Power spectrum of stochastic wave and diffusion equations in the warm inflation models

We discuss dissipative stochastic wave and diffusion equations resulting from an interaction of the inflaton with an environment in an external expanding metric. We show that a diffusion equation well approximates the wave equation in a strong friction limit. We calculate the long wave power spectrum of the wave equation under the assumption that the perturbations are slowly varying in time and the expansion is almost exponential. Under the assumption that the noise has a form invariant under coordinate transformations we obtain the power spectrum close to the scale invariant one. In the diffusion approximation we go beyond the slow variation assumption. We calculate the power spectrum exactly in models with exponential inflation and polynomial potentials and with power-law inflation and exponential potentials.


I. INTRODUCTION
The standard consequence of the inflationary paradigm is an almost scale invariant spectrum [1][2] which is confirmed by WMAP observations [3] [4]. The power spectrum results from quantization of the quadratic fluctuations around the homogeneous solution [5][6] [7][8] [9]. The formalism is explicitly gauge invariant [10] [11]. It treats quantum gravitational fluctuations and inflaton fluctuations on the same footing. Nevertheless, it has been shown [12] [13] [14] that if the quantum inflaton fluctuations are expressed by stochastic fluctuations in e-fold time then the inflaton fluctuations already contain the gravitational fluctuations leading to the same power spectrum as in [11]. The scalar field models can be considered as effective field theories of scalar cosmological perturbations. In fact, the power spectrum close to the scale invariant one has also been obtained in warm inflation models [15] [16] on the basis of a stochastic wave equation and some thermodynamics arguments. The model of warm inflation is treated as an effective field theory of an inflaton interacting with a large number of other fields [17] [18]. In this paper we study in detail the stochastic wave equation in the form derived from an interaction of scalar fields with an environment [19] [20] [21]. We repeat a calculation of the power spectrum in an extended model of the inflaton-environment interaction [21] on the basis of previously developed methods [22] [23] under the assumption that the perturbations of the non-linear wave equation are slowly varying in time (are almost constant). The main objective of this paper is a development of another tool for a computation of the power spectrum based on the approximation of the dissipative wave equation by a diffusion equation. In the case of a random diffusion the calculation of the power spectrum is much simpler. Its dependence on the evolution law can be seen in a more transparent way. We are able to calculate the power * Electronic address: zbigniew.haba@uwr.edu.pl spectrum with parameters which can substantially vary in time.
The plan of the paper is the following. In sec.2 we discuss a dissipative stochastic wave equation, which has been derived in [21], for an inflaton interacting with an environment by means of a potential U . In sec.3 we show that at strong friction the solutions of the wave equation tend to the solutions of a diffusion equation. In sec.4 we calculate the power spectrum of the stochastic wave equation under the assumption that the evolution of the scale factor is almost exponential and the variables in this equation can be treated as constants. This is a repetition of the standard calculations [22][23] but in a model with different potentials and a different noise. In sec.5 we calculate the power spectrum of the stochastic diffusion equation assuming again that the expansion of the metric is almost exponential and that the variables in this equation can be treated as constants. We obtain the same power spectrum as in the case of the wave equation,i.e., an almost scale invariant spectrum, which is shown to be a consequence of the form of the noise. In sec.6 we discuss solutions of the diffusion equation with almost exponential expansion but with varying potentials. We obtain a shift in the formula for the spectral index. Then, we study exponential potentials in a power-law expanding metric when the method of constant parameters does not apply. The diffusion equation reveals a sensitive dependence of the power spectrum on the potentials. In Appendix A we show in a simple way that the form of the noise that leads to the scale invariant spectrum follows from its invariance under coordinate transformations. In Appendix B we give a simplified derivation of the scale invariant spectrum showing the crucial role of the noise and the exponential expansion.

II. RANDOM WAVE EQUATION
In a flat FLWR expanding metric we consider the wave equation where β −1 is the temperature of the environment and H = a −1 ∂ t a (H may depend on time but we do not write it down explicitly, we do not assume its dependence on φ). U (φ) describes a coupling of the inflaton to the environment. The thermal noise is defined by the covariance Eq.(1) has been derived from an interaction of the inflaton with an environment in [19][20] [21]. Its simplest scheme U (φ) = φ is the basis of the warm inflation approach to cosmology [24] [17] . We consider a linearized form of eq.(1) resulting from an expansion around its homogeneous (spaceindependent) solution The initial conditions are contained in φ c , so we assume zero as the intitial condition for φ. The linearization of eq.(1) expanded about φ c reads We can transform eq.(4) to another form . Let Then where The wave equation with friction is transformed into a wave equation with a complex mass Ω. Note that large 3H + γ 2 (U ′ ) 2 means large Ω.

