String Excitation by Initial Singularity of Inflation

We discuss excitation of string oscillation modes by an initial singularity of inflation. The initial singularity of inflation is known to occur with a finite Hubble parameter, which is generally lower than the string scale, and hence it is not clear that stringy effects become significant around it. With the help of Penrose limit, we find that infinitely heavy oscillation modes get excited when a singularity is strong in the sense of Krolak's classification. We demonstrate that the initial singularities of Starobinsky and hill top inflation, assuming the slow roll inflation to the past infinity, are strong. Hence stringy corrections are inevitable in the very early stage of these inflation models. We also find that the initial singularity of the hill top inflation could be weak for non-slow roll case.


Introduction
The presence of an inflationary stage of the Universe is strongly supported by the cosmic microwave background observations [1]. The observation can access the period of the last 60 e-folding in inflationary era and we hardly know about a stage of universe before, or in the very beginning of the inflation. From the theoretical view point, the Universe must be in another phase before the inflation. In Ref. [2], it is shown that the spacetime region that is covered by the world lines of comoving observers who feel accelerated expansion is past incomplete. The past incompleteness suggests the presence of a boundary for inflationary spacetime. Then if the boundary is inextendible, the inflation spacetime is said to have an initial singularity, otherwise it is a coordinate singularity.
Since a singularity is expected to be a sign of violation of classical gravity, it is important to ask whether the past boundary of an inflationary universe is a singularity or not. In Ref. [3], the past extendibility of accelerated flat Friedmann-Lemaître-Robertson-Walker (FLRW) universes is investigated by checking the presence of a parallelly propagated curvature singularity [4,5], which is defined by the divergence of the components of the Riemann tensor with respect to the basis which is parallel transportation along the incomplete geodesics. Then it was found that the extendibility depends on whether lim t→−∞Ḣ /a 2 is finite or not, where a is the scale factor, H is the Hubble parameter and t is the proper time of the comoving observers. By applying this method, for example, the Starobinsky's inflation model is found to have an initial parallelly propagated curvature singularity assuming inflation persists on the slow roll trajectory toward t → −∞. See also Refs. [6,7] where the initial singularity of other type inflation like a power law was studied. The analysis of extendibility in Ref. [3] has been then extended to the universe with a spatial curvature or anisotropies recently in Ref. [8].
Interestingly, the singularity of inflationary universe is quite different from the big-bang singularity in the sense that there is no divergence with the curvature invariants. Thus this singularity occurs even when the typical energy scale read from the Hubble parameter is much smaller than the string scale. Therefore it is not clear that this kind of singularity leads the violation of low energy effective theory description of inflation model. The purpose of this paper is to study the propagation of strings on inflationary universe to answer this question.
The equations of motion for string in de Sitter spacetime was solved exactly by Refs. [9,10]. Recently, this analysis is revisited to see the consistency between the Higuchi bound and the modified Regge trajectory [11][12][13][14]. String propagation in an expanding universe has been studied in many contexts so far [15][16][17][18][19]. Contrary to the exact de Sitter case, one need to take some approximation to study the string on general expanding universe. In this paper, we will focus on the method of Penrose limit [20], as did in [19].
The Penrose limit is a way to obtain a plane wave spacetime spreading an infinitesimal neighborhood of a null geodesic outward. It has been studied actively as a way to construct nontrivial supersymmetric solutions [21] from known supersymmetric spacetimes. This is because the Penrose limit does not decrease a number of symmetry [22,23]. A class of spacetime with a singularity is expected to have a kind of universality [24,25]. As we will see later, an initial singularity of an inflationary universe is out of this universality.
String propagation in a plane wave spacetime with a singularity has actively studied so far [26][27][28][29][30][31]. An important feature found in Ref. [30] is that the quantum string propagation is possibly ill-defined even when a singularity is weak in the sense of Tipler's classification. Actually, we will see later, that consistency of string propagation can be judged by Krolak's classification, not the Tipler's one. Note that the singular behavior becomes milder for a point-like shock wave spacetime [32][33][34][35][36][37], not a plane wave. This paper is organized as follows. In Sec. 2, we examine the presence/absence of an initial singularity of inflationary Universe based on Ref. [3] and discuss its past completion and the Penrose limit. As a result of Penrose limit, singular/non-singular inflationary Universe respectively reduces to singular/non-singular plane wave spacetime. In the Sec. 3, the propagation of test string in a plane wave spacetime is discussed. There we find that string can propagate through the initial singularity of the Universe if the strength of the singularity is sufficiently weak. In the Sec. 4, we study the strength of the singularity for three models of inflation; Starobinsky inflation and general and quadratic hill top inflation. The final section is devoted to summary and discussion.

