Energy-momentum tensor of bouncing gravitons

In models of the Universe with extra dimensions gravity propagates in the whole space-time. Graviton production by matter on the brane is significant in the early hot Universe. In a model of 3-brane with matter embedded in 5D space-time conditions for gravitons emitted from the brane to the bulk to return back to the brane are found. For a given 5-momentum of graviton falling back to the brane the interval between the times of emission and return to the brane is calculated. A method to calculate contribution to the energy-momentum tensor from multiple graviton bouncings is developed. Explicit expressions for contributions to the energy-momentum tensor of gravitons which have made one, two and three bounces are obtained and their magnitudes are numerically calculated. These expressions are used to solve the evolution equation for dark radiation. A relation connecting reheating temperature and the scale of extra dimension is obtained. For the reheating temperature $T_R\sim 10^6 GeV$ we estimate the scale of extra dimension $\m$ to be of order $10^{-9} GeV\,\,\, (\m^{-1}\sim 10^{-5} cm )$.


Introduction
Brane-world scenarios with the observable Universe located on a 3-brane embedded in a higherdimensional space-time have attracted considerable interest recently. Such models with matter on the brane can reproduce the main cosmological data [1,2,3,4].
A general property of extra-dimensional models is that although ordinary matter is supposed to be confined to a brane, gravity propagates in the whole space-time. This entails the effect that gravitons produced in reactions of particles on the brane can escape to the bulk. Graviton production is strong in the early hot Universe, and can alter the time evolution of matter on the brane and, in particular, the primordial nucleosynthesis.
In this paper we calculate graviton production in a model of five-dimensional Universe with one large extra dimension. Matter is supposed to be confined to the 3D brane. Time evolution of matter in this model is described by the generalized Friedmann equation H 2 = ρ 2 + 2µρ + · · · [5,6,7]. We consider the period of early cosmology, in which the term quadratic in energy density is dominant, µ/ρ ≪ 1 (ρ is the normalized energy density on the brane defined in (4), µ = (−Λ/6) 1/2 , and Λ is 5D cosmological constant).
Because the space-time is curved, a part of gravitons emitted in the bulk can return back to the brane [8,10,11] and bounce again to the bulk. In paper [12] an analytical method to show that a bounce is possible was developed. In the present paper we investigate further conditions of a bounce. Solving the combined system of equations of trajectories of the brane and of emitted graviton, we find conditions at which graviton can fall back to the brane. We derive an equation for the interval of time between graviton emission and its return to the brane and develop a scheme to calculate times of returns of graviton to the brane for multiple bounces. We show that in the period of early cosmology the ratio of times t 0 /t 1 of graviton emission t 0 and its return to the brane t 1 to a good approximation can be expressed as a function of x = m(t 1 )/E(t 1 ), where (E(t 1 ), m(t 1 ), p) are the components of graviton 5-momentum at the time t 1 when graviton returns to the brane.
As an application of the above results, using the distribution function of emitted gravitons of paper [11], we calculate the the components of the energy-momentum tensor of bouncing gravitons T in,(k) nn (n A is transverse to the brane, k is a number of a bounce). For the first three bounces we obtain explicit expressions for T in,(k) nn and estimate their numerical magnitudes. The expressions for the energy-momentum tensor of bouncing gravitons are used to solve the evolution equation of dark radiation [11,13]. Solving this equation, we find a relation connecting the reheating temperature of the Universe T R and the scale of the extra dimension µ. Qualitative constraints on T R and µ are discussed.
In Sect.2 we review two approaches to the 5D model. In Sect.3 we solve geodesic equations for gravitons propagating in the bulk. In Sect.4 we solve the combined system of equations for graviton and brane trajectories and find conditions for return of graviton to the brane. We calculate the interval of times between graviton emission and detection as a function of graviton momentum at the time of detection.
In Sect. 5 consider multiple graviton bounces. We calculate the (nn) components of the energymomentum tensor of gravitons falling to the brane.
In Sect. 6 we present qualitative numerical analysis of the energy-momentum tensor and discuss solution of evolution equation for dark radiation.

