Scalar Perturbation Produced at the Pre-inflationary Stage in Eddington-inspired Born-Infeld Gravity

We investigate the scalar perturbation produced at the pre-inflationary stage driven by a massive scalar field in Eddington-inspired Born-Infeld gravity. The scalar power spectrum exhibits a peculiar rise for low $k$-modes. The tensor-to-scalar ratio can be significantly lowered compared with that in the standard chaotic inflation model in general relativity. This result is very affirmative considering the recent dispute on the detection of the gravitational wave radiation between PLANCK and BICEP2.


I. INTRODUCTION
The Eddington-inspired Born-Infeld (EiBI) gravity is described by the action [1], where κ is the only additional parameter of the theory to the gravitational constant G (in this work, we set 8πG = 1), λ is a dimensionless parameter, and S M (g, ϕ) is the action for the matter which is coupled only to the gravitational field g µν . The cosmological constant is related by Λ = (λ − 1)/κ which we will set to zero in this paper. The theory follows the Palatini formalism in which the metric g µν and the connection Γ ρ µν are treated as independent fields. The Ricci tensor R µν (Γ) is evaluated solely by the connection.
The inflationary model with a massive scalar field in this theory was investigated in Ref. [2]. The matter action is given by which is the same form as for the chaotic inflation model [3] in general relativity (GR). Due to the square-root type of the EiBI action, there is an upper bound in pressure beyond which the theory is not defined. In the maximal pressure state (MPS), the scale factor exhibits an exponential expansion of the Universe. The MPS is the past attractor from which all the evolution paths of the Universe originate [2]. At early times, the energy density is very high at the MPS (the magnitudes of the field ϕ and its velocity are very large). The Hubble parameter becomes H MPS ≈ 2m/3, and thus the curvature scale remains constant. In describing the high-energy state of the early universe, therefore, quantum gravity is not necessary .
The MPS is known to be unstable under the global perturbation (zero-mode scalar perturbation) [2]. With a small perturbation, the Universe evolves to the near-MPS stage, and then enters into the intermediate stage which is followed by an inflationary attractor stage. The inflationary feature at the attractor stage was found to be the same as the ordinary chaotic inflation in GR [2].
As a whole, there are two inflationary stages in the EiBI inflation model, the near-MPS stage and the attractor stage. (Since the inflationary feature at the attractor stage is similar to the standard inflation, we call the near-MPS stage as the pre-inflationary stage for convenience.) If the initial conditions drive the Universe to evolve in such a way to acquire the sufficient 60 e-foldings at the attractor stage, the cosmological situation must be very similar to that of the standard chaotic inflation. However, if they do not, the supplementary e-foldings should be provided at the near MPS-stage to solve cosmological problems. In this case, the story of the density perturbation may be altered in the long wave-length modes.
The tensor perturbation in the EiBI inflation model was investigated in Ref. [4]. For short wave-length modes, the perturbation is very similar to that of the standard chaotic inflation in GR, with a small EiBI correction. The tensor power spectrum is smaller than that in GR as P T ≈ P GR T /(1 + κm 2 ϕ 2 i /2), where ϕ i is the value of the scalar field at the beginning of the inflationary attractor stage. For long wave-length modes, there exists a peculiar rise in the power spectrum P T originated from the near-MPS stage.
The scalar perturbation produced at the inflationary attractor stage was investigated in Ref. [5] in the limit of κm 2 ≪ 1. The scalar power spectrum is smaller than that in GR as P ATT From the results of the tensor and the scalar spectra, the tensor-to-scalar ratio r becomes smaller than that in GR as r ≈ [(1 + 4κm 2 /3)/(1 + κm 2 ϕ 2 i /2)]r GR . In this paper, we investigate the scalar perturbation produced at the near-MPS stage in the very early universe. One question is whether or not the peculiar phenomenon arises for long-wave modes as in the tensor mode, which may leave a signature in the cosmic microwave background radiation. The other is how much different the tensor-toscalar ratio would be from that for the perturbation produced at the attractor stage. The results are summarized as following. The peculiar rise in the scalar power spectrum P R for long-wave length modes appears exactly in the same way as the tensor spectrum P T in EiBI theory. The ratio r is not different from that for the perturbation produced at the attractor stage because the EiBI correction turns out to be exactly the same.
This paper is organized as following. In Sec. 2, we present the background field equations and the perturbation equations in the literature following Refs. [1,[4][5][6]. In Sec. 3, we apply the near-MPS approximation on the equations, and obtain the solution for the scalar perturbation. In Sec. 4, following the solution-matching technique used in Ref. [4], we derive the power spectrum and obtain the tensor-to-scalar ratio. In Sec. 5, we conclude.

II. FIELD EQUATIONS
In this section, we present the background field equations, the scalar perturbations and their field equations introduced in Ref. [5].

A. Background Field Equations
When there is no cosmological constant (λ = 1), the EiBI action (1) is equivalent to a bimetric-like theory action where g µν is the physical metric and q µν is the auxiliary metric by which the affine connection Γ is determined. From this action, the equations of motion are obtained as where T µν is the energy-momentum tensor in the standard form. The ansätze for the metrics are where t and η are the cosmological and the conformal time, respectively. For the derivatives, we denote asˆ≡ d/dt, ′ ≡ / . dη, H ≡ a ′ /a, H ≡â/a, and h ≡ b ′ /b. The components of Eq. (4) give where the subscript 0 stands for the unperturbed background fields, i.e., ρ 0 = ϕ ′2 0 /2a 2 + V (ϕ 0 ) and p 0 = ϕ ′2 0 /2a 2 − V (ϕ 0 ). From Eq. (8), we get z = (1 + κρ 0 )/(1 − κp 0 ). The dynamical equations for the metric coefficients are obtained Here, V i is the ith-order potential from V = V 0 (ϕ 0 ) + V 1 (χ) + V 2 (χ). Using ρ 0 and p 0 , S 3 can be rewritten as where (Please note that there was a typo in the definition for Y in Ref. [5]; the signature "−" was missing.) Denoting nine perturbation fields as F l (l = 1 ∼ 9), we introduce the Fourier modes as The gauge conditions for the scalar-field type and the perfect-fluid type have been rigorously studied in Ref. [6]. For the scalar-field type, one element of two sets given below can be fixed, We fix the gauge conditions as Performing the variation for S 2 and S 3 with respect to φ 2 , ψ 2 , E 2 , and B 2 , we get and for S 1 and S 2 with respect to φ 1 and B 1 , we get From Eqs. (23) and (24), we get E 2 = 0. From Eqs. (22) and (24), we get and from Eqs. (25) and (27), we get Then we finally get from Eqs. (23) and (28), which is expressed only by the matter-field perturbation χ and the background fields. This result for ψ 2 will be used in evaluating the comoving curvature later.
Using the results of Eqs. (22)-(30), S s [χ] is expressed only by χ and the background fields in the Fourier space as where and Here, where We can construct a perturbation field Q in the canonical form from the action (31) by a transformation Q = ωχ with introducing a new time coordinate τ by dτ = (ω 2 /f 1 )dη. The field equation then becomes where˙≡ d/dτ and σ 2 s ≡ f 1 f 2 /k 2 ω 4 . We consider a Bunch-Davies vacuum in the k → ∞ limit. Requiring σ 2 s → 1 in this limit determines ω to be The normalization condition for the canonical field is given by

III. PRODUCTION OF PERTURBATION AT NEAR-MPS STAGE
In this section, we investigate the scalar perturbation produced at the near-MPS stage by solving the perturbation equation (37). In order to do that, we need to know f 1 , f 2 , and ω as a function of τ . First, we evaluate them in terms of a and z, and then get the expressions in τ .
f 1 (a, z) and ω(a, z) are obtained in general, while f 2 (a, z) is obtained only in the near-MPS approximation. Let us get f 1 (a, z) and ω(a, z) first. The equation (9) can be rewritten as Using Eq. (8), we get .

A. Perturbation Equation in Near-MPS Approximation
In this subsection, we introduce the near-MPS solutions investigated in Ref. [2], and apply the approximation using them in order to get f 2 (a, z). Then we get a(τ ) and ω(τ ) in this approximation, and finally get the perturbation equation in terms of τ .
The maximal pressure state (MPS) is achieved when p 0 = 1/κ beyond which the theory is not well defined, The MPS solution for the background scalar field is obtained from this condition [2], where we considered the scalar field initially rolling down the potential in the region of ϕ 0 < 0 (soφ 0 > 0). At the MPS, the Friedmann equation reduces to H =â/a = −(2/3)dU/dϕ 0 , where U ≡ 2[1/κ + V (ϕ 0 )] = 2/κ + m 2 ϕ 2 0 , and the scale factor is solved as [2] where a 0 is an integration constant. The MPS was found to be unstable under the global perturbation in Ref. [2]. With a small perturbation, the Universe leaves the MPS and evolves to the near-MPS. Introducing small perturbations ψ(t) and γ(t) for the velocities of the scalar field and the scale factor at the near-MPS aŝ the solutions were found as [2] ψ = ψ 0 U −4/3 e t/tc , where t c = 3κ/8. Then at the near-MPS, using the above results, we have which give Therefore, for the near-MPS approximation, we assume z ≫ 1. From Eqs. (8) and (51)-(53), we have Using Eq. (54), h in Eq. (40) can be expressed as and X 2 in Eq. (41) can be approximated as where we used the relation a = a 0 U −2/3 in Eq. (46). Using Eqs. (45) and (46), we get Differentiating Eq. (9) with respect to η and using Eq. (10), we get Plugging the approximations (56)-(58) into Eqs. (33)-(36), and keeping the k-dependent term and the highest-order terms in z (z 0 terms), we get Using Y/X ≈ −ma from Eqs. (56) and (57), H = aH ≈ 2ma/3 from Eq. (46), and h ≈ a/ √ 6κ from Eq. (55), one can show that the last four terms in Eq. (59) cancel, and f 2 can be further approximated as Using the results of f 1 (a, z), f 2 (a, z), and ω(a, z), the perturbation equation (37) is approximated at the near-MPS stage as Now let us express this perturbation equation in terms of τ . From Eqs. (46), (53), and (49), at the near-MPS stage, we have The time coordinates are transformed for z ≫ 1 as where we set τ = 0 for t → −∞. In terms of τ , we have With these results of a(τ ) and z(τ ), we get and from Eq. (43) for z ≫ 1, we get Finally, the perturbation equation (61) at the near-MPS stage becomes The κ-dependent EiBI corrections in σ 2 s k 2 andω/ω cancel each other, and the resulting equation is in the same form with the near-MPS equation for the tensor perturbation studied in Ref. [4].

