Bouncing cosmology in a curved braneworld

One of the major challenges in modern theoretical cosmology is to explain the early stage of the universe, in particular, whether the universe emerged from an initial singularity (also known as the Big-Bang singularity) or the universe underwent a non-singular bounce leading to a possible singularity free expansion of the universe. Some of the early universe scenarios, proposed so far, that can generate an almost scale invariant power spectrum and hence confront the observational constraints are the inflationary scenario [1, 2, 3, 4, 5, 6, 7, 8], the bouncing universe [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 70, 56, 68, 69, 57], the emergent universe scenario [71, 72, 73, 74, 75, 76] and the string gas cosmology [77, 78, 79, 80, 81, 82, 83].

In this work, we study the bouncing scenario in a non-flat warped braneworld model. The bouncing scenario consists of two eras$-$an era of contraction and an era of expansion of the scale factor, both the eras being connected by a non-singular bounce [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 70, 56, 68, 69, 57]. Beside producing an observationally compatible primordial power spectrum, the bouncing scenario has the merit to give rise to a singularity free evolution of the early universe. Although it has been argued that the Big-Bang singularity may be avoided through a suitable quantum generalization of gravity, the absence of a consistent quantum theory of gravity makes the bouncing description of the universe a promising scenario. In this context it may be metioned that string theory [86, 87, 84, 85] inherently incorporates the quantum nature of gravity and is associated with several extra spatial dimensions. Although originally intended to unify the known forces of nature, it turns out that extra dimensions can also provide plausible resolution to the gauge-hierarchy problem or the finetuning problem in particle physics arising due to large quantum corrections of the Higgs mass [88, 86, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97]. In this context warped geometry models are particularly relevant. In particular, the warped geometry models due to Randall-Sundrum (RS) [94] earned a lot of attention since it resolves the gauge hierarchy problem without introducing any intermediate scale (between Planck and TeV scale) in the theory.

The RS scenario consists of an extra spatial dimension (over the usual four dimensional spacetime) with $S^{1}/Z_{2}$ orbifold symmetry, enclosed between two 3-branes which are considered to be flat. The distance $r_{c}$ between the two branes governs the magnitude of the brane warping and therefore plays the crucial role in resolving the gauge-hierarchy problem. The assumption of flat branes which gives rise to a vanishing brane cosmological constant in the RS scenario can be relaxed in a generalized warped braneworld model [98], which allows the branes to be non-flat giving rise to de Sitter (dS) or anti-de Sitter (AdS) branes. The interplay of the brane warping (which depends on the interbrane distance $r_{c}$) and the magnitude of the brane cosmological constant leads to the resolution of the gauge-hierarchy problem in such models. The cosmological, astrophysical and phenomenological implications of warped braneworld models (with flat or non-flat branes) have been discussed in [99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121]. Since the resolution of the finetuning problem essentially depends on the interbrane distance $r_{c}$, the stabilization of $r_{c}$ to the appropriate value becomes crucial. The stabilization is achieved in the RS scenario by introducing a scalar field in the bulk [123, 124], which leads to a potential for $r_{c}$ in the four dimensional effective action whose minima can be suitably adjusted to address the finetuning problem. The origin of the bulk scalar however remains unexplored. This problem can evaded in the non-flat warped braneworld scenario with dynamical branes such that the interbrane distance attains the status of a field (the so called radion or the modulus). Such a framework enables the radion to generate its own potential along with a non-canonical kinetic term in the four dimensional effective action which in turn can stabilize the modulus to the suitable value [125, 120], without invoking any additional scalar field in the theory.

The non-canonical scalar kinetic term becomes negative for certain values of the modulus which endows the radion a phantom-like behavior where the null energy condition is violated. Such a violation is a generic feature observed in a bouncing universe, which motivates us to explore the prospect of the non-flat warped braneworld model in addressing bouncing cosmology. We investigate the cosmological evolution of the radion field in the FRW background and the subsequent evolution of the primordial fluctuations which allows us to understand the viability of the model in purview of the Planck 2018 constraints.

The paper is organized as follows: in 2, we briefly describe the non-flat warped braneworld model and its four dimensional effective theory. Having set the stage, 3 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 4. We conclude with a summary of our results and a discussion of our findings in 5.

Randall & Sundrum (RS)[94] proposed the warped braneworld scenario to address the fine-tuning problem in particle physics. The RS model consists of a 5-dimensional AdS bulk bounded by two 3-branes, namely the visible brane (where our 4-d universe resides) and the hidden brane. The extra dimension denoted by $\phi$ is associated with a $S^{1}/Z_{2}$ orbifold symmetry and the hidden brane resides at $\phi=0$ while the visible brane is located at $\phi=\pi$. In the RS scenario, the bulk metric is described by

from which it is evident that the branes are flat which is ensured by the exact cancellation of the brane tension and the cosmological constant induced on the brane [126]. In 1 the warp factor is represented by $e^{-2A}$ with $A=k_{0}r_{c}|\phi|$ where $r_{c}$ is the compactification radius and $k_{0}=\sqrt{-\Lambda/24M^{3}}$ such that $\Lambda$ and $M$ denote the five dimensional cosmological constant and Planck mass respectively. The presence of the exponential warp factor in the metric ensures that $r_{c}\sim 12$ is sufficient to bring down the Higgs mass from the Planck scale to the TeV scale on the visible brane without introducing any new energy scale in the theory.

A generalization of the RS model to incorporate non-flat branes is important since the non-flatness of our universe is often evident from the physical situations, e.g., an expanding universe, the presence of black holes etc. This is achieved by replacing the brane metric $\eta_{\mu\nu}$ with $g_{\mu\nu}$ in the above metric ansatz. This induces a non-zero cosmological constant $\Omega$ on the brane inherited from the bulk which can be both positive or negative. In the situation where the branes are de-Sitter, the warp factor is given by [98]:

where, $\omega=(\Omega/3k_{0}^{2})$ is a dimensionless constant directly proportional to the brane cosmological constant $\Omega$ and $c_{2}=1+\sqrt{1+\omega^{2}}$. It can be shown that the above warp factor can give rise to the requisite warping of the Higgs mass on the visible brane (i.e., $k_{0}r_{c}\pi\sim 16\ln 10$) as in the RS scenario while keeping $\Omega\sim 10^{-124}$ (the magnitude of the present day cosmological constant in Planckian units).

For AdS branes, it can be shown that the warp factor is given by $e^{-A^{\prime}}=\omega cosh(ln\frac{\omega}{c_{1}}+k_{0}r_{c}|\phi|)$, with $c_{1}=1+\sqrt{1-\omega^{2}}$ [98]. We further note that in the event $\omega\rightarrow 0$, we retrieve the RS warp factor describing the flat braneworld scenario, for both the dS and AdS branes. 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 and investigate its role in bouncing cosmology.

In this section we explore the role of the radion field in triggering a bouncing universe which can potentially avoid the big bang singularity. The bouncing scenario often invokes a scalar field with a potential and there exist plenty of models in the literature which can give rise to such a scenario (see [11]). In most of the cases the scalar potentials are reconstructed to explain the observations and often their origin remains unexplained. The merit of the non-flat warped braneworld model lies in the fact that the the radion field naturally arises from compactification in the effective four-dimensional theory and generates its own potential and non-canonical kinetic term. Here we consider the implications of the radion field in inducing a bouncing universe.

Since we are interested to study early universe cosmology with the radion field we consider the metric in the Einstein frame to be described by the FRW spacetime in the spatially flat form,

with $a(t)$ is known as the scale factor of the universe. In the generalized RS scenario, the 3-branes can be Minkowskian, de-Sitter or anti de-Sitter depending on the values of the induced brane cosmological constant [98]. Recall, in the present work, we consider the branes to be de-Sitter, i.e our visible universe is a 3-brane (i.e having three spatial dimension along with the time coordinate), described by the spatially flat FRW metric. Since the FRW metric is curved irrespective of its spatial curvature, the visible 3-brane is non-flat. At this stage it deserves mentioning that the spatially flat FRW metric is more consistent over the closed or open FRW universe from latest Planck 2018 data through TT, TE, EE + lowE + lensing + BAO data, where TT means temperature temperature cross-correlation of CMB data, TE means cross-correlation between temperature and electric type polarization of CMB data and finally BAO stands for Baryon Acoustic Oscillation [122]. Due to the time dependency of the scale factor, the metric in 21 indicates a non-zero curvature on the four dimensional brane geometry and moreover the brane curvature is characterized by the corresponding Ricci scalar given by $R=6\frac{\ddot{a}}{a}+6\frac{\dot{a}^{2}}{a^{2}}$. As we will see from 25 that the kinetic as well as the potential energy of radion field contributes to the on-brane Ricci scalar via the effective four dimensional Freidmann equation. Furthermore, as shown in 19, the potential energy of the radion field is proportional to the induced brane cosmological constant and thus one may argue that the radion potential energy is generated entirely due to the presence of the non-zero brane cosmological constant. Thereby the brane cosmological constant affects the evolution of the scale factor through the potential energy density of the radion field. Below, we will show that the metric ansatz of 21 is consistent with the field equations of motion and moreover it will lead to a non-singular bounce on our visible brane.

