Pre-inflation and Trans-Planckian Censorship
Yong Cai Yun-Song Piao


































    (
            [ x kφ̃_kunsrt
            
            
            
            ^1 School of Physics, University of Chinese Academy of
            Sciences, Beijing 100049, China^2 Institute of Theoretical Physics,
            Chinese Academy of Sciences, Beijing 100190, China
            
            
            
            We investigate the implication of Trans-Planckian Censorship Conjecture (TCC) for the initial state of primordial perturbations. It is possible to set the state of perturbation modes in the infinite past as the Minkowski vacuum, only if the pre-inflationary era is past-complete. We calculate the evolution of the perturbation modes in such a pre-inflationary era and show that at the beginning of inflation the perturbation modes with wavelengths much shorter than the Hubble scale (but still larger than the Planck length scale) will behave as they are in the Bunch-Davis state. Therefore, a past-complete pre-inflationary evolution may automatically prepare the initial state required for the inflationary perturbations at the CMB window while obeying the TCC.
            
            98.80.-k, 98.80.Cq, 04.50.Kd Pre-inflation and Trans-Planckian Censorship
                Yun-Song Piao^1,2[]
                December 30, 2023
            =============================================
            
            
            
            § INTRODUCTION
            
            
            
            Inflation
            <cit.>
            is the standard paradigm of the early universe. It predicted the
            nearly Gaussian and scale-invariant primordial perturbations, which
            are consistent with the observations of cosmic microwave background
            (CMB) anisotropies <cit.>. However,
            inflation is not the final story of the early universe, since it
            is past-incomplete, as shown in
            Refs. <cit.>.
            
            
            
            Usually, the initial state of the inflationary perturbations is
            set as the Bunch-Davis (BD) vacuum <cit.>, which is
            the lowest-energy state of de Sitter spacetime. However, due to
            the past-incompleteness of inflation <cit.>,
            initially, the background and fluctuations of spacetime will be
            inevitably involved in the cosmological singularity, where the
            length scale is smaller than the Planck scale or the cutoff scale
            of the effective field theory (EFT). This is the so-called
            “trans-Planckian" problem of inflation
            <cit.>. In this situation,
            inflation does not automatically pick out the BD state as the
            initial state of perturbation modes.[Non-BD initial
            states may significantly affect the power spectrum and
            Non-Gaussianity of primordial perturbations, e.g.,
            <cit.>.]
            
            Recently, Bedroya and Vafa proposed the Trans-Planckian
            Censorship Conjecture (TCC) <cit.>, which states
            that “a field theory consistent with a quantum theory of
            gravity does not lead to a cosmological expansion where any
            perturbation with length scale greater than the Hubble radius
            trace back to trans-Planckian scales at an earlier time", see
            also its application to inflation models <cit.>.
            
            
            Actually, since inflation is past-incomplete, the
            inflationary perturbations inevitably originated from the
            sub-Planckian scale fluctuations. It is well-known that the BD
            initial state of perturbations is quite essential for acquiring
            the nearly scale-invariant power spectrum of density
            perturbations. Therefore, how to set the BD state at the beginning
            of the observable inflationary era (i.e., the CMB window) is a
            question to be answered, see also <cit.>. In a
            certain sense, the TCC indicates that the existence of a
            pre-inflationary era might be required.
            
            
            One possibility of such pre-inflationary scenarios is a
            nonsingular bounce, i.e., the so-called bounce inflation
            <cit.>, which not only avoids
            the past-incompleteness problem of inflation but also reserves the
            advantages of inflation. Recently, the instability problems of
            perturbations in flat nonsingular cosmologies (which is inevitable
            <cit.> in Horndeski theory, see
            also <cit.>) have been solved in the
            “beyond Horndeski" theory, as proved within the EFT of
            cosmological perturbations
            <cit.>, see also
            <cit.>
            (see <cit.> for reviews).[The earlier attempts (but with the instabilities
            of perturbations) on nonsingular bounce, as well as G-bounce, can
            be found in, e.g., Refs.
            <cit.>.]
            
            
            
            
            In the pre-inflationary era, initially the universe is slowly
            contracting with ϵ=-Ḣ/H^2≫ 1. Thus, in the
            infinite past, the spacetime is nearly Minkowskian.[See also <cit.>
            for the scenario with a slow expansion (ϵ≪-1) preceding
            inflation.] In this case, it is natural to pick out the Minkowski
            vacuum state as the initial state of the primordial perturbations.
            However, at present, the pre-inflationary physics is still in
            speculation and exploration. It is possible that the
            pre-inflationary era might consist of multiple different phases.
            The question is: provided the primordial perturbation modes
            originate from the Minkowski vacuum  in the infinite past
            and obey the TCC throughout, can the sub-horizon modes at the
            beginning of inflation be in the BD state?
            In this paper, we will investigate the relevant issues.
            
            
            The outline of this paper is as follows:
            In Sec.
            <ref>, we briefly revisit the “trans-Planckian" problem
            within the EFT of cosmological perturbations. In Sec.
            <ref>, we discuss the TCC and its implication. We point
            out that it is possible to set the state of perturbations in the
            infinite past as the Minkowski vacuum only if the spacetime is
            past-complete. In Sec. <ref>, we calculate the evolution
            of the perturbation modes in the past-complete pre-inflationary era to
            see whether the corresponding modes approximate to the BD state at
            the beginning of inflation. Sec. <ref> is the
            conclusion.
            
            
            
            
            
            
            
            §  TRANS-PLANCKIAN PROBLEM
            
            
            
            In this section, we will focus on the evolution of the primordial perturbations. In unitary gauge, the
            quadratic action of scalar perturbation ζ is 
            S^(2)_ζ =∫  a^3 Q_s[ ζ̇^2 -c_s^2
            (∂_iζ)^2/a^2]d^3 x d t . It is convenient to define
            u_k=zζ_k with z=a √(2 Q_s), where ζ_k is the
            scalar perturbation mode. The Mukhanov-Sasaki equation for u_k
            is  u_k^''+(c_s^2 k^2-z^''/z) u_k=0 , where '=d/dτ,
            c_s is the sound speed. In general relativity (GR), c_s^2=1 and Q_s=
            M_P^2ϵ with ϵ=-Ḣ/H^2.
            
            In the slow-roll inflation, z^''/z∼
            a^''/a≃ (2+ O(ϵ))/τ^2.
            Thus the solution of Eq. (<ref>) is 
            u_k(τ)=√(-πτ) 2[α(k)
            H_ν^(1)(-kτ)+β(k) H_ν^(2)(-kτ) ] , 
            where the k-dependent coefficients α(k) and β(k)
            obey the Wronskian condition |α|^2-|β|^2=1,
            ν≈ 3/2, H_ν^(1) and H_ν^(2) are the first and
            second kind Hankel functions of the ν-th order, respectively.
            
            
            
            
            One must select the initial state of perturbations to calculate the
            coefficients α(k) and β(k). Requiring the
            perturbation modes coincide with the Minkowski solution (u_k≃1√(2k)e^-ikτ) in the infinite past
            gives[Note that there could be an undetermined phase
                difference in α (see, e.g., <cit.> for a
                review).] |α|=1 ,     |β|=0 , 
            where we have used that
             
            H^(1)_ν(-kτ)≈√(2 -π
                    kτ)e^-ikτ e^-i(ν2+14)π
            in the limit -kτ≫1.
            Particularly, we have ν=3/2 for de Sitter inflation. Using H^(1)_3/2(-kτ)=-√(2 -π kτ)e^-ikτ(1-i kτ), one gets 
            u_k=1√(2k)e^-ikτ(1-i kτ), 
            which is just the lowest-energy state in de Sitter spacetime (i.e., the BD
            vacuum).
            
