Stochastic emergence of inflaton fluctuations in a SdS primordial universe with large-scale repulsive gravity from a 5D vacuum
Luz Marina Reyes, Jose Edgar Madriz Aguilar, Mauricio Bellini
aa r X i v : . [ g r- q c ] M a y Stochastic emergence of inflaton fluctuations in a SdS primordialuniverse with large-scale repulsive gravity from a 5D vacuum. L. M. Reyes ∗ , Jos´e Edgar Madriz Aguilar † , and , Mauricio Bellini ‡ Departamento de F´ısica, DCI, Universidad de Guanajuato,Lomas del Bosque 103, Col. Lomas del Campestre,C.P. 37150 Le´on Guanajuato, M´exico. Departamento de F´ısica, Facultad de Ciencias Exactas y Naturales,Universidad Nacional de Mar del Plata,Funes 3350, C.P. 7600, Mar del Plata, Argentina. Instituto de Investigaciones F´ısicas de Mar del Plata (IFIMAR),Consejo acional de Investigaciones Cient´ıficas y T´ecnicas (CONICET), Argentina.
Abstract
We develop a stochastic approach to study scalar field fluctuations of the inflaton field in an earlyinflationary universe with a black-hole (BH), which is described by an effective 4D Schwarzschild-de Sitter (SdS) metric. This effective 4D metric is the induced metric on a 4D hypersurface, hererepresenting our universe, which is obtained from a 5D Ricci-flat SdS static metric, after implementa planar coordinate transformation. On this background we found that at the end of inflation, thesquared fluctuations of the inflaton field are not exactly scale independent and result sensitive tothe mass of the BH. ∗ E-mail address: luzreyes@fisica.ugto.mx † E-mail address: jemadriz@fisica.ugto.mx ‡ E-mail address: [email protected], [email protected] . INTRODUCTION In stochastic inflation the dynamics of the inflaton field is described by a second orderstochastic equation, where the emergence of a long-wave classical field that drives inflationis subject to a short-wave classical noise. Starobinsky [1] has noted that under certainassumptions, the splitting of the scalar field into long-wavelength and short-wavelengthcomponents leads to a quantum Langevin equation that could become classical stochasticdynamics for the long-wavelength modes of the scalar field. This approach emphasizes therole of the quantum fluctuations as the driving forces of the inflation. It considers as amain ingredient the set of long-wavelength modes as a whole, from which the coarse-grainedfield emerges. This coarse-grained field is assumed to have a highly classical behavior, butthe inflow of short-wavelength modes alters its evolution in a random way. Furthermore,the quantum fluctuations of the short-wavelength field, give place to cosmological densityperturbations that could be the origin of the structure of the universe [2]. The coarse-graining representation of the inflaton field has played an important role in 4D standardinflationary cosmology [3], in 5D inflationary cosmology from modern Kaluza-Klein theory[4], and in extensions to vectorial fields more recently implemented in the framework ofGravitoelectromagnetic Inflation [5].On the other hand, in the last years theories with extra dimensions have become quitepopular in the