A Two-Dimensional String Cosmology

We study two-dimensional string theory on a time-dependent background, whose worldsheet description consists of Liouville theory at central charge c = 1 and Liouville theory at central charge c = 25, together with the conformal ghosts. We compute the tree-level three-point and four-point cosmological wavefunctions in string perturbation theory. The latter is evaluated numerically by decomposing the Liouville four-point correlation functions into Virasoro conformal blocks and three-point function coefficients and integrating over the moduli space of the four-punctured sphere string diagram. This computation numerically confirms a surprisingly simple conjectural result for the four-point wavefunction whose physical interpretation remains to be clarified.

More precisely, we investigate a closed string time-dependent background that admits an exact worldsheet conformal field theory (CFT) description.Concretely, the worldsheet CFT of the two-dimensional string cosmology model is the following where the Liouville sector at central charge c = 1, suitably continued to Lorentzian signature as explained in Section 2.3, is interpreted as time, and where the Liouville sector at central charge c = 25 is interpreted as space as in the more familiar (time-independent) two-dimensional bosonic or type 0B/0A string theories.
While in this paper we will adopt the bootstrap approach to describe the worldsheet theory (1.1) as an abstract CFT in terms of its operator spectrum and three-point function coefficients, it is useful to consider the worldsheet action of Liouville theory in order to understand the spacetime interpretation of the string theory.The action for Liouville theory is given by where ϕ is the Liouville field, R is the Ricci scalar of the two-dimensional worldsheet of the string with metric g, and µ is a constant2 .The central charge of the theory is related to the background charge Q of the linear dilaton term by c = 1 + 6Q 2 , which in turn is related to the parameter b that controls the Liouville exponential potential by Q = b + b −1 .
For the c = 25 Liouville sector of the worldsheet CFT (1.1) we have that b = 1, Q = 2.The string coupling g s is controlled by the linear dilaton term that multiplies the Ricci scalar R, and thus goes as g s ∼ e 2ϕ .Since the field ϕ is interpreted as the spatial coordinate of the two-dimensional spacetime, the strength of the string coupling varies exponentially in space.The potentially uncontrolled region of strong string coupling as ϕ → ∞, however, is shielded by the exponential "Liouville wall" 4πµe 2ϕ that reflects closed strings back to the asymptotic region of weak string coupling at ϕ → −∞.
On the other hand, for the c = 1 Liouville sector of (1.1) the relevant value for the Liouville parameter is b = i.To distinguish this c = 1 sector, let us denote its Liouville field by φ.The first immediate consequence is that Q = 0 in this case and therefore, since this c = 1 sector will be interpreted as the time coordinate of the spacetime, the string coupling g s is independent of time.Furthermore, in order to interpret the worldsheet field φ as a Lorentzian time coordinate in target spacetime we perform a Wick rotation in field space such that φ = iχ 0 , and the action becomes 3) The last term of (1.3) represents a time-dependent potential in spacetime that grows exponentially large in the infinite past χ 0 → −∞.In the far future, the potential vanishes and we recover the familiar two-dimensional bosonic string vacuum, whose c = 1 sector is a time-like free boson.
The spacetime interpretation of the two-dimensional string theory background described by the worldsheet CFT (1.1) is pictured in Figure 1.The spatial dimension parametrized by the c = 25 Liouville field ϕ runs horizontally and the time direction parametrized by the c = 1 Liouville field χ 0 runs vertically.The region shaded with a gradient in grey represents the space-like Liouville wall that shields the strong string coupling region, and the region shaded with a gradient in blue represents the time-dependent exponential potential in (1.3).
Worldsheet string perturbation theory in this case does not compute an S-matrix element of in-and out-states in the Hilbert space of perturbative string states.Instead, we propose that it computes the overlap of an initial state of the two-dimensional spacetime in the infinite past, labeled as "Big Bang" in Figure 1, with an out-state in the Hilbert space of string states -a particular component of the cosmological wavefunction in the basis of perturbative string states in the far future.
In this paper, we compute the simplest nontrivial cosmological wavefunction components in the time-dependent background (1.1), the three-and four-point wavefunction components at tree-level in string perturbation theory.We will not, however, make use of the semiclassical Liouville action (1.2).The two formal manipulations we performed in order to arrive at (1.3), namely the continuation of the Liouville parameter b → i and the rotation in field space to Lorentzian time in target space φ → iχ 0 , will be described in terms of the abstract CFT structure of c = 1 Liouville theory.
Although the string-theoretic calculation of cosmological wavefunction components is technically complicated, particularly the four-point string diagram discussed in Section 3.2, we find strikingly simple expressions for the three-point and four-point cosmological wavefunction components in terms of the outgoing energies of the asymptotic closed strings in the infinite future.The tree-level cosmological wavefunction components (3.1) and (3.11) are the main result of this paper, but both their simplicity and a more fundamental understanding of their physical interpretation remain to be clarified.We will leave this to future work.
The rest of the paper is organized as follows.In Section 2, we review the bootstrap results of Liouville CFT at c = 1 and at c = 25, and then describe the asymptotic string states (in the far future) of the worldsheet string theory (1.1).In Section 3, we present the computation of the three-and four-point cosmological wavefunction components of the two-dimensional string cosmology.We conclude in Section 4 with a discussion of our results and future directions.ϕ χ 0 "Big Bang" Figure 1: Spacetime interpretation of the two-dimensional string cosmology background.The region shaded in gray represents the space-like Liouville potential wall, whereas the region shaded in blue represents the time-dependent Liouville potential e −2χ 0 .Worldsheet string diagrams in string perturbation theory compute the overlap of an initial state in the infinite past, denoted by "Big Bang", and an out-state in the Hilbert space of perturbative string states, i.e. a particular component of the cosmological wavefunction.