            Generally, setting the initial state of perturbation
            modes as the BD state is essential for inflation to predict the
            nearly Gaussian, scale-invariant primordial perturbations. As for
            the non-BD states, we will have |α|>1 and |β|> 0,
            where |β|^2 could be interpreted as the number density of
            particle excitations <cit.>.
            
            
            
            The choice of the initial BD state seems natural. However, it is
            not the case due to the well-known “trans-Planckian" problem
            <cit.>. In the following, based
            on the EFT of cosmological perturbations, we will revisit this
            problem.
            
            
            As mentioned, one will get the BD state by requiring the
            perturbation modes u_k at -kτ≫ 1 to coincide with the Minkowski
            solution. Usually, it is thought that in the infinite past, the
            perturbation modes are deep inside the horizon, so that
            Eq. (<ref>) equals the equation of free wavepacket in
            flat Minkowski space. However, the so-called Minkowski spacetime
            is only “conformal" Minkowski, which is actually expanding. In
            such a spacetime, since the wavelength of perturbation modes
            λ∼ a, which will blueshift back to the past, we will inevitably have λ≲ 1/Λ before some
                τ=τ_Λ, where Λ≲ M_P is the cutoff
            scale below which the higher-order derivative operators are
            negligible.
            
            
            When λ≃ O(1/Λ), the EFT
            responsible for (<ref>) is no longer robust, and the
            neglected higher-order derivative operators, such as (see, e.g., <cit.>),
             L/M_P^2∼λ_4Λ^2(R^(3))^2+λ_6Λ^4(∇_iR^(3))^2+⋯,  will be
            no longer negligible, where R^(3) is the Ricci scalar on the
            3-dimensional spacelike hypersurface and λ_4,6≃ O(1) are constants. One is also allowed to take account of the
            operators R^(3)_ijR^(3)ij and
            ∇_iR^(3)_jk∇^iR^(3)jk, which we have omitted for
            simplicity. We have put such operators and the higher-order
            spatial derivative operators of R^(3) into the ellipsis.
            
            Generally,[One also has considered the operators
                ∼∇_μδ K_νσ∇^μδ K^
                νσ, which contribute
                k^6ζ_k^2 as well <cit.>.] (R^(3))^2∼
            k^4ζ_k^2,  (∇_iR^(3))^2∼ k^6ζ_k^2. The
            higher-order derivative operators in (<ref>) do not
            affect the background. Hence, the effective theory of inflation may be
            still valid at the background level when the perturbation mode satisfies  λ≲
                1/Λ. However, in this situation, Eq. (<ref>) will
            be modified to  u_k^''+(∼∑_p k^2pΛ^2p)k^2u_k=0, where p⩾ 0
            is some integer. In this case, what the perturbation u_k feels
            will be a regime dominated by the higher-order momentum operators
            instead of a Minkowski space, since Eq. (<ref>) is no
            longer the equation of free wavepacket in flat Minkowski space.
            Therefore, it is not sufficiently reasonable to set the state of
            the perturbation modes in the infinite past as the Minkowski
            vacuum.
            
            
            
            
            
            
            
            
            § IMPLICATIONS OF TCC FOR PRE-INFLATION 
            
            
             §.§ On TCC
            
            
            
            
            Recently, Bedroya and Vafa proposed the Trans-Planckian Censorship
            Conjecture (TCC) <cit.>, which requires
            that the sub-Planckian fluctuations should never cross their
            Hubble scale to become classical, otherwise the corresponding EFT
            belongs to the swampland
            <cit.>. The TCC
            suggests that inflation can only last for a limited e-folding
            number, i.e., ∫_t_i^t_fH dt<lnM_P
            H_f , where the subscript “i" and
            “f" represent the beginning and ending of inflation,
            respectively. The TCC might be also argued as follows.
            
            
            
            Generally, neglecting the higher-order operators in
            (<ref>), we have |u_k|= 1/√(2k) for the
            sub-horizon perturbation modes (noting that it is also
            approximately valid for λ≲ 1/H, see also
            Sec. <ref>). Thus one will have |ζ_k|∼ 1/a for such a
            perturbation mode, since |ζ_k|∼ |u_k|/a on sub-horizon scale. It can be found
            that at the beginning of inflation, the spectrum of the sub-horizon
            perturbation (λ≪ 1/H) is 
            P_ζ^1/2(t_i,k)=P_ζ^1/2(t_f,k)a_f a_i∼H_f√(ϵ_f)M_Pa_f a_i, 
            provided this perturbation mode crossed the horizon (k≃ a_f
            H_f) at t_f. The validity of perturbation theory requires that
            P_ζ^1/2(t_i,k)<1, otherwise the spacetime will be distorted by
            lots of black holes on the sub-Hubble scale. According to (<ref>),
            we have 
            ∫_t_i^t_fH dt< ln√(ϵ_f) M_P
            H_f, 
            which, for ϵ_f≲ 1, is
            consistent with the TCC constraint (<ref>).[It is
            interesting that for the tensor perturbation, (<ref>) is
            exactly same as the TCC constraint (<ref>).]
            
            
            The TCC might suggest the existence of a pre-inflationary
            evolution. The question is whether the pre-inflationary evolution can prepare the required initial condition for observational
            (i.e., at the CMB window) inflationary perturbations.
            The relevant issue is closely related to the past-incompletion (or
            singularity) of inflation <cit.>. In inflationary
            spacetime, the affine length ∫ dη_ζ∼∫ c_s^2Q_s
            a dt of scalar perturbation mode must be finite. As a result, back to the
            past, the perturbation modes will inevitably hit the singularity,
            as depicted in Fig. <ref>. 
            Although it seems acceptable
            to set the BD state as the initial state of perturbation modes
            below the scale Λ≲ M_P,[The decay of non-BD state into
            BD state through the self-interaction of the perturbations ∼ϵ^2ζ(ζ^')^2 is very inefficient
            <cit.>, since ϵ≪ 1.
            Nevertheless, see also <cit.> for further discussions.]
            it is not justified either.
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
             §.§ Minkowski in past-complete pre-inflation 
            
            
            
            
            Recalling the Raychaudhuri equation for the null geodesics, which
            is simplified as (assuming the null vector k^μ is orthogonal
            to the hypersurfaces, see e.g., <cit.>) d
            θ d η⩽ -θ^2
            2-R_μνk^μk^ν ,  where η is the
            affine parameter of the null geodesics and
            θ=▿_μ k^μ is the expansion parameter.
            When the null convergence condition (NCC)
            R_μνk^μk^ν≥0 is satisfied, one has
            dθ/dη+θ^2/2<0. In this case,  back along η the
            null rays will converge, which indicates that the wavelengths of
            perturbation modes will be blueshifted, hence λ≲
            1/Λ is inevitable. In Ref. <cit.>, the NCC has
            been applied to argue the past-incompletion of inflation. Thus,
            only if the pre-inflationary era is
            past-complete,[Necessary condition but not sufficient,
            see e.g., <cit.>.] it is possible to set the initial
            state of perturbations in the infinite past (as the Minkowski
            vacuum).
            
