Bouncing cosmology from warped extra dimensional scenario
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Abstract
From the perspective of four dimensional effective theory on a two brane warped geometry model, we examine the possibility of “bouncing phenomena”on our visible brane. Our results reveal that the presence of a warped extra dimension lead to a nonsingular bounce on the brane scale factor and hence can remove the “bigbang singularity”. We also examine the possible parametric regions for which this bouncing is possible.
1 Introduction
Over the last two decades models with extra spatial dimensions [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] have been increasingly playing a central role in physics beyond the standard model of particle [14] and cosmological [15] physics. Apart from the phenomenological approach, higher dimensional scenarios occur naturally in string theory. Depending on different possible compactification schemes for the extra dimensions, a large number of models have been constructed, and their predictions are yet to be observed in the current experiments. In all these models, our visible universe is identified as one of the three branes embedded within a higher dimensional spacetime. The low energy effective description [16, 17, 18] of the dynamical three brane turned out to be a very powerful tool in studying the dynamics ranging from particle to cosmology. In our present work we will take this ansatz to understand cosmological bouncing phenomena in the early universe cosmology considering the Randall–Sundrum two brane model.
Among various extra dimensional models proposed over the last several years, the Randall–Sundrum (RS) warped extra dimensional model [6] earned special attention since it can resolve the gauge hierarchy problem without introducing any intermediate scale (between Planck and TeV) in the theory. The RS model is a five dimensional AdS space with \(S^1/Z_2\) orbifolding along the extra dimension while two three branes are placed at the orbifold fixed points. The bulk negative cosmological constant along with appropriate boundary conditions generate exponentially warped geometry along the extra dimension. Due to this exponential warping, the Planck scale on one brane gets suppressed along the extra dimension and emerges as TeV scale [6] on the visible brane. In the RS model the interbrane separation (known as modulus or radion) is \(\sim \) Planck length and generates the required hierarchy between the branes. Subsequently, Goldberger and Wise (GW) proposed a modulus stabilzation mechanism [19] by introducing a massive scalar field in the bulk with appropriate boundary conditions. Different variants of the RS model and its modulus stabilization are extensively studied in [21, 22, 23, 24, 25, 26, 27, 28]. In this paper we will consider a specific variant of RS scenario and study the cosmological dynamics from the perspective of low energy effective field theory induced on the visible brane.
It is well known that standard Big Bang scenario is quite successful in explaining many aspects of cosmological evolution of our universe. However, the bigbang model is plagued with a singularity (known as “cosmological singularity”) in the finite past. Resolving this time like cosmological singularity is an important issue which is a subject of great research in theoretical cosmology for the last several decades. It is widely believed that quantum theory of gravity, if any, should play very important role in resolving this singularity. One of the important aspects of all the known nonsingular cosmological models is the existence of pre BigBang universe [29]. In terms of effective theory, different models of nonsingular cosmologies, such as ekpyrotic universe [30, 31], loop quantum cosmology [32, 33], Galileon genesis [34, 35, 36], or the classical bouncing model, can be described by gravity coupled to a scalar field which generically violates the null energy condition at the background level. Therefore, the scale factor of the universe undergoes a nonsingular bounce from a preexisting universe to the present universe. This fact resulted in a reasonable amount of work on classical bouncing cosmology [37, 38, 39, 40, 41, 42, 43], with/without the presence of matter components (see also [44, 45, 46, 47]).

Can the effect of extra dimension trigger a nonsingular bounce on the brane scale factor and allow one to remove the “bigbang singularity” ?
The aim of this paper is to address the aforementioned question in the backdrop of a generalized scenario of the RS model proposed in [16]. The effective onbrane action we used in this paper has been formulated by Kanno and Soda in [16] by the method of a “low energy expansion scheme”.
Our paper is organized as follows: in Sect. 2, we briefly describe the generalized RS model and its effective action on the visible brane. In Sect. 3, we present the cosmological solutions of the effective Friedmann equations. The stabilization mechanism of the radion field is discussed in Sect. 4 and finally we end the paper with some conclusive remarks.
2 Low energy effective action on the visible brane
In the RS model, the Einstein equations are derived for a fixed interbrane separation as well as for flat three branes. However, the scenario changes if the distance between the branes becomes a function of spacetime coordinates and the brane geometry is curved. These generalizations are incorporated while deriving the effective action on the TeV brane via the “low energy expansion scheme” proposed in [16].