3-brane in 5D bulk
We consider the 5D model with one 3D brane embedded in the bulk. Matter is confined to the brane, gravity extends to the bulk. In the leading approximation we neglect graviton emission from the brane to the bulk. The action is taken in the form where x 4 ≡ y is coordinate of the infinite extra dimension, κ 2 = 8π/M 3 . The 5D model can be treated in two alternative approaches. In the first approach metric is non-static, and the brane is located at a fixed position in the extra dimension [5,6]. We consider the class of metrics of the form The brane is spatially flat and located at y = 0. The freedom of parametrization of t, allows to set n(0, t) = 0. The energy-momentum tensor of matter on the brane is taken in the form For the following it is convenient to introduce the normalized expressions for energy density, pressure and cosmological constant on the brane which all have the same dimensionality [GeV ] Reduction of the metric (2) to the brane is The function a(t) = a(0, t) satisfies the generalized Friedmann equation [6] where H(t) =ȧ(t)/a(t) and ρ w (t) is the Weyl radiation term [1], which below is set to zero.
In the second approach the brane separates two static 5D AdS spaces attached to both sides of the brane. The metrics of the AdS spaces are solutions of the Einstein equations of the form where Below we consider the case µ 1 = µ 2 and P i = 0. Trajectory of the moving brane in the R, T plane is given by parametric equations dot is derivative over t. Reduction of the 5D metric to the brane is The function r b (t) satisfies the generalized Friedmann equation [14,15,16] Equations (10) and (6) with ρ w (t) = 0 are of the same form, and a 2 (0, t) can be identified with r 2 b (t). Below we consider the case σ = µ [4], so that (10) takes a form The normalized velocity vector of the brane and the normal vector to the brane are Hereτ where ǫ = ±.
In the following we choose n A with the sign (-) Let λ be parameter along a geodesic. Geodesic equations in the metric (7) are We consider solutions of the geodesic equations even in λ. Integrating the geodesic equations, one obtains [13] dT where (C T , C a , C R ) are integration parameters. Tangent vectors to a null geodesic satisfy the relation from which it follows that C R = 0. Eqs. (14)- (17) were solved with the initial condition thatT (0) and R(0) are located on the brane world sheet: . Here t 0 is the proper time of the point on the brane world sheet τ b (t 0 ), r b (t 0 ) at which the geodesic begins, i.e. the time of the graviton emission. The components of momentum of a graviton propagating along a null geodesic are proportional to the tangent vector to a null geodesic where ǫ R = ±. Also we define ǫ T as C T = ǫ T |C T |. Expanding the graviton momentum p A in the basis (v A (t), n A (t), e Ā a (t)), where e Ā a = δ Ā a /µa, we have where The components of p A in the two bases are connected as The components E and m depend on t through r b (t). Introducing and expressing γ through E and m, we have If at a time t graviton is on the brane world sheet R = r b (t) (t is a time of emission, t 0 , or a time of return of the graviton to the brane, t 1 , we take the basis (v A , n A , e A )(t) at the time t and obtain γ in a form

Bounce of massless particles in the period of early cosmology
We consider the radiation-dominated period of the early cosmology, when ρ/µ ≫ 1 or,equivalently, µt ≪ 1. Supposing that the energy loss from the brane to the bulk is sufficiently small to comply with the observational data, we neglect in the conservation equation for the energy-momentum tensor the energy flow in the bulk. In the period of early cosmology, in the model with extra dimension, from the expression for energy density of relativistic degrees of freedom (g * (T ) is a total number of relativistic degrees of freedom [18,19] ) it follows that For times t 1 and t in the region of early cosmology, from the Friedman equation one obtains Following [11], with the use of (8), the equation for the brane trajectory can be written as Integrating Eq. (30) with the boundary conditions , we obtain the equation for trajectory of the brane From the first integrals of the null geodesic equations (17) we obtain Integrating Eq. (32) with the initial conditions R = r b (t 0 ),T = τ b (t 0 ), we obtain the equation for for a null geodesic (graviton trajectory) If graviton returns to the brane at time t 1 , we have R = r b (t 1 ). Combining Eqs. (31) and (33) and using Friedmann equation, H 2 = ρ 2 + 2µρ, we obtain an equation for r b (t 1 ) Eq. (34) can be interpreted as an equation which determins the time of return of graviton to the brane t 1 for a given time of emission t 0 . It is seen that Eq.(34) admits solution only if Expanding the integrand of (34) in powers of µ/ρ, we have The series in (36) is convergent. Introducing and substituting ρ 0 = ρ 1 z −4 , where ρ 0 = ρ(t 0 ) and ρ 1 = ρ(t 1 ), we transform (36) to a form The function is monotone increasing with the maximum at the point z = 1 equal to 1.

Conditions of the fall of graviton on the brane
The sign of the graviton momentum component m(t) at the time t 1 at which graviton returns to the brane is opposite to that at the time of emission t 0 . From (23) we have Condition (35) is satisfied in the following cases: Let us find in which case is realized one of the possibilities: Here E and m are Analogously to the case (i)(b) in the case (ii)(a) there are no solutions.
E and m are E and m are Because m is negative and does not change its sign, there are no solutions. Analogously in the case (iv) m is always positive, and no solution exists.
To conclude, we are left with the solutions of the types (i)(a) and (ii)(b), which are physically equivalent, because Eqs. (17) with ǫ T = ǫ R = + transform to equations with ǫ T = ǫ R = − under the change λ → −λ. In the following we consider the case (i)(a).