            In the pre-inflationary era, the past-completion requires that the
            converging null rays must be reversed
            (i.e., dθ/dη>0) at least for some period.
            According to (<ref>), during this period the NCC must be
            violated, i.e.,
             R_μνk^μk^ν<0, 
            or equivalently, Ḣ>0, since R_μνk^μk^ν=-2Ḣ(k^0)^2 for the spatially flat FLRW metric.
            
            
            
            One possibility that satisfies (<ref>) is the pre-inflationary
            bounce, where Ḣ>0 regardless of whether the gravity is GR
            or not.
            
            In the corresponding bounce inflation scenario, before the
            nonsingular bounce, one may have  a∼1
            1+(τ_cτ)^n,
             for 0<n≪ 1, where
            τ_c is negative. In the infinite past τ→
            -∞, a∼ 1, we have an asymptotically Minkowski
            spacetime. When |τ|≪ |τ_c|, a∼ (τ/ τ_c)^n,
            the pre-inflationary universe is slowly contracting (i.e., the ekpyrotic
            phase <cit.>) until a nonsingular bounce occurred.
            Hereafter it will begin to inflate. The corresponding causal patch
            is displayed in Fig. <ref>.
            
            
            In such a bounce inflation scenario, all perturbation modes
            are in a flat Minkowski space in the infinite past. The
            wavelengths of the perturbation modes that are interested for
            observations satisfy λ≫ 1/Λ, so that the
            k/Λ-suppressed higher-order derivative operators in
            (<ref>) are negligible. In particular, such a
            scenario is also consistent with the TCC, since the sub-Planckian
            perturbation modes may never have the opportunity to cross their
            horizon,[Strictly, the so-called “horizon" is actually
            “the horizon of the perturbations", which is defined by
            1/H_per=a√(|z/z^''|). It will be significantly
            different from 1/H, if Q_s is rapidly changing, see also
            <cit.>.] as long as a_f/a_b<M_P/H_f and a_b
            is sufficiently large (see also <cit.>),
            
            
            
            
            where the subscript “b" corresponds to the bounce point. 
            
            
            
            
            
            § BUNCH-DAVIS AT THE BEGINNING OF INFLATION
            
            
             §.§ Analytical evaluation
            
            
            In this section, we will focus on the bounce inflation scenario and
            evaluate the states of perturbation modes at the beginning of
            inflation. The result is also applicable for other scenarios with a
            past-complete pre-inflationary era.
            
            Generally, the evolution of the perturbation modes satisfies
            Eq. (<ref>) with c_s^2 replaced by C_s^2, where
             C_s^2=c_s^2+(∼∑_p
            k^2pΛ^2p) .
            In the pre-inflationary era, we can safely neglect higher-order
            derivative operators and have C_s^2=c_s^2, provided the wavelengths of the perturbation modes
            λ≪ 1/Λ (see also <cit.> for related discussions). One can always set Q_s>0 and c_s^2=1, as in
            Refs. <cit.>,
            so that the ghost or gradient instabilities are avoided.
            
            
            
            
            Considering the pre-inflationary era consists of different phases
            with n=n_j(j≥ 0) in Eq. (<ref>), we have  z_j=√(2 a^2 Q_s)∼
            (τ_R,j-τ)^n_j for the j-th phase, where
            n_j≃ const. During the bounce, a is almost
            constant. However, since Q_s is rapidly changing,
            
            
            
            
            we still could have
            Eq. (<ref>), see Ref. <cit.>. Hence, in the phase j,
            z_j^''/z_j can be written as z_j^''/z_j=ν_j^2-1/4(τ-τ_R,j)^2 , where
            ν_j=|n_j-1/2|. Regarding the phases j and j+1 as adjacent
            phases, we can write down the solutions to Eq. (<ref>) in
            the corresponding phases as  u_k,j(τ)   =   √(π
                    (τ_R,j-τ))
                2{α_jH_ν_j^(1)[k(τ_R,j-τ)]+β_jH_ν_j^(2)[k(τ_R,j-τ)]
            } ,   (τ<τ_j+1) ,
            
            
            u_k,j+1(τ)   =   √(π (τ_R,j+1-τ))
                2{α_j+1H_ν_j+1^(1)[k(τ_R,j+1-τ)]
            
                                 
            +β_j+1H_ν_j+1^(2)[k(τ_R,j+1-τ)] }
            ,             (τ>τ_j+1) ,  where
            α_j(j+1) and β_j(j+1) are k-dependent
            coefficients, τ_j+1 denotes the conformal time of the juncture of these two phases.
            
            Using the matching conditions
            u_k,j(τ_j+1)=u_k,j+1(τ_j+1) and
            u_k,j'(τ_j+1)=u_k,j+1'(τ_j+1) (see e.g., <cit.>), we have (
            [ α_j+1; β_j+1 ])
               =   ([ M_11 M_12; M_21 M_22 ])
            ([ α_j; β_j ]) ,  with 
                   
                M_11=ikπ√(τ̃_j+1,1τ̃_j+1,2)4[
            H^(1)_ν_j(kτ̃_j+1,1)H^(2)_ν_j+1-1(kτ̃_j+1,2)
            -H^(1)_ν_j-1(kτ̃_j+1,1)H^(2)_ν_j+1(kτ̃_j+1,2)
            
                                             + (2ν_j-1
                2kτ̃_j+1,1-2ν_j+1-1 2kτ̃_j+1,2)
            H^(1)_ν_j(kτ̃_j+1,1)H^(2)_ν_j+1(kτ̃_j+1,2)
            ] ,
            
            
                   
                M_12=ikπ√(τ̃_j+1,1τ̃_j+1,2)4[
            H^(2)_ν_j(kτ̃_j+1,1)H^(2)_ν_j+1-1(kτ̃_j+1,2)
            -H^(2)_ν_j-1(kτ̃_j+1,1)H^(2)_ν_j+1(kτ̃_j+1,2)
            
                                             + (2ν_j-1
                2kτ̃_j+1,1-2ν_j+1-1 2kτ̃_j+1,2)
            H^(2)_ν_j(kτ̃_j+1,1)H^(2)_ν_j+1(kτ̃_j+1,2)
            ] ,
            
            
                   
                M_21=-ikπ√(τ̃_j+1,1τ̃_j+1,2)4[
            H^(1)_ν_j(kτ̃_j+1,1)H^(1)_ν_j+1-1(kτ̃_j+1,2)
            -H^(1)_ν_j-1(kτ̃_j+1,1)H^(1)_ν_j+1(kτ̃_j+1,2)
            
                                             + (2ν_j-1
                2kτ̃_j+1,1-2ν_j+1-1 2kτ̃_j+1,2)
            H^(1)_ν_j(kτ̃_j+1,1)H^(1)_ν_j+1(kτ̃_j+1,2)
            ] ,
            
            
                   
                M_22=-ikπ√(τ̃_j+1,1τ̃_j+1,2)4[
            H^(2)_ν_j(kτ̃_j+1,1)H^(1)_ν_j+1-1(kτ̃_j+1,2)
            -H^(2)_ν_j-1(kτ̃_j+1,1)H^(1)_ν_j+1(kτ̃_j+1,2)
            
                                             + (2ν_j-1
                2kτ̃_j+1,1-2ν_j+1-1 2kτ̃_j+1,2)
            H^(2)_ν_j(kτ̃_j+1,1)H^(1)_ν_j+1(kτ̃_j+1,2)
            ] ,  where we have defined
            τ̃_j+1,1=τ_R,j-τ_j+1 and
            τ̃_j+1,2=τ_R,j+1-τ_j+1.
            
