\left(a^{3}\Phi H^{\mu\nu\lambda}\right)$		(32)

which are the Friedmann equation, the scalar field equation and the KR field equation respectively. A little bit of playing with the above equations lead to the following solutions of KR field energy density:

$\displaystyle h_{\mathrm{4}}h^{\mathrm{4}}=h_{\mathrm{0}}/a^{\mathrm{6}}~{},$

(33)

with $h_{0}$ being an integration constant which is taken to be positive to ensure a real valued solution for $h^{4}(t)$. Furthermore, we consider an ansatz for the scalar field solution in terms of the scale factor as,

$\displaystyle\Phi(t)=-\frac{1}{\kappa}\sqrt{\frac{3}{2}}\left(\frac{1}{a^{\mathrm{n}}(t)}\right)~{},$

(34)

with $n>0$. Eq.(34) indicates that the scalar field acquires negative values during the cosmological evolution of the universe, which actually proves to be useful to get a non-singular bounce. With the above solutions for $\Phi(t)$ and $h_{4}(t)$, Eq.(30) provides a two branch solution of $H=H(a)$:

$\displaystyle H(a)=\pm a^{\mathrm{-3n^{2}/4}}\left\{C_{\mathrm{1}}+\frac{2\kappa^{2}h_{\mathrm{0}}a^{\mathrm{\left(3n^{2}-2n-12\right)/2}}}{\left(3n^{2}-2n-12\right)}\right\}^{1/2}~{},$

(35)

where $C_{1}$ is an integration constant. Therefore the evolution of $H(a)$ becomes different depending on whether $3n^{2}-2n-12<0$ or $3n^{2}-2n-12>0$. However both the cases will be proved to lead a non-singular bounce irrespective of the values of $n>0$. Here we discuss the case when $3n^{2}-2n-12<0$ and its consequences, while the other case can be described by a similar fashion Paul:2022mup .

Here we consider $3n^{2}-2n-12=-2q$, with $q>0$, and consequently, the solution of $H(a)$ in Eq.(35) can be expressed as Paul:2022mup ,

$\displaystyle H(a)=\pm\sqrt{\frac{\kappa^{2}h_{\mathrm{0}}}{q}}~{}a^{\mathrm{-3n^{2}/4}}\left\{\frac{1}{a_{\mathrm{0}}^{q}}-\frac{1}{a^{q}}\right\}^{1/2}$

(36)

where the integration constant $C_{1}$ is replaced by $a_{0}$ as $C_{\mathrm{1}}=\kappa^{2}h_{\mathrm{0}}/\left(qa_{\mathrm{0}}^{q}\right)$. The above expression of $H(a)$ satisfies the following two conditions at $a=a_{0}$,

$\displaystyle H(a=a_{0})=0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathrm{and}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{dH}{dt}\bigg{|}_{a_{0}}=\left(aH\right)\frac{dH}{da}\bigg{|}_{a_{0}}=\frac{\kappa^{2}h_{0}}{2a_{0}^{6+n}}~{},$

(37)

which clearly depicts that the universe experiences a non-singular bounce at $a=a_{0}$. Here it deserves mentioning that in absence of the KR field, the model is not able to predict a non-singular bounce of the universe. In particular, the Hubble parameter evolves as $H(a)\propto a^{-3n^{2}/4}$ when the KR field is absent, which does not lead to a bouncing scenario at all. It is important to realize that the KR field which has negligible footprints at present epoch of the universe, plays a significant role during the early universe to trigger a non-singular bounce.

Due to the ekpyrotic condition $n>2$, the comoving Hubble radius diverges to infinity at the deep contracting phase, which in turn indicates that the primordial perturbation modes generate during the contracting phase far away from the bounce when all the perturbation modes lie within the Hubble horizon. Moreover, since the model involves two fields, apart from the curvature perturbation, isocurvature perturbation also arises. In this regard, the ratio of the Kalb-Ramond to the scalaron field energy density is given by,

$\displaystyle\frac{\rho_{KR}}{\rho_{\Phi}}=\left[\frac{\frac{1}{2}\sqrt{\frac{2}{3}}\kappa\Phi~{}h_{\mathrm{4}}h^{\mathrm{4}}}{\frac{3}{4\kappa^{2}}\left(\frac{\dot{\Phi}}{\Phi}\right)^{2}+V(\Phi)}\right]~{}~{}.$

