Quantum fluctuations of Cosmological Perturbations in Generalized Gravity

Transcription

1 Quantum fluctuations of Cosmological Perturbations in Generalized Gravity Jai-chan wang Department of Astronomy and Atmospheric Sciences, Kyungpook National University, Taegu, Korea (July 6, 996 Recently, we presented a unified way of analysing classical cosmological perturbation in generalized gravity theories. In this paper, we derive the perturbation spectrums generated from quantum fluctuations again in unified forms. We consider a situation where an accelerated expansion phase of the early universe is realized in a particular generic phase of the generalized gravity. We take the perturbative semiclassical approximation which treats the perturbed parts of the metric and matter fields as quantum mechanical operators. Our generic results include the conventional power-law and exponential inflations in Einstein s gravity as special cases. PACS numbers: Sq, h, v, Cq The highly isotropic cosmic microwave background radiation permits a linear paradigm of treating the structure formation process in the evolving universe. It is widely accepted that the currently observable large scale structures are developed from remnants of the quantum fluctuations imprinted during early accelerated expansion stage. In the context of Einstein s gravity both the quantum generation and classical evolution processes can be rigorously handled; studies in [,] can be compared with the previous order of magnitude analyses [3], or analyses in differnt gauges [4]. For a scalar field, self consistent and rigorous analyses are possible mainly due to a special role of particular gauge condition (or equivalently, gauge invariant combination which suits the problem: the uniform-curvature gauge. Variety of theoretical reasons allude possible generalization of the gravity sector due to quantum correction in the high energy limit, and thus in the early universe [5,6]. There were many studies on the classical evolution of structures in some favored generalized gravity theories [7]. owever, again, in the classes of generalized gravity involving the scalar field and scalar curvature, we found that the uniform-curvature gauge suits the problem allowing simple and unified treatment possible [8]. For the growing mode, the large scale solution known in the minimally coupled scalar field remains valid even in a wide class of generalized gravity. In this paper we investigate the quantum generation process in the context of generalized gravity. The properly chosen gauge condition again allows us to present the generated power spectrum in generic forms which are applicable to various generalized gravity theories and underlying background evolutions. We consider the gravity theories with an action S = d 4 x [ g f(φ, R ] ω(φφ;a φ,a V (φ, ( where φ is a scalar field and R is a scalar curvature; f is a general algebraic function of φ and R, andv and ω are general functions of φ. It includes diverse classes of generalized gravity theories as subsets, [9,0]. As a metric describing the universe we consider a spatially homogeneous, isotropic and flat background, and general perturbations of a scalar type ds = ( + α dt χ,α dtdx α + a δ αβ ( + ϕ dx α dx β, ( where a(t is a cosmic scale factor; α(x,t, χ(x,t, and ϕ(x,t are perturbed order quantities. For the scalar field we let φ(x,t= φ(t+δφ(x,t, (3 where the background quantities are indicated by overbars which will be neglected unless necessary. We have not chosen the temporal gauge condition which can be used as an advantage in handling problems; all perturbed order variables are spatially gauge invariant, see Sec. IV C of [9]. We introduce a gauge invariant combination δφ ϕ δφ φ φ ϕ ϕ δφ, (4 where ȧ/a. δφ ϕ is a gauge invariant variable which is the same as δφ in the uniform-curvature gauge which chooses ϕ = 0 as the gauge condition; Eq. ( shows that ϕ = 0 implies that spatial curvature vanishes (uniform in general, thus justifying the name. Ignoring the surface terms, the action valid to the second order in the perturbation variables is derived in [0] as δs = { a 3 δ φ ϕ α δφ a ϕ δφ ϕ,α [ ( ] } + φ a a 3 φ 3 δφ ϕ dtd 3 x. (5 The non-einstein nature of the theory is present in a parameter which is defined as