scientific community [6]. In particular, some brane scenarios [7], and theinduced matter (IM) theory of gravity [8], have been subject of a great amount of research.Even when both theories have different physical motivations for the introduction of a largeextra dimension, they are equivalent each other, and predict identical non-local and localhigh energy corrections to general relativity in 4D, and usual matter in 4D is a consequenceof the metric dependence on the fifth extra coordinate [9].On the basis of the IM theory, we have recently shown in [10] that there exists a 5D SdSBH solution of the theory, from which we can derive a 4D cosmological model where gravitymanifests itself as attractive on small (planetary and astrophysical) scales, but repulsive onvery large (cosmological) scales. This behavior of gravity derived from this 5D framework,leave us to put on the desk the following question: can repulsive gravity be considered as astrong candidate for explaining the large-scale accelerated expansion of the universe in thepast and today? To answer this question let us to start by defining the physical vacuum via2he 5D Ricci-flat metric [11]: dS = (cid:18) ψψ (cid:19) (cid:20) c f ( R ) dT − dR f ( R ) − R ( dθ + sin θdφ ) (cid:21) − dψ . (1)Here, f ( R ) = 1 − (2 Gζ ψ /Rc ) − ( R/ψ ) is a dimensionless function, { T, R, θ, φ } arethe usual local spacetime spherical coordinates employed in general relativity and ψ is thenon-compact space-like extra dimension. In this line element ψ is an arbitrary constantwith length units, c denotes the speed of light, and the constant parameter ζ has units of( mass )( length ) − . This static metric is a 5D extension of the 4D SdS metric. In order toget this metric written on a dynamical chart coordinate { t, r, θ, φ } , we use the coordinatetransformation given by [12] R = ar (cid:20) Gζ ψ ar (cid:21) , T = t + H Z r dR Rf ( R ) (cid:18) − Gζ ψ R (cid:19) − / , ψ = ψ, (2) a ( t ) = e Ht being the scale factor, and H the Hubble constant. Thus the line element (1)can be written in terms of the conformal time τ as dS = (cid:18) ψψ (cid:19) (cid:2) F ( τ, r ) dτ − J ( τ, r ) (cid:0) dr + r ( dθ + sin θdφ ) (cid:1)(cid:3) − dψ , (3)where the metric functions F ( τ, r ) and J ( τ, r ) are given by F ( τ, r ) = a ( τ ) (cid:20) − Gζ ψ a ( τ ) r (cid:21) (cid:20) Gζ ψ a ( τ ) r (cid:21) − , J ( τ, r ) = a ( τ ) (cid:20) Gζ ψ a ( τ ) r (cid:21) , (4)with dτ = a − ( τ ) dt and a ( τ ) = − / ( Hτ ), so that the Hubble parameter is a constant givenby H = a − dadτ . As it was shown in [11], for certain values of ζ and ψ , both metrics (1)and (3) have two natural horizons. The inner horizon is the analogous of the Schwarzschildhorizon and the external one is the analogous of the Hubble horizon.Now we consider a 5D massless scalar field which is free of any interactions: (5) (cid:3) ϕ = 0. Weassume that ϕ ( τ, r, θ, φ, ψ ) can be separated in the form ϕ ( τ, r, θ, φ, ψ ) ∼ Φ( τ, r ) G ( θ, φ )Ω( ψ ),so that the expression (5) (cid:3) ϕ = 0 leaves to (cid:18) ψψ (cid:19) − ddψ "(cid:18) ψψ (cid:19) d Ω dψ + M Ω = 0 , (5)1 √ F J ∂∂τ r J F ∂ Φ ∂τ ! − (cid:18) F ∂F∂r + 1
