Dynamical Casimir effect for gravitons in bouncing braneworlds

TL;DR

This paper investigates the creation of gravitons in a five-dimensional braneworld scenario with bouncing branes, analyzing their spectrum, energy density, and implications for cosmology, including constraints from nucleosynthesis and dark matter considerations.

Contribution

It provides a detailed calculation of graviton production in a bouncing braneworld, highlighting the spectrum, energy density, and the role of Kaluza-Klein modes, with implications for cosmology.

Findings

Massless gravitons have a blue spectrum.

Energy density satisfies nucleosynthesis bounds.

Kaluza-Klein modes unlikely to be dark matter.

Abstract

We consider a two-brane system in a five-dimensional anti-de Sitter spacetime. We study particle creation due to the motion of the physical brane which first approaches the second static brane (contraction) and then recedes from it(expansion). The spectrum and the energy density of the generated gravitons are calculated. We show that the massless gravitons have a blue spectrum and that their energy density satisfies the nucleosynthesis bound with very mild constraints on the parameters. We also show that the Kaluza-Klein modes cannot provide the dark matter in an anti-de-Sitter braneworld. However, for natural choices of parameters, backreaction from the Kaluza-Klein gravitons may well become important. The main findings of this work have been published in the form of a Letter [R. Durrer and M. Ruser, Phys. Rev. Lett. 99, 071601 (2007), arXiv:0704.0756].

Figures

Equations

$$d{s}^2=g_{AB}d{x}^{A}d{x}^B=\frac{L^2}{y^2}\left[-d{t}^2+{\delta }_{ij}d{x}^{i}d{x}^j+\mathrm{d}{y}^2\right].$$

$$d{s}^2$$

$$a(\eta )=\frac{L}{y_b(t)}$$

$$d\eta =\sqrt{1-{\left(\frac{d{y}_b}{dt}\right)}^2}dt\equiv {\gamma }^{-1}dt.$$

$$v$$

$$\gamma$$

$$H\equiv \dot{a}/a^2\equiv a^{-1}\mathcal{H}=-L^{-1}\gamma v,$$

$${\kappa }_{{}_5}(\rho +\mathcal{T})$$

$${\kappa }_{{}_5}(\rho +P)$$

$$\dot{\rho }=-3Ha(\rho +P),$$

$$H^2=\frac{{\kappa }_{{}_5}^2}{18}\mathcal{T}\rho \left(1+\frac{\rho }{2\mathcal{T}}\right)+\frac{{\kappa }_{{}_5}^2{\mathcal{T}}^2}{36}-\frac{1}{L^2}.$$

$$\frac{{\kappa }_{{}_5}^2{\mathcal{T}}^2}{12}=\frac{3}{L^2}\text{and we set}{\kappa }_{{}_4}=8\pi G_4=\frac{{\kappa }_{{}_5}^2\mathcal{T}}{6}.$$

$${\kappa }_5={\kappa }_{4}L.$$

$${\kappa }_5=\frac{1}{M_5^3}=L_s^3,{\kappa }_4=\frac{1}{M_{\mathrm{Pl}}^2}=L_{\mathrm{Pl}}^2,$$

$$\frac{L}{L_s}={\left(\frac{L_s}{L_{\mathrm{Pl}}}\right)}^2.$$

$$P=\frac{1}{3}\rho \text{mbox}and|v|\ll 1\text{mbox}sothat\gamma \simeq 1,d\eta \simeq dt.$$

$$a(t)$$

$$y_b(t)$$

$$v(t)$$

$$a_b=a(0)=\frac{1}{\sqrt{v_b}},|v(0)|\equiv v_b=\frac{L^2}{t_b^2}.$$

$$\mathrm{d}{s}^2=\frac{L^2}{y^2}\left[-\mathrm{d}{t}^2+({\delta }_{ij}+2h_{ij})\mathrm{d}{x}^i\mathrm{d}{x}^j+\mathrm{d}{y}^2\right].$$

$$h_{ij}(t,\mathbf{x},y)=\int \frac{d^{3}k}{(2\pi {)}^{3/2}}\sum _{\bullet =+,\times }{e}^{i\mathbf{k}\cdot \mathbf{x}}{e}_{ij}^{\bullet }(\mathbf{k})h_{\bullet }(t,y;\mathbf{k}),$$