III. DIFFUSION APPROXIMATION
In this section we show that the diffusion approximation to eq.(4) , i.e., the omission of ∂ 2 t φ, is equivalent to the neglect of fast decaying modes (for a large Ω) in the solution of eqs.(5)- (6). The diffusion approximation to eq.(4) in the momentum space reads (from now on we denote the Fourier transformφ(k) by the same letter as its spatial form φ(x)) On the other hand we may express the solution of eq.(6)(momentum space) with zero initial condition at t 0 by means of the Green function G where the approximate Green function (for large slowly varying ω) is with Expanding ω in powers of (3H +γ 2 ) −1 wherẽ we obtain in the lowest order of the expansion where We compare solutions of the wave equation (4) with solutions of the diffusion equation (7). The solution of the diffusion equation (7) is We compare the solution (14) with (8)- (9). In the Green function (9) we have we neglect the second term as quickly vanishing (for X > 0) and in eq.(9) ω(s) − 1 2 is approximated by ( 3 2 H+ 1 2γ 2 ) − 1 2 (and the same approximation for ω(t) − 1 2 ) then a simple comparison of eqs.(8)- (9) and (14)- (15) leads to the conclusion that for large 3H +γ 2 the solutions of the wave equation and the diffusion equation (with zero initial conditions) coincide.

IV. POWER SPECTRUM OF THE LINEARIZED WAVE EQUATION
We have calculated the power spectrum in the Einstein-Klein-Gordon system in [23] in the case U (φ) = φ. The changes corresponding to the replacement φ → U (φ) are the following: We repeat here the main steps of [23] in order to fix the stage for the discussion of the extended model. We still rewrite the correspondence in a different way. Let replacing Γ from [23], and replacing η from [23]. If in eq.(3) we applied the slow roll approximation then we could express ∂ t φ c in Q by derivatives of the potential U . The power spectrum ρ of fluctuations φ is defined by or in Fourier transform The spectral index 2κ is defined by the low k = |k| be- In [23] we have calculated the spectrum under the assumption that δ andΓ are almost constant. If H is varying in time then estimates by means of the methods [23] (based on [22]) are not reliable for varying parameters. For this reason in the next section we discuss the diffusion approximation when the time evolution can be treated in a more controllable way. We define Without the thermal noise (γ = 0) ǫ can be expressed from Friedmann equations as 1 8πG (V ′ ) 2 V −2 (where G is the Newton constant). With the thermal noise and the interaction U the formula for ǫ in terms of potentials is more involved (see [21],eq.(89)). We keep (21) as a definition of ǫ and do not attempt to express it by potentials.
We introduce the conformal time With a slowly varying H we have approximately Eq. (22) can be obtained by an integration of the identity and the assumption that the last term on the rhs of eq. (24) is small in comparison with the first term.
In terms of τ eq.(4) for the Fourier transform φ(k) reads (k = |k|) and (the term − 3 2 ǫ in eq. (27) is replaced by + 9 2 ǫ in the corresponding formula in [23] owing to the contribution of gravitational modes as expressed by scalar perturbations).
The calculation of the expectation value of the solution of eq.(25) over the noise η leads to for small k (as the Bessel function Y ν (ζ) ≃ ζ −ν for small ζ).Here In eq.(28) the time t in φ c (t) must be replaced by τ then τ is expressed as ζ k . For a smallΓ and U ′ ≃ const we have in a linear approximation in the indices describing the interaction corrections: From eqs.(28)- (30) and (31) if U ′ ≃ const The power spectrum of the stochastic wave equation has been calculated also in [15] [16] in the limit of large friction and under some assumptions of equilibrium thermodynamics.