It is evident from 18 that the energy momentum tensor $T^{\mu}_{\nu}$ due to the radion field is given by,

such that

represents the energy density while

corresponds to the pressure due to the radion field. We note that the radion field $\xi$ depends only on time since the background metric given by 21 is only time dependent.

Using 23 the Friedmann equation obtained from the temporal component of the Einstein’s equations assume the form,

while the Friedman equation derived from the spatial component of the Einstein’s equations is given by,

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

Using 23 and 24, 27 can be written as,

25, 26 and 28 are the background equations, although it is important to note that 28 is not independent but can be derived from 25 and 26.
At this stage it deserves mentioning that in a canonical scalar tensor theory, the Friedmann equation becomes $\dot{H}\propto-\dot{\xi}^{2}$ (26) and thus the Hubble parameter decreases monotonically with cosmic time. Therefore a bounce phenomena is impossible in a canonical scalar tensor model as it cannot give rise to $\dot{H}>0$ which is one of the necessary conditions to get a bounce. On the contrary, in a non-canonical scalar tensor theory where the scalar field has non-canonical kinetic term (as $G(\xi)$ in the present context), the Friedmann equations are modified due to the presence of $G(\xi)$ and the modified equations are given by 25 and 26 respectively. 26 clearly indicates that in a non-canonical scalar tensor model, the sign of $G(\xi)$ actually controls the energy condition, in particular $G(\xi)<0$ leads to a violation of null energy condition which in turn may ensure a bouncing phase in our visible universe. We have already noted in 3 that in the present scalar-tensor model the non-canonical kinetic term exhibits a transition from a normal to a phantom regime (i.e from $G(\xi)>0$ to $G(\xi)<0$) where the null energy condition is violated. Therefore it is important to investigate the prospect of bouncing cosmology with the present non-flat warped braneworld model which we explore next. In particular, we first present the background evolution of $H(t)$ and $\xi(t)$ (governed by 25 and 26) in the next section and subsequently study the evolution of the perturbations in 4.

In general, a non-singular bounce is characterized by the conditions $H(t_{b})=0$ and $\dot{H}(t_{b})>0$ where $t_{b}$ is the cosmic time when the bounce occurs. Keeping these conditions in mind, if we look into 25 and 26, then it is evident 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. The analysis in 3 reveals that $G(\xi)$ is indeed 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 previously found analytic solutions.
In particular we consider,

with $\delta(t)\ll 1$. Due to the above form of $\xi(t)$, $h(\xi)$ in 12 simplifies to

such that $V(\xi)$ is given by,

while $G(\xi)$ can be approximated as,

The above simplifications in the form of $V(\xi)$ and $G(\xi)$ hold only in the regime where $\xi(t)$ is given by 29 with $\delta(t)\ll 1$. With these simplifications the evolution equations for the Hubble parameter $H(t)$ and the radion field (i.e 25 and 26) turn out to be,

and

respectively. 33 can be solved to obtain the time evolution of the Hubble parameter which assumes the form,

such that $\dot{H}$ is given by,

Using 36 in 34 we obtain,

with $A=\frac{4+2ln\big{(}\frac{c_{2}}{\omega}\big{)}}{ln\big{(}\frac{c_{2}}{\omega}\big{)}}$ and solving the above equation yields the following time evolution for $\delta(t)$,