            
            
            
            
            
            When kτ̃_j+1,1 and kτ̃_j+1,2≫1, we have 
                    M_11≈(1-ix )e^iθ_j+1 ,
            
            
              M_12≈ xe^-iθ̃_j+1 ,
            
            
              M_21≈ xe^iθ̃_j+1 ,
            
            
              M_22≈(1+ix) e^-iθ_j+1,  
            where
            x=1-2ν_j 4kτ̃_j+1,1-1-2ν_j+1
                    4kτ̃_j+1,2,
            θ_j+1=k(τ̃_j+1,1-τ̃_j+1,2)-π/2
                (ν_j-ν_j+1), θ̃_j+1=k(τ̃_j+1,1+τ̃_j+1,2)-π/2
                (ν_j+ν_j+1). Thus, considering the perturbation modes
            what we concern satisfy kτ̃_j+1,1 and kτ̃_j+1,2≫1,
            
            
            we have (
            [ α_j+1; β_j+1 ])
               ≈   (
            [          e^iθ_j+1[1+ O(1 kτ̃)]                       O(1 kτ̃);                       O(1 kτ̃) e^-iθ_j+1[1+
                    O(1 kτ̃)] ])
            ([ α_j; β_j ]) ,  where τ̃
            correspond to τ̃_j+1,1 or τ̃_j+1,2 and j≥0.
            
            
            Define the past-infinite phase and the inflation as the phases
            j=0 and j=i, respectively, with Eq. (<ref>), we
            straightforwardly find that (
            [ α_i; β_i ])
               ≈   (
            [  e^i∑_j=1^iθ_j[1+ O(1 kτ̃)]                    O(1 kτ̃);                    O(1 kτ̃) e^-i∑_j=1^iθ_j[1+ O(1 kτ̃)] ])
            ([ α_0; β_0 ]) ,  where ∑_j=1^iθ_j=-π2(ν_0-ν_i)+k∑_j=1^i(τ̃_j,1-τ̃_j,2),
            for the perturbation modes satisfying kτ̃≫ 1 (i.e.,
            kτ̃_j,1≫ 1 and kτ̃_j,2≫ 1 for j=1,2,⋯, i),
            which actually equals requiring that the corresponding
            perturbation modes have not ever exited its horizon 1/H_per in the
            pre-inflationary era. Therefore, for the perturbation modes with
            λ≪ 1/H_i at the beginning of inflation,  α_i(k) and β_i(k) acquire only some phase
            shift, compared
            with their initial values α_0(k) and β_0(k) in the infinite
            past.
            
            In the past infinity, if the universe is flat and Minkowskian,
            naturally the perturbation mode is in its Minkowski vacuum, i.e.,
            |α_0|=1 and β_0=0.  Hence,  |α_i|≈
            1+ O(1 kτ̃) ,     |β_i|≈ O(1 kτ̃)  for kτ̃≫1. Therefore, at
            the beginning of inflation, the perturbation modes with the
            wavelengths λ≪ 1/H_i (the Hubble scale of inflation)
            behave themselves as they are in the BD state, which will be
            responsible for the nearly scale-invariant density perturbations. As
            for the modes with λ≳ 1/H_i, they are in the non-BD
            state at the beginning of inflation, hence possibly explaining the
            power deficit of the CMB TT-spectrum at low multipoles (e.g., <cit.>) and the dip
            at multipole l∼ 20 (see e.g., <cit.> and
            references therein). A similar analysis can be carried out for the primordial tensor perturbations.
            
            
            It should be pointed out that the TCC's prediction of an extremely small tensor-to-scalar ratio (see e.g., <cit.>) implicitly requires that the perturbation modes are in the BD states at the beginning of inflation.
            Therefore, in our pre-inflation model, 
            the tensor-to-scalar ratio should also be bounded similarly due to the upper bound on the inflationary energy scale put by the TCC, provided the power spectrum of tensor perturbations takes the standard form.
            
            
            
            
            
            
            
            
            
            
            
             §.§ Numerical simulation
            
            
            
            Defining
             C(k)=k  2( u_ku_k^*+1
            k^2u_k'u_k^*')+1 2 ,     (τ≈τ_i) ,
            
            we have 
            C(k)=|α_i|^2+ O(1(-k/
            H_i)^2)≃ |α_i|^2 
            for ν_i≈ 3/2 and -kτ_i≫ 1, where H_i=aH_i=τ_i and the Wronskian condition has been used. We will use |C(k)|=1 as the criterion to
            check whether the perturbation modes with λ≪ 1/H_i
            at the beginning of inflation are in the BD state.
            
            
            
            
            
            We consider the bounce inflation model proposed in
            Ref. <cit.> (see Appendix <ref>), where around the bounce we
            have C_s^2=c_s^2=1 and  Q_s=A_Q[B-tanh( t
                t_*) ]  with constant A_Q and B. As
            mentioned, in Eq. (<ref>), z=√(2a^2Q_s) depends not
            only on a but also on Q_s. During the ekpyrotic contraction,
            z^''/z≈ 0/τ^2, since both a and Q_s
            are nearly constant. During the inflation,
            z^''/z≈ 2/τ^2, which is well-known. Around
            t_*, Q_s has a rapid change, but H≃ H_b^2(t-t_b) around
            the bounce point t_b, hence  a∼ e^H_b^2(t-t_b)^2/2≃
            1+H_b^2(t-t_b)^2/2 is nearly constant for t-t_b≪ 1/H_b.
            As a result, we have z^''/z≈ -(τ-τ_R)^-2/4  with τ_R being some reference time.
            Therefore, all phases are actually consistent with
            (<ref>). As expected, small deviation will not make any
            qualitative difference for our result.
            
            
            With the background of bounce inflation in Ref. <cit.>
            (see Fig. <ref> in Appendix <ref>), we plot
            |C(k)| in Fig. <ref> while A_Q=3 and B=2. We can see that for sufficiently large
            k (i.e., k^2≫ z^''/z≈ a^2H^2_i), we will have |C(k)|= 1.
            Thus, the perturbation modes with
            wavelengths λ≪ 1/H_i at the beginning of inflation will be naturally in the BD state.
            
            
            
            As for the perturbation modes
            with wavelengths comparable or even larger than the Hubble horizon
            (i.e., λ≳ 1/H_i) at the beginning of inflation,
            they would be in the non-BD state. These modes exit horizon
            during the pre-inflationary era and carry the distinctive
            pre-inflationary signatures that can be tested by the CMB
            observations. It should be pointed out that if inflation lasted
            more than 65 e-folds, it would become a challenge to keep the
            observational signatures of primordial perturbations generated in the
            pre-inflationary era. However, due to the bound of TCC, inflation of less
            e-folds is favored. As a result, the large scale CMB window will be sensitive to testing the pre-inflationary evolution.
            Theoretically, the particular range of the scale of such a window
            is model-dependent, as can be inferred from Eq. (<ref>).
            