Inflationary Universe and its past completion
Here, we review the presence/absence of an initial singularity of inflationary Universe in Sec. 2.1. Then we construct a past completion in Sec. 2.2 and take the Penrose limit to obtain singular/non-singular plane wave spacetime in Sec. 2.3.

Initial singularity of inflation
We consider 3 + 1 dimensional flat FLRW universe, Here a is the scale factor and t represents the comoving time while η represents the conformal time, dt = adη. We assume that the scale factor a(t) approaches to zero toward a initial time t i : lim t→t i a(t) = 0. To discuss the (in)completeness and extendibility of null geodesics at a → 0, we consider null geodesics defined by η + r = constant. The affine parameter u of such geodesics can be defined by du = a 2 dη = adt. (2.2) Thus the incompleteness can be checked by In the following, we focus on the incomplete case and fix the integration constant by u(t i ) = 0. The incompleteness of the null geodesics implies two possibilities. One is that the whole of the spacetime is not spanned by the flat FLRW coordinates (2.1) and the end point of null geodesics corresponds to a coordinate singularity. A simple example of the extendible case is the exact de Sitter space, where flat chart only covers a half of the entire closed de Sitter space. The other possibility is that the end point of null geodesics corresponds to a true singularity. The tangent vector k µ = dx µ /du of such null geodesics is given by Because of the relation ∇ [µ k ν] = 0, we can introduce a scalar potential for k by k µ dx µ = −dv, that is, we introduce v by Following, we fix an integration constant by v = η + r. Note that a curve v = constant corresponds to a null geodesic. By using u and v coordinates instead of t and r, we can rewrite (2.1) as In this coordinates, the Ricci tensor can be represented as where the prime stands for derivatives with respect to u and H is the Hubble parameter defined by H :=ȧ/a = a . The expression appeared in R uu plays an important role in this paper and let us call it A(u): One can see that the null tetrad basis e u := du − 1 2 a 2 dv, e v := dv, e θ := dθ, e φ := sin θdφ, (2.9) are parallelly transportation along null geodesics. By the definition, the divergence of the components of the Riemann tensor with respect to parallelly propagated basis corresponds to a parallelly propagated curvature singularity [4,5]. Since e u du in the limit a → 0, we obtain, Thus a → 0 corresponds to a parallelly propagated curvature singularity if A(u) or 3H 2 +Ḣ diverges.
In the following, we focus on an inflationary universe which approaches flat de Sitter Universe in the limit t → −∞, a(t) eH t + · · · . (2.11) This scale factor satisfies the assumptions which we have assumed above: vanishing scale factor at t → t i := −∞ and the incompleteness of the null geodesics. The timelike comoving geodesics are past complete because t is the proper time of these geodesics. A typical Penrose diagram of this kind of universe is shown in Fig. 1.
The assumption (2.11) means that H andḢ are finite as a → 0. Thus only the dangerous component in Eq. (2.10) is the one with A(u). Therefore, with the assumption (2.11), the past boundary (the dashed line in Fig. 1) is a parallelly propagated curvature singularity if and only if (2.12) The limit (2.12) depends on the next to leading order of the scale factor omitted in Eq. (2.11) and hence depends on the detail of inflationary model. For example, A(u) = 0 for the exact flat de Sitter space a ∝ eH t and hence the past boundary is extendible as expected. The metric has at least continuous extension beyond the past boundary, though the curvature could be a parallelly propagated curvature singularity depending on the scale factor. We note that this is a diagram with future complete null geodesics, though the future structure is irrelevant to our discussion in this paper.
An interesting property of this Universe is that the metric components are well defined in the limit u → 0, Thus the metric has continuous extension beyond the boundary, even though it may not be smooth. We can define the further past of inflationary Universe beyond an "initial" singularity.

An example
Let us consider an explicit example of an universe with the assumption (2.11), which can be understood as Thus we obtain a(u) =Hu + · · · . (2.16) As pointed above, the property of the boundary is sensitive to the next to leading term.
Here let us assume that the next to leading order term is power of u, with constants β and κ. In order for this term to be sub-leading, β is assumed to satisfy β < 2. We will see in Sec. 4 that this scale factor corresponds to that of the quadratic hill top inflation. Note that this scale factor can be expressed by the comoving time t as From the expression (2.17), the curvature component A(u) can be evaluated as Thus for a positive β, the boundary u → 0 becomes a parallelly propagated curvature singularity while it is a regular boundary for a non-positive β. Since the sign of A coincides with that ofḢ, a positive κ corresponds to the expansion of the universe with the matter contents satisfy the null energy condition. Let us check the implication of the parameter β to the strength of the singularity formulated by Tipler and Krolak [38][39][40][41]. The strong/weak singularity in the sense of Tipler is defined by the divergence/convergence of the limit 20) while that in the sense of Krolak is defined with (2.21) The classification of the strength of the singularity can be summarized in Table 1. With the expression of the conformal time, one can express the metric (2.6) near u = 0 as The metric components are well defined in the limit u → 0 as expected.