2 (t ω + 3Ḟ φ F ( F +, (6 F where F f/( R. becomes unity in Einstein s gravity where F ==ω. Equation (5 leads to an equation of motion δ φ ϕ + (a3 a 3 δ φ ϕ { a + a 3 [ ( ] } φ a φ 3 δφ ϕ =0. (7 In the large scale limit, ignoring the Laplacian term, we have a general integral form solution δφ ϕ (x,t= φ [ t ] C(x D(x 0 a 3 φ dt, (8 where C(x andd(x are coefficients of the growing and the decaying modes, respectively. Notice that the growing mode is not affected by the non-einstein nature of the theory. The growing mode of ϕ δφ is conserved as C(x, whereas the decaying mode is higher order in the large scale expansion; see Sec. VI A of [9]. Thus, C(x, which encodes the spatial structure, can be interpreted as ϕ δφ in the large scale limit. The solution in Eq. (8 is valid considering general V (φ, ω(φ, and f(φ, R in various subsets of generalized gravity theories described in [9,0]. Introducing v(x,t z Ḣ φ δφ ϕ, Eq.(7canbewrittenas v ( + z z z(t a φ, (9 v =0, (0 where a prime denotes a derivative with respect to the conformal time η, dη dt/a. In the small scale limit (z /z k wehave δφ ϕ (k,η= [ a c (ke ikη + c (ke ikη]. ( k In an approach called a perturbative semiclassical approximation, [], we regard the perturbed part of the field and metric as quantum mechanical operators, meanwhile the background parts are considered as classical. Instead of the classical decomposition we replace the perturbed order variables with the quantum (eisenberg representation operators as φ(x,t= φ(t+δˆφ(x,t, ϕ(x,t ˆϕ(x,t, etc., δ ˆφ ϕ δ ˆφ φ ˆϕ. ( An overhat indicates the quantum operator. The background order quantities are considered as classical variables. This approach has a different spirit compared with the quantum field theory in curved spacetime, where in the latter case the metric sector is regarded as classical and given (sometimes considering some prescribed backreaction [6,]. Our approach considers the field and the metric in equal footing []. Since we are considering a flat three-space background we may expand δ ˆφ(x,tin the following mode expansion δ ˆφ ϕ (x,t= d 3 k [ â (π 3/ k δφ ϕk (te ik x +â k δφ ϕk (te ik x]. (3 The annihilation and creation operators â k and â k satisfy the standard commutation relations: [â k, â k ]=0, [â k, â k ]=0, [â k, â k ]=δ3 (k k. (4 δφ ϕk (t is a mode function, a complex solution of the classical mode evolution equation. Equation (7, with δ ˆφ ϕ replacing δφ ϕ, leads to an equation for the mode function δφ ϕk which satisfies the same form as Eq. (7. From the action for δφ ϕ in Eq. (5 we can derive the conjugate momenta and the commutation relation. Using S = Ldtd 3 x we have δπ ϕ (x,t L/( δ φ ϕ = a 3 δ φ ϕ (x,t. Thus, the equal-time commutation relation [δ ˆφ ϕ (x,t,δˆπ ϕ (x,t] = iδ 3 (x x leadsto [δˆφ ϕ (x,t,δ ˆφϕ (x,t] = i a 3 δ3 (x x. (5 In order for Eq. (4 to be in accord with Eq. (5, the mode function δφ ϕk (t should follow the Wronskian condition Assuming δφ ϕk δ φ ϕk δφ ϕk δ φ ϕk = i a 3. (6 z /z = n/η, n = constant, (7 Eq. (7 becomes a Bessel equation with a solution π η [ δφ ϕk (η = c (k ν ( (k η a ] + c (k ν ( (k η, ν n+ 4. (8 The coefficients c (k andc (k are arbitrary functions of k which are normalized according to Eq. (6 as c (k c (k =. (9 Imposition of the quantization condition in Eq. (6 does not completely fix the coefficient. The remaining freedom is related to the choice of the vacuum state. The