            
            
            
             §.§ Trans-Planckian effect
            
            
            So far we have neglected the higher-order momentum corrections in
            (<ref>). However, it is interesting to see what will
            happen if they are not negligible.
            
            Generally, with (<ref>), the solution to
            Eq. (<ref>) can be given as  u_k,j(τ)   =   √(π
                    (τ_R,j-τ)) 2{α_jH_ν_j^(1)[
                C_s,j k(τ_R,j-τ)]
            
                                +β_jH_ν_j^(2)[ C_s,j
            k(τ_R,j-τ)] } ,    (τ<τ_j+1) .
            By matching phase j with phase j+1, we can
            get (<ref>) with 
                   
                M_11=ikπ√(τ̃_j+1,1τ̃_j+1,2)4[
            (2ν_j-1 2kτ̃_j+1,1-2ν_j+1-1
                2kτ̃_j+1,2) H^(1)_ν_j( C_s,j
            kτ̃_j+1,1)H^(2)_ν_j+1( C_s,j+1kτ̃_j+1,2)
            
                                             + C_s,j+1
            H^(1)_ν_j( C_s,j
            kτ̃_j+1,1)H^(2)_ν_j+1-1( C_s,j+1
            kτ̃_j+1,2)
            
                                             -
                C_s,jH^(1)_ν_j-1(
                C_s,jkτ̃_j+1,1)H^(2)_ν_j+1(
                C_s,j+1kτ̃_j+1,2) ] ,
            
            
                   
                M_12=ikπ√(τ̃_j+1,1τ̃_j+1,2)4[
            (2ν_j-1 2kτ̃_j+1,1-2ν_j+1-1
                2kτ̃_j+1,2) H^(2)_ν_j( C_s,j
            kτ̃_j+1,1)H^(2)_ν_j+1( C_s,j+1 kτ̃_j+1,2)
            
                                             + C_s,j+1
            H^(2)_ν_j( C_s,j
            kτ̃_j+1,1)H^(2)_ν_j+1-1( C_s,j+1
            kτ̃_j+1,2)
            
                                             -
                C_s,jH^(2)_ν_j-1( C_s,j
            kτ̃_j+1,1)H^(2)_ν_j+1( C_s,j+1 kτ̃_j+1,2)
            ] ,
            
            
                   
                M_21=-ikπ√(τ̃_j+1,1τ̃_j+1,2)4[
            (2ν_j-1 2kτ̃_j+1,1-2ν_j+1-1
                2kτ̃_j+1,2) H^(1)_ν_j( C_s,j
            kτ̃_j+1,1)H^(1)_ν_j+1( C_s,j+1 kτ̃_j+1,2)
            
                                             + C_s,j+1
            H^(1)_ν_j( C_s,j
            kτ̃_j+1,1)H^(1)_ν_j+1-1( C_s,j+1
            kτ̃_j+1,2)
            
                                             -
                C_s,jH^(1)_ν_j-1( C_s,j
            kτ̃_j+1,1)H^(1)_ν_j+1( C_s,j+1 kτ̃_j+1,2)
            ] ,
            
            
                   
                M_22=-ikπ√(τ̃_j+1,1τ̃_j+1,2)4[
            (2ν_j-1 2kτ̃_j+1,1-2ν_j+1-1
                2kτ̃_j+1,2) H^(2)_ν_j( C_s,j
            kτ̃_j+1,1)H^(1)_ν_j+1( C_s,j+1 kτ̃_j+1,2)
            
                                             + C_s,j+1
            H^(2)_ν_j( C_s,j
            kτ̃_j+1,1)H^(1)_ν_j+1-1( C_s,j+1
            kτ̃_j+1,2)
            
                                             -
                C_s,jH^(2)_ν_j-1( C_s,j
            kτ̃_j+1,1)H^(1)_ν_j+1( C_s,j+1 kτ̃_j+1,2)
            ] .   Thus, for the perturbation modes satisfying
            kτ̃_j+1,1 and kτ̃_j+1,2≫1, we have (
            [ α_j+1; β_j+1 ])
               ≈   (
            [         e^iθ_j+1[
                 C_s,j+ C_s,j+1 + O(1 kτ̃)  2√( C_s,j C_s,j+1)] -e^-iθ̃_j+1[
                  i( C_s,j- C_s,j+1) + O(1 kτ̃)  2√( C_s,j C_s,j+1)];   e^iθ̃_j+1[
                  i( C_s,j- C_s,j+1) + O(1 kτ̃)  2√( C_s,j C_s,j+1)]        e^-iθ_j+1[
                 C_s,j+ C_s,j+1 + O(1 kτ̃)  2√( C_s,j C_s,j+1)] ])
            ([ α_j; β_j ]),  which reduces to
            (<ref>) when C_s^2=1, where θ_j+1=k(
                C_s,jτ̃_j+1,1- C_s,j+1τ̃_j+1,2)-π/2 (ν_j-ν_j+1) and θ̃_j+1=k(
                C_s,jτ̃_j+1,1+ C_s,j+1τ̃_j+1,2)-π/2 (ν_j+ν_j+1).
            
            Consider such a pre-inflationary era, in which C_s^2=(
            k/Λ)^2p≫ 1 for the phase j, while C_s^2=1 in
            other phases. According to (<ref>), we straightforwardly have (
            [ α_j+1; β_j+1 ])
               ≈   k^p 2Λ^p(
            [     e^iθ_j+1 -ie^-iθ̃_j+1;   ie^iθ̃_j+1    e^-iθ_j+1 ])
            ([ α_j; β_j ]) .
            This suggests that even if we have |α_0|=1 and
            β_0=0 initially or in the infinite past,  both
            α_i and β_i will be altered significantly as 
            |α_i|∼k^p 2Λ^p ,     |β_i|∼k^p 2Λ^p for kτ̃≫1, after passing
            through one Planckian regime (k≫Λ). Thus, in this case,
            the initial state of perturbation modes on sub-horizon scale will
            be no longer (actually far away from) the BD state at the
            beginning of inflation. This is actually equivalent to the
            “trans-Planckian problem" of inflation. The initial state of
            perturbation modes at the beginning of inflation being the
            BD state requires that the wavelengths of the perturbation modes
            (at the CMB window) must satisfy λ≫ 1/Λ at any
            moment in the pre-inflationary era, so that the higher-order
            spatial derivative operators may be neglected. This is
            consistent with the condition of TCC. In certain sense, such a
            pre-inflationary era should be past-complete, otherwise the phase
            with λ< 1/Λ is inevitable.
            
            
            
            
            
            
            
            § CONCLUSION
            
            
            
            
            Due to the past-incompleteness of inflation, the pre-inflationary
            physics must be considered to set the initial states for the
            inflationary perturbations.
            
            
            
            We discussed the implication of TCC for the initial states
            of the primordial perturbations and pointed out that it is possible to
            consistently set the initial state of perturbation modes in the
            infinite past, only if a past-complete pre-inflationary era
            exists.
            
            In the past-complete pre-inflationary era, the spacetime may be Minkowskian in the infinite past. Hence, it is natural to
            pick out the Minkowski vacuum state as the initial state of
            perturbations. Moreover, in such a scenario, the
            sub-Planckian perturbation modes can never cross their horizon,
            which is also consistent with the TCC. We calculate the evolution
            of the perturbation modes in such a pre-inflationary era, and show
            that the perturbation modes will behave as they are in the BD
            state at the beginning of inflation, as long as their wavelengths
            λ≪ 1/H at that time. Therefore, a past-complete
            pre-inflationary evolution may automatically prepare the initial
            state required for the inflationary perturbations.
            