Contracting Universe as a past completion
So far, we have seen that the metric components have at least continuous extension beyond the past boundary of the quasi de Sitter universe. In this section we provide an extension of the universe beyond the past boundary. Let us consider a contracting FLRW universe which has a scaleã(t) with FLRW coordinates (t,r, θ, φ). The metric is given by and we assume the scale factor satisfies where a is the scale factorã(t) of the expanding Universe. Similar to the case of expanding Universe, we introduce the conformal timeη by dt =ãdη, the affine length of a future out-going null geodesic byũ and the coordinateṽ byṽ =η −r. Then the metric (2.24) can be written as where we used the relation,ã(ũ) = a(−ũ),η(ũ) = −η(−ũ). Based on these coordinates, we consider a contracting Universe as a past completion of the inflationary Universe, by identifying u < 0 asũ and v asṽ. Thus the metric of the entire spacetime can be written as A typical Penrose diagram of this universe is given in Fig. 2. The Ricci tensor in whole of the Universe can be evaluated as

An example
Again, let us focus on the example (2.17). In this case the scale factor of the contracting universe is given bỹ Now, we rewriteã andη as a function ofũ: By introducing {ũ,ṽ} coordinates, the metric of the contracting universe can be written as whereũ ∈ (−∞, 0) andṽ ∈ (−∞, ∞).
By identifying u < 0 asũ and v asṽ, we can joint the expanding and contracting universe as The curvature component A(u) is given by (2.35)

Penrose limit of FLRW Universe
The purpose of this paper is to discuss the property of the past boundary of inflationary universe by investigating a test string on the past completed inflationary universe (2.27). However, it is hard to perform the quantization of the string directly on the FLRW universe, because the wave equations is in general nonlinear equation of motion. A way to see the effect of past boundary is taking the Penrose limit [20], which is the limit to expand an infinitesimal neighborhood of a null geodesic outward. As a result, a general spacetime reduces to a plane wave spacetime. Let us take the Penrose limit of (2.27) with respect to a null geodesic v = 0. By introducing new coordinates (u,v, x, y) as v = 2v , θ = θ 0 + x, φ = φ 0 + 1 sin θ 0 y, the metric (2.27) can be expressed as (2.36) The Penrose limit is then defined by g P µν dx µ dx ν := lim →0 1 2 g µν dx µ dx ν . Thus we obtain, This is a plane wave spacetime in the Rosen coordinates. To discuss the string propagation in the next section, we will use other coordinates called the Brinkmann coordinates. The coordinate transformation to the Brinkmann coordinates is given by and we obtain the expression where A(u) is defined by Eq. (2.8) in each coordinate patch. Note that any of curvature invariants vanishes for the plane wave, R P = R P µν R P µν = R P µνρσ R P µνρσ = · · · = 0.
(2.40) Nonetheless, the curvature tensor has a non-vanishing component, As clear from this expression, the plane wave spacetime g P µν has a parallelly propagated curvature singularity if the original universe (2.27) does. Thus we can discuss the effect of this singularity even after taking the Penrose limit. It is worth to mention that for the example (2.19) the behavior of A(u) is beyond the universality of Penrose limit proposed in [24,25], where the Penrose limit of a physically reasonable spacetime with a singularity is conjectured to have A(u) ∝ u −2 .

Excitation of string oscillation modes on plane wave spacetime
In the previous section, we obtain the plane wave metric (2.39) by taking the Penrose limit of past completed inflation universe (2.27). Excitation of oscillation modes of a string passing through a plane wave singularity has been studied in Refs. [26][27][28][29][30]. Here we revisit this analysis with aiming an application to our inflationary universe. In particular, we will clarify the relation between the strength of the singularity and the excitation of string oscillations.