Worldsheet description of two-dimensional string cosmology
In Sections 2.1 and 2.2, we describe Liouville theory at central charges c = 25 and c = 1 as abstract conformal field theories, in terms of their operator spectrum and their respective three-point function coefficients subject to the consistency conditions of crossing symmetry of the four-point function on the sphere and the modular covariance of the one-point function on the torus.In Section 2.3, we combine these ingredients to describe the time-dependent string theory background of Figure 1.

Liouville CFT at c = 25
The complete operator spectrum of Liouville theory at central charge c = 25 consists of a continuum of scalar Virasoro primaries V P , labeled by the "Liouville momentum" P ∈ R ≥0 and with conformal weights h = h = 1 + P 2 , together with its Virasoro descendants.In this paper, we will follow the normalization convention of [28] in which the vertex operators V P are delta-function normalized 3 , The three-point function coefficients, defined by (where h i ≡ 1 + P 2 i ) were bootstrapped in [29,30] and are given by the well-known DOZZ formula which takes the following form for the particular value of the central charge c = 25, 3 In this convention, the wavefunctional of the vertex operator V P in the asymptotic region ϕ → −∞ of Liouville target space is identified with the free field expression where the reflection phase S(P ) = −(Γ(2iP )/Γ(−2iP )) 2 .With this convention, there are no additional "leg-pole" factors when comparing perturbative string amplitudes in c = 1 string theory and amplitudes in the c = 1 matrix quantum mechanics [28].
where the function Υ 1 (x) is a special case of Barnes double Gamma function, and is defined as and where Γ 1 (x) is related to the Barnes G-function G(x) by Γ 1 (x) = (2π) (x−1)/2 (G(x)) −1 [31].Υ 1 (x) is an entire function with zeros at x = n with multiplicity (n − 1) for n ∈ Z ≥2 and at x = −m with multiplicity (m + 1) for m ∈ Z ≥0 .Note that the three-point function coefficient (2.4) is real-valued for real Liouville momenta P i .
The four-point function of c = 25 Liouville vertex operators on the sphere admits a decomposition in terms of the three-point function coefficients (2.4) and c = 25 Virasoro conformal blocks as ) where F c (h 4 , h 3 , h 2 , h 1 ; h|z) is the sphere four-point holomorphic Virasoro conformal block at central charge c with external weights h i for i = 1, . . ., 4, intermediate weight h, evaluated at the cross-ratio z.Similarly, the one-point function on the torus with modulus τ can be decomposed into Virasoro conformal blocks as C(P ext , P, P )F c=25 (h ext ; 1 + P 2 ; q)F c=25 (h ext ; 1 + P 2 ; q), (2.7) where F c (h ext ; h int ; q) is the torus one-point Virasoro conformal block at central charge c with external weight h ext , internal weight h int , and q = e 2πiτ .Our conventions for the sphere and torus Virasoro conformal blocks are given in Appendix B. In Appendix A, we verify numerically the crossing symmetry of sphere four-point function (2.6) and the modular covariance of torus one-point function (2.7).