            The past-complete pre-inflationary era might have left the imprint
            on CMB. One possibility of such scenarios is the bounce inflation
            <cit.>. As showed recently in
            <cit.>, the pre-inflationary bounce might be
            responsible for the power deficit of the CMB TT-spectrum at low
            multipoles and the dip at l∼ 20. Relevant issues are
            interesting for further study. In addition, it is also possible that the
            universe, as well as the initial state of perturbation modes, was
            created quantumly <cit.>. In such a
            scenario, how the perturbation is in its lowest-energy state after the creation is still under study, see recent
            Refs. <cit.>.
            
            
            
            
            
            
            
            Y. C. is funded by the China Postdoctoral Science Foundation (Grant No. 2019M650810) and NSFC (Grant No. 11905224). Y. S. P. is supported by NSFC (Grant Nos. 11575188 and 11690021).
            This project is supported by the University of Chinese Academy of Sciences.
            
            
            
            
            
            
            § BOUNCE INFLATION MODEL
            
            
            
            The stable nonsingular bounce inflation model has been implemented
            in the beyond Horndeski theory, as inspired by the EFT of
            nonsingular cosmologies
            <cit.>. In
            this Appendix, we briefly review it, see <cit.> for
            the details.
            
            
            Our effective Lagrangian for the bounce inflation is
             L∼      M_P^2
                    2R-M_P^2 2X-V(ϕ)
                  Contraction+Inflation
               +       P̃(ϕ,X)             +L_δ
                    g^00 R^(3)    +L_δ K δ g^00 , 
                  (Ghost free) Bounce      Removing  c_s^2<0  where
              L_δ g^00 R^(3)    =   f_1(ϕ) 2δ g^00 R^(3)
               =   f 2R -X 2∫ f_ϕϕd ln X
            -(f_ϕ+∫f_ϕ 2dln X)ϕ
                    + f 2X[ϕ_μνϕ^μν-(ϕ)^2]-f-2Xf_X
                X^2[ϕ^μϕ_μρϕ^ρνϕ_ν-(ϕ)ϕ^μϕ_μνϕ^ν]  ,
             L_δ K δ
                g^00   =   g_1(ϕ)2δ Kδ g^00
               =   g
                21√(-X)(ϕ^μϕ_μνϕ^ν X-ϕ) -32g ·  h ,R^(3)δ g^00 and δ Kδ g^00 are the EFT
            operators which will modify c_s^2 and Q_s in
            Eq. (<ref>), ϕ_μ=∇_μϕ,
            ϕ^μ=∇^μϕ,
            ϕ_μν=∇_ν∇_μϕ, X=ϕ_μϕ^μ
            and ϕ=ϕ^μ_ μ. Here, the coefficients f and g
            depend on ϕ and X, while h and f_2 depend only on
            ϕ. In certain conditions (see <cit.> for
            details), f, g and h can be set to be vanishing at the
            background level.
            
            As shown in Ref. <cit.>, with the operators
            R^(3)δ g^00 and δ Kδ g^00, we can have
            c_s^2=1 and Q_s satisfying Eq. (<ref>). In
            <cit.>, the background solution to bounce inflation
            has been displayed. Here, we only show the evolution of the background
            (the Hubble parameter H) in Fig. <ref>, which will be used
            in Sec. <ref>.
            
            99
            Guth:1980zm
            A. H. Guth,
            
            Phys. Rev. D  23, 347 (1981).
            
            
            
            Linde:1981mu
            A. D. Linde,
            
            Phys. Lett.  108B, 389 (1982).
            
            
            
            Albrecht:1982wi
            A. Albrecht and P. J. Steinhardt,
            
            Phys. Rev. Lett.  48, 1220 (1982).
            
            
            
            Starobinsky:1980te
            A. A. Starobinsky,
            
            Phys. Lett.  91B, 99 (1980).
            
            
            
            Ade:2015xua
            P. A. R. Ade  et al. [Planck Collaboration],
            
            Astron. Astrophys.  594, A13 (2016)
            
            [arXiv:1502.01589 [astro-ph.CO]].
            
            
            Ade:2015lrj
            P. A. R. Ade  et al. [Planck Collaboration],
            
            Astron. Astrophys.  594, A20 (2016)
            
            [arXiv:1502.02114 [astro-ph.CO]].
            
            
            Borde:1993xh
            A. Borde and A. Vilenkin,
            
            Phys. Rev. Lett.  72, 3305 (1994)
            
            [gr-qc/9312022].
            
            
            Borde:2001nh
            A. Borde, A. H. Guth and A. Vilenkin,
            
            Phys. Rev. Lett.  90, 151301 (2003)
            
            [gr-qc/0110012].
            
            
            Bunch:1978yq
            T. S. Bunch and P. C. W. Davies,
            
            Proc. Roy. Soc. Lond. A  360, 117 (1978).
            
            
            
            Brandenberger:2000wr
            R. H. Brandenberger and J. Martin,
            
            Mod. Phys. Lett. A  16, 999 (2001)
            
            [astro-ph/0005432].
            
            
            Martin:2000xs
            J. Martin and R. H. Brandenberger,
            
            Phys. Rev. D  63, 123501 (2001)
            
            [hep-th/0005209].
            
            
            Ashoorioon:2004vm 
            A. Ashoorioon, A. Kempf and R. B. Mann,
            
            Phys. Rev. D  71, 023503 (2005)
            
            [astro-ph/0410139].
            
            
            Ashoorioon:2005ep 
            A. Ashoorioon, J. L. Hovdebo and R. B. Mann,
            
            Nucl. Phys. B  727, 63 (2005)
            
            [gr-qc/0504135].
            
            
            Chen:2013tna
            X. Chen and Y. Wang,
            
            JCAP  1407, 004 (2014)
            
            [arXiv:1306.0609 [hep-th]].
            
            
            Ashoorioon:2013eia
            A. Ashoorioon, K. Dimopoulos, M. M. Sheikh-Jabbari and G. Shiu,
            
            JCAP  1402, 025 (2014)
            
            [arXiv:1306.4914 [hep-th]].
            
            
            Ashoorioon:2014nta
            A. Ashoorioon, K. Dimopoulos, M. M. Sheikh-Jabbari and G. Shiu,
            
            Phys. Lett. B  737, 98 (2014)
            
            [arXiv:1403.6099 [hep-th]].
            
            
            Ashoorioon:2018sqb 
            A. Ashoorioon,
            
            Phys. Lett. B  790, 568 (2019)
            
            [arXiv:1810.04001 [hep-th]].
            
            
            Bedroya:2019snp
            A. Bedroya and C. Vafa,
            
            arXiv:1909.11063 [hep-th].
            
            Bedroya:2019tba
            A. Bedroya, R. Brandenberger, M. Loverde and C. Vafa,
            
            arXiv:1909.11106 [hep-th].
            
            DiTucci:2019xcr 
            A. Di Tucci, J. Feldbrugge, J. L. Lehners and N. Turok,
            
            Phys. Rev. D  100, no. 6, 063517 (2019)
            
            [arXiv:1906.09007 [hep-th]].
            