Equations of motion of a string
Let us consider the Polyakov action with four dimensional curved target space, where X µ (µ = 0, 1, 2, 3) are the embeddings of the world sheet on the spacetime, h ab (a, b = 0, 1) is the world sheet metric and α is the Regge slope parameter. g µν (X) is the metric of the target space, which we identify as an inflation spacetime (2.27) or the plane wave spacetime (2.39) after taking the Penrose limit. The complete set of the equations of motion, in the conformal gauge h ab = e 2ω(τ,σ) η ab with the world sheet Minkowski metric η ab , are given by Since the trace part of the Eqs. (3.3) vanishes automatically, only two components are independent. Let us solve the equations of motion by taking the Penrose limit. Plugging the plane wave metric (2.39) into the equations of motion, and fixing the remaining gauge degree of freedom by the light-cone gauge u = α pτ , we obtain following wave equations for X i = {X, Y }, and for V , as well as two equations from Eq (3.3), Eqs. (3.5), (3.6) and (3.7) can be solved by where v 0 is constant. Thus Eq. (3.8), as well as the light-cone gauge condition u = α pτ , reduces the dynamical variables to X i = {X, Y }. The important point here is that the remaining equations of motion (3.4) are linear in X i and we can quantize this system by similar way to the flat spacetime. In particular, we can separately treat each Fourier modes of X i , By plugging this expression into (3.4), we obtain This is nothing but the Schrödinger equation with the potential V(τ ) and the energy E n .

Excitation of oscillation modes
Let us evaluate the excitation of oscillation modes of a string passing through u = 0. To pick up the effect of a singularity at u = 0, we replace A(u) withÂ(u) defined bŷ where C is a positive constant, and solve the Schrödinger equation (3.10) with the replaced potentialV(τ ) := α 2 p 2Â (α pτ ) (See Fig. 3). Since the replaced potential dumps rapidly in each asymptotic region τ → ±∞, our Schrödinger equation has positive and negative frequency solution e ±inτ there. We define (f in ) i n by the solution of Schrödinger equation with the boundary condition (f in ) i n τ →−∞ e −inτ , while (f out ) i n by that with the boundary condition (f out ) i n τ →+∞ e −inτ . By using these basis, general solution of the Schrödinger equation can be written as following two ways,  Each mode operators (α in ) i n , (α in ) i n , (α out ) i n , (α out ) i n satisfies the canonical commutation relation, 13) and so on. We consider the quantum vacuum |0 in with no oscillation modes in the initial time: To discuss the excitation of oscillation modes, let us define Bogoliubov coefficients by The operators (α out ) i n , (α out ) i n are related to (α in ) i n , (α in ) i n by the Bogoliubov transformation The mass squared in the region τ → ∞ is defined by where m 2 0 is the zero point energy. Therefore, the expectation value in our vacuum state |0 in is expressed by the Bogoliubov coefficients as Let us derive a useful formula to express the Bogoliubov coefficients B i n . By using the Wronskian (3.20) one can write the Bogoliubov coefficients B i n as .

(3.21)
Since the Wronskian is constant in time τ , we can evaluate the value in the limit τ → +∞, Thus, we obtain the expression of B i n as follows, Under the Born approximation (f in ) i n (τ ) e −inτ , B i n can be evaluated as a Fourier transformation of the potentialÂ, (3.25)

An example
Again let us consider the example (2.17), where the potential A(u) is given by A(u) = −κ/|u| β . Then we obtain where the coefficient B is independent from the cut off parameter C and given by that is finite because the infinite sum of n −s is finite for s > 1. Since B i n itself are diverge for 1 ≤ β < 2, the behavior of the mass squared can be summarized as (3.29) Thus, the propagation of a string is well defined only when β < 1. In other words, The excitation of infinitely heavy modes can be avoidable only when the past boundary is a regular (β < 0) or weak singularity (0 < β < 1) in the sense of Krolak's classification. Before closing this section, we give some comments on the behavior for 1 ≤ β < 2 case. Since the above analysis is based on the Born approximation, one may think that our result, B i n → ∞, is possibly an artifact of this approximation. To clarify this point, let us reveal an ill-behavior of the Schrödinger equation for 1 ≤ β case without using the approximation. As the discussion of the junction condition for wave function at a point where the potential has discontinuity, the τ integration of our Schrödinger equation (3.10) gives the following condition, To obtain a well-defined solution, the right hand side must be finite. For 1 ≤ β case, the right hand side diverges unless X i n τ →0 ∼ 0. Since our boundary condition X i n τ →−∞ ∼ e −inτ has nothing to do with the condition X i n τ →0 ∼ 0 in general, we can conclude that this system is ill-defined for 1 ≤ β. As an example, let us focus on the case β = 1 case, where the Schrödinger equation (3.10) can be solved exactly in terms of hypergeometric function. One can check that the solution which satisfied the condition X i n τ →0 ∼ 0 has the asymptotic property, which is a mixing of positive and negative frequency modes.