3 adiabatic vacuum (in de Sitter space it is often called as Bunch-Davies vacuum, [3] chooses c (k and c (k 0 which corresponds to the positive frequency solution in the Minkowski space limit. The power spectrum becomes P δ ˆφ ϕ (k, t k3 π δ ˆφ ϕ (x + r,tδˆφ ϕ (x,t vac e ik r d 3 r = k3 π δφ ϕk(t, (0 where we used a k vac 0 for every k. Assuming the adiabatic vacuum, the two-point function becomes G(x,x δˆφ ϕ (x δˆφ ϕ (x vac = ( 4 ν sec(πν F ( 3 + ν, 3 ν;;+ η x 4η η 6πa a η η, ( which is valid for ν< 3 ;x (x,t, η (η η and x (x x. In the small scale limit, thus kη, (8 becomes δφ ϕk (η = [ a c (ke ikη i(ν+ π k + c (ke ikη+i(ν+ π ]. ( In the large-scale limit we have ( ν η Γ(ν k η [ ] δφ ϕk (η =i a c (k c (k, π (3 and the power spectrum becomes P / δ ˆφ (k, η = Γ(ν ( 3/ ν k η c (k c ϕ π 3/ (k. a η (4 In Eqs. (8-4 no additional dependence on k arises from the generalized nature of the theory. Let us see the implication of the condition in Eq. (7. Introduce the following notations ɛ Ḣ, ɛ φ φ, ɛ 3 F F, ( ɛ 4 Ė E, E F ω+ 3 F. (5 φ F For ɛ =0,wehave η=, (6 a +ɛ and for ɛ i = 0 we have [for general ɛ i s, see Eq. (88 of [9]] z z = n η = ( ɛ η ( + ɛ + ɛ ɛ 3 + ɛ 4 (+ɛ ɛ 3 +ɛ 4. (7 For ɛ i = 0 we have a t /ɛ, φ a ɛ (thus, φ t ɛ/ɛ for ɛ ɛ,andφ ln t for ɛ = ɛ, F a ɛ3, and E a ɛ4. For a t q with q = 3(+w,wehave ɛ = /q and a η /(+3w. In the limit of Einstein s gravity, thus =, the solutions in Eqs. (8-4 reduce to the ones derived in III of [], [3,4]. For a power-law expansion stage supported by the scalar field in Einstein s gravity, we have ɛ 3 =0=ɛ 4 and φ/ = constant. Thus, ɛ = ɛ = /q and Eqs. (8,7 lead to ν = 3q 3(w = (q (3w +. (8 w < corresponds to q>. The exponential expansion stage corresponds to w, thus ν 3 ;inthis case we have η = /(a wherebecomes a constant. 3 Now, we derive some observationally relevant classical power spectrums generated from quantum fluctuations as the initial seeds. Ignoring the transient mode, from Eq. (8 we have P / C (k, t = φ P/ δφ ϕ (k, t, (9 where the classical power spectrum of a fluctuating field f(x,t is defined as P f (k, t k3 f(x + r,tf(x,t π x e ik r d 3 r. (30 We have in mind a generic scenario where the classical structures arise from the quantum fluctuations pushed outside horizon and classicalized during accelerated expansion stage. As an ansatz we identify P δφϕ (k, t =Q(k, t P δˆφϕ (k, t, (3 where P δφ and P δ ˆφ are based on the classical volume average and the quantum vacuum expectation value, respectively, Eqs. (30,0. Q(k, t isafactorwhichmay take into account of the possible modification of the spectrum due to the classicalization process of the quantum field fluctuations. We may call it a classicalization factor. Ordinarily it is taken to be unity, however, the decoherence, noise and nonlinear field effects may affect its value, particularly its amplitude, [5]. Assuming Eq. (7 we have derived the quantum fluctuations in the large scale limit in Eq. (4. Thus, combining Eqs. (9,3,4 we have P / C (k, η = ( 3/ ν Γ(ν k η φ π 3/ a η Q(k c (k c (k, (3 (η LS,GGT 3

4 where quantities in the right hand side should be evaluated when the scale we are considering was in the large scale limit (LS during an expansion stage supported by a generalized gravity theory (GGT. In the Einstein gravity limit ( = and an exponential expansion (EXP stage [ν = 3 and η = /(a] Eq. (3 reduces to the well known result P / C (k, η = π c (k c (k Q(k. (33 φ LS,EXP We note that in general the power spectrum depends on the choice of a vacuum state and possibly on the classicalization factor Q(k. Now, in Eq. (3 we have the quantum fluctuation generated power spectrum imprinted in a conserved quantity C(x. From the power spectrum of C(x we can derive the spectrums of observable quantities in the present universe, e.g., density fluctuations, velocity fluctuations, potential fluctuations, and temperature fluctuations in the cosmic microwave background radiation in the matter dominated era in Einstein s gravity; these are [8] δϱ ϱ = ( k C, δv = ( k C, 5 a 5 a δφ = 3 5 C, δt T = C. (34 5 Notice that we are considering a linear theory. In