Metric fluctuations and its evolution during inflation

W discuss the evolution of the fluctuations in a symmetric $\phi_c$-exponential potential which provides a power-law expansion during inflation using both, the gauge invariant field $\Phi$ and the Sasaki-Mukhanov field.


I. INTRODUCTION AND MOTIVATION
It is now widely accepted that dominant cause of structure in the Universe is a spatial perturbation. This perturbation is present on cosmological scales a few Hubble times before these scales enter the horizon, at which stage it is time-independent with an almost flat spectrum. One of the main objectives of theoretical cosmology is to understand its origin [1]. The usual assumption is that the curvature perturbation originates during inflation of the slow-rolling inflaton field. As cosmological scales leave the horizon, the quantum fluctuation is converted to a classical gaussian perturbation with an almost flat spectrum, generating inmediately the required curvature perturbation which is constant until the approach of horizon entry [2]. This idea has the adventage that prediction for the spectrum is independent of what goes on between the end of inflation and horizon entry. The spectrum depends only on the form of the potential and on the theory of gravity during inflation, provinding, therefore, a direct probe of conditions during this era.
Stochastic inflation has played an important role in inflationary cosmology in the last two decades. It proposes to describe the dynamics of this quantum field on the basis of two pieces: the homogeneous and inhomogeneous components [3][4][5][6][7][8][9][10]. Usually the homogeneous one is interpreted as a classical field φ c (t) that arises from the vacuum expectation value of the quantum field. The inhomogeneous component φ( x, t) are the quantum fluctuations. The field that take into account only the modes with wavelengths larger than the now observable universe is called coarse-grained field and its dynamics is described by a second order stochastic equation [10,11]. Since these perturbations are classical on super Hubble scales, in this sector one can make a standard stochastic treatment for the coarse-grained matter field [10]. The IR sector is very important because the spatially inhomogeneities in super Hubble inflationary scales would explain the present day observed matter structure in the universe.
In this work we consider gauge-invariant fluctuations of the metric in the early inflationary universe [2]. Metric fluctuations are here considered in the framework of the linear perturbative corrections. The scalar metric perturbations are spin-zero projections of the graviton, which only exists in nonvacuum cosmologies. The issue of gauge invariance becomes critical when we attempt to analyze how the scalar metric perturbations produced in the early universe influence a background globally flat isotropic and homogeneous universe. This allows us to formulate the problem of the amplitude for the scalar metric perturbations on the evolution of the background Friedmann-Robertson-Walker (FRW) universe in a coordinate-independent manner at every moment in time. On the other hand, the Sasaki-Mukhanov (SM) field takes into account both, metric and inflaton fluctuations [12]. One of the aims of this work is the study of the evolution of the SM field during inflation to make a comparison with gauge-invariant metric fluctuations.

II. FLUCTUATIONS
Matter field fluctuations are responsible for metric fluctuations around the background FRW metric. When these metric fluctuations do not depend on the gauge, the perturbed globally flat isotropic and homogeneous universe is described by [2] where a is the scale factor of the universe and (ψ, Φ) are the gauge-invariant perturbations of the metric. In the particular case where the tensor T αβ is diagonal, one obtains: Φ = ψ [2]. We consider a semiclassical expansion for the inflaton field ϕ( x, t) = φ c (t) + φ( x, t) [10], with expectation values 0|ϕ|0 = φ c (t) and 0|φ|0 = 0. Here, |0 is the vacuum state. Due to 0|Φ|0 = 0, the expectation value of the metric (1) gives the background metric that describes a flat FRW spacetime: After linearizing the Einstein equations in terms of φ and Φ, one obtains where β = 0, 1, 2, 3, a is the scale factor of the universe and the prime denotes the derivative with respect to φ c . The dynamics of φ c is given by the equations and H =ȧ/a is the Hubble parameter. Furthermore, the scalar potential can be written in terms of the Hubble parameter The equation (2) can be simplified by introducing the field This field can be expanded in terms of the modes where α k and α † k are the annihilation and creation operators that complies with the commutation relations The equation for the modes Q k isQ where ω 2 k = a −2 (k 2 − k 2 0 ) is the squared time dependent frequency and k 0 separates the infrared and ultraviolet sectors, and is given by Since the field Q satisfy a Klein-Gordon like equation on a FRW background metric ds 2 = dt 2 − a 2 dx 2 , also satisfy the commutation relationship This implies that the modes Q k are renormalized by the expressioṅ A

. Particular solutions
If the inflaton field oscillates around the minimum of the potential at the end of inflation the particular solutions whenφ c = 0 andφ c = 0 are very important.
On the pointsφ c = 0 we obtain that Q k = 0. However, the solutions for Φ k are nonzero where φ 0 k is the initial amplitude for Φ k , for each wavenumber k. This means that the amplitude of each mode decreases exponentially with time.
Other interesting particular solution is located at the pointsφ c = 0, when the field is at the minumum of the potential. In these points the equation (11) adopts the form where Φ k = a −1/2 Q k .