            Piao:2003zm
            Y. -S. Piao, B. Feng and X. -m. Zhang,
            
            Phys. Rev. D  69, 103520 (2004)
            [hep-th/0310206];
            Y. -S. Piao,
            
            Phys. Rev. D  71, 087301 (2005)
            [astro-ph/0502343]; 
            
            Y. S. Piao,
            
            Phys. Rev. D  70, 101302 (2004)
            
            [hep-th/0407258].
            
            
            Liu:2013kea
            Z. G. Liu, Z. K. Guo and Y. S. Piao,
            
            Phys. Rev. D  88, 063539 (2013)
            
            [arXiv:1304.6527 [astro-ph.CO]].
            
            
            Qiu:2015nha
            T. Qiu and Y. T. Wang,
            
            JHEP  1504, 130 (2015)
            
            [arXiv:1501.03568 [astro-ph.CO]].
            
            
            Libanov:2016kfc 
            M. Libanov, S. Mironov and V. Rubakov,
            
            JCAP  1608, 037 (2016)
            
            [arXiv:1605.05992 [hep-th]].
            
            
            Kobayashi:2016xpl
            T. Kobayashi,
            
            Phys. Rev. D  94, no. 4, 043511 (2016)
            
            [arXiv:1606.05831 [hep-th]].
            
            
            Ijjas:2016vtq
            A. Ijjas and P. J. Steinhardt,
            
            Phys. Lett. B  764, 289 (2017)
            
            [arXiv:1609.01253 [gr-qc]].
            
            
            Dobre:2017pnt
            D. A. Dobre, A. V. Frolov, J. T. G. Ghersi, S. Ramazanov and A. Vikman,
            
            JCAP  1803, 020 (2018)
            
            [arXiv:1712.10272 [gr-qc]].
            
            
            Cai:2016thi
            Y. Cai, Y. Wan, H. G. Li, T. Qiu and Y. S. Piao,
            
            JHEP  1701, 090 (2017)
            
            [arXiv:1610.03400 [gr-qc]].
            
            
            Creminelli:2016zwa
            P. Creminelli, D. Pirtskhalava, L. Santoni and E. Trincherini,
            
            JCAP  1611, no. 11, 047 (2016)
            
            [arXiv:1610.04207 [hep-th]].
            
            
            Cai:2017tku
            Y. Cai, H. G. Li, T. Qiu and Y. S. Piao,
            
            Eur. Phys. J. C  77, no. 6, 369 (2017)
            
            [arXiv:1701.04330 [gr-qc]].
            
            
            Cai:2017dyi
            Y. Cai and Y. S. Piao,
            
            JHEP  1709, 027 (2017)
            
            [arXiv:1705.03401 [gr-qc]].
            
            
            Kolevatov:2017voe
            R. Kolevatov, S. Mironov, N. Sukhov and V. Volkova,
            
            JCAP  1708, no. 08, 038 (2017)
            
            [arXiv:1705.06626 [hep-th]].
            
            
            Ijjas:2016tpn
            A. Ijjas and P. J. Steinhardt,
            
            Phys. Rev. Lett.  117, no. 12, 121304 (2016)
            
            [arXiv:1606.08880 [gr-qc]].
            
            
            Kolevatov:2016ppi
            R. Kolevatov and S. Mironov,
            
            Phys. Rev. D  94, no. 12, 123516 (2016)
            
            [arXiv:1607.04099 [hep-th]].
            
            
            Ye:2019sth
            G. Ye and Y. S. Piao,
            
            Phys. Rev. D  99, no. 8, 084019 (2019)
            
            [arXiv:1901.08283 [gr-qc]].
            
            
            Ye:2019frg
            G. Ye and Y. S. Piao,
            
            Commun. Theor. Phys.  71, no. 4, 427 (2019)
            
            [arXiv:1901.02202 [gr-qc]].
            
            
            Mironov:2018oec
            S. Mironov, V. Rubakov and V. Volkova,
            
            JCAP  1810, no. 10, 050 (2018)
            
            [arXiv:1807.08361 [hep-th]].
            
            
            
            Boruah:2018pvq
            S. S. Boruah, H. J. Kim, M. Rouben and G. Geshnizjani,
            
            JCAP  1808, 031 (2018)
            
            [arXiv:1802.06818 [gr-qc]].
            
            
            Mironov:2019haz
            S. Mironov, V. Rubakov and V. Volkova,
            
            JETP Vol. 156 (4) (2019)
            
            [arXiv:1906.12139 [hep-th]].
            
            Akama:2019qeh
            S. Akama, S. Hirano and T. Kobayashi,
            
            arXiv:1908.10663 [gr-qc].
            
            
            
            Langlois:2018dxi
            D. Langlois,
            
            Int. J. Mod. Phys. D  28, no. 05, 1942006 (2019)
            
            [arXiv:1811.06271 [gr-qc]].
            
            
            Kobayashi:2019hrl
            T. Kobayashi,
            
            Rept. Prog. Phys.  82, no. 8, 086901 (2019)
            
            [arXiv:1901.07183 [gr-qc]].
            
            Cai:2007qw
            Y. F. Cai, T. Qiu, Y. S. Piao, M. Li and X. Zhang,
            
            JHEP  0710, 071 (2007)
            
            [arXiv:0704.1090 [gr-qc]].
            
            
            Cai:2008qw
            Y. F. Cai, T. Qiu, R. Brandenberger and X. m. Zhang,
            
            Phys. Rev. D  80, 023511 (2009)
            
            [arXiv:0810.4677 [hep-th]].
            
            
            Qiu:2011cy
            T. Qiu, J. Evslin, Y. F. Cai, M. Li and X. Zhang,
            
            JCAP  1110, 036 (2011)
            
            [arXiv:1108.0593 [hep-th]].
            
            
            Easson:2011zy
            D. A. Easson, I. Sawicki and A. Vikman,
            
            JCAP  1111, 021 (2011)
            
            [arXiv:1109.1047 [hep-th]].
            
            
            
            Cai:2012va
            Y. F. Cai, D. A. Easson and R. Brandenberger,
            
            JCAP  1208, 020 (2012)
            
            [arXiv:1206.2382 [hep-th]].
            
            
            Cai:2013kja 
            Y. F. Cai, E. McDonough, F. Duplessis and R. H. Brandenberger,
            
            JCAP  1310, 024 (2013)
            
            [arXiv:1305.5259 [hep-th]].
            
            Qiu:2013eoa
            T. Qiu, X. Gao and E. N. Saridakis,
            
            Phys. Rev. D  88, no. 4, 043525 (2013)
            [arXiv:1303.2372 [astro-ph.CO]].
            
            
            Koehn:2013upa
            M. Koehn, J. L. Lehners and B. A. Ovrut,
            
            Phys. Rev. D  90, no. 2, 025005 (2014)
            
            [arXiv:1310.7577 [hep-th]].
            
            
            Battarra:2014tga
            L. Battarra, M. Koehn, J. L. Lehners and B. A. Ovrut,
            
            JCAP  1407, 007 (2014)
            
            [arXiv:1404.5067 [hep-th]].
            
            
            Wan:2015hya
            Y. Wan, T. Qiu, F. P. Huang, Y. F. Cai, H. Li and X. Zhang,
            
            JCAP  1512, no. 12, 019 (2015)
            
            [arXiv:1509.08772 [gr-qc]].
            