where $C_{1}$ is an integration constant which can be determined by demanding,

which implies that $\xi(t)$ (i.e the radion field) monotonically decreases with time and asymptotically goes to the value $\frac{\omega}{c_{2}}$ which is the minimum possible value of $\xi$ from the requirement of positive warp factor, as discussed after 9. From the condition given by 39, the form of $C_{1}$ turns out to be,

which when substituted in 38 gives,

With this, we arrive at the background solution for $H(t)$ and $\delta(t)$ in the regime $\xi(t)=\frac{\omega}{c_{2}}(1+\delta(t))$ with $\delta(t)\ll 1$,

where, $A=\frac{4+2ln\big{(}\frac{c_{2}}{\omega}\big{)}}{ln\big{(}\frac{c_{2}}{\omega}\big{)}}$. 42 clearly indicates $H(0)=0$ and $\dot{H}>0$ at $t=0$ (corresponding to the bounce time) which are the necessary conditions for a non-singular bounce. Therefore, the present non-flat warped braneworld model predicts a bouncing universe in the visible brane when the radion field lies within the phantom regime, in particular near $\xi\sim\omega$. In the phantom regime, due to the negative kinetic energy of the radion field, the effective null energy condition (NEC) is violated and makes the bounce possible at a certain finite time, in particular at $t=0$. Here we would like to mention that such NEC violation occurs irrespective of any values of $\frac{\omega}{c_{2}}$ and $k_{0}$ (i.e the model parameters), and moreover the initial conditions of the solution of 42 is free from fine tuning of the model parameters. Thereby, we may argue that the bounce solution in the present context is a generic feature and does not require any fine-tuned values of the model parameters.

At this stage, it is important to check whether the radion field, starting from a value in the normal regime, will reach to the phantom regime by its $dynamical~{}evolution$. For this purpose, we solve the coupled equations for $H(t)$ and $\xi(t)$ (i.e 25 and 26) 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 42, in particular the boundary conditions are given by $H(0)=0$ 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 2a and 2b respectively (the radion field plot is magnified 1000 times i.e $\xi(t)\times 1000$). In the inset of 2b, the magenta curve denotes the time evolution of $G(\xi)$ while the blue curve represents the zoomed-in version of $\xi(t)\times 1000$ near the zero crossing of $G(\xi)$. From 2b it is evident that $G(\xi)$ exhibits a transition from a normal regime (where $G(\xi)>0$) to a phantom regime (where $G(\xi)<0$) with its zero crossing occurs at a finite time before the bounce at $t=0$. Moreover 2b demonstrates that there is no divergence in the dynamical evolution of $\xi(t)$ as $G(\xi)$ transits from normal to the phantom regime. However on the other hand, as evident from 26, $\frac{d\xi}{dt}=\dot{\xi}$ diverges at the time when the non-canonical kinetic coupling $G(\xi)$ makes the zero crossing. Here we would like to mention that such divergence of $\dot{\xi}$ does not lead to any pathology to the radion field equation of motion i.e to 28 and the reason is following: 28 can be equivalently expressed as $\frac{d}{dt}\bigg{(}\frac{3M^{3}c_{2}^{2}}{k_{0}}G(\xi)\dot{\xi}^{2}+2M^{3}k_{0}V(\xi)\bigg{)}+\frac{18M^{3}c_{2}^{2}}{k_{0}}HG(\xi)\dot{\xi}^{2}=0$ which includes $G(\xi)\dot{\xi}^{2}$ and its derivative with respect to cosmic time. Now 26 evidents that $G(\xi)\dot{\xi}^{2}$ is proportional to $\dot{H}$ which, along with its derivative, is indeed finite for all possible cosmic time (see 2a). Thereby the term $G(\xi)\dot{\xi}^{2}$ and its derivative with respect to $t$ are finite everywhere even at $G(\xi)\rightarrow 0$, and thus we may argue that the radion field equation of motion does not lead to any inconsistency in the present context.