Liouville CFT at c = 1
The CFT data of Liouville theory at c = 1 is similar to that of c = 25 presented in the previous section.Naively, Liouville theory at central charge c ≤ 1 is obtained by the continuation of the Liouville parameter b to imaginary values b = iβ, with β ∈ R. The analytic continuation of b to complex values, however, fails precisely along the imaginary axis.For instance, the upsilon function Υ b (x) in terms of which the structure constants are written diverges in this limit [32].Consequently, the bootstrap of the structure constants along this branch of b = iβ has to be reanalyzed.This has been done in [32][33][34][35] 4 , and we will present the result for the particular case of c = 1 (β = 1).
The operator spectrum of Liouville CFT at c = 1 again consists of a continuum of scalar Virasoro primaries V P , labeled by a momentum P ∈ R and now with conformal weights h = h = P 2 , together with its Virasoro descendants. 5We will also assume the vertex operators V P are delta-function normalized,6 In turn, the three-point function coefficients that appear in are given by Note that (2.10) is real-valued for real or purely imaginary Liouville momenta P i ∈ R, iR, and that (2.10) is an even function of any of the momenta P i .
The four-point function of Liouville vertex operators in c = 1 Liouville CFT admits the following Virasoro conformal block decomposition, where the contour of integration C over the intermediate state with Liouville momentum P is specified as follows.In contrast with the DOZZ structure constant (2.4), C(P 1 , P 2 , P ) viewed as a function over the complex P -plane only has poles (not necessarily simple) at ) is chosen to run parallel to the real axis and shifted vertically by a small ϵ > 0 amount to avoid the poles as shown in Figure 2 [32].In Appendix A, we verify numerically that the four-point function (2.11) satisfies crossing symmetry with this contour prescription.(2.11) in Liouville CFT at c = 1.Poles of the P -integrand are marked with crosses.The contour C runs parallel to the real axis and shifted vertically by a small ϵ > 0 amount in the imaginary direction in order to avoid the poles.The integration over intermediate states in the Virasoro conformal block decomposition of the torus one-point function (2.12) has the same pole structure, without counting multiplicities, and hence we use the same contour prescription.
Likewise, the one-point function on a torus with modulus τ in c = 1 Liouville CFT admits the conformal block decomposition, C(P ext , P , P )F c=1 ( h ext ; P 2 ; q)F c=1 ( h ext ; P 2 ; q).(2.12) In particular, the poles of the integrand on the RHS of (2.12) coming from C(P ext , P , P ) again occur at P = m 2 for m ∈ Z ̸ =0 , whereas those coming from the torus one-point conformal block F c=1 ( h ext ; P 2 ; q) occur at P = n 2 for n ∈ Z.Therefore, the integrand on the RHS of (2.12) has the same pole structure as shown in Figure 2 (not counting multiplicities of the poles), and the same contour prescription C is used in (2.12) as well.In Appendix A, we verify numerically that the torus one-point function (2.12) with the contour prescription shown in Figure 2 is modular covariant. 7

Two-dimensional string cosmology
The full worldsheet CFT description of the two-dimensional string cosmology background consists of the Liouville CFTs at c = 1 and c = 25, together with the b, c conformal ghosts.
The Liouville CFT at central charge c = 1 described in Section 2.2 describes a theory in Euclidean signature.In order to have a description of the time-dependent background in Lorentzian signature shown in Figure 1, we will make one further modification to the worldsheet CFT.We will analytically continue the c = 1 Euclidean Liouville momenta P of external on-shell string states to imaginary values, such that ω ∈ R ≥0 has the interpretation of a Lorentzian energy of the string asymptotic state.With this continuation of the c = 1 Liouville CFT sector, on-shell asymptotic states in the two-dimensional string cosmology background corresponding to BRST cohomology classes of the full worldsheet CFT are represented by the following vertex operators, As in other (time-independent) two-dimensional string theories, the on-shell condition that V ω is a weight (1, 1) vertex operator leads to a dispersion relation of a massless scalar in 1 + 1 dimensions: −ω 2 + P 2 = 0 where the spatial c = 25 Liouville momentum takes the value P = ω.
The continuation (2.13) in c = 1 Liouville CFT can be justified at the level of the sphere four-point function as follows.Consider again the Virasoro conformal block decomposition in (2.11).We want to continue the external c = 1 Liouville momenta P i → −iω i for i = 1, . . ., 4, while maintaining analyticity of the CFT four-point function, deforming the contour of integration C over the intermediate state with momentum P accordingly, if necessary.The key property of the c = 1 structure constant C(P 1 , P 2 , P ) is that the positions of its poles in the complex P -plane are independent of the values of the external momenta P 1 and P 2 , and hence the analytic continuation (2.13) to imaginary external c = 1 momenta is trivial8,9 [39].Similarly, at the level of the torus one-point function (2.12), since the poles in the P -integrand coming from the three-point coefficient C(P ext , P , P ) and from the torus one-point conformal blocks are independent of the external c = 1 Liouville momenta P ext , the continuation to an imaginary value is trivial as well.
Although clear from this discussion, in Appendix A we verify numerically that the sphere four-point function (2.11) and the torus one-point function (2.12) for imaginary external Liouville momenta continue to satisfy crossing symmetry and modular covariance, respectively.These results show that the continuation (2.13) to Lorentzian energies is well-defined for any n-point function on all oriented Riemann surfaces in c = 1 Liouville CFT.
The spacetime interpretation of the string theory background (1.1), with the continuation to Lorentzian signature described above, is depicted in Figure 1.Our proposal is that worldsheet string perturbation theory computes a component of the cosmological wavefunction defined as the overlap between the initial state of the universe (denoted by "Big Bang") with an asymptotic out-state in the Hilbert space of perturbative string states, whose one-particle vertex operator representatives are given by (2.14), The nature of the initial state of the universe is tied to the specific worldsheet CFT background under consideration.At present, it appears that string perturbation theory, starting from the background (1.1), is insufficient to explore different possibilities for this initial state.Our goal in this paper will be more modest, and we will study components of the cosmological wavefunction corresponding to the initial state of the universe implicitly defined by the specific worldsheet CFT (1.1).

Cosmological wavefunction components in string perturbation theory
In this section, we compute the three-and four-point components of the cosmological wavefunction at tree level in string perturbation theory.

Three-point string diagram
The first nontrivial cosmological wavefunction component computed in string perturbation theory is the tree-level three-point wavefunction component.Fixing the position of the three vertex operators, the three-punctured sphere diagram has no remaining moduli and is evaluated in terms of the three-point coefficients of the matter sector reviewed in the previous section.We obtain that, ) where C S 2 is the normalization constant associated with the sphere topology.In timeindependent string perturbation theory, perturbative unitarity (factorization of string scattering amplitudes) fixes the value of C S 2 and is proportional to g −2 s .In this paper, since at present we do not know the precise implications of unitarity on cosmological wavefunctions we will not fix C S 2 and leave it as an undetermined constant.
It is interesting to compare the three-point cosmological wavefunction component (3.1) to the 1 → 2 decay S-matrix element in c = 1 string theory [28], which is also proportional to the product of the external energies.In c = 1 string theory, whose matter sector consists of a time-like free boson and c = 25 Liouville theory only, energy conservation is crucial in order for the DOZZ structure constant C(ω, ω 1 , ω 2 ) to reduce to the simple product of the energies.In the two-dimensional string cosmology background the energies ω i for i = 1, 2, 3 are completely arbitrary, yet after multiplying the structure constants of both the c = 1 and c = 25 Liouville CFTs the wavefunction still simplifies drastically to the result (3.1).

Four-point string diagram
Next, we compute the four-point cosmological wavefunction component in two-dimensional string cosmology.This is computed by the four-punctured sphere diagram which, after fixing the positions of three vertex operator insertions, has one remaining modulus corresponding to the position of the last vertex operator in the complex plane.The four-point wavefunction component takes the form where the Liouville CFT four-point functions at c = 25 and c = 1 admit the Virasoro conformal block decompositions in (2.6) and (2.11), repectively.
The moduli integral on the RHS of (3.3) possibly diverges in regions of moduli space where the vertex operators collide, as is familiar in string perturbation theory.For instance, near z = 0 the moduli integral of (3.3) takes the form (3.4) where we have expanded both the c = 1 and c = 25 Virasoro conformal blocks to leading order in the small z expansion using F c (h i ; h|z) ≃ z h−h 1 −h 2 , in our conventions.For small z, the integrals over the intermediate Liouville momenta P and P are dominated by their values near P = 0 and P = 0. Using Laplace's method to approximate these integrals, we obtain that the moduli integral as z → 0 has the following behavior, and is therefore convergent.The limits as z approaches 1 and infinity lead to same type of convergent behavior (3.5), as can be easily seen by first using the crossing relations (A.3) and (A.4) to map each point to 0, respectively, and expanding the conformal blocks to leading order.
In this paper, however, as detailed in Appendix C we will follow closely the numerical integration strategy presented in [28,31,40].In particular, we will switch the order of integrations and perform the moduli integral over z first, and the integrals over the intermediate Liouville momenta P and P last, which was a more convenient strategy of numerical integration in the computation of closed string scattering amplitudes in c = 1 string theory [28].Following this order of integrations, and parametrizing the contour of integration C of Figure 2 by P = p + iϵ for p ∈ R and small ϵ > 0, we observe from (3.4) that the four-point wavefunction component (3.3) has a power divergence in z whenever P 2 + p 2 < ϵ 2 .The region in the (P, p) plane for which we have a divergence (and the divergence itself) can be made arbitrarily small as we take ϵ → 0. In this sense, the contour prescription depicted in Figure 2 serves as an "iϵ prescription" to regulate the cosmological wavefunction component (3.3), when performing the z moduli integral first.In practice, we will compute the four-point cosmological wavefunction component numerically for a small but fixed value of ϵ, and hence we will need to properly regularize these divergences.We will do so following the procedure employed in [28,31,40] of counterterm subtraction to the moduli integral.The counterterms that remove the power divergences in the moduli z-integral are The fully regularized four-point cosmological wavefunction in two-dimensional string cosmology is given by, (3.7) In the direct numerical evaluation of the regularized cosmological wavefunction component (3.7) we will make the following choices for the outgoing energies of the asymptotic closed string states: (ii) and (v) (3.9) The strategy we employ in the numerical calculation of the four-point cosmological wavefunction component follows [28].We compute the four-point Virasoro conformal blocks numerically using Zamolodchikov's recursion relations [41], reviewed in Appendix B, truncated to a sufficiently high order in the elliptic nome q series expansion, which is related to the cross-ratio z by where K(z) = 2 F 1 (1/2, 1/2; 1|z) is the complete elliptic integral of the first kind.In order to obtain accurate numerical results with truncated conformal blocks, we first use crossing symmetry of the c = 1 and c = 25 Liouville four-point functions to reduce the moduli z-integration to a finite domain near the origin of the q-disc, or of the z-plane.The explicit expression for the four-point cosmological wavefunction component that we compute numerically is given by (C.1).Further details of the calculation are given in Appendix C.
The numerical results for the four-point cosmological wavefunction component (3.7) for the choices of outgoing closed string energies (3.8) and (3.9), following the strategy outlined in Appendix C, are shown in Figures 3 and 4. We find to a high level of accuracy that a good fit to the numerical results is given by10 , with α = 7.513 ± 0.001, β = 15.99 ± 0.01. (3.11) Note added: A more precise calculation has been performed in [42], resulting in the values α = 8 and β = 16 in the conventions of the present paper.
Figures 3 and 4 show numerical results in pink dots and the fit (3.11) in a solid blue curve.The discrepancy between the numerical evaluation of (C.1) and the fit (3.11) is at most 0.2%, but is typically much smaller than that.Further discussion of the possible sources of error in our numerical calculation of the regularized four-point cosmological wavefunction component is given in Appendix C.
The wavefunction component (3.11) is the main result of this paper.The fact that the three-point component (3.1) and the four-point component (3.11) of the cosmological wavefunction take a very simple form is indicative of a dual description in terms of a matrix quantum mechanics, as is the case for the bosonic and type 0B two-dimensional string theories.We hope that our result for the cosmological four-point wavefunction component (3.11) serves to identify the precise dual description of the cosmological background (1.1).

Discussion
In this paper, we have studied in detail the time-dependent closed string background defined by the worldsheet CFT (1.1), after analytic continuation of the c = 1 Liouville CFT to Lorentzian signature.
In contrast to earlier work in the literature in the context of cosmological backgrounds in two-dimensional string theory 11 , the main novelty of the present work is that the bootstrap solution of c = 1 Liouville theory as a conformal field theory allowed for a description of the two-dimensional cosmological background that is exact in the string scale α ′ (which was set  to α ′ = 1 throughout this paper).Hence, we were able to directly study components of the cosmological wavefunction using string perturbation theory.The main results of this paper are the explicit calculation of the tree-level three-point cosmological wavefunction component (3.1) and the conjectural result (3.11), verified numerically, for the four-point wavefunction component.
The simplicity of the results (3.1) and (3.11) is suggestive of the existence of a dual matrix quantum mechanics (MQM) description.Indeed, several time-dependent solutions of the dual MQM, which describe a time-dependent Fermi surface, have been studied in [52][53][54] purely from the matrix side of the duality.One of the most interesting future directions is to identify the correct MQM dual to the string cosmological background described in this paper.This would provide an example of an exact holographic duality of a string theory in a cosmological background, a subject for which we have limited understanding and examples are scarce. 12nother important question is to understand the nature of the initial state of the twodimensional universe, which appears to be tied to the specific background (1.1).The classical worldsheet action (1.3) of the time coordinate χ 0 has an exponentially large potential in the infinite past.This suggests that the worldsheet of the string is prevented from going into the infinite past, and thus the initial state of the two-dimensional string universe looks empty.This property looks reminiscent of the Hartle-Hawking state. 13Starting from the time-independent two-dimensional bosonic ("c = 1 string theory") background, whose c = 1 sector is a time-like free boson, a generic marginal deformation in general may correspond to a distinct initial state of the universe.Such deformations may not lead to an integrable worldsheet conformal field theory (and perhaps even to a local worldsheet theory).A special property of the initial state of the cosmological background studied in this paper is that it corresponds to a solvable worldsheet CFT.
Another interesting direction is to understand more generally the implications of unitarity for string theory in cosmological spacetimes.In time-independent asymptotically flat spacetimes, unitarity is directly manifested as the unitarity of the string S-matrix.For example, string amplitudes in string perturbation theory satisfy unitary factorization into lower order amplitudes [58].At present, we do not have a complete non-perturbative understanding of the implications of unitarity on cosmological wavefunctions (see [59,60] for a recent survey of research in this direction).Indeed, one motivation for the present work was to discover a simple example in string theory, hence ultra-violet complete, where the implications of unitarity for quantum field theory in cosmological spacetimes can be studied in detail.[61,62].

A.1 Crossing symmetry of the sphere four-point function
The first consistency condition on Liouville CFT is the crossing symmetry of the four-point function on the sphere.Our convention for the sphere four-point Virasoro conformal block decomposition of four scalar Virasoro primaries is such that where z = z 12 z 34 z 13 z 24 is the cross-ratio, h i for i = 1, . . ., 4 are the external weights, and h is the internal weight.For Liouville CFT at c = 25 and at c = 1, we define 2) where C is the contour depicted in Figure 2, h i = 1 + P 2 i , and h i = P 2 i .The crossing relation obtained by exchanging operators 1 ↔ 3 takes the form This relation is obeyed at the level of individual Virasoro conformal blocks and hence it is not sensitive to the structure constants (2.4) and (2.10) of Liouville CFT at c = 25 and c = 1, respectively.On the other hand, the crossing relation obtained by exchanging operators 2 ↔ 3 takes the form and is only satisfied after integration over the intermediate states against the three-point function coefficients.
Although clear from the analytic structure of the P -integrand of G(4321|z) as discussed in Section 2.2, we have also checked that the crossing relations (A.3) and (A.4) continue to hold for imaginary values of the external c = 1 Liouville momenta, P i = −iω i for i = 1, . . ., 4.
Figure 5 shows a sample verification of the crossing symmetry relation (A.4) in c = 1 Liouville CFT, for a purely real and a purely imaginary assignment of the external c = 1 Liouville momenta P i .For a range of positive real values for the cross-ratio z, the "crossed" channel 13 → 24 calculated with increasing truncation order in the q expansion of the conformal blocks (red to blue) can be seen to converge to the "direct" channel 12 → 34 result shown in black. ) Figure 5: Numerical test of the crossing symmetry of ⟨ V P 4 (∞) V P 3 (1) V P 2 (z, z) V P 1 (0)⟩ c=1 Liouv.for a range of real values of the cross-ratio z, at external c = 1 Liouville momenta (i) (P 1 , P 2 , P 3 , P 4 ) = ( 1 3 , 1 4 , 1 5 , 3  7 ) and (ii) (P 1 , P 2 , P 3 , P 4 ) = (− i 3 , − i 4 , − i 5 , − 3i 7 ).The crossed channel (RHS of (A.4)) computed with the sphere four-point conformal blocks truncated to order q L is shown with a color scheme from red to blue for increasing L from 2 to 8. The direct channel (LHS of (A.4)) computed with conformal blocks truncated to order q 8 is shown in black.Data points are joined with straight lines for visualization.

A.2 Modular covariance of the torus one-point function
The second consistency condition on Liouville CFT is the modular covariance of the onepoint function of a Liouville vertex operator V Pext with external momentum P ext on the torus with modulus τ , ) and (2.12), written again here for convenience = ∞ 0 dP π C(P ext , P, P )F c=25 (h ext ; 1 + P 2 ; q)F c=25 (h ext ; 1 + P 2 ; q), V Pext (0) Figure 6 shows a sample verification of the modular covariance (A.6) of the torus onepoint function in c = 1 Liouville CFT, for a real and a purely imaginary value of the external Liouville momenta.For a range of τ with increasing imaginary values, the "S-transformed" channel (LHS of (A.6)) computed with increasing truncation order in the q expansion of the conformal blocks (red to blue) can be seen to converge to the "direct" channel result (RHS of (A.6)) shown in black.

B Recursive representations for the sphere 4-point and torus 1-point Virasoro conformal blocks
In this section we provide the recursion relations that we use to efficiently compute the sphere four-point and the torus one-point Virasoro conformal blocks, originally derived in [41] and in [63,64], respectively.
We parametrize the central charge of the Virasoro algebra as c = 1+6Q 2 with Q = b+b −1 , and the holomorphic weights of external primaries as (i) for a range of values of τ with Re τ = 1 3 and varying Im τ , at external c = 1 Liouville momenta (i) P ext = 1 3 and (ii) P ext = − i 2 .The S-transformed channel (LHS of (A.6)) computed with the torus onepoint conformal blocks truncated to order q L is shown with a color scheme from red to blue for increasing L from 2 to 8. The direct channel (RHS of (A.6)) computed with conformal blocks truncated to order q 8 is shown in black.Data points are joined with straight lines for visualization.
We also define h r,s = 1 4 The holomorphic sphere four-point Virasoro conformal block F c (h i ; h|z) with i = 1, . . ., 4, defined in (A.1), can be expressed as [41] F 3) where θ 3 (q) is a Jacobi theta function, and the elliptic nome q is related to the cross-ratio z by The so-called "elliptic conformal block" H c (h i ; h|q) admits a power series expansion in q and satisfies the following recursion relation, where the "fusion polynomials" P r,s are given by where the notation p ∈ {1 − r, r − 1, 2} stands for p ranging from 1 − r to r − 1 with step 2, and λ i are related to the external weights h i by (B.1).
Similarly, the holomorphic tours one-point Virasoro conformal block F c (h ext ; h int |q) can be expressed as [63] where now q is related to the torus modulus τ by q = e 2πiτ .The elliptic conformal block H c (h ext ; h int |q) admits a power series expansion in q and obeys the recursion relation, q rs A r,s P r,s (h ext , h r,s + rs)P r,s (h ext , h r,s ) (B.8) where again the arguments of the fusion polynomials P r,s are related to the quantities λ appearing in (B.6) by (B.1).In this case, the product of the fusion polynomials may be written as P r,s (h ext , h r,s + rs)P r,s (h ext , h r,s ) = k∈{1,2r−1,2} l∈{1,2s−1,2} (B.9) A Mathematica notebook implementing these recursions relations is attached to this paper.

C Details of the numerical evaluation of the four-point cosmological wavefunction component
In this section, we describe the strategy we employ for the numerical evaluation of the regularized four-point cosmological wavefunction component (3.7).It follows closely that of the four-point scattering amplitude in c = 1 string theory of [28].
First, we make use of the crossing symmetry relations generated by (A.3) and (A.4) of the four-point correlation functions in Liouville CFT at c = 1 and c = 25 in order to reduce the moduli z-integration over the complex plane C to a finite domain near z = 0 as follows (see Appendix C.2 of [65]).We divide the complex plane C into six regions:   For the numerical evaluation of (C.1), we further cut out a disc of small radius δ > 0 around z = 0, D δ ≡ {|z| ≤ δ}.Inside the region D δ ∩ I, we truncate the Virasoro conformal blocks to next-to-leading order in z and perform the z-integral analytically, including the regulator counterterms.Outside this cut-out disc, for D δ ∩ I we perform the z-integral numerically with the conformal blocks truncated to a sufficiently high order in the q-recursion expansion.The results shown in Figures 3 and 4 were computed with conformal blocks truncated to order q 8 , and with a choice of δ = 10 −2 .
As mentioned in the main text, in the numerical evaluation of (C.1) we switch the order of integrations, following [28].We first perform the moduli z-integral over region (I) with the strategy described above for a fixed value of the c = 1 and c = 25 intermediate Liouville momenta P and P .Second, we perform the c = 1 Liouville P -integral over the contour C parametrized by P = p + iϵ; the results reported in Figures 3 and 4 were computed with ϵ = 10 −1 .Lastly, we perform the c = 25 Liouville P -integral.Sample plots of (i) the P -integrand and (ii) the P -integrand are shown in Figure 7.The discontinuity in the Pintegrand visible in Figure 7 (ii) is due to the regulator counterterms that are only nonzero for P ≤ ϵ = 10 −1 .and ω 4 = 2 5 .(i) shows the P -integrand of (C.1) for a fixed value of P = 1 2 after having performed the z-integration over region (I) with the strategy outlined above with δ = 10 −2 .(ii) shows the P -integrand after having performed both the z moduli integration and the P integration over the contour C parametrized by P = p + iϵ with ϵ = 10 −1 .
As we approach the limit of c → 25 and c → 1, the recursion relations for the Virasoro conformal blocks involve delicate cancelations.In order to avoid loss of precision when computing the conformal blocks recursively, we work with sufficiently close values (but not exactly equal) to the Liouville central charges.For the results reported in Figures 3 and 4, we set c = 0.995 and c = 25.00001.In fact, this appears to be the largest source of numerical error in our calculation of (C.1).
To a lesser extent, another source of error is the numerical integration in the cross-ratio z over region (I), and the numerical interpolation and subsequent integration over the Liouville intermediate momenta P and P , as in the sample plots in Figure 7.In addition, for larger values of the outgoing closed strings energies ω i , both Liouville momenta integrations have wider non-vanishing support.The results presented in Figures 3 and 4 were calculated by integrating up to a maximum value of |p| = 2 and P = 2.5, which could be another source of small error for larger values of the outgoing energies ω i .

Figure 2 :
Figure 2: Contour of integration C over the intermediate states in the Virasoro conformal block decomposition of the four-point function(2.11)  in Liouville CFT at c = 1.Poles of the P -integrand are marked with crosses.The contour C runs parallel to the real axis and shifted vertically by a small ϵ > 0 amount in the imaginary direction in order to avoid the poles.The integration over intermediate states in the Virasoro conformal block decomposition of the torus one-point function (2.12) has the same pole structure, without counting multiplicities, and hence we use the same contour prescription.

6 ( 4 )Figure 3 :
Figure 3: Shown in dots are the numerical results for the regularized four-point cosmological wavefunction component in two-dimensional string cosmology, explicitly given by (C.1), with the energy assignment (3.8) for the outgoing asymptotic closed string states.The fit (3.11) is shown in the solid curve.

Figure 4 :
Figure 4: Shown in dots are the numerical results for the regularized four-point cosmological wavefunction component in two-dimensional string cosmology, explicitly given by (C.1), with the energy assignment (3.9) for the outgoing asymptotic closed string states.The fit (3.11) is shown in the solid curve.

T 2
(τ ) , (A.6) where h ext = h ext denote the conformal weights of the Liouville operator V Pext .The first relation (A.5) is a property of the torus one-point conformal blocks and thus insensitive to the Liouville three-point function coefficients.The second relation (A.6) can be verified numerically in both c = 25 and c = 1 Liouville CFT by expanding the torus one-point function in terms of Virasoro conformal blocks as in (2.7 ext , P , P )F c=1 ( h ext ; P 2 |q)F c=1 ( h ext ; P 2 |q), (A.7) where C is the contour depicted in Figure 2. Using the recursion relations for the torus one-point Virasoro conformal blocks reviewed in Appendix B, we verified the relation (A.6) for various values of the external Liouville momenta P ext and of the torus modulus τ in both c = 1 and c = 25 Liouville CFT.Furthermore, and although clear from the analytic structure of the P -integrand of the torus one-point function (2.12) as discussed in Section 2.2, we have also verified modular covariance (A.6) for imaginary values of the external c = 1 Liouville momenta, P ext = −iω.