J ∂J∂r (cid:19) ∂ Φ ∂r − r ∂∂r (cid:18) r ∂ Φ ∂r (cid:19) − (cid:18) l ( l + 1) r − M J (cid:19) Φ = 0 , (6)where M > l is an integer dimensionlessparameter related with the angular momentum.3 I. THE DYNAMICS OF ϕ ON THE 4D HYPERSURFACE Σ Assuming that the 5D spacetime can be foliated by a family of hypersurfaces Σ : ψ = ψ ,from the metric (3) we obtained that the 4D induced metric on every leaf Σ is given by dS = F ( τ, r ) dτ − J ( τ, r )[ dr + r ( dθ + sin θdφ )] , (7)where the metric functions F ( τ, r ) and J ( τ, r ) can be now written in terms of the physicalmass m = ζ ψ (introduced by the first time in [11]), in the form F ( τ, r ) = a ( τ ) (cid:20) − Gm a ( τ ) r (cid:21) (cid:20) Gm a ( τ ) r (cid:21) − , J ( τ, r ) = a ( τ ) (cid:20) Gm a ( τ ) r (cid:21) . (8)The induced metric (7) has a Ricci scalar (4) R = 12 H . describes a black hole in an expadinguniverse, where the expansion is driven by a kind of cosmological constant, whose value ingeneral depends of the value of ψ . According to [11], this metric also indicates that thereexists a length scale that separates regions on which gravity changes from attractive torepulsive. This length scale is called the gravitational-antigravitational radius, which in thecoordinates ( T, R ) is given by R ga = ( Gmψ ) / . With the help of (2), in the new coordinates( τ, r ) this radius must obey the relation r ga = 12 a ( τ ) h R ga − Gm ± q R ga − GmR ga i , (9)where r ga is denoting the gravitational-antigravitational radius in the new coordinates andthe solution with the minus sign is not physical. In order to r ga to be a real value quantity,we require the condition R ga − GmR ga ≥ m ≤ ψ G . Thus, if we consider the foliation ψ = c /H and the fact that for c = ~ = 1 the Newtonian constant is G = M − p , this condition leaves now to the restriction ǫ = mHM p ≤ √ ≃ . ǫ = GmH ≪
1. For these values, R ga is smaller than the sizeof our universe horizon. The same restriction has been used in [13] with different motivation.Now, from (5) and (6), the 4D induced field equation reads1 √ F J ∂∂τ "r J F ∂ ¯ ϕ∂τ − (cid:18) F J ∂F∂r + 1 J ∂J∂r (cid:19) ∂ ¯ ϕ∂r − J ∇ ¯ ϕ + M ¯ ϕ = 0 , (10) M p = 1 . × GeV is the Planckian mass. ϕ ( τ, r, θ, φ ) = ϕ ( τ, r, θ, φ, ψ ) is the effective scalar field induced on the generic hy-persurface Σ, which we shall identify with the inflaton field. It can be easily seen from (10)that M here corresponds to the physical mass of ¯ ϕ . We can expand the field ¯ ϕ as¯ ϕ ( ~r, τ ) = Z ∞ dk X lm h a klm ¯Φ klm ( ~r, τ ) + a † klm ¯Φ ∗ klm ( ~r, τ ) i , (11)where ¯Φ klm ( ~r, τ ) = k j l ( kr ) ¯Φ kl ( τ ) Y lm ( θ, φ ), Y lm ( θ, φ ) are the spherical harmonics, j l ( kr )are the spherical Bessel functions and the annihilation and creation operators obey: h a klm , a † k ′ l ′ m ′ i = δ ( k − k ′ ) δ ll ′ δ mm ′ , [ a klm , a k ′ l ′ m ′ ] = h a † klm , a † k ′ l ′ m ′ i = 0. Hence, using theaddition theorem for spherical harmonics, we obtain for the mean squared fluctuations (cid:10) (cid:12)(cid:12) ¯ ϕ ( ~r, τ ) (cid:12)(cid:12) (cid:11) = Z ∞ dkk X l l + 14 π k j l ( kr ) (cid:12)(cid:12) ¯Φ kl ( τ ) (cid:12)(cid:12) . (12)Now, if we assume that ¯ ϕ l ( τ, r, θ, φ ) = ¯Φ l ( τ, r ) ¯ G l,m ( θ, φ ), then the equation for ¯Φ l ( r, τ ) onthe hypersurface Σ can be written as ∂ ¯Φ l ∂τ − τ ∂ ¯Φ l ∂τ − r ∂ ¯Φ l ∂r − ∂ ¯Φ l ∂r − (cid:20) l ( l + 1) r − M a ( τ ) (cid:21) ¯Φ l = (cid:18) − JF (cid:19) ∂ ¯Φ l ∂τ − " τ + 1 √ F J ∂∂τ (cid:18) J F (cid:19) / ∂ ¯Φ l ∂ τ − M ( J −