B. The Sasaki-Mukhanov field
A manner to study both, metric and inflaton fluctuations, can be made by means of the SM field [12]: S = φ +φ c Hc Φ. The modes of this field obeys the following equation where the modes S k complies with the renormalization conditioṅ

C. Power spectrum
One can estimate the power spectrum of the fluctuations for the fields Φ and S. The spectrum of the fluctuations for Φ are whilst the power spectrum for the SM field is the same that of the inflation field It is well known from experimental data [13] that the universe has a scale invariant power spectrum on cosmological scales.

III. AN EXAMPLE: SYMMETRIC EXPONENTIAL φ C -POTENTIAL: POWER-LAW INFLATION
As a first example we consider a scalar potential given by V (φ c ) = V 0 e 2α|φc| , where α 2 = 4π M 2 p p gives the relationship between α and the power of the expansion p. This potential is related to a scale factor that evolves as a ∼ t p (with constant power p), which corresponds to a Hubble parameter H(t) = p/t, which can be written in terms of the scalar field and H e = p/t e is the value of the Hubble parameter at the end of inflation. The temporal evolution for |φ c (t)| is given by where t ≥ t 0 . Sinceφ c = −sgn(φ c ) 1 αt andφ c = sgn(φ c ) 1 αt 2 (we assume sgn(φ c ) = ±1 for φ c positive and negative, respectively), the equation that describes the evolution for Φ results After make the transformation Q = Φe (p+2)t −1 dt , we obtain the differential equation for Q The general solution for the modes Q k (t) is where (C 1 ,C 2 ) are constants, (H (1) ) are the Hankel functions of (first, second) kind with x(t) = t p o k a 0 (p−1)t p−1 and ν 1 = p+1 2(p−1) . Using the renormalization conditionQ * k Q k −Q k Q * k = i, we obtain the Bunch-Davis vacuum [14] In the UV sector the function H (2) ν 1 [x] adopts the asymptotic expression (i.e., for x ≫ 1) whilst on the IR sector (i.e., for x ≪ 1) it tends asymptotically to The Φ-squared field fluctuations on the IR sector are ( Φ 2 ) IR = 1 2π 2 ǫk 0 (t) 0 dkk 2 |Φ k | 2 , and becomes where ǫ = k (IR) max /k p ≪ 1 is a dimensionless constant, k (IR) max = k 0 (t * ) at the moment t * when the horizon entry and k p is the Planckian wavenumber (i.e., the scale we choose as a cut-off of all the spectrum). The power spectrum on the IR sector is P Φ | IR ∼ k 3−2ν 1 . Note that ( Φ 2 ) IR increases for p > 2, so that to the IR squared Φ-fluctuations remain almost constant on cosmological scales we need p ≃ 2. We find that a power close to p = 2 give us a scale invariant power spectrum (i.e., with ν 1 ≃ 3/2 for ( Φ 2 ) IR . Furthermore, density fluctuations for matter energy density are given by δρ/ρ = −2Φ, so that δρ 2 1/2 / ρ ∼ Φ 2 1/2 .
On the other hand, in the UV sector these fluctuations are given by

IV. FINAL COMMENTS
In this paper we have studied the evolution of the fluctuations in a symmetric φ cexponential potential which provides a power-law expansion using both, the gauge invariant field Φ and the Sasaki-Mukhanov field. This last takes into account simultaneously, the inflaton and metric fluctuations. The results obtained from the evolution of ( Φ 2 ) and ( S 2 ) are different in both treatments. The reason can be explained from the fact that the field S = φ +φ c H Φ is not gauge-invariant and hence do not describes correctly the fluctuations for φ and Φ. The fluctuations are well described by the field Φ which is gauge invariant and predicts a scale invariant power spectrum on the IR sector for p → 2. Note that we have not considered back-reaction effects which are related to a second-order metric tensor fluctuations. This topic was considered by Abramo and Nambu, who investigated a renormalization-group method for an inflationary universe [15,16]. A different approach to describe the metric fluctuations was considered more recently by Lyth and Wands [17] (see also [18]), who suggested that curvature perturbation could be generated by a light scalar field named curvaton.