V. THE POWER SPECTRUM OF DIFFUSION
The solution φ η = φ c + φ of eq.(1) with a given initial condition is a sum of the solution φ c of the homogeneous equation (3) with this initial condition and φ with 0 as an initial condition at t 0 . From the diffusion approximation (14) we obtain We can have ρ t ≃ k −2κ with κ > 0 if t 0 = −∞ (otherwise the integral (33) would be finite at k = 0). For a = exp( t t0 H(t ′ )dt ′ ) this means that the initial condition is at a(t 0 = −∞) = 0. If we introduce the e-fold time (it is assumed that in φ c (s) the cosmic time has been expressed by the e-fold time). We assume in this section thatΓ, δ ,U ′ (φ c ) , H and (U ′ ) −2 U ′′ U are slowly varying in time, so that we may approximate them by a constant.
We introduce the variable u = exp(−2ν) (36) and assume that H(ν) ≃ const then The result of integration in eq.(37) assuming that H,U ′ Γ and q are approximately constant can be expressed by the incomplete Γ function We have for x << 1 and for x >> 1 If ν 0 → −∞(u(ν 0 ) → +∞) then the second term in eq.(39) is vanishing. In such a case for a small k where in a linear approximation in the indices δ andΓ we get 3) (normalized so that the scalar modes behave as plane waves at large k(aH) −1 ). It follows from eq.(39) that the amplitude of thermal fluctuations is determined by H, κ (known from CMB measurements [30][31]), β and γ (which this way would be fixed by ρ t (k)). On the other hand the friction γ is related (depending on the model) to other measurable quantities as ,e.g., the diffusion constant [32] or the density of radiation at the end of inflation in the warm inflation scenario [24]. In this way the amplitude of stochastic thermal fluctuations depends on many parameters, whereas the virtue of the quantum result consists in the prediction of its 10 −5 magnitude [5][6] [7][8] [9] in agreement with observations. The theory shows that under the assumption of almost exponential expansion both the quantum fluctuations and the thermal fluctuations of the inflaton lead to a small deviation from the scale invariant spectral index ( this index is crucial for distinguishing various inflation models on the basis of observational data [30] [31]. The assumption that thermal fluctuations are of quantum origin does not change essentially the results as at high temperatures at the early stage of the universe quantum theory is well approximated by the classical one. We could derive the corresponding stochastic wave equation with quantum thermal noise, as briefly discussed in [21] ( see also [29] ) , but at higher temperature this quantum noise tends to the classical noise. Although CMB shows the quantum Planck spectrum (at all wave lengths) the perturbations of the homogeneous solutions at large wave lengths exhibit no quantum effects.
In the next section we show that the potential U describing an interaction of the inflaton with the environment can shift the spectral index. It may be difficult on the basis of a study of the power spectrum to determine whether the deviation from the scale invariant spectrum discovered in WAMP observations comes from quantum or thermal fluctuations. If the initial state of the universe is Gaussian then further evolution of quantum cosmological perturbations proceeds in a squeezed state with a classical evolution as shown in [35]. In such a case it would be difficult to discover whether the origin of the universe is of quantum nature. The eventual observation of non-Gaussian correlations [33] in CMB could show that a decoherence of quantum superpositions really takes palace.

VI. BEYOND THE SLOWLY VARYING CORRECTIONS
In the calculations of the spectrum of the stochastic wave equation in sec.4 as well as of the spectrum of the diffusion equation in sec.5 we assumed that H,δ andΓ vary so slowly that we can approximate them by constants in the calculation of the power spectrum. The slow variation is consistent with the slow roll approximation usually made for inflation. The calculations in sec.4 relied heavily on the assumption of the slow variation. are much easier then the study of the corresponding wave equation. We rewrite the formula (33) for the spectrum in the form In this formula we admit that V ′′ (φ c (t)) and U ′ (φ c (t)) have a substantial variation in time. This property depends on the potentials as well as on H. First, we assume that H is (almost) constant. In the damped wave equation the large time behaviour does not depend on the term ∂ 2 t φ. For small time in order to neglect ∂ 2 t φ we must make the assumptions that (V ′ V −1 ) 2 and V ′′ V −1 are small and slowly varying in time (this is the slow-roll approximation). Neglecting ∂ 2 t φ in eq.(3) we can represent this equation in an integral form (3) will have solutions decaying to zero (or to a constant) as from eq.(45) ∂ t φ is negative if V ′ + 3 2γ 2 is positive. If the decay is exponential so that then, as follows from the estimates of sec.5 after an insertion of (47) in eq.(44) Hence, the decay (47) leads to a shift of the spectral index. The behaviour (48) can really happen as we can see assuming that V ′ is negligible and γ 2 (U ′ ) 2 >> 3H. Then, eq.(45) hast the solution If U ≃ φ n then r = 3H(n−1) 2n . An exponential decay will be a common behaviour for polynomial V and As a next example if V = g 4 φ 4 , U = φ then b = 1 2 γ 2 , but r = 0 (no effect on the power spectrum in eq.(48)). The decay can be non-exponential as can be seen if V = g 4 φ 4 and U = 1 2 φ 2 then In such a case b = r = 0 and the power spectrum is changed only by logarithmic corrections. As a different class of models let us consider the powerlaw inflation a = t α , H = α t . Consider the potentials and U (φ) = Λ exp(uφ). Eq.
(3) has a solution of the form if the parameters satisfy the relation We require α > 1, hence u 2 γ 2 Λ 2 < 1 The s-integral in eq.(44) reads (we choose t 0 = 0 so that a(t 0 ) = 0 ) where K is a certain function bounded for a small k, Performing the integral (55) we obtain the power spectrum (42) with For h = 1 the result is the same as in the case of a power spectrum of quantum fields [26] in a metric a ≃ t α