B. Near-MPS Solution
Since the perturbation equation (68) is in the same form with that for the tensor perturbation at the near-MPS stage, we manipulate the solution here in the same way as in Ref. [4]. The solution to Eq. (68) is given by Here, c 1 and c 1 are complex, where c and R are real. One arbitrariness was fixed by imposing c Im 1 = 0, and c Im 2 was determined from by normalization condition (39). R and c are to be determined by imposing the initial condition at the moment of the production of the perturbation. The initial condition is given by minimizing the energy, The production moment τ * of the perturbation is determined as following. As it was studies in Ref. [2], the curvature scale H is finite at the beginning of the Universe at τ = 0 (t → −∞ and ϕ → −∞), so the quantum gravity is not necessary. However, the wave-length scale of the perturbation becomes smaller than the Planck scale. To treat the perturbation in a classical way, for which the classical treatment of the perturbation is questioned. we consider the production of the perturbation when the wave-length scale λ phys is comparable to the Planck scale l p , For high k-modes, the perturbation is produced after the solution Q in Eq. (69) is relaxed to the oscillatory behavior (kτ * ≫ 1). Using the asymptotic formulae for kτ ≫ 1, it was found in Ref. [4] that the energy E in Eq. (71) is minimized when R = 0 and c 2 = π/4. Then the perturbation solution for high k-modes becomes which is the plane-wave solution with only the positive energy mode selected. For low k-modes, the perturbation is produced before the solution Q is relaxed to the oscillatory behavior (kτ * < 1). (Please see Fig. 1 in Ref. [4].) In this case, the energy E in Eq. (71) is minimized when where , and Y 0,1 ≡ Y 0,1 (kτ * ). (Please see Ref. [4] for detailed calculations.) With these c and R, the solution (69) for low k-modes becomes As a whole, the near-MPS solutions were obtained as Eq. (73) for high-k modes, and as Eq. (75) low-k modes. The coefficients were fixed by imposing the minimum-energy condition at the production moment τ * . These two perturbation modes produced at the near-MPS stage evolve to the intermediate stage which is connected to the inflationary attractor stage.

