Supercritical String Cosmology drains the Swampland

The First and Second Swampland Conjectures (FSC&SSC) are substantially modified in non-critical string cosmology, in which cosmic time is identified with the time-like Liouville mode of the supercritical string. In this scenario the Friedmann equation receives additional contributions due to the non-criticality of the string. These are potentially important when one seeks to apply the Bousso bound for the entropy of states that may become light as the dilaton takes on trans-Planckian values, as in a de Sitter phase, and restore consistency with the FSC and in at least some cases also the SSC. The weak gravity conjecture (WGC) for scalar potentials is saturated in the supercritical string scenarios discussed in this work, but only if one uses the dilaton as appears in the string effective action, with a kinetic term that is not canonically normalised. In the case of a non-critical Starobinsky potential, the WGC is satisfied by both the canonically-normalised dilaton and the dilaton used in the string effective action.


I. INTRODUCTION
One of the outstanding challenges for string theory is to identify experimental measurements that could test some characteristic prediction derived from the theory. Confronted with the tremendous gap between the primary formulation of string theory in multiple dimensions at a distance scale beyond the direct reach of observation, the conventional approach to this experimental challenge has been to formulate and probe some effective field theory that describes four-dimensional physics at accessible energy scales that is derived from (or at least motivated by) string theory. Unfortunately, this approach has long been stymied by the enormous ambiguity in the choice of possible effective field theory.
A promising way forward has been provided by the swampland conjectures [1][2][3][4][5], which postulate some very general properties of the possible effective field theory, whose violation would lead to the conclusion that (our current understanding of) string theory does not describe Nature. This line of attack is particularly intriguing because some aspects of the swampland conjectures appear to be in tension with the appearance of a cosmological constant in present-day cosmology as well as the commonly-studied models of inflation in the early Universe, and may raise questions about the formulation of spontaneous symmetry breaking in the Standard Model of particle physics [6].
One possible way to resolve these issues may be to abandon the conventional effective field theory description of physics in the infra-red limit at energies far below the putative string scale. Effective field theory emerges naturally from critical string theory via its formulation in terms of theory on the string world sheet. We have long argued that conformal field theory is just one aspect of the underlying string theory, and that critical string theory should be regarded as embedded in a broader framework that includes supercritical string scenarios in which there is a time-like Liouville mode that can be identified with cosmic time in scenarios for an evolving Universe [7][8][9][10][11].
As we have discussed previously, the non-trivial dynamics of such a Liouville mode modifies the equations of motion of a conventional effective field theory. In particular, as we emphasise in this paper, in the cosmological context the supercritical dynamics introduces extra terms into the Friedmann equation describing the evolution of the Universe, reopening the possibility of a stringy description of inflation in the past and/or dark energy today.
We recall in this context that an essential feature of perturbative critical string theory is the existence of a perturbative S-matrix. This requirement appears to exclude a de Sitter background, since it does not accommodate asymptotic states because of the presence of a horizon [12]. On the other hand [8,11,13], supercritical strings are formulated without requiring the existence of an S-matrix [13], and hence can accommodate both an early nearde Sitter epoch from which the Universe exits smoothly and another near-de Sitter epoch at late times. 1 The outline of this paper is as follows. In Section II we outline relevant aspects of the swampland conjectures (SC). Then, in Section III we examine the swampland from the point of view of non-critical string theory. Section IV provides a parallel discussion of the weak gravity conjecture (WGC) in the framework of non-critical string theory.

II. THE SWAMPLAND CONJECTURES
There are several Swampland Conjectures (SC) that bear upon the possible embedding of a de Sitter (dS) solution in a low-energy effective field theory arising from a microscopic string theory model that is supposed to provide a self consistent UV completion.
• The Distance in Field Space Conjecture (DFSC) [1] When considering a scalar field φ within the effective low-energy field theory (EFT) of some string theory, which is displaced by a 'distance' ∆φ in field space from some initial value, the validity of the EFT is guaranteed provided where c 1 is a positive constant of order O(1), and κ = 1 M Pl is the gravitational constant in four space-time dimensions, where M Pl = 2.4 × 10 18 GeV is the reduced Planck mass. In general, when (1) is violated towers of string states descend from the UV, becoming light and thus 'contaminating' the EFT, jeopardising any physