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Comments on Cosmological Singularities in String Theory

We compute string scattering amplitudes in an orbifold of Minkowski space by a boost, and show how certain divergences in the four point function are associated with graviton exchange near the singularity. These divergences reflect large tree-level backreaction of the gravitational field. Near the singularity, all excitations behave like massless fields on a 1+1 dimensional cylinder. For excitations that are chiral near the singularity, we show that divergences are avoided and that the backreaction is milder. We discuss the implications of this for some cosmological spacetimes. Finally, in order to gain some intuition about what happens when backreaction is taken into account, we study an open string rolling tachyon background as a toy model that shares some features with R^{1,1}/Z.


Introduction
Time dependent solutions in string theory are of interest, both for describing the early universe, and for studying various dynamical issues. In particular, the behavior near cosmological singularities has recently been studied by many researchers, including .
The first step in such studies typically involves determining the wavefunctions of particles in the singular geometry, and in particular the manner in which these wavefunctions are to be continued through the singularities. This can be done e.g. by using orbifold [5,8,11] or coset conformal field theory [9,11] techniques.
The second step involves an analysis of stability of the spacetime under small perturbations. Very general arguments lead one to believe that already classically a large backreaction of the geometry to small perturbations is to be expected, since any non-zero stress tensor gets infinitely amplified near a singularity [29]. Quantum mechanical effects generically lead to further large backreaction [30,18].
The classical backreaction in a class of models which can be described near the singularity by certain Lorentzian orbifolds has recently been studied in [8,13,16]. It was found that, as expected, these orbifolds suffer from instabilities associated with the divergent stress tensor of matter near the singularity. This is reflected in certain new divergences of the 2 → 2 tree level scattering amplitude of particles in the geometry. These divergences are associated with the region near the singularity, and signal the fact that, when backreaction is included, the curvature and string coupling in these backgrounds grow without bound (while without backreaction, the string coupling is weak everywhere, and the curvature vanishes everywhere except at the singularity).
We here will examine a different class of models, which reduce near a cosmological singularity to where the orbifold generator acts as a boost in IR 1,1 . This type of singularity, which is usually referred to in the literature as Milne or Misner spacetime, arises in a number of examples of recent interest: (1) The Nappi-Witten model [31,9], in which this singularity appears at the intersection of two copies of a closed, big-bang/big-crunch universe, and certain non-compact static regions (the "whiskers" of [9]).
(2) The big-bang/big-crunch cosmology described in [11], corresponding to a circle shrinking from finite to zero size, and then back to the original size.
(4) IR 1,1 /Z Z is also of independent interest, and has been studied as such in [5,7] (in the latter paper with the Rindler wedges omitted).
The plan of the paper is as follows. In section 2 we review the structure of the orbifold (1.1), the wavefunctions on it, and the relation between the wavefunctions and the choice of vacuum for quantum fields on IR 1,1 /Z Z.
In section 3 we perform a string calculation of the tree level 2 → 2 scattering amplitude in the spacetime (1.1), and study its singularities. We are particularly interested in singularities of the scattering amplitude that are associated with the cosmological singularity (the fixed set of the boost action).
In section 4 we compare the results of the string calculation to a gravity analysis of the same amplitude. This helps one identify the part of the amplitude associated with the backreaction to the growing stress tensor of perturbations near the orbifold singularity. We find that in general the dilaton remains finite near the singularity, while the curvature of the metric (including backreaction) blows up and causes the divergence found in section 3.
We also find that one can fine-tune the initial conditions in such a way that the large backreaction is avoided.
The physical picture is the following. Minkowski spacetime, the backreaction is always large, in another natural choice of vacuum one often finds a small backreaction. In the Nappi-Witten model, the backreaction to the modes studied in [9] is small. In the model of [11] one finds a large backreaction. We also briefly discuss the case of the non-rotating BTZ black hole; for a more comprehensive recent discussion see [34].
In situations where the classical backreaction is large, it would be interesting to understand what happens to the background when backreaction is taken into account. To gain insight into this interesting problem, we discuss in section 6 an open string toy model, which exhibits some of the features described for IR 1,1 /Z Z above. The toy model is the dynamics of open strings on an unstable D-brane, in the background of a homogenous rolling tachyon [35].
We show that an effective field theory, which should provide a good description of the dynamics of the tachyon at late times [36], exhibits features similar to those found for the closed string systems in sections 3 -5. Generic perturbations lead to "large backreaction" on the tachyon background at late times. By fine-tuning the solution in a way similar to that in section 4, one can arrange for the backreaction to be small. Moreover, by thinking of the D-brane as a collection of D0-branes, the fine-tuning in question is precisely that needed to keep the D0-branes at rest relative to each other at late times. The large backreaction in the general case is associated with D0-branes approaching each other and interacting in a non-trivial way via open strings stretched between them. We discuss the role of these interactions in the full open string problem, and the possible implications for the cosmological singularities studied in sections 3 -5.
In section 7 we summarize the results, and discuss some open issues. Three appendices contain some useful technical results.

Wavefunctions and vertex operators in
so that the metric reads 3) IR 1,1 /Z Z is obtained by orbifolding with the group Z Z generated by the boost The resulting spacetime consists 1 of four cones (times IR D−2 , which we will suppress in some of the formulae below), touching at the spacelike singularity X ± = 0. We will refer to the four cones as the "early time region" or "past Milne wedge" (X + , X − < 0), the "late time region" or "future Milne wedge" (X + , X − > 0), and the "regions with closed timelike curves" or "Rindler wedges" (X + X − < 0). In the early and late time regions, it is useful to define coordinates (t, x), 5) in terms of which the metric and identification are It will also be convenient to define the conformal time coordinate η by (e.g. in the early time region) so that the metric becomes

Wavefunctions
Wavefunctions on the orbifold (1.1) are wavefunctions on Minkowski space that are invariant under (2.4). A convenient basis of wavefunctions is given by [5] ψ(X + , The normalization has been chosen such that the Klein-Gordon norm is 1. For p + , p − ≥ 0 (which we will assume in what follows), these wavefunctions are superpositions of negative 1 We neglect one dimensional pieces of spacetime that come from moding out the X + X − = 0 locus. frequency plane waves in Minkowski space, so they describe excitations over the (adiabatic) vacuum inherited from Minkowski space (see e.g. [37]). The mass shell condition is 2p + p − = m 2 (m being the two-dimensional mass, which includes contributions from momenta in any additional dimensions), so by shifting w and multiplying the wavefunction by a phase we can put (2.10) Using the integral representation [38] of the Hankel function H (1) ν , valid for 0 < argz < π or for argz = 0, −1 < Re ν < 1, the wavefunction (2.9) can be brought to the form (for X + , X − ≥ 0) where we have usedz We see that l is the momentum along the x circle (2.6). The expression of the wavefunction (2.9) in the other regions of IR 1,1 /Z Z is described in appendix C.
The Hankel function H (1) ν (z) can be written as where the Bessel function J −il (z) has the power series expansion .
Note that together with their complex conjugates. For t → 0 (η → −∞), (2.17) become purely negative frequency with respect to conformal time η, as can be seen using (2.15): (2.18) As a consequence, the complex conjugate modes annihilate a state which in the limit m = 0 is called the conformal vacuum [37]. This is to be contrasted with the modes (2.12), which for t → 0 involve both positive and negative frequencies with respect to η: Using the identity (2.20) and writing Γ(1 + il) = e iφ l πl sinh(πl) , with φ l = −φ −l , we can rewrite (2.19) as We see that, as mentioned in the introduction, wavefunctions of all particles behave near the Milne singularity like those of massless 1 + 1 dimensional fields on the cylinder labelled by (η, x). We will return to this fact later.
Finally, one might want to consider wavefunctions which are Fourier transforms of (2.9):

22)
Since l is the momentum along the circle labelled by x (2.6), σ can be thought of as a position along this circle. Using (2.9), we have so Ψ(σ) is a plane wave in Minkowski space superposed with all its images under the orbifold group. As η → −∞, ψ l , and hence Ψ, splits into left and right-moving parts, The coefficients a ± n (σ) can be extracted from (2.21): The Klein-Gordon norm of these wavefunctions diverges, so we will only consider (2.9) and (2.17) in this paper.

2 → 2 scattering in string theory
We are interested in the behavior of strings near the singularity X ± = 0 of the orbifold (2.4). Therefore we will compute classical string scattering amplitudes involving the vertex operators (2.9), following [8] who studied a different time-dependent orbifold. We will focus on the 2 → 2 scattering amplitude, since this is the simplest case in which gravitational backreaction is expected to play a role.
Two and three point functions are studied in appendix A. It turns out that although the three point functions exhibit an interesting structure, this structure is associated to the asymptotic regions in the Milne wedges, not to the singularity. A very similar behavior was found in [8].
In the present section, we compute the four point function of the tachyon vertex operators (2.9). We will study the process 1 + 2 → 3 + 4. The corresponding amplitude is (3.1) The coefficients ǫ i are 1 for the incoming particles 1 and 2, and −1 for the outgoing particles 3 and 4. The mass shell condition is (setting α ′ = 1) As before, m 2 is the effective two-dimensional mass squared, and we assume it to be positive, so we can define m to be positive as well. Define Then the Mandelstam invariants are given by We can reduce the expression for the four-point function to a single integral as follows.
We first perform the z integral 2 in (3.1): .
) and performing the w 1 and X ± integrals, we can write the four point function as (2π) 24 4 The z integral runs, as usual, over the (Euclidean) worldsheet -the sphere or plane. This might seem like a problem since we are studying an inherently Minkowski signature spacetime, (1.1), and thus should take the worldsheet to have Minkowski signature as well. However, the Euclidean calculation is only used here to arrive at the Shapiro-Virasoro amplitude, which can be taken to be the starting point of the analysis, and used directly in Minkowski spacetime.
We now perform the v 2 , v 3 integrals. Setting the arguments of the delta functions to zero amounts to solving for v 2 , v 3 . The solutions are where (3.9) Note that v 2 , v 3 should be positive, so one should retain only the positive solutions among (3.8). Including a Jacobian factor from the delta functions and plugging in the solutions (3.8), the four point function can be reduced to the following single integral: where the sum runs over the positive solutions among (3.8). We are interested in the divergences of the four-point function, since they can potentially teach us something about the singularity.
Consider the v 4 → ∞ region of the integral. In this limit the positive solution among so that the Mandelstam variables t and s are This is the Regge limit s → ∞, t fixed, in which . (3.14) The v 4 → ∞ limit of the four-point function is therefore The v 4 → 0 limit is equivalent to the v 4 → ∞ limit (they are complex conjugates).
In the above expression, we have set α ′ = 1. For later comparison with the gravity calculation, it is also useful to consider the limit α ′ → 0, or more precisely the limit α ′ t → 0 (α ′ s and α ′ u cannot go to zero at the same time since s + t + u = −16/α ′ ; this is an irrelevant complication, which is due to the fact that the mass of the tachyon is of order the string scale. Similar results would be obtained for fields with masses well below the string scale). Near x , so that the α ′ → 0 limit of the four-point function in the Regge limit becomes (2π) 24 16 There are additional divergences from other regions of the integral (3.11). Some of them may be understood as different versions of the above. For example, m 2 2 < m 2 3 , m 2 1 < m 2 4 and B a small positive number corresponds to large s, fixed u. In this regime the four-point function diverges whenever ( This is an IR effect which is not associated with the singularity (see appendix B).

Gravity analysis
In this section we will show that the divergence (3.16) is due to exchange of gravitons near the singularity, and signals a large gravitational backreaction in that region. We will also see that in some situations the backreaction is milder than for generic kinematics, and the tree level 2 → 2 scattering amplitude is finite.
In subsection 4.1 we will compute the massless exchange contribution to the four point function (3.1) in an alternative way, and show that, in a certain kinematic regime, the dominant contribution to (3.16) comes from graviton exchange near the singularity. In subsection 4.2 we will study the backreaction in dilaton-gravity coupled to a scalar field, and relate these calculations to the scattering amplitudes studied earlier.

4.
1. An alternative calculation of the 2 → 2 scattering amplitude in the α ′ t → 0 limit In the α ′ t → 0 limit, with t a Mandelstam variable (3.4), (3.5) can be replaced by In this limit, (3.5), and as a result the four point function studied in section 3, is dominated by a massless exchange in the t-channel. Thus, computing the 2 → 2 scattering of the wavefunctions (2.9) (with t small in string units) amounts to computing (3.1), with the last integral replaced by the r.h.s. of (4.1): One way of computing (4.2) is to repeat the analysis of section 3, i.e. first perform the integrals over X ± , which give delta functions of momentum conservation on the covering space, and then integrate over w i . The result is the α ′ t → 0 limit of (3.11), which reduces in the v 4 → ∞ limit to (3.16). Note that in this limit one has t ≃ −( p 1 − p 3 ) 2 (see (3.13)).
Also, to relate (4.2) to (3.16) one uses s ≈ m 1 m 2 v 2 (see (3.12) and (3.13)); this amounts to p 1 · p 2 ≈ −p + 1 p − 2 . These observations will be relevant for comparison with a second way of computing (4.2), to which we turn next.
Another way of computing (4.2) is to perform the w i integrals first. Then one obtains The large v 4 divergences found in section 3 come from the term (see the previous paragraph). Now consider the contribution to (4.4) from the wedge X ± > 0; the other three wedges are discussed in detail in appendix C. We will see shortly that for a certain range of the transverse momenta, the integral (4.4) is dominated by the region of small X + X − . Let us assume for now that the integral (4.4) is dominated by the region of small X + X − , and that it is consistent to use the leading behavior (2.19) of the wavefunctions, Under these assumptions, the 1/∂ 2 operator in (4.4) acts only on the exp(iǫ j p j · X) parts of the full wavefunctions. Thus, one has and (4.4) becomes (4.7) Using X ± = 1 √ 2 e η±x , we can write the integral in the above expression as Performing the x integral, and defining v = 2e −2η , we can simplify this to There is a small subtlety in the preceding discussion, which we would like to mention at this point. In (4.6), we assumed that in evaluating the 1/∂ 2 operator, we can use the asymptotic form of the wavefunctions ψ m,l near the singularity X + X − = 0. In fact, if one is very close to the t-channel graviton pole, the exchanged particle can propagate for a large distance and the amplitude is not dominated by the behavior of the wavefunctions near the singularity. However, one can show that if the momentum transfer ( p 2 − p 4 ) 2 is small compared to the string scale, but large compared to the two-dimensional masses squared, the approximation leading to (4.6) is valid at large v 4 , and that subleading corrections in m 2 /( p 2 − p 4 ) 2 correspond to subleading corrections in 1/v 4 in the analysis of section 3.
Thus, we conclude that the divergence (3.16) of the four point function is associated with exchange of massless particles near the singularity. An interesting by-product of this analysis is that it points to particular kinematical situations where this divergence is absent. Consider two incoming particles 1 and 2 whose wavefunctions are purely left moving (or purely right moving) near the singularity, i.e. they only depend on X + (or only on X − ). Then it is clear that the contribution (4.4) responsible for the divergence vanishes. Examples of wavefunctions that are chiral near the singularity are given by (2.17). In section 5, we discuss some situations in which they are physically relevant.

Backreaction in dilaton gravity
In this subsection we would like to compute the classical backreaction of the graviton and dilaton to an incoming tachyon perturbation, and relate it to the amplitude calculations of the previous subsection.
Consider the action where g µν is the string frame metric, Φ the dilaton, and T a scalar field of mass m. It is convenient to transform to Einstein frame, by defining The action is now (4.12) In this action, Φ is essentially decoupled fromg and so we can treat them separately. The equation of motion for Φ, to leading order in Φ, is Thus, we see that the dilaton is only sensitive to the mass term, and in particular, it does not seem to diverge for the wavefunctions we consider in this paper (e.g., if the two T 's on the r.h.s. of (4.13) are combinations of (X ± ) ∓il j , see (4.5)).
Now we turn to a discussion of the backreaction of the metric. We expand Plugging into the action (4.12) we get where Lorentz indices are raised and lowered with the flat metric η µν and we defined h ≡ h µ µ . The stress tensor that enters the action is We would like to fix the de Donder gauge We do this in a way analogous to the way Feynman gauge is introduced in QED. We add to the Lagrangian a term which leads to the following form of the action: Varying with respect to h µν , we find the equation of motion It is useful in verifying this to write also the trace of this equation: Note that this gives the Ricci tensor of the perturbed metric, since in the gauge (4.17) (see e.g. [39], eq. (6.6)). In particular, we see that there is large backreaction when the field T has both X + and X − dependent pieces near the singularity. If the field T is chiral, say only a function of X + , then only R ++ will be large near the singularity, and the problem seems much milder behaved (for instance, powers of the Ricci scalar are finite, and α ′ corrections to the Einstein action constructed out of powers of the Ricci tensor can be neglected). This is in agreement with the observation made at the end of subsection 4.1, that divergences of four point functions are absent if the incoming fields are chiral.
To study the effect of the backreaction (4.13), (4.22) on the scalar field T , one classically integrates out h µν , Φ, by plugging the solutions of their equations of motion back into the classical action (4.12). To order T 4 , one finds The second term on the r.h.s. of (4.23) is despite appearances non-singular. By integrating by parts, one can show that for on-shell tachyons it equals −T 2 ∂ 2 T 2 /8. The first term is of the form (4.3), which was used to compute the 2 → 2 scattering amplitude of T 's in subsection 4.1.

Applications
The picture emerging from the analysis in sections 3 and 4 is that generic small In this section we will discuss some qualitative implications of this observation for a few spacetimes that look locally like the Milne orbifold (1.1).

Milne orbifold
When quantizing fields on the Milne orbifold, one has to choose a vacuum. One natural choice of vacuum is the one inherited from the Minkowski space prior to orbifolding.
Excitations of this vacuum are described by the wavefunctions (2.12), which near the singularity involve both chiralities (see (2.19)). Thus, the analysis of sections 3 and 4 implies that this vacuum exhibits large classical backreaction to any perturbation.
On the other hand, as we discussed around (2.18), in another natural vacuum state (see e.g. [37]) the positive energy wavefunctions are given by (2.17). Near the singularity, they depend only on X − for positive momentum l, and only on X + for negative momentum.
So we conclude that in this state, there is large backreaction if one has incoming particles of both positive and negative momentum. Large backreaction is avoided if all incoming particles are moving in the same direction on the cylinder near the singularity (and similarly for outgoing particles).

Big crunch/big bang cosmology of [11]
In [11], an orbifold of a coset CFT was studied which describes a spacetime with a big crunch/big bang singularity, and with a number of asymptotic regions as well as compact regions with closed timelike curves (see also [40,41]). Close to the big crunch/big bang singularity, the spacetime looks like a Milne orbifold (1.1). Natural in and out vacua were identified, and the amount of particle creation was computed in string theory. It was found that, due to the presence of different asymptotic regions and in particular the singularities connecting them, particle creation of any given mode did not decay with energy for large energies, unlike the situation in smooth spacetimes where it is know to decay exponentially with energy. This raises the suspicion that there should be large backreaction in this model.
The modes annihilating the natural incoming vacuum of [11] turn out to involve both chiralities near the Milne singularity, so there is large backreaction to any perturbation of this vacuum. It would be interesting to see if and how this is related to the large amount of particle creation in this model found in [11].

Nappi-Witten model
In [9], a coset CFT was studied which describes a four-dimensional spacetime containing a few copies of a closed big-bang/big-crunch universe, which was originally studied in [31], as well as a number of non-compact static regions which extend to spatial infinity ("whiskers"). The closed cosmological regions are attached to the whiskers at a singularity, which looks locally like the Milne orbifold (1.1).
As discussed in [9], it is natural to study scattering amplitudes of n to m particles in a given whisker. The incoming state corresponds to particles sent in from infinity in the whisker. Their wavefunctions are given by (3.29) in [9]; they have the property that near The situation is similar to that described in [42] for black holes (see also [20]). The full geometry contains two disconnected boundaries, at infinity in regions 1 and 1 ′ , on which asymptotic states are defined. Amplitudes in whisker 1 can be computed using a density matrix, corresponding to tracing out the degrees of freedom in whisker 1 ′ . One can also compute correlations between whiskers 1 and 1 ′ , which are non-zero because the states of the two whiskers are entangled.
The amplitudes of the modes described above do not get contributions from the cosmological big bang/big crunch regions. Thus, it is not very surprising that they do not give rise to large backreaction of the geometry. Generic incoming modes correspond to particles coming in from infinity in whisker 1, as well as from the cosmological region I.
Such modes correspond to wavefunctions which contain both chiralities near the Milne singularity, and thus lead to large backreaction. More generally, amplitudes that probe dynamics in the compact, cosmological regions of spacetime are expected to suffer from the divergences discussed in sections 3 and 4, from one or both of its big bang and big crunch Milne singularities.

BTZ black hole
As mentioned in the introduction, non-rotating BTZ black holes have a spacelike singularity of Milne type, and one might wonder whether our analysis sheds any light on its fate in string theory.
In asymptotically AdS spacetimes, one is interested in computing boundary correlations functions, which are the AdS analogues of S-matrix elements. Thus, the question is whether these boundary correlation functions are sensitive to physics near the BTZ singu-

Lorentzian correlation functions with insertions on both boundaries (in regions I and II).
Assuming that these boundary correlation functions probe local physics in the full extended BTZ spacetime, and in particular near the singularity, they can be used to resolve the singularity; see [42] and [34] for recent discussions.
From the perspective of our discussion here, this second continuation is more puzzling.
The wavefunctions (in the Hartle-Hawking state) that one gets in this continuation (e.g. eq. (2.7) in [42]) diverge near the BTZ singularity like in the coordinates used in sections 2 -4. Thus, the contribution to these amplitudes from the vicinity of the singularity is divergent, as in the discussion of sections 3 and 4 above. This divergence signals a large backreaction of the metric, as in section 4. Since the full amplitude obtained by continuation from Euclidean spacetime is finite, it must be that from the point of view of the discussion in sections 3 and 4, the divergences due to different singularities cancel [34]. However, this is a non-local cancellation, and there is some tension between the statement that it occurs in all correlation functions, and the expectation that boundary correlation functions can be used to probe local physics in the bulk of the full extended BTZ spacetime. Also, it is not completely clear in what sense one can neglect the backreaction near a particular singularity, which appears to be large.
This issue clearly requires a better understanding.

Open string toy model
In this section we will discuss an open string model which shares some features with the closed string systems discussed in the previous sections. Consider a D-string in bosonic string theory. 4 As is well known, the lowest lying excitation of the D-string is tachyonic.
The condensation of this open string tachyon, T , leads to the disappearance of the brane.
Condensation of spatially dependent modes of the tachyon leads to lower dimensional D-branes, which are also unstable and can further decay by tachyon condensation. the open string field. This leads to a pressureless fluid "tachyon matter" state [35,43].
In conformal field theory on the strip (i.e. classical open string theory), this "rolling tachyon" background is described by adding the boundary interaction δS = dτ e x 0 (τ ) (6.1) to the worldsheet Lagrangian of a D-brane 5 . This boundary CFT has not been analyzed in detail (see [44] for a recent discussion), but it has been argued that the endpoint of the time evolution is highly unstable; for example, if one turns on an arbitrarily small coupling to closed strings, one expects the tachyon matter to decay into closed strings and disappear.
A closer analogue of the instabilities of cosmological spacetimes to small perturbations at early times discussed in previous section, would in this case be instabilities of the rolling tachyon background (6.1) to small open string perturbations at early times. We will next see that such instabilities indeed do arise and discuss their physical interpretation. 4 A similar analysis can be performed for non-BPS branes and brane-antibrane systems in type II string theory. 5 More precisely, (6.1) is the form of the interaction at early times, x 0 → −∞, where the perturbation is small. The precise form of the perturbation at large x 0 depends on the renormalization prescription.
We will use a field theoretic effective action which has been argued [36] to give a good description of tachyon dynamics for large T . It is useful for our purposes, since this is the region which we are most interested in.
The action is S = dxdx 0 L ; where the potential is taken to be be similar for other potentials which go exponentially to zero at large T , such as those that arise in boundary string field theory [45,46,47] (see [48,49,50] for discussions of more general potentials).
The energy density corresponding to (6.2) is In terms of the momentum conjugate to T , the Hamiltonian is H = dx T 00 ; and the Hamilton equations of motion arė

(6.7)
A homogeneous solution of the equations of motion is given by which for late times approaches Note, in particular, that the leading asymptotic behavior of T is independent of the energy density E, while that of Π does depend on E. More generally, there is a class of solutions of the equations of motion that goes for late times like [36] T ∼ x 0 ; where f (x) is an arbitrary function of the spatial coordinates, and the corrections to (6.10) are exponentially small at late times. Eq. (6.6) implies that the energy density of such solutions is T 00 = |f (x)|. In fact, it has been argued in [51] that the most general solution of the equations of motion (6.7) approaches at late times a solution of the first order The solutions (6.10) are indeed of this form.
We would like next to perform a classical stability analysis of the homogenous rolling tachyon solution (6.8). Thus, we expand the tachyon field T as T = T 0 + φ, where T 0 is the homogenous solution (6.8) and φ a small fluctuation (for early times). We would like to check whether φ remains a small perturbations as t → ∞, or whether it grows to dominate the solution.
Expanding the action (6.2) to second order in φ, we find The order φ 0 term is (times the length of space, which we suppress -the problem is translation invariant in x).
The linear term in φ, as well as the last term in brackets in (6.12) are total derivatives and can be neglected. The quadratic effective action for φ is thus [52] S = E 2 dxdx 0 cosh 2 (x 0 )φ 2 − ( ∇φ) 2 , (6.14) which leads to the equations of motion Substituting u = tanh(x 0 ), and φ(u, x) = e ipx F (u) the equation of motion simplifies to The solutions are hypergeometric functions: We are particularly interested in the large time behavior of the solution (i.e. in the behavior as u → 1). The two solutions approach constants, . (6.18) C 1 and C 2 are generically non-zero, which implies that the approximation that φ is a small perturbation (in general) fails. Indeed, if F approaches a constant at late times, the second term in (6.14) makes an infinite contribution to the action, which overwhelms the 0'th order term (6.13), and one can check that the expansion of the square root that led from (6.2) to (6.12) is not justified in this case (at late times). Thus, generically, small perturbations create "large backreaction" to the original solution (6.8), much like in the discussion of sections 3 and 4. Another similar aspect of the two problems is that like there we can choose a linear combination of the two solutions (6.17), for which the "backreaction" is small. Indeed, the linear combination (6.19) goes to zero exponentially at late times, and for it the quadratic action (6.14) is finite and the perturbative expansion (6.12) is well behaved. Thus, for roughly half of the possible initial states, we find mild backreaction.
All this is consistent with the general picture presented in [51] and in equation (6.11) above. For the fine-tuned initial data sets corresponding to (6.19), the late time behavior is of the form (6.10), with f (x) determined by the initial conditions. This late time behavior corresponds to a small perturbation of the original solution (6.9). Generic initial conditions lead to a generic solution of (6.11). In particular, it is no longer true that the tachyon behaves like T ∼ x 0 at late times, and the perturbation is not small there. Moreover, the authors of [51], who analyzed the general solutions of (6.11), showed that for generic initial conditions, the solutions develop caustics and become ill defined beyond a certain finite (but late) time, which depends on the initial conditions. This is another aspect that is reminiscent of the problem of backreaction near a cosmological singularity.
It is natural to ask what is the origin of the large backreaction for generic initial data, and what is the fate of the system once it is taken into account. We will next propose a possible physical picture which explains these phenomena, leaving a more complete understanding for future work.
Specifying generic initial data at early times corresponds to perturbing the solution More generally, the D0-branes are in relative motion, and the late time dynamics is more complicated. In particular, when two D0-branes approach each other, the strings that connect them become light and can no longer be integrated out. We believe that this is the origin of the singularities associated with caustics found in [51], and the growth of fluctuations found earlier in this section. The effective action (6.2) appears to be well suited for describing the collective dynamics of well separated D0-branes; it breaks down when the interactions between different D0-branes become important at late times.
It should also be pointed out in this context that the analysis of caustics in [51] used the method of characteristics to study solutions of (6.11). In this method, the solution is obtained by following the behavior of certain worldlines of massive particles; caustics arise when different worldlines collide. The resulting picture is very reminiscent of what one would get by analyzing the motion of the D0-branes out of which the D1-brane is composed. It would be interesting to make the connection more precise.
Armed with a qualitative understanding of the origin of the singularities associated with generic perturbations of the solution (6.8), one can ask how these singularities are resolved in the full open string theory. As we saw, to do that one has to take into account the interaction between nearby D0-branes. In particular, the fact that the D0-branes can form bound states is not taken into account in the description (6.2). The analysis of [51] seems to suggest that for generic initial conditions, inhomogeneities in the tachyon field grow with time and give rise to localized clusters of D0-branes. These bounds states of D0-branes are quantum mechanical objects that need to be analyzed in the full D0-brane quantum mechanics. We will leave a more detailed analysis of this for future work. When D0-branes approach each other, one can no longer ignore effects due to open strings stretched between them. The analogous objects in the cosmological Milne singularity are twisted strings, which become light near the singularity and can no longer be neglected (see [26] for related comments). These twisted strings give rise to interactions between the incoming particles, the outcome of which has not been analyzed so far. A natural guess is that these interactions are a first step in the process of creation of black holes out of these colliding particles, as in [18]. This is similar to the creation of bound states of D0-branes in the time evolution of the tachyon in the open string problem. In fact, the analysis of the tachyon Lagrangian (6.2) in [51] gives rise to a picture quite reminiscent of the time evolution of matter under the influence of gravity. The nonlinearity of the equations of motion has a similar effect in both cases: initial inhomogeneities are magnified, and the matter tends to cluster in different places in space.

Summary and discussion
We have computed string scattering amplitudes in the presence of a spacelike orbifold singularity (1.1), and found divergences similar to those of [8]. We argued that these divergences are due to graviton exchange near the singularity, and that they reflect large tree level gravitational backreaction. Interestingly, divergences can be avoided for special perturbations which behave like chiral two-dimensional fields near the singularity, and we discussed the extent to which such special perturbations are natural in some cosmological and black hole models. We also briefly discussed an open string rolling tachyon model, which seems to share some features with the cosmological backgrounds studied earlier, and might help understand the backreaction near the singularity.
It would be interesting to refine and extend this work in various directions, including: (1) We have argued that for fields that are chiral near the singularity, the gravitational backreaction is milder. Indeed, only the (++) component of the Ricci tensor blows up, and four point functions are free of the usual divergences associated to the singularity.
The same is expected to be true for higher point functions. It would be interesting to find an exact CFT description of the orbifold with chiral perturbations, and study its properties.
(2) The system discussed in section 6 would be worth understanding in more detail. In particular, we proposed to think of the time evolution of inhomogeneous tachyon profiles in terms of the dynamics of a collection of D0-branes, and pointed out that this is very reminiscent of the worldlines of auxiliary massive particles used in [51] to solve the equations of motion of an effective field theory. It would be interesting to try and make this analogy more precise, and understand the late time behavior of the system.

A.1. Two point function
The two point function of two wavefunctions (2.9) is computed as follows: We assume p + 1 , p − 1 , p + 2 , p − 2 > 0. Using (2.10), and writing w 1 = w + + w, w 2 = w + − w, this becomes The first integral gives 2πδ l 1 ,l 2 if we only integrate over values of w + that cannot be identified by the action of (2.4) on the wavefunctions. (Otherwise it would give 2πδ(l 1 −l 2 ).) We finally obtain We now would like to know which region of spacetime gives the dominant contribution to We first perform the w 1 , w 2 integrals: where, as in (2.13), and appendix C, where a divergence is analyzed that comes from the singularity. In that computation, the dominant contribution comes from the region of spacetime near the singularity, and the four wedges make a comparable contribution.

A.2. Three point function
The three point function reads .
where we have used (2.10). The w 1 integral gives . (A.9) Writing w 2 = w + + w, w 3 = w + − w, this becomes (A.10) The w integral gives rise to a sum of over the roots w = ±w 0 (w 0 ≥ 0) of the argument of the delta function: We finally obtain the following expression for the three point function with m 1 > m 2 + m 3 (the three point function vanishes if m 1 < m 2 + m 3 ): Here e iφ and e iφ ′ are phases: (A.14) For later convenience, let us note that when m 1 ≈ m 2 + m 3 , the three-point function can be simplified, because in this limit, w 0 ≈ 0: To see where this non-analyticity in the masses comes from, it is useful to analyze the three point function using the explicit form of the wavefunctions in terms of Hankel functions. We have The t integral can be performed by rotating the integration contour, clockwise or counter-

Appendix B. Operator product expansion and infrared divergences
In this appendix, we compute the OPE of the operators (2.9), and indicate how it can be used to explain infrared divergences in four point functions.
We have, using (2.10), The worldsheet scaling dimension of the operator ψ m 2 ,l is (−m 2 /4, −m 2 /4) (we have set We can perform the OPE of the plane waves in (B.1) and then integrate over the w's. The leading term in the OPE is We would like to write the r.h.s. as (B.6) One can write the integral dw 1 dw 2 as d(m 2 )dwF (m 2 , w) where F is the absolute value of the Jacobian of the transformation (B.6). This Jacobian is obtained by using (B.6) to express w 1,2 as functions of m, w, and then computing the partial derivatives: This divergence is an infrared effect, and can be understood using the OPE we have just described. Assume for simplicity m 2 ≥ m 3 , and consider a process where particle 2 turns into particle 3, thereby emitting an intermediate (u-channel) particle with m 2 = (m 2 − m 3 ) 2 − ǫ. The OPE gives a factor 1/ √ ǫ from (B.7). The intermediate particle is now absorbed by particle 1, which turns into particle 4. From the OPE, this process is Now there are two possibilities: either there is a strict inequality, in which case the relevant coefficient in the second OPE is ǫ-independent; or the inequality is saturated, in which case the second OPE coefficient also scales like 1/ √ ǫ, so that the four point function has a dǫ/ǫ divergence.

Appendix C. Comparing the string theory and gravity computations
In this appendix we complete the computation in section 4 of the four-point function in dilaton-gravity, and compare it with the string theory result derived in section 3. In section 4, we computed the contribution of one of the four regions of the spacetime IR 1,1 /Z Z.
In order to obtain the contribution of the other three regions, we first need to find useful expressions for the wavefunctions (2.9) in those regions.

C.1. The wavefunctions in the four regions
We need expressions for the behavior near the singularity of the wavefunctions (2.9).
For the future Milne wedge, the result is given in (2.19). This can be extended to the other regions using the fact that (2.9) is built from purely negative frequency modes in Minkowski space, which are analytic on the lower half-planes of the complexified horizons.
However, in this appendix we will derive the explicit expressions directly in the four regions of spacetime. We label the regions of IR 1,1 /Z Z according to the sign of X + , X − as follows: I) X + , X − < 0 ; II) X + , X − > 0 ; III) X + > 0, X − < 0 ; IV) X + < 0, X − > 0 .
The wavefunction in all regions is (2.9) ψ(X + , X − )e i p· X = 1 2 √ 2πi dwe i(p + X − e −w +p − X + e w +lw) e i p· X , (C.1) with 2p + p − = m 2 . As before, we will assume that all the m's and p's are positive, and set The leading behavior of the wavefunction near the origin is This reproduces (2.19).

(C.20)
This expression is identical to the α ′ → 0 limit of the string theory result (3.16) provided we rescale where the coefficient A(l i ) is independent of the masses m i .