Relation between emission and detection times
Relation (42) is valid at the endpoints of graviton trajectory, at emission point and at points where graviton hits the brane. Because, as discussed in preceding subsection, x 1 < 0, the relation (42) written at time t 1 takes the form In the period of early cosmology expression (43) approximately is The advantage of this form of γ is that we have extracted the factor (µ/ρ 1 ) 2 . Now the Eq. (38) can be written as Eqs. (38) define z 01 ≃ (t 0 /t 1 ) 1/4 through the ratio x 1 = m 1 /E 1 and ρ 1 = ρ(t 1 ), or, equivalently, the emission time t 0 through the "mass" and "energy" of the graviton at the time of return of the graviton to the brane t 1 . In the following, for practical calculations, in the region µ/ρ 1 < 1 we use a simplified equation Domain of applicability and corrections to this equation are discussed in Sect.6.

Multiple bounces
To consider multiple reflections from the brane of bouncing gravitons we use matrix notations. Introducing we have The case with multiple bouncings is illustrated by the scheme The left column corresponds to the emission time t 0 . In the next brackets, in the left columns are momenta of ingoing particle, in the right ones are the outgoing. Under reflection from the brane momentum p a parallel to the brane and energy E are conserved, transverse momentum to the brane m changes its sign: Momentap T,R ∼ C T,R /r b (t) are are rescaled when moving from one bracket to the next Transformation from "in" to "out" components within a bracket is given by Let us consider the first two brackets in (51). Using (50) we express E 0 and m 0 through E 1 and m 1 and obtain In the period of early cosmology, from the Friedman equation it follows that H/µ ≃ 1/(4µt). For small µt we can simplify the expressions in (54) as Introducing where ψ ± 01 = ψ ± (z 01 ) For multiple bouncings we have where z 0n = z 01 z 12 z 23 · · · z n−1,n , Here or explicitly u 02 = z −1 01 z 12 , u 03 = z 01 z −1 12 z 23 , u 04 = z −1 01 z 12 z −1 23 z 34 , · · · From (60) and (61) it follows that u 0,2k+1 < 1 and u 0,2k > 1 . In the latter case and It should be noted that the functions z n−1,n in processes with different number of bounces are different. If z (k) n−1,n refers to the process with k − 1 bounces, different z (k) n−1,n are connected by the following relations The distribution function of non-interacting gravitons in the bulk satisfies the Liouville equation without the collision term. If coordinates and momenta gravitons along a geodesic are parametrized by parameter λ, i.e. f (x A (λ), p A (λ)), we have In the case of the metric (7) relation (64) can be written as [11] f

First fall of gravitons to the brane
We suppose that the distribution function of emitted gravitons f out depends on E 0 = m 2 0 + p 2 , m 0 and temperature T 0 , i.e. f out = f out (E 0 , m 0 , T (t 0 )). The distribution function of gravitons emitted at time t 0 and falling back for the first time on the brane at time t 1 is where In the period of early cosmology from (28) and (29) it follows that T 1 /T 0 ≃ ρ b (t 0 )/ρ b (t 1 ) = z 01 . We obtain the distribution function at time t 1 as where z 01 is determined as a function of x 1 = m 1 /E 1 by Eq.(45). For µ/ρ 1 ≪ 1 the terms O(µ/ρ 1 ) could be neglected and z 01 is defined via (47).
Condition that x 0 = m 0 /E 0 > 0 takes the form Because the nominator of this ratio is positive, this condition is equivalent to ψ − (z 01 ) −|x 1 |ψ + (z 01 ) > 0, or To show that this inequality is satisfied, we wright