$$h_{\bullet }^{\ast }(t,y;\mathbf{k})=h_{\bullet }(t,y;-\mathbf{k}).$$

$$\left[{\partial }_t^2+k^2-{\partial }_y^2+\frac{3}{y}{\partial }_y\right]h_{\bullet }(t,y;\mathbf{k})=0.$$

$${\left[LH{\partial }_t{h}_{\bullet }-\sqrt{1+L^2{H}^2}{\partial }_y{h}_{\bullet }\right]|}_{y_{\mathrm{b}}}$$

$$-{\gamma \left(v{\partial }_t+{\partial }_y\right)h_{\bullet }|}_{y_{\mathrm{b}}}$$

$${\left(v{\partial }_t+{\partial }_y\right)h_{\bullet }|}_{y_b(t)}=0.$$

$${\mathcal{S}}_h=2\frac{L^3}{2{\kappa }_5}\sum _{\bullet }\int dt\int d^{3}k{\int }_{y_b(t)}^{y_s}\frac{dy}{y^3}\times$$

$$\displaystyle\times\Big{[}|\partial_{t}h_{\bullet}|^{2}-|\partial_{y}h_{\bullet}|^{2}-k^{2}|h_{\bullet}|^{2}\Big{]}~{},$$

$$\rho \mathcal{T}\gg {\rho }^2\mathrm{and}\mathrm{therefore}|v|\ll 1.$$

$${{\partial }_y{h}_{\bullet }|}_{y_{\mathrm{b}}}={{\partial }_y{h}_{\bullet }|}_{y_s}=0.$$

$$\left[-{\partial }_y^2+\frac{3}{y}{\partial }_y\right]{\varphi }_{\alpha }(t,y)$$

$$2\sum _{\alpha }{\varphi }_{\alpha }(t,y){\varphi }_{\alpha }(t,\tilde{y})=\delta (y-\tilde{y})y^3,$$

Full text

I Introduction

In recent times, the possibility that our observed Universe might represent a hypersurface in a higher-dimensional space-time has received considerable attention. The main motivation for this idea is the fact, that string theory Polchinski (1998a, b), which is consistent only in ten spac-etime dimensions (or 11 for M–theory) allows for solutions where the standard model particles (like fermions and gauge bosons) are confined to some hypersurface, called the brane, and only the graviton can propagate in the whole space-time, the bulk Polchinski (1998b, 1995). Since gravity is not well constrained at small distances, the dimensions normal to the brane, the extra dimensions, can be as large as 0.1mm.

Based on this feature, Arkani-Hamed, Dimopoulos and Dvali (ADD) proposed a braneworld model where the presence of two or more flat extra-dimensions can provide a solution to the hierarchy problem, the problem of the huge difference between the Planck scale and the electroweak scale Arkani-Hamed et al. (1998); Arkani:1999 .

In 1999 Randall and Sundrum (RS) introduced a model with one extra dimension, where the bulk is a slice of five-dimensional anti de-Sitter (AdS) space. Such curved extra dimensions are also referred to as warped extra dimensions. While in the RS I model Randall and Sundrum (1999a) with two flat branes of opposite tension at the edges of the bulk the warping leads to an interesting solution of the hierarchy problem, it localizes four-dimensional gravity on a single positive tension brane in the RS II model Randall and Sundrum (1999).

Within the context of warped braneworlds, cosmological evolution, i.e., the expansion of the Universe, can be understood as the motion of the brane representing our Universe through the AdS bulk. Thereby the Lanczos-Sen-Darmois-Israel-junction conditions Lanczos (1924); Sen (1924); Darmois (1927); Israel (1966), relate the energy-momentum tensor on the brane to the extrinsic curvature and hence to the brane motion which is described by a modified Friedmann equation. At low energy, however, the usual Friedmann equations for the expansion of the Universe are recovered Kraus (1999); Binetruy et al. (2000).