the linear theory, all perturbed order quantities are linearly related with each other which is true even between variables in different gauges. C(x is a temporally constant, but spatially varying, coefficient of the growing mode. The spatial curvature (or potential fluctuation in the uniform-field gauge (which coincides with the comoving gauge [CG] in a minimally coupled scalar field, see Sec. IV C of [9] is conserved as C(x; i.e., ϕ δφ (x,t=c(x= ϕ CG (x,t. C(x encodes the spatial structure of the fluctuations and is conserved during the linear evolution in the large scale limit. Indeed, this linearity is one basic underlying reason why we were successful in tracing the structure evolution in a simple and unified way. owever, apparently, there arises no structure formation in the linear theory [structures are preserved in C(x], and one should not miss that the gravity theories are highly nonlinear. In this paper we have not used the conformal transformation which relates the generalized gravity in Eq. ( to Einstein s gravity in classical level, [0]. Quantum fluctuations are derived in the original frame of the generalized gravity. Still, the underlying conformal symmetry with Einstein s gravity can be regarded as an important factor which allows the unified and simple analyses possible in the classical level, [0]. We have implictly assumed the existence of an accelaration stage supported by the generalized gravity with the condition in Eq. (7. Constructing specific models with applications will be considered elsewhere. We thank Dr.. Noh for useful discussions. This work was supported in part by the Korea Science and Engineering Foundation, Grants No and No , and through the SRC program of SNU-CTP. [] J. wang, Phys. Rev. D 48, 3544 (993. [] J. wang, J. Korean Phys. Soc. 8, S50 (995 [3] A.. Guth and S. Pi, Phys. Rev. Lett. 49, 0 (98; S. W. awking, Phys. Lett. B5, 95 (98; A. A. Starobinsky, Phys. Lett. B7, 58 (98. [4] J. M. Bardeen, P. J. Steinhardt and M. S. Turner, Phys. Rev. D 8, 679 (983. R.. Brandenberger and R. Kahn, Phys. Rev. D 9, 7 (984; J. J. alliwell and S. W. awking, Phys. Rev. D 3, 777 (985; D.. Lyth, Phys. Rev. D 3, 79 (985; V. F. Mukhanov, JETP Lett. 4, 493 (985; M. Sasaki, Prog. Theor. Phys. 76, 036 (986; V. F. Mukhanov,. A. Feldman and R.. Brandenberger, Phys. Rep. 5, 03 (99; S. W. awking, R. Laflamme and G. W. Lyons, Phys. Rev. D 47, 534 (993. [5]B.S.DeWitt,Phys.Rev.6 95 (967; G. t ooft and M. Veltman, Ann. Inst. enri Poincaré XX, 69 (974; S. L. Adler, Rev. Mod. Phys. 54, 79 (98; M. Green, J. Schwarz and E. Witten, Superstring Theory, Vol and (Cambridge Univ. Press: Cambridge, 987; Y. M. Cho, Phys. Rev. Lett. 68, 333 (99. [6] N. D. Birrell and P. C. W. Davies, Quantum fields in curved space (Cambridge, Cambridge University Press, 98; [7] D. S. Salopek, J. R. Bond and J. M. Bardeen, Phys. Rev. D 40, 753 (989; E. W. Kolb, D. S. Salopek and M. S. Turner, Phys. Rev. D 4, 395 (990; J. wang, Class. Quantum Grav. 8, 95 (99; N. Deruelle, C. Gundlach and D. Langlois, Phys. Rev. D 46, 5337 (99; R. Fakir, S. abib and W. Unruh, Astrophys. J. 394, 396 (99; A.. Guth and B. Jain, Phys. Rev. D 45, 46 (99; S. Mollerach and S. Matarrese, Phys. Rev. D 45, 96 (99; V. F. Mukhanov,. A. Feldman and R.. Brandenberger, Phys. Rep. 5, 03 (99; R. Brustein, et al,, Phys. Rev. D 5, 6744 (995. [8] J. wang, Phys. Rev. D 53, 76 (996. [9] J. wang and. Noh, Phys. Rev. D 54, 460 (996. [0] J. wang, unpublished (996 gr-qc/ [] J. wang, Class. Quantum Grav., 305 (994. [] M. J. Duff, in Quantum Gravity : A Second Oxford Symposium, Eds.C.J.Isham,R.PenroseandD.W.Sciama (Oxford, Oxford University Press, 98 8p. [3] T. S. Bunch and P. C. W. Davies, Proc. R. Soc. A 360, 7 (978. [4] L.. Ford and L. Parker, Phys. Rev. D 6, 45 (977; L. F. Abbott and M. B. Wise, Nucl. Phys. B 44, 54 (984; C. Pathinayake and L.. Ford, Phys. Rev. D 4

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