From the background solution of the KR and the scalaron field (that we have obtained earlier), one may argue that the Kalb-Ramond to the scalaron field energy density during the late stage of the universe goes by $1/a^{q}$ which tends to zero. This results to a weak coupling between the curvature and the isocurvature perturbations during the late evolution of the universe. However the ratio between $\rho_{KR}$ and $\rho_{\Phi}$ becomes comparable at the bounce, which in turn leads to a considerable coupling between the curvature and the isocurvature perturbations at the bounce. In the present context, we are interested to examine the curvature perturbation during the contracting universe (away from the bounce), and thus, we can safely ignore the coupling between the curvature and the isocurvature perturbations and solely concentrate on the curvature perturbation.

The Mukhanov-Sasaki (MS) variable of the curvature perturbation follows the equation (in the Fourier space and conformal time coordinate):

$\displaystyle v_{k}^{\prime\prime}(\eta)+\left(k^{2}-\frac{4\left(8-3n^{2}\right)}{\left(3n^{2}-4\right)^{2}\eta^{2}}\right)v_{k}(\eta)=0~{},$

(40)

on solving which for $v_{k}(\eta)$, we get,

$\displaystyle v(k,\eta)=\frac{\sqrt{\pi|\eta|}}{2}\left[c_{1}(k)H_{\nu}^{(1)}(k|\eta|)+c_{2}(k)H_{\nu}^{(2)}(k|\eta|)\right]\,,$

(41)

with $\nu=\sqrt{\frac{4\left(8-3n^{2}\right)}{\left(3n^{2}-4\right)^{2}}+\frac{1}{4}}$. Moreover $c_{1}$, $c_{2}$ are integration constants, $H_{\nu}^{(1)}(k|\eta|)$ and $H_{\nu}^{(2)}(k|\eta|)$ are the Hermite functions (having order $\nu$) of first and second kind, respectively. Considering the Bunch-Davies initial condition at the deep sub-Hubble scales, the curvature power spectrum in the super-Hubble scale becomes Paul:2022mup ,

$\displaystyle\mathcal{P}(k,\eta)=\left[\left(\frac{1}{2\pi}\right)\frac{1}{z\left|\eta\right|}\frac{\Gamma(\nu)}{\Gamma(3/2)}\right]^{2}\left(\frac{k|\eta|}{2}\right)^{3-2\nu}\,.$

(42)

Consequently the scalar spectral tilt becomes,

$\displaystyle n_{s}=\frac{9n^{2}-4}{3n^{2}-4}~{}.$

(43)

As demonstrated in Eq.(39), the parameter $n$ is constrained by $2<n<\frac{1}{3}\left(1+\sqrt{37}\right)$, which immediately leads to the following range of the spectral tilt: $3.6<n_{s}<4$. This indicates a blue-tilted curvature power spectrum, and thus is not consistent with the Planck 2018 results.

In order to get a scale invariant curvature power spectrum in the present context, we consider a quasi-matter dominated pre-ekpyrotic phase where the scale factor behaves as $a_{p}(t)\sim t^{2m}$ with $m<1/2$. Such a quasi-matter dominated phase can be realized by introducing a perfect fluid having constant EoS parameter $\approx 0$, in which case, the energy density grows as $\approx a^{-3}$ during the contracting phase. Thereby after some time, the KR field energy density (that grows as $a^{-{6+n}}$ with the universe’s contraction, see Eq.(33)) dominates over that of the perfect fluid and leads to an ekpyrotic phase of the universe.
In this modified cosmological scenario, the scale factor of the universe is:

	$\displaystyle a_{p}(t)=a_{\mathrm{1}}\left(\frac{\eta-\eta_{0}}{\eta_{e}-\eta_{0}}\right)^{2m/\left(1-2m\right)}\quad$	$\displaystyle\mbox{with}\quad m<1/2\,,\quad\mathrm{for}~{}\left|\eta\right|\geq\left|\eta_{e}\right|\,,$
	$\displaystyle a(t)=a_{2}\left(-\eta\right)^{4/\left(3n^{2}-4\right)}\quad$	$\displaystyle\mbox{with}\quad 2<n<\frac{1}{3}\left(1+\sqrt{37}\right)\,,\quad\mathrm{for}~{}\left|\eta\right|\leq\left|\eta_{e}\right|\,.$		(44)