IV. POWER SPECTRUM
The power spectrum P R is evaluated at the end of the inflationary attractor stage, while the coefficients of the mode solution at the attractor stage are determined from the initial perturbation produced at the near-MPS stage. In order to determine the coefficients of the mode solution Q ATT at the attractor stage from the near-MPS solution Q MPS , we assume that the perturbation evolves adiabatically from the near-MPS stage through the intermediate stage till the attractor stage. The adiabatic period spans from the late near-MPS stage to the early attractor stage, and is described by the WKB solution Q WKB . In order to determine the coefficients, we need the solution matching between Q MPS and Q WKB at some moment τ 1 at the late near-MPS stage, and between Q WKB and Q ATT at some moment τ 2 at the early attractor stage.
The tensor perturbations at all the stages were investigated in Ref. [4]. In Ref. [5], the scalar perturbation at the attractor stage was investigated, and the solution Q ATT was found. Both of Q ATT and Q MPS (obtained in this paper) are exactly in the same form respectively with those of the tensor perturbation obtained in Ref. [4]. In this section, therefore, we shall follow the solution-matching technique exactly in the same manner as for the tensor perturbation. (For details, please see Ref. [4].) We shall focus on the low k-modes, then the result is applied for the high k-modes simply by setting R = 0 and c 2 = π/4 in the end.
The near-MPS solution is where c 1 and c 2 are given by Eqs. (70) and (74). In the adiabatic period, the solution to the perturbation equation Q + Ω 2 k (τ )Q = 0 is given by the WKB approximation, for which the adiabatic condition is Ω −3 k dΩ 2 k /dτ ≪ 1. The attractor solution was obtained in Ref. [5], where A ′ 1 = (A 1 − iA 2 )/2, A ′ 2 = (A 1 + iA 2 )/2, τ 0 ≡ τ i − √ 6/ϕ i ma i , and the subscript i stands for the values at the beginning of the attractor stage. Now we match Q's andQ's at τ 1 for MPS and WKB and at τ 2 for WKB and ATT. Then from the results in Ref. [4], the coefficients are determined as At the end of inflation, the perturbation is approximated as , and with the aid of above equations (79) and (80), one can get Now let us discuss the power spectrum evaluated at the end of inflation. The comoving curvature at the attractor stage is given by where we used the approximations, H ≈ −mϕ i / √ 6 and ψ 2 ≈ κm 2 ϕ i χ ATT /2 obtained in Ref. [5] for ψ 2 in Eq. (30). With 4 and a(τ ) = a i (τ i − τ 0 )/(τ − τ 0 ) at the attractor stage obtained in Ref. [5], the power spectrum becomes Here, P GR R ≡ m 2 ϕ 4 i /96π 2 is the power spectrum in GR, and E S κ ≡ (1 − κm 2 ) 2 /(1 − 4κm 2 /3) 1/2 is the EiBI correction which is the same as that for the perturbation produced at the attractor stage obtained in Ref. [5], P ATT R = E S κ P GR R . The coefficient D k ≡ (2/π)(c 2 + R 2 + π 2 /16c 2 ) is the k-dependence factor which is the same form obtained for the tensor perturbation in Ref. [4]. As seen in Fig. 1, D k exhibits a peculiar rise at low k, while it becomes unity, D k → 1, at high k. As a result, compared with the power spectrum for the scalar perturbation produced at the attractor stage, P R for the perturbation produced at the near-MPS stage is the same for high k-modes, but exhibits a peculiar peak for low k-modes.
The tensor power spectrum in EiBI gravity was obtained in Ref. [4] as Here, P GR T is the spectrum for the chaotic inflation model in GR, and E T κ is the EiBI correction. Since the factor D k is common both for the tensor and the scalar perturbations, the tensor-to-scalar ratio for the perturbation produced at the near-MPS stage is given by which is exactly the same with that for the perturbations produced at the attractor stage obtained in Ref. [5]. Here, r GR ∼ 0.131 for 60 e-foldings. As ϕ i ∼ O(10), the EiBI correction of the tensor spectrum is dominant and the value of r is lowered. While the scalar perturbation has been investigated in the limit of κm 2 ≪ 1, the result of the tensor perturbation investigated in Ref. [4] does not restrict the value of κm 2 ϕ 2 i much. Therefore, within the accuracy of this work, κm 2 O(10 −2 ), and with ϕ i ∼ O(10), the EiBI correction can be considerably large and the value of r can be significantly suppressed. In this paper, we investigated the scalar perturbation produced at the near-MPS (maximal pressure state) which arises at the early stage of the Universe driven by a massive scalar field in Eddington-inspired Born-Infeld gravity. This work is a completion of studying the scalar perturbation in this model, combined with the work [5] for the scalar perturbation produced at the inflationary attractor stage.
The scalar perturbation pattern is very similar to that [5] at the attractor stage for high k-modes which mimics the standard chaotic inflation in GR, with a small EiBI correction. The pattern for low k-modes is very similar to that [4] of the tensor perturbation. The power spectrum exhibits a peculiar rise, which can distinguish this EiBI model from the standard chaotic inflation model in GR. It also possesses the same EiBI correction as for the high k-modes.
Because of the similarity, we could buy the results of Refs. [4] and [5]. The tensor-to-scalar ratio r was obtained for all k-modes. The result was the same with that obtained at the attractor stage. The scalar perturbation increases the value of r, while the tensor spectrum decreases it. The tensor contribution is dominant, so r can be lowered significantly. Recent observational results of BICEP2 [28] and PLANCK [29] for the tensor perturbation have attracted much attention, and are still in dispute. The direction is that the value of r is to be lowered ( 0.09) from the one that BICEP2 observed. In this sense, our result is very affirmative since it can decrease r to a very low value.
In this work, we considered the perturbation in the limit of κm 2 ≪ 1. From the star formation study in Refs. [11,12,14], the theory parameter is very mildly constrained as κ < 10 −2 m 5 kg −1 s −2 ∼ 10 77 in Planck unit. Considering the consistency of our model with inflation, the parameter is constrained more strongly as κ ≪ m −2 ∼ 10 10 .