2a reveals that the Hubble parameter becomes zero and increases with respect to cosmic time at $t=0$, which confirms a non-singular bounce at $t=0$. Before demonstrating the dynamics of the radion field, we recall that for $\omega=10^{-3}$, the zero crossing of $G(\xi)$ occurs at $\xi=\xi_{f}\simeq 0.00148$, as shown in 1b i.e. $G(\xi)$ exhibits the normal to phantom transition as $\xi$ crosses $\xi_{f}=0.00148$ from higher values. Numerical solution of 25 and 26 indicates that the radion field starts its journey from the normal regime (i.e $\xi>\xi_{f}$) and dynamically moves to the phantom era (i.e $\xi<\xi_{f}$) with time by monotonically decreasing in magnitude and asymptotically stabilizes to the value $\xi_{i}\to\omega/c_{2}$ which for $\omega=10^{-3}\sim\xi_{i}=5\times 10^{-4}$. This is in accordance with the analytical results obtained in 42 and the time evolution of the background radion field is explicitly illustrated in 2b. As the radion field asymptotically tends to $\xi_{i}$ (i.e., $\xi\to\xi_{i}+\varepsilon$), the warp factor $e^{-A}\simeq c_{2}\varepsilon$ which in turn resolves the gauge-hierarchy problem for $\varepsilon\simeq 10^{-16}$ while the stabilized inter-brane separation $k_{0}\pi\langle T\rangle\to\ln{\big{(}\frac{c_{2}}{\omega}\big{)}}$ [125]. Therefore, in the non-flat warped braneworld scenario with dynamical branes, the radion generates its own potential which in turn stabilizes the modulus dynamically in the FRW background. Further, the presence of the phantom era enables violation of the null energy condition for the radion field which makes this a promising model to explore the bouncing scenaio.

At this stage, it may be mentioned that a holonomy improved non-canonical scalar tensor model may rescue the energy condition in a bouncing scenario. In the holonomy generalized model, the squared Hubble parameter (i.e $H^{2}$ in 25) is proportional to the linear as well as quadratic power of energy density, unlike the usual Friedmann equations where $H^{2}$ is proportional only to the linear power of energy density. Such difference in the field equations may play a significant role to rescue the null energy condition necessary for a non-singular bounce. This investigation is expected to be carried out soon in a future work.

In this section, 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 ($A_{s}$). In a bouncing universe, the Hubble parameter becomes zero and consequently the comoving Hubble radius diverges at the bouncing point. However, the asymptotic behaviour of the Hubble radius differentiates various bouncing models which can be broadly classified into two scenarios. In the first case, the comoving Hubble radius decreases and goes to zero asymptotically with time, which corresponds to a late time accelerating universe. In this case, the perturbation modes generate near the bounce, because at that time, the horizon has an infinite size and all the perturbation modes lie within the horizon. In the second situation the Hubble radius diverges asymptotically with time, which indicates a decelerating universe at late time and consequently the primordial perturbation modes relevant for the present era generate at a distant past far away from the bounce. More explicitly, in the latter case, the comoving wave number $k$ begins its journey from the infinite past in the contracting universe, within the sub-Hubble scale, exits the horizon as it contracts, and again re-enters the horizon in the low curvature regime of the expanding phase and becomes relevant for present time observations. Therefore, depending on the asymptotic behavior of the Hubble radius, the perturbation modes in a bounce model generate either near the bounce or far away from the bounce deeply in the contracting regime.

Based on the above arguments, before moving to the perturbation calculations, we would like to investigate the asymptotic behaviour of the comoving Hubble radius (defined by $\frac{1}{aH}$) in the context of present model. Using the background solution of the Hubble parameter from 2a, we give the evolution of $\frac{1}{aH}$ with respect to cosmic time in 3a which clearly demonstrates that the comoving Hubble radius monotonically decreases with time and goes to zero asymptotically on both sides of the bounce. Here it may be mentioned that unlike to the comoving Hubble radius, the inverse Hubble parameter does not go to zero asymptotically but reaches to a constant value at late stage of the universe, which is depicted in 3b showing the behaviour of $H^{-1}$ vs. $t$. This corresponds to a late time accelerating universe. The asymptotic evolution of the comoving Hubble radius leads to the perturbation modes generate near the bouncing regime where the Hubble radius has an infinite size such that all the perturbation modes are contained inside the horizon. In this regard the present scenario is different from the usual matter bounce models where the Hubble radius diverges asymptotically and the perturbation modes generate far away from the bounce. Therefore in the next section we solve the perturbation equations near the bouncing point $t=0$.