1) ¯Φ l + 12 (cid:18) F ∂F∂r + 1
J ∂J∂r (cid:19) ∂ ¯Φ l ∂r . (13)Next,using the fact that ǫ is a small parameter, we propose the following expansion for ¯Φ l in orders of ǫ : ¯Φ l ( r, τ ) = ¯Φ (0) l + ¯Φ (1) l + ¯Φ (2) + .... (14)If we expand the right hand side of the equation (13) as powers of ǫ ≪ − (cid:16) ǫτ r (cid:17) " ∂ ¯Φ (0) l ∂τ − τ ∂ ¯Φ (0) l ∂τ − M H τ ¯Φ (0) l − (cid:16) ǫτ r (cid:17) " ∂ ¯Φ (1) l ∂τ −
115 1 τ ∂ ¯Φ (1) l ∂ ¯Φ l − M H τ ¯Φ (1) l + ... (15)Thus, the spectrum (12) can be written using the expansion (14) as P k ( τ ) = X l (2 l + 1)4 π k j l ( kr ) h ¯Φ (0) kl + ¯Φ (1) kl + ... i h(cid:16) ¯Φ (0) kl (cid:17) ∗ + (cid:16) ¯Φ (1) kl (cid:17) ∗ + ... i = k π (cid:12)(cid:12)(cid:12) ¯Φ (0) kl =0 (cid:12)(cid:12)(cid:12) + H π ǫ ∞ X l =1 (2 l + 1) j l ( kr ) ∆ (1) kl + ... , (16)5here∆ (1) kl = (cid:18) π H ǫ (cid:19) k π (cid:12)(cid:12)(cid:12) ¯Φ (0) kl (cid:16) ¯Φ (1) kl (cid:17) ∗ + ¯Φ (1) kl (cid:16) ¯Φ (0) kl (cid:17) ∗ i = 2 πH ǫ k Re h ¯Φ (1) kl (cid:16) ¯Φ (0) kl (cid:17) ∗ i . (17)Notice that the first term in (16) corresponds to l = 0, so that the zeroth order approximationin ǫ is due only to isotropic fluctuations. Terms with l = 1 correspond to dipoles and l ≥ III. COARSE-GRAINING OF ¯ ϕ As it was shown in [11], the metric (7) written in the static coordinate chart (
T, R ), de-scribes an spherically symmetric object having properties of attractive and repulsive grav-ity, under the election of ψ = H − . Specifically, at scales larger than the gravitational-antigravitational radius R ga , gravity manifests itself as repulsive in nature. On the contrary,on scales smaller than R ga gravity recovers its usual attractive behavior. In this section ourgoal is to study the evolution of the effective scalar field ¯ ϕ under the presence of such anobject but in the dynamical coordinate chart ( τ, ~r ).To study the evolution of the effective field ¯ ϕ ( τ, ~r ) on scales larger than the gravitational-antigravitational radius r ga we introduce the field¯ ϕ L ( τ, ~r ) = Z k Sch k H dk X l,m Θ L ( σk ga − k ) h a klm ¯Φ klm ( τ, ~r ) + a † klm ¯Φ ∗ klm ( τ, ~r ) i , (18)where Θ L is denoting the heaviside function, and the wave number associated to the Hubblehorizon is k H ( τ ) ≃ π/ [ a ( τ ) r H ] = − (2 π ) Hτ /r H . (19)Furthermore, the time dependent wavenumber k ga ( τ ) = [2 π/ ( a ( τ ) r ga )][(2 a ( τ ) r ga ) / (2 a ( τ ) r ga + Gm )] , (20)is the wave number associated to the gravitational-antigravitational radius r ga , and σ is adimensionless parameter that during inflation ranges in the interval 10 − − − .Similarly, the evolution of the effective scalar field ¯ ϕ ( τ, ~r ) on small scales: scales be-tween the Schwarzschild radius r Sch and the gravitational-antigravitational radius r ga , canbe described by the field¯ ϕ S ( τ, ~r ) = Z k Sch k H dk X l,m Θ S ( k − σk ga ) h a klm ¯Φ klm ( τ, ~r ) + a † klm ¯Φ ∗ klm ( τ, ~r ) i , (21)6here Θ S denotes the heaviside function and k Sch ≃ πa ( τ ) r Sch / ( Gm ) = − πr Sch / [ Hτ ( Gm ) ] is the wave number associated to the Schwarzschild radius r Sch .From the expressions (18) and (21) it can be easily seen that ¯ ϕ ( τ, ~r ) = ¯ ϕ L ( τ, ~r ) + ¯ ϕ S ( τ, ~r ). IV. SCALAR FIELD FLUCTUATIONS AT ZEROTH ORDER IN ǫ At zeroth order in the expansion (14), the equation (13) reduces to ∂ ¯Φ (0) l ∂τ − τ ∂ ¯Φ (0) l ∂τ − r ∂ ¯Φ (0) l ∂r − ∂ ¯Φ (0) l ∂r − (cid:20) l ( l + 1) r − M a ( τ ) (cid:21) ¯Φ (0) l = 0 , (22)where for the zeroth approximation we must restrict to l = 0. Now in order to simplify thestructure of (22), let us to introduce the field χ (0) l =0 ( τ, r ), with ¯Φ (0) l =0 ( τ, r ) = τ χ (0) l =0 ( τ, r ), sothat the equation (22) can be written in the form ∂ χ (0) l =0 ∂τ − r ∂χ (0) l =0 ∂r − ∂ χ (0) l =0 ∂r − m eff ( τ ) χ (0) l =0 = 0 , (23)where m eff ( τ ) = 2 /τ − M / ( H τ ) is the effective mass of the inflaton field. By means ofthe Bessel transformation χ (0) l =0 ( τ, r ) = Z ∞ dkk j l =0 ( kr ) ξ (0) kl =0 ( τ ) (24)we derive from (23) the next equation for the modes ξ k : ∂ ξ (0) k ∂τ + (cid:2) k − m eff ( τ ) (cid:3) ξ (0) k = 0 , (25)such that the modes of ¯Φ (0) l =0 are given by ¯Φ (0) k = τ ξ (0) k . Thus solving (25) the normalizedsolution for the modes ¯Φ (0) k has the form¯Φ (0) k ( τ ) = A ( − τ ) / H (1) ν [ − k τ ] + A ( − τ ) / H (2) ν [ − k τ ] , (26)Here, H (1 , ν [ − kτ ] are respectively the first and second kind Hankel functions, ν = − M H ,and the normalization constants are given by A = − √ πH e − iνπ/ , A = 0 . (27)7ow we introduce the fields h χ (0) L i l =0 ( τ, r ) = Z k Sch k H dk Θ L ( σk ga − k ) h a k j ( kr ) ξ (0) k ( τ ) + a † k j ∗ ( kr ) ξ (0) k ∗ ( τ ) i , (28) h χ (0) S i l =0 ( τ, r ) = Z k Sch k H dk Θ S ( k − σk ga ) h a k j ( kr ) ξ (0) k ( τ ) + a † k j ∗ ( kr ) ξ (0) k ∗ ( τ ) i , (29)where χ (0) l =0 ( τ, r ) = h χ (0) L i l =0 ( τ, r ) + h χ (0) S i l =0 ( τ, r ) and ξ (0) k ( τ ) = τ − ¯Φ (0) k ( τ ). The equationof motion for h χ (0) L i l =0 is given by h ¨ χ (0) L i l =0 − m eff ( τ ) h χ (0) L i l =0 = σ ¨ k ga η (0) l =0 ( τ, r ) + σ ˙ k ga λ (0) l =0 ( τ, r ) + 2 σ ˙ k ga γ (0) l =0 ( τ, r ) , (30)where the stochastic operator fields η (0) l =0 , λ (0) l =0 and γ (0) l =0 are defined as η (0) l =0 ( τ, r ) = Z k Sch k H dk δ ( k − σk ga ) h a k j ( kr ) ξ (0) k ( τ ) + a † k j ∗ ( kr ) ξ (0) k ∗ ( τ ) i , (31) λ (0) l =0 ( τ, r ) = Z k Sch k H dk ˙ δ ( k − σk ga ) h a k j ( kr ) ξ (0) k ( τ ) + a † k j ∗ ( kr ) ξ (0) kl ∗ ( τ ) i , (32) γ (0) l =0 ( τ, r ) = Z k Sch k H dk δ ( k − σk ga ) h a k j ( kr ) ˙ ξ (0) k ( τ ) + a † k j ∗ ( kr ) ˙ ξ (0) k ∗ ( τ ) i , (33)with the dot denoting ∂/∂τ . The field equation (30) can be expressed in the form h ¨ χ (0) L i l =0 − m eff ( τ ) h χ (0) L i l =0 = σ (cid:20) ddτ ( ˙ k ga η (0) l =0 ) + 2 ˙ k ga γ (0) l =0 (cid:21) . (34)This is a Kramers-like stochastic equation, that with the help of the auxiliary field: u (0) l =0 = h ˙ χ (0) L i l =0 − σ ˙ k ga η (0) l =0 , can be written as the first order stochastic system˙ u (0) l =0 = m eff h χ (0) L i l =0 + 2 σ ˙ k ga γ (0) l =0 , (35) h ˙ χ (0) L i l =0 = u (0) l =0 + σ ˙ k ga η (0) l =0 , (36)The role that the noise γ (0) l =0 plays in this system, can be minimized in the system (35)and (36) when the condition ˙ k ga D ( γ (0) l =0 ) E ≪ ¨ k ga D ( η (0) l =0 ) E is valid. This condition can beexpressed as ˙ ξ (0) k ( ˙ ξ (0) kl ) ∗ ξ (0) k ( ξ (0) k ) ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k = σk ga ≪ ¨ k ga ˙ k ga ! , (37)which is valid on large scales i.e. scales where k ga ( τ ) < k < k H ( τ ), for k H and k ga givenrespectively by (19) and (20) . If this is the case, the system (35) and (36) can be approxi- When the background is an exact de Sitter space-time and the field is free, this condition is analogous toone obtained already by Mijic[14] in a different approach. u (0) l =0 = m eff h χ (0) L i l =0 , (38) h ˙ χ (0) L i l =0 = u (0) l =0 + σ ˙ k ga η (0) l =0 . (39)This is an stochastic two-dimensional Langevin equation with a noise η (0) l =0 which is gaussianand white in nature, as it is indicated by the following expressions: D η (0) l =0 E = 0 , (40) D ( η (0) l =0 ) E = 4 πσ k ga ˙ k ga j ( kr ) j ∗ ( kr ) ξ (0) k ξ (0) k ∗ (cid:12)(cid:12)(cid:12) k = σk ga δ ( τ − τ ′ ) . (41)The correlation functions of η (0) l =0 and γ (0) l =0 have the same structure, similarto the momenta of a Gaussian white noise. The dynamics of the probabil-ity transition P (0) l =0 h(cid:2) χ IL (0) (cid:3) l =0 , (cid:2) u I (0) (cid:3) l =0 | h χ (0) L i l =0 , u (0) l =0 i from an initial configuration( (cid:2) χ IL (0) (cid:3) l =0 , (cid:2) u I (0) (cid:3) l =0 ) to a configuration (cid:16) χ (0) L , u (0) (cid:17) , is given by the Fokker-Planck equa-tion: ∂ P (0) l =0 ∂τ = − u (0) ∂ P (0) l =0 ∂ h χ (0) L i l =0 − m eff h χ (0) L i l =0 ∂ P (0) l =0 ∂u (0) l =0 + 12 D (0)11 ∂ P (0) l =0 ∂ h χ (0) L i l =0 , (42)where D (0)11 = R ( σ ˙ k ga ) D ( η (0) l =0 ) E dτ is the diffusion coefficient related to h χ (0) L i l =0 . By using(41) the diffusion coefficient D (0)11 becomes D (0)11 = 4 πσ ˙ k ga k ga j ( kr ) j ∗ ( kr ) ξ (0) k ξ (0) k ∗ (cid:12)(cid:12)(cid:12) k = σk ga . (43)Hence, the dynamics of (cid:28)(cid:16)h χ (0) L i l =0 (cid:17) (cid:29) = R d h χ (0) L i l =0 du (0) l =0 (cid:16)h χ (0) L i l =0 (cid:17) P (0) l =0 is given bythe equation ddτ (cid:28)(cid:16)h χ (0) L i l =0 (cid:17) (cid:29) = 12 D (0)11 ( τ ) . (44)Now, in order to return to the original zeroth order scalar field, let us to use the expression¯Φ (0) l =0 ( τ, r ) = τ χ (0) l =0 ( τ, r ) in (44) to obtain ddτ (cid:28)(cid:16)h ¯ φ (0) L i l =0 (cid:17) (cid:29) = 2 τ (cid:28)(cid:16)h ¯ φ (0) L i l =0 (cid:17) (cid:29) + 12 τ D (0)11 ( τ ) . (45)The general solution of (45), is then (cid:28)(cid:16)h ¯ φ (0) L i l =0 (cid:17) (cid:29) = 12 τ (cid:20)Z τ D (0)11 ( τ ′ ) dτ ′ + C (cid:21) , (46)9ith C an integration constant. Next, we employ the relation (cid:28)(cid:16) ¯ ϕ (0) L (cid:17) (cid:29) = (cid:18) π (cid:19) (cid:28)(cid:16)h ¯Φ (0) L i l =0 (cid:17) (cid:29) , (47)where we have used the addition theorem of the spherical harmonics, to derive the equation D ( ¯ ϕ (0) L ) E = (cid:18) π (cid:19) τ (cid:20)Z D (0)11 ( τ ) dτ + C (cid:21) . (48)This equation, give us in principle the squared fluctuations of ¯ ϕ L on large scales. Employing(26) and (43), the expression (48) with C = 0, can be approximated on the IR sector as (cid:28)(cid:16) ¯ ϕ (0) L (cid:17) (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) kr ≪ ≃ (cid:18) H π (cid:19) ν − σ − ν Γ ( ν ) ( − τ ) − ν Z dk ga k ga k − νga (49)where we have used the asymptotic expansion j ( kr ) | kr ≪ ≃
1. The spectrum derived from(49) at zeroth order (i.e. for l = 0), has the form P (0) k ga ( τ ) ≃ ν − Γ ( ν ) (cid:18) H π (cid:19) [ σ ( − τ ) k ga ] − ν , (50)which results scale invariant when ν = 3 / k ga and therefore with the mass of the BH. Furthermore, theamplitude of this spectrum tends to zero (as τ → ν < / V. FINAL COMMENTS
We have developed a stochastic approach to study scalar field fluctuations of the inflatonfield in an early inflationary universe, which is described by an effective 4D SdS metric. Thecosmological metric was obtained using planar coordinate transformations on a 5D Ricci-flatSchwarzschild-de Sitter (SdS) static metric (1), for a SdS BH. From the dynamical pointof view, the effective 4D cosmological metric (7) describes the collapse of the universe onscales k ≫ k ga and an accelerated expansion for scales much bigger than the gravitational -antigravitational radius r ga , which is related with the wavenumber k ga .10he main difference with earlier stochastic approaches to inflation where the windowfunction is defined on the Hubble horizon is that, in our approach [see eq. (18)], the coarse-grained field is defined using a window function Θ L ( σk ga − k ), which takes into accountonly modes with wavelengths larger than the gravitational - antigravitational radius r ga .This fact indicates the scale for which the universe is starting to expand accelerated. Onsmaller scales the universe is collapsing due to the attraction of the BH. However, on largerscales gravitation is repulsive and drives inflation. For the limit case in which this massis very small, Gm/ (2 ar ga ) ≪
1, we obtain that k ga | Gm/ (2 ar ga ) ≪ → k H , and our resultagrees completely with whole of the squared field fluctuations of a de Sitter expansionduring the inflationary stage when the horizon entry. For r → ∞ J and F approach to a ( τ ) and the metric (7) describes a de Sitter expansion. However, for very large (but finite)cosmological scales the spectrum is not exactly scale independent, because becomes sensitiveto the wavenumber k ga . For ν < / n s = 3 − ν is positive and theamplitude decreases as τ → φ l in the equation (13), which takes into account multipolar expansion due to non-gaussiannoises. Acknowledgements
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