Since gravity probes the extra dimension, gravitational perturbations on the brane, i.e. in our Universe, carry five-dimensional effects in form of massive four-dimensional gravitons, the so-called Kaluza-Klein (KK) tower. Depending on the particular brane trajectory, these perturbations may be significantly amplified leading to observable consequences, for example, a stochastic gravitational wave background. (For a review of stochastic gravitational waves see mm .) This amplification mechanism is identical to the dynamical Casimir effect for the electromagnetic field in cavities with dynamical walls (moving mirrors); see Ruser (2005a); Ruser:2005xg ; Ruser:2006xg and references therein. In the quantum field theoretical language, such an amplification corresponds to the creation of particles out of vacuum fluctuations. Hence, in the same way a moving mirror leads to production of photons, the brane moving through the bulk causes creation of gravitons. Thereby, not only the usual four-dimensional graviton might be produced, but also gravitons of the KK tower can be excited. Those massive gravitons are of particular interest, since their energy density could dominate the energy density of the Universe and spoil the phenomenology if their production is sufficiently copious.

The evolution of cosmological perturbations under the influence of a moving brane has been the subject of many studies during recent years. Since one has to deal with partial differential equations and time-dependent boundary conditions, the investigation of the evolution of perturbations in the background of a moving brane is quite complicated. Analytical progress has been made based on approximations like the “near brane limit” and a slowly moving brane Battye:2004a ; Battye:2004b ; Easther:2003 ; Kobayashi:2004ana .

The case of de Sitter or quasi-de Sitter inflation on the brane has been investigated analytically in Gorbunov et al. (2001); Kobayashi:2003 ; Maartens:2000 ; Langlois:2000 ; Frolov and Kofman (2002). In Langlois:2000 it is demonstrated that during slow-roll inflation (modeled as a period of quasi-de Sitter expansion) the standard four-dimensional result for the amplitude of perturbations is recovered at low energies while it is enhanced at high energies.

However, most of the effort has gone into numerical simulations Hiramatsu:2004 ; Hiramatsu:2005 ; Hiramatsu:2006 ; Koyama:2004cf ; Ichiki:2004a ; Ichiki:2004b ; Kobayashi:2005 ; Kobayashi:2006a ; Kobayashi:2006b ; Seahra:2006 , in particular in order to investigate the high-energy regime. Thereby different coordinate systems have been used for which the brane is at rest, and different numerical evolution schemes have been employed in order to solve the partial differential equation.

In this work we chose a different way of looking at the problem. We shall apply a formalism used to describe the dynamical Casimir effect to study the production of gravitons in braneworld cosmology. This approach and its numerical implementation offers many advantages. The most important one is the fact that this approach deals directly with the appearing mode couplings by means of coupling matrices. (In Battye:2004b a similar approach involving coupling matrices has been used. However, perturbatively only, and not in the complexity presented here.) Hence, the interaction between the four-dimensional graviton and the KK modes is not hidden within a numerical simulation but can directly be investigated making it possible to reveal the underlying physics in a very transparent way.

We consider a five-dimensional anti-de Sitter spacetime with two branes in it; a moving positive tension brane representing our Universe and a second brane which, for definiteness, is kept at rest. This setup is depicted in Fig. 1.

For this model we have previously shown that in a radiation dominated Universe, where the second, fixed brane is arbitrarily far away, no gravitons are produced Cartier et al. (2005).

The particular model which we shall consider is strongly motivated by the ekpyrotic or cyclic Universe and similar ideas Khoury:2001 ; Kallosh:2001 ; Neronov:2001 ; Steinhardt:2002 ; Khoury:2002a ; Khoury:2002b ; Khoury:2003 ; Khoury:2004 ; Tolley:2004 . In this model, roughly speaking, the hot big bang corresponds to the collision of two branes; a moving bulk brane which hits “our” brane, i.e. the observable Universe. Within such a model, it seems to be possible to address all major cosmological problems (homogeneity, origin of density perturbations, monopole problem) without invoking the paradigm of inflation. For more details see Khoury:2001 but also Kallosh:2001 for critical comments.