 The effective four dimensional action from the constant part of \(R^{(5)}\) iswhere we use the expression of \(A(\varphi ,x)\) (see Eq. (6) below) and the fact that the spacetime has \(S^1/Z_2\) orbifolding along \(\varphi \).$$\begin{aligned} S_{1}= & {} \frac{1}{2\kappa ^2} \int \mathrm{d}^4x\nonumber \\&\times \int ^{\pi }_{\pi } \mathrm{d}\varphi \sqrt{G} \bigg [\frac{16}{bl}\big [\delta (\varphi )\delta (\varphi \pi )\big ]  \frac{20}{l^2}\bigg ]\nonumber \\= & {} \frac{3}{\kappa ^2l} \int \mathrm{d}^4x \sqrt{h} \bigg [1  e^{4\frac{b}{l}\pi }\bigg ] \end{aligned}$$(11)
 The effective action from the bulk cosmological constant is$$\begin{aligned} S_{2}= & {} \frac{1}{2\kappa ^2} \int \mathrm{d}^4x \int ^{\pi }_{\pi } \mathrm{d}\varphi \sqrt{G} \frac{12}{l^2}\nonumber \\= & {} \frac{3}{\kappa ^2l} \int \mathrm{d}^4x \sqrt{h} \bigg [1  e^{4\frac{b}{l}\pi }\bigg ]. \end{aligned}$$(12)
 The effective action from the brane tensions iswhere we use the expressions of the brane tensions obtained in Eq. (10).$$\begin{aligned} S_{3}= & {}  \int \mathrm{d}^4x \sqrt{h} \bigg [V_{\mathrm{hid}} + e^{4\frac{b}{l}\pi }V_{\mathrm{vis}}\bigg ]\nonumber \\= & {} \frac{6}{\kappa ^2l} \int \mathrm{d}^4x \sqrt{h} \bigg [1  e^{4\frac{b}{l}\pi }\bigg ] \end{aligned}$$(13)
3 Cosmological solution for effective onbrane theory
3.1 Step 1: solution for \(y=y(\eta )\)
3.2 Step 2: solution for \(\Phi =\Phi (\eta )\)
3.3 Step 3: solution for \(a=a(\eta )\)
3.4 Step 4: solution for \(a=a(t)\) and \(\Phi =\Phi (t)\)
However, it can be checked from Eq. (29) that \(\Phi (t)\) has a positive asymptotic value as \(t\rightarrow \infty \) and goes to zero at \(t=\frac{\sqrt{BD}}{2}\). Using Eq. (29) and the relation \(\Phi (t)=[\exp {\big (2\pi \frac{b(t)}{l}\big )} 1]\), we obtain Fig. 1, demonstrating the variation of the interbrane separation (b(t)) with time.
Figure 1 clearly reveals that the branes collapse into each other within a finite time \(t=\frac{\sqrt{BD}}{2}\), which indicates the instability of the entire setup. Thus we need a suitable mechanism to stabilize the modulus. Following the procedure adopted in [19, 20], the stabilization method for the present setup is discussed in the next section.
4 Radion stabilization
In order to address the stabilization of the time dependent radion field, one needs to consider a dynamical stabilization mechanism, which can be achieved by a time dependent generalization of the Goldberger–Wise (GW) mechanism [19]. Earlier, a similar approach was adopted in [20]. Introducing a time dependent scalar field (with quartic brane interactions) in the bulk [19, 20], we address the dynamics of modulus stabilization without sacrificing the conditions necessary to resolve the gauge hierarchy problem.
4.1 Determination of the constants: \(f_0\) and \(E_0\)
Figure 2 clearly depicts \(b_{\mathrm{min}}(t)\), showing it to be nonvanishing for the entire range of t (\(\infty<t< \infty \)) and saturating at the Goldberger–Wise value (\(b_{\mathrm{GW}}\)) at large time. Thus the time dependent modulus can be stabilized by imposing a time dependent massive scalar field in the bulk. Moreover, we fix the integration constants (\(f_0\), \(E_0\)) in such a way that the solution of the gauge hierarchy problem is ensured.
Figure 3 clearly demonstrates that the feature of the bouncing phenomena remains unaffected due to the effect of the stabilizing scalar field.
Figure 4 clearly demonstrates that the free kinetic energy density of the radion field remains always positive while the interaction energy density (between radion and gravitational field) becomes negative for a certain regime of time. The negativity of \(\rho _{\mathrm{coup}}\) along with the positive value of \(\rho _{\mathrm{cur}}\) and \(\rho _{\mathrm{rad}}\) cancel each other to generate a zero effective energy density at \(t=0\), which causes the bounce.
5 Conclusion
We consider a five dimensional AdS compactified warped geometric model with two three branes residing at the orbifold fixed points. Our universe is identified with the visible brane. Instead of considering the five dimensional dynamics of the brane under gravity, we studied the low energy effective theory induced on our brane following Ref. [16]. In the high bulk curvature limit, the induced four dimensional effective theory appeared to be a Brans–Dicke type theory where the scalar field is playing the role of the distance modulus between the two branes. In this paper, we investigate the possibility of having a classical bouncing solution in the visible three brane (i.e. our universe). Out of three possible spatial curvatures of the Friedmann–Robertson–Walker brane, the bouncing solution exists only for hyperbolic spatial curvature \((\kappa = 1)\). Following the procedure as mentioned in Sect. III, it can be shown easily that, for \(\kappa = 0\) and for \(\kappa = +1\), one cannot have any bouncing solution. While finding the solution for \(\kappa =1\), we also introduce the stabilization mechanism to make sure that the two branes do not collapse, and maintain the hierarchy of scale in the asymptotic limit. In addition, we also need to satisfy the specific constraint \(0<D< 1\) to ensure the real valued bouncing solution for the scale factor.
As the solution of the radion field presented in Eq. (29) clearly implies, in the epoch after the bouncing, (depicted in Fig. 1), the two branes would collapse, leading to instability. Therefore, in order to stabilize this, a time dependent massive scalar field is introduced in the bulk. Thus we have a dynamical stabilization of the RS model, where in the asymptotic past the hierarchy of scales was larger than that of the present Goldberger–Wise value, which is achieved in the asymptotic future. This is clearly demonstrated in Fig. 2. We have determined the stabilization condition in Eq. (39), and finally taking this into account, we numerically solve the Hubble parameter as shown in Fig. 3. This clearly reveals that the “bouncing” phenomenon is not affected by the stabilizing scalar field.
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