            
            Koehn:2015vvy
            M. Koehn, J. L. Lehners and B. Ovrut,
            
            Phys. Rev. D  93, no. 10, 103501 (2016)
            
            [arXiv:1512.03807 [hep-th]].
            
            
            Odintsov:2014gea
            S. D. Odintsov and V. K. Oikonomou,
            
            Phys. Rev. D  90 (2014) no.12,  124083
            
            [arXiv:1410.8183 [gr-qc]];
            S. Nojiri, S. D. Odintsov and V. K. Oikonomou,
            
            Phys. Rev. D  93 (2016) no.8,  084050
            
            [arXiv:1601.04112 [gr-qc]].
            
            
            Banerjee:2016hom
            S. Banerjee and E. N. Saridakis,
            
            Phys. Rev. D  95, no. 6, 063523 (2017)
            
            [arXiv:1604.06932 [gr-qc]].
            
            
            Creminelli:2006xe
            P. Creminelli, M. A. Luty, A. Nicolis and L. Senatore,
            
            JHEP  0612, 080 (2006)
            
            [hep-th/0606090].
            
            
            Pirtskhalava:2014esa
            D. Pirtskhalava, L. Santoni, E. Trincherini and P. Uttayarat,
            
            JHEP  1412, 151 (2014)
            
            [arXiv:1410.0882 [hep-th]].
            
            
            Kobayashi:2015gga
            T. Kobayashi, M. Yamaguchi and J. Yokoyama,
            
            JCAP  1507, no. 07, 017 (2015)
            
            [arXiv:1504.05710 [hep-th]].
            
            
            Liu:2014tda
            Z. G. Liu, H. Li and Y. S. Piao,
            
            Phys. Rev. D  90, no. 8, 083521 (2014)
            
            [arXiv:1405.1188 [astro-ph.CO]].
            
            
            Wang:2013eqj
            Y. Wang,
            
            Commun. Theor. Phys.  62, 109 (2014)
            
            [arXiv:1303.1523 [hep-th]].
            
            
            Ford:1997hb
            L. H. Ford,
            “Quantum field theory in curved space-time,”
            In *Campos do Jordao 1997, Particles and fields* 345-388
            [gr-qc/9707062].
            
            
            Ashoorioon:2018uey
            A. Ashoorioon, R. Casadio, M. Cicoli, G. Geshnizjani and H. J. Kim,
            
            JHEP  1802, 172 (2018)
            
            [arXiv:1802.03040 [hep-th]].
            
            
            Obied:2018sgi 
            G. Obied, H. Ooguri, L. Spodyneiko and C. Vafa,
            
            arXiv:1806.08362 [hep-th].
            
            
            Agrawal:2018own
            P. Agrawal, G. Obied, P. J. Steinhardt and C. Vafa,
            
            Phys. Lett. B  784, 271 (2018)
            
            [arXiv:1806.09718 [hep-th]].
            
            
            Garg:2018reu 
            S. K. Garg and C. Krishnan,
            
            arXiv:1807.05193 [hep-th].
            
            
            Ooguri:2018wrx
            H. Ooguri, E. Palti, G. Shiu and C. Vafa,
            
            Phys. Lett. B  788, 180 (2019)
            
            [arXiv:1810.05506 [hep-th]].
            
            
            ArmendarizPicon:2006pd
            C. Armendariz-Picon,
            
            JCAP  0702, 031 (2007)
            
            [astro-ph/0612288].
            
            
            Jiang:2015hfa
            H. Jiang and Y. Wang,
            
            Phys. Lett. B  760, 202 (2016)
            
            [arXiv:1507.05193 [hep-th]].
            
            
            Jiang:2016nok
            H. Jiang, Y. Wang and S. Zhou,
            
            JCAP  1604, no. 04, 041 (2016)
            
            [arXiv:1601.01179 [hep-th]].
            
            
            Carroll:2004st
            S. M. Carroll,
            
            San Francisco, USA: Addison-Wesley (2004) 513 p
            Cai:2015nya
            Y. Cai, Y. T. Wang and Y. S. Piao,
            
            Phys. Rev. D  92, no. 2, 023518 (2015)
            
            [arXiv:1501.01730 [astro-ph.CO]].
            
            
            Khoury:2001wf
            J. Khoury, B. A. Ovrut, P. J. Steinhardt and N. Turok,
            
            Phys. Rev. D  64, 123522 (2001)
            
            [hep-th/0103239].
            
            
            Liu:2011ns
            Z. G. Liu, J. Zhang and Y. S. Piao,
            
            Phys. Rev. D  84, 063508 (2011)
            
            [arXiv:1105.5713 [astro-ph.CO]].
            
            
            Brandenberger:2002ty
            R. H. Brandenberger, S. E. Joras and J. Martin,
            
            Phys. Rev. D 66, 083514 (2002)
            
            [arXiv:hep-th/0112122 [hep-th]].
            
            
            Cai:2017pga
            Y. Cai, Y. T. Wang, J. Y. Zhao and Y. S. Piao,
            
            Phys. Rev. D  97, no. 10, 103535 (2018)
            
            [arXiv:1709.07464 [astro-ph.CO]].
            
            
            Joras:2008ck
            S. E. Joras and G. Marozzi,
            
            Phys. Rev. D 79, 023514 (2009)
            
            [arXiv:0808.1262 [gr-qc]].
            
            
            Ni:2017jxw
            S. Ni, H. Li, T. Qiu, W. Zheng and X. Zhang,
            
            Eur. Phys. J. C 78, no.8, 608 (2018)
            
            [arXiv:1707.05570 [astro-ph.CO]].
            
            Mizuno:2019bxy
            S. Mizuno, S. Mukohyama, S. Pi and Y. Zhang,
            
            [arXiv:1910.02979 [astro-ph.CO]].
            
            
            
            
            Vilenkin:1982de
            A. Vilenkin,
            
            Phys. Lett.  117B, 25 (1982).
            
            
            
            Hartle:1983ai
            J. B. Hartle and S. W. Hawking,
            
            Phys. Rev. D  28, 2960 (1983).
            
            
            
            
            Feldbrugge:2017fcc
            J. Feldbrugge, J. L. Lehners and N. Turok,
            
            Phys. Rev. Lett.  119, no. 17, 171301 (2017)
            
            [arXiv:1705.00192 [hep-th]].
            
            
            DiazDorronsoro:2017hti
            J. Diaz Dorronsoro, J. J. Halliwell, J. B. Hartle, T. Hertog and O. Janssen,
            
            Phys. Rev. D  96, no. 4, 043505 (2017)
            
            [arXiv:1705.05340 [gr-qc]].
            
            
            Vilenkin:2018dch
            A. Vilenkin and M. Yamada,
            
            Phys. Rev. D  98, no. 6, 066003 (2018)
            
            [arXiv:1808.02032 [gr-qc]].
            
            
            Bojowald:2018gdt 
            M. Bojowald and S. Brahma,
            
            Phys. Rev. Lett.  121, no. 20, 201301 (2018)
            
            [arXiv:1810.09871 [gr-qc]].
            
            
            DiTucci:2019dji
            A. Di Tucci and J. L. Lehners,
            
            Phys. Rev. Lett.  122, no. 20, 201302 (2019)
            
            [arXiv:1903.06757 [hep-th]].