Next falls of gravitons to the brane
The distribution function of gravitons emitted at time t 0 , which bounced off the brane at time t 1 and fall on the brane the second time at time t 2 is Tracing the propagation of graviton, we obtain where u 02 = z 01 z −1 12 and The time t 1 of the bounce is determined from Eq. (47) with z = z 12 ≃ (t 1 /t 2 ) 1/4 , where where x 1 = (m 1 /E 1 ) in . Expressing x 1 through x 2 , we have Substituting (74) in (73), we obtain where the distribution function of infalling gravitons is E 0 (m 1 , E 1 , z 01 ) and m 0 (m 1 , E 1 , z 01 ) are defined by (66)-(67). Substituting these expressions, we have First, we integrate over E 1 in the limits (0, ∞), and below we consider integration taking into account lower and upper bounds. For T (1) nn we have For the energy-momentum tensor of gravitons which have made one bounce we obtain where z 02 = z For the energy-momentum tensor of gravitons which have made two bounces we have 01 . Integration over E in the integrals T in,(k) nn is performed for E > T min . Taking taking g * ∼ 200 2 and assuming that we find that for µ ≃ 10 −13 GeV and for 10 −10 ÷ 10 −9 GeV the bound is satisfied for T min > 2.5 · 10 3 and (1.5 ÷ 5) · 10 4 GeV correspondingly.
Because of the high power of E in the integrals for T (k),in , the main contribution to the integrals is produced from the region near the upper limit of integration T max . Provided T min ≪ T max , we set T min = 0. The functions z k−1,k (x) and the integrands I (k) for characteristic values of x are given in Table 1 and Fig. 1. For the following it is convenient to introduce the notations .438 2 The ambiguity in g * and in A in (82) is due to incomplete knowledge of the contribution of dark matter. We assume that the mass of particles which form dark matter is in the interval (20 ÷ 100)GeV . In the period of early cosmology these particles are relativistic. Phenomenologically the acceptable number of dark matter particles with the mass in the above interval is g * ∼ 100 [19]. With the number of particle species in the non-supersymmetric Standard model g * ∼ 100, the total number is ∼ 200. Because of the high power of T estimates weakly depend on variations of this number.
The limiting temperature at which the emission begins is the reheating temperature T R . The emission energy E 0 is bounded by T R . Using the expression of Sect.
, where z 0n (x) and ψ ± 0n = ψ ± (u 0n ) are defined in (59) and (60) correspondingly. For T n,(k) nn we obtain where γ is incomplete gamma-function. If T /T R 1, the region producing the main contribution to the integral is 1 > z 0n > T /T R . The corresponding region of x is 0 < x < O(1 − T /T R ), where 1 − T /T R ≪ 1. At small x the integrand decreases as a power of x, and contribution from this region is strongly suppressed.
The energy density of dark radiation satisfies the evolution equation [9,11,13] where T in = T in,(k) and In the period of early cosmology from the Friedmann and approximate conservation equations, ρ + 4Hρ ≃ 0, it follows that Here we substituted M 3 5 ≃ µM 2 pl [4]. Eq. (95) is transformed as Explicitly we have . It is seen that as k increases the region of x, which produces the main contribution to T in,(k) is shifted to smaller x and larger z(x).
Integrating (99) with the boundary condition ρ D (T R ) = 0, we obtain Here we substituted M 3 5 ≃ µM 2 pl [4]. For T R /T > 20 the integral is practically constant and independent of T . The main contribution to the integral is produced by integration over the region of y near the lower limit. Taking T R /T = 20 and performing integration over y, we obtain 0.224 For the ratio of energy density of dark radiation to energy density of matter we have A typical order of constraint on magnitude of the ratio ρ D /ρ in the period of early cosmology, which follows from primordial nucleosynthesis, is |ρ D /ρ| 0.07 [20]. For µ in the interval 10 −13 ÷ 10 −12 GeV the estimate (104) gives T R ∼ (0.7 ÷ 2.3) · 10 4 GeV . This value of T R is significantly lower than usually accepted T R ∼ 10 5 ÷10 7 GeV, indicating that large extra dimensions r extr ∼ 1/µ appear with low reheating temperature. The estimate can be improuved, if there is more complete cancellation between two terms in (101), or for larger values of µ. Because of strong dependence of ρ D /ρ on µ, the possibility of larger µ seems more plausible. For reheating temperature T R ∼ 10 6 GeV the scale of extra dimension obtained from (104) is µ ∼ 2 · 10 −9 GeV . For larger µ the lower bound of the region of early cosmology, E min , increases resulting in smaller magnitudes of the integrals T in nn . For T R > 10 6 GeV this does not change the above results significantly, but for smaller T R the effect of the lower bound of integration must be taken into account.

Conclusion
The key point of the present paper is solution of the system of equations for the brane trajectory and geodesic equation for graviton trajectory. Given graviton 5-momentun at detection time, we calculated the time of graviton emission. We obtained the recursion relations enabling, in principle, calculate the energy-momentum tensor of gravitons falling back to the brane which have made an arbitrary number of bounces. For the first three returns of graviton to the brane we obtained the explicit expressions for the energy-momentum tensor of the gravitons falling back to the brane and made their numerical estimates. Solving the evolution equation for energy density of dark radiation, we obtained a relation connecting the reheating temperature and the scale of extra dimension.