Here $\eta_{e}$ represents the conformal time when the transition from the pre-ekpyrotic to the ekpyrotic phase occurs, and $\eta_{0}$ is a fiducial time. Moreover the exponent $m<1/2$ so that the comoving Hubble radius diverges at the distant past and the perturbation modes generate at the deep contracting phase within the sub-Hubble regime. As a whole, in this modified cosmological scenario, the scale factor of the universe is:

	$\displaystyle a_{p}(t)=a_{\mathrm{1}}\left(\frac{\eta-\eta_{0}}{\eta_{e}-\eta_{0}}\right)^{2m/\left(1-2m\right)}\quad$	$\displaystyle\mbox{with}\quad m<1/2\,,\quad\mathrm{for}~{}\left|\eta\right|\geq\left|\eta_{e}\right|\,,$
	$\displaystyle a(t)=a_{2}\left(-\eta\right)^{4/\left(3n^{2}-4\right)}\quad$	$\displaystyle\mbox{with}\quad 2<n<\frac{1}{3}\left(1+\sqrt{37}\right)\,,\quad\mathrm{for}~{}\left|\eta\right|\leq\left|\eta_{e}\right|\,.$		(45)

The continuity of the scale factor as well as of the Hubble parameter at the transition time $\eta=\eta_{e}$ result to the following expressions,

	$\displaystyle a_{1}$	$\displaystyle=$	$\displaystyle a_{2}\left(-\eta_{e}\right)^{4/\left(3n^{2}-4\right)}~{},$
	$\displaystyle\left(\frac{2m}{1-2m}\right)\frac{1}{\left(\eta_{e}-\eta_{0}\right)}$	$\displaystyle=$	$\displaystyle\left(\frac{4}{3n^{2}-4}\right)\frac{1}{\eta_{e}}~{},$		(46)

respectively. In effect of the pre-ekpyrotic phase of contraction, the large scale perturbation modes cross the horizon either during the pre-ekpyrotic or during the ekpyrotic stage depending on whether the transition time ($\eta_{e}$) is larger than the horizon crossing instant of the large scale modes ($\eta_{h}$) or not. For a scale invariant power spectrum, here we consider the first case, i.e when the large scale modes cross the horizon during the pre-ekpyrotic phase. Thus the horizon crossing instant for $k$th mode is given by,

$\displaystyle\left|\eta_{h}\right|=\left(\frac{2m}{1-2m}\right)\frac{1}{k}\,.$

(47)

Consequently the horizon crossing instant for the large scale modes, in particular $k=0.002\mathrm{Mpc}^{-1}$, is estimated as,

$\displaystyle\left|\eta_{h}\right|\approx\left(\frac{2m}{1-2m}\right)\times 13\,\mathrm{By}\,.$

(48)

Thus one may argue that the transition from the pre-ekpyrotic to the ekpyrotic phase occurs at $\left|\eta_{e}\right|<13\mathrm{By}$ so that the large scale modes cross the horizon during the pre-ekpyrotic era. Following the same procedure as of the previous section, we calculate the spectral tilt for the curvature perturbation in the modified scenario where the ekpyrotic phase is preceded by a period of a pre-ekpyrotic contraction:

$\displaystyle n_{s}=\frac{5-14m}{1-2m}\,.$

(49)

Clearly for $m=1/3$ which describes a matter dominated epoch before the ekpyrotic phase, the spectral tilt becomes unity – i.e an exact scale invariant power spectrum is predicted when the curvature perturbations over the large scale modes generate during a matter dominated pre-ekpyrotic era. However the observations according to the Planck data depict that the curvature power spectrum should not be exactly flat, but a has a slight red tilt. For this purpose, we give a plot of $n_{s}$ with respect to $m$ in Fig.[4]. The figure clearly demonstrates that the theoretical prediction of $n_{s}$ becomes consistent with the Planck 2018 data if the parameter $m$ lies within $0.3341\lesssim m\lesssim 0.3344$. Therefore the spectral index for the primordial curvature perturbation, on scales that cross the horizon during the pre-ekpyrotic stage with $0.3341\lesssim m\lesssim 0.3344$, is found to be consistent with the recent Planck observations.