VII. SUMMARY AND OUTLOOK
The stochastic wave equation of warm inflation is usually considered as a phenomenological effective field theory of an inflaton. We have investigated its long wave power spectrum on a basis of some well-controlled approximations. We have shown that the scale invariant spectrum is related to the coordinate-independent form of the noise and to the accelerated expansion of the metric. The diffusion approximation derived in this paper allows to study the inflaton power spectrum beyond the assumption of an almost exponential expansion and small variation of the potentials. We considered a potential U describing interaction with an environment which in the case of an almost exponential expansion shifted the spectral index. In an example of a power-law inflation we have obtained a power spectrum close to the scale invariant one in models with exponential potentials. For a small friction and an almost exponential expansion the departure from the scale invariant spectrum is determined by the same formula as the one obtained from quantization of the scalar field on an external expanding spacetime. If there is a friction then Hamiltonian quantum mechanics is not well-defined. However, we suppose that the proper formulation as a dissipative Lindblad theory would lead the same formula for the power spectral index. Our stochastic methods suggest that the spectral long wave index cannot distinguish between quantum inflaton fluctuations and classical thermal fluctuations. The time evolution of cosmological perturbations has been studied in [34] [35] with the conclusion that if the inflation starts from a Gaussian state then it quickly becomes classical (decoherence without the environment). The CMB spectrum satisfying the Planck law is certainly quantum but we could not see this in the long wave limit. If we admitted non-Gaussian states then a complete decoherence theory based on the Lindbald equation would be needed [36] in order to explain the structure formation and detect when the classical behaviour begins. From the formula for ρ t (k) in this paper we could conclude that ρ t (k) ≃ k −1 for large k. This is a quantum behaviour of φ 2 . However, for large k the stochastic equation discussed in this paper is not reliable. One should rather study the interaction with an environment at high momenta initiated in [21].

VIII. APPENDIX A:INVARIANCE UNDER A CHANGE OF COORDINATES
We give a simple proof that the stochastic wave equation without friction (friction comes from an interaction of φ with an environment as in [21], the low momentum approximation is not invariant under change of coordinates) where ∂x ∂y | is the Jacobian.So On the other hand g(x) = g(y)| ∂x ∂y | 2 (61) Hence, g − 1 4 η is invariant (in a flat expanding metric this is a − 3 2 η).

IX. APPENDIX B:EXACT FORMULA FOR THE EXPONENTIAL EXPANSION
When a(t) = exp(Ht) then the solution of the diffusion can be obtained explicitly. Let us denote and assume that M 2 can be approximated by a constant. The solution of the linear diffusion equation (7) with zero initial condition at t 0 = −∞ ( a(−∞) = 0) is Let u(s) = exp(−2Hs) R = 3H 2 (1 + 1 3γ 2 H −1 ) = 3H 2 (1 +Γ) Then The integral can be expressed by the incomplete Γ From eq.(67) If γ = 0 then with δ = M 2 3H 2 . This is exactly the index resulting from a quantization of the scalar field in an exponentially expanding universe [26][27] [28].