String Excitation by initial singularity of inflation
Let us apply the above analysis to various models of single field inflation and see the consistency of the effective field theory description. We consider the Einstein-Hilbert action with an inflaton φ with a potential V (φ), Assuming the flat FLRW ansatz for the metric (2.1) and the homogeneity of inflaton φ = φ(t), this action yields following equations of motion, Under the slow roll condition |φ| 3H|φ| andφ 2 /2 V (φ), the equations of motion reduce to By using equations of motion, slow roll condition can be rewrite as 1 and |η| 1 with To see the strength of the singularity, we need to evaluate A =Ḣ/a 2 as a function of u.
Recall that we are interested in inflationary solutions with the property a eH t in the limit t → −∞. Then we rewrite the potential V (φ) as and we focus on specific solutions As mentioned before, the leading term of a can be expressed as On the other hand, by the equations of motion under the slow roll condition, a can be expressed as a function of φ as a = a e e −N (φ) , where a e is a constant and N (φ) is the e-folding number defined by . (4.8) Here we assume 1 + δV ∼ 1 in the final expression. By comparing two expressions, we can obtain a relation between u and φ, which can be used to express φ as a function of u.
For the equations of motion under the slow roll condition, we can write dH/dt as a function of φ, Thus the component of the curvature tensor, A(u), can be expressed as To summarize, for a given inflaton potential V (φ), we can get the expression φ = φ(u) from Eq.(4.9), and then we can obtain A(u) from (4.11).

Starobinsky type model
Let us consider a Starobinsky-type inflation model with a potential (4.12) in the large field region φ µ. This model includes the α attractor model [42][43][44] and pole inflation model [45][46][47][48] in with a pole of order 2. We are interested in the solution that the slow roll conditions are satisfied t → −∞, otherwise the kinetic energy of inflaton dominates over the vacuum energy in the past [49][50][51], which leads to a scalar curvature singularity.
We can evaluate δV as Then Eq. (4.9) can be evaluated as (4.14) Thus we can write φ as a function of u by and A(u) can be represented as with constants κ, c 1 and c 2 . We observe that the divergence of A(u) in the limit u → 0 is stronger than 1/u, while it is weaker than 1/u 2 . Therefore, we conclude that this is a strong singularity in the sense of Krolak and the infinitely heavy oscillation modes are excited.

General hill top inflation
Let us consider a general hill top potential, Then the slow roll conditions are satisfied in the limit φ → 0. We focus on the specific solution that the inflaton had been rolling down the potential in an infinite time: The derivative of δV can be evaluated as δV (φ) = −nφ n−1 /µ n and hence Eq. (4.9) reduces to Thus φ can be represented as (4.19) Therefore, A(u) can be evaluated as (4.20) Again, the divergence of A(u) is stronger than 1/u. Hence, we conclude that this is a strong singularity in the sense of Krolak and the infinitely heavy oscillation modes are excited.

Quadratic hill top Inflation
Let us consider a slow roll inflation caused by an inflaton with a quadratic hill top potential, whereη is a negative parameter which represents the slow roll parameter η in the limit φ → 0. Actually the slow roll parameters can be evaluated as Let us focus on the case |η| 1 first. In this case, the slow roll conditions are satisfied and we can use Eq. (4.9) and Eq. (4.11). Since δV ∼ −|η|φ/M 2 pl , Eq.(4.9) reduces to where the slow roll parameter |η| can read |η| = M 2 pl /(2f 2 ). Then the condition for the effective description of very early stage of inflation to be well-defined can be written as This relation looks similar with the axionic weak gravity conjecture f < O(M pl ) [52][53][54], see also Ref. [55]. Thus it is interesting to clarify the relation of the violation of effective description of inflation models due to the spacetime singularity to that based on the general property of quantum gravity. We mainly focused on the non-scalar curvature singularity here. It is also known that yet another type of singularity occurs inflation spacetime when the spatial topology is nontrivial [56,57]. This can be understood as a de Sitter space analogue of well known Misner singularity [58]. For this singularity, an infinite number of winding modes of string, not oscillation modes, will be excited [59]. Thus it is interesting to study the excitation of winding modes in inflationary case. Because of the difficulty to quantize strings directly in curved spacetime, it will be useful to use a method like the Penrose limit or develop an alternative method to treat winding modes in expanding universe such as the double field theory [60][61][62][63][64] and its massive extension [65,66].
In the present work, we evaluated the excitation of string at a future time far enough from the singularity. It may be interesting to evaluate string excitation near the singularity depending on time. Then we can evaluate the effect of back reaction to the spacetime and discuss the resolution of the initial singularity by string excitation. We leave above interesting directions for future work.