One important difference between the ekpyrotic model and standard inflation is that in the latter one tensor perturbations have a nearly scale invariant spectrum. The ekpyrotic model, on the other hand, predicts a strongly blue gravitational wave spectrum with spectral tilt $n_{T}\simeq 2$ Khoury:2001 . This blue spectrum is a key test for the ekpyrotic scenario since inflation always predicts a slightly red spectrum for gravitational waves. One method to detect a background of primordial gravitational waves of wavelengths comparable to the Hubble horizon today is the polarization of the cosmic microwave background. Since a strongly blue spectrum of gravitational waves is unobservably small for large length scales, the detection of gravitational waves in the cosmic microwave background polarization would falsify the ekpyrotic model Khoury:2001 .

Here we consider a simple specific model which is generic enough to cover important main features of the generation and evolution of gravitational waves in the background of a moving brane whose trajectory involves a bounce. First, the physical brane moves towards the static brane, initially the motion is very slow. During this phase our Universe is contracting, i.e. the scale factor on the brane decreases, the energy density on the brane increases and the motion becomes faster. We suppose that the evolution of the brane is driven by a radiation component on the brane, and that at some more or less close encounter of the two branes which we call the bounce, some high-energy mechanism which we do not want to specify in any detail, turns around the motion of the brane leading to an expanding Universe. Modeling the transition from contraction to subsequent expansion in any detail would require assumptions about unknown physics. We shall therefore ignore results which depend on the details of the transition. Finally the physical brane moves away from the static brane back towards the horizon with expansion first fast and then becoming slower as the energy density drops. This model is more similar to the pyrotechnic Universe of Kallosh, Kofman and Linde Kallosh:2001 where the observable Universe is also represented by a positive tension brane rather than to the ekpyrotic model where our brane has negative tension.

We address the following questions: What is the spectrum and energy density of the produced gravitons, the massless zero mode and the KK modes? Can the graviton production in such a brane Universe lead to limits, e.g. on the AdS curvature scale via the nucleosynthesis bound? Can the KK modes provide the dark matter or lead to stringent limits on these models? Similar results could be obtained for the free gravi-photon and gravi-scalar, i.e. when we neglect the perturbations of the brane energy momentum tensor which also couple to these gravity wave modes which have spin-1 respectively spin-0 on the brane.

The reminder of the paper is organized as follows. After reviewing the basic equations of braneworld cosmology and tensor perturbations in Sec. II, we discuss the dynamical Casimir effect approach in Sec. III. In Sec. IV we derive expressions for the energy density and the power spectrum of gravitons. Thereby we show that, very generically, KK gravitons cannot play the role of dark matter in warped braneworlds. This is explained by the localization of gravity on the moving brane which we discuss in detail. Section V is devoted to the presentation and discussion of our numerical results. In Sec. VI we reproduce some of the numerical results with analytical approximations and we derive fits for the number of produced gravitons. We discuss our main results and their implications for bouncing braneworlds in Sec. VII and conclude in Sec. VIII. Some technical aspects are collected in appendices.

The main and most important results of this rather long and technical paper are published in the Letter letter .

II Gravitons in moving braneworlds

II.1 A moving brane in AdS5

We consider a AdS-5 spacetime. In Poincaré coordinates, the bulk metric is given by

[TABLE]

The physical brane (our Universe) is located at some time dependent position $y=y_{b}(t)$, while the 2nd brane is at fixed position $y=y_{s}$ (see Fig. 1). The induced metric on the physical brane is given by

[TABLE]

where

[TABLE]

is the scale factor and $\eta$ denotes the conformal time of an observer on the brane,

[TABLE]

We have introduced the brane velocity

[TABLE]

Here $H$ is the usual Hubble parameter,

[TABLE]

and an overdot denotes the derivative with respect to conformal time $\eta$. The bulk cosmological constant $\Lambda$ is related to the curvature scale $L$ by $\Lambda=-6/L^{2}$. The junction conditions on the brane lead to CR ; Cartier et al. (2005)

[TABLE]

Here ${\cal T}$ is the brane tension and $\rho$ and $P$ denote the energy density and pressure of the matter confined on the brane. Combining (8) and (9) results in

[TABLE]

while taking the square of (8) leads to

[TABLE]

These equations form the basis of brane cosmology and have been discussed at length in the literature (for reviews see Maartens (2004); Durrer:2005dj ). The last equation is called the modified Friedmann equation for brane cosmology Binetruy et al. (2000). For usual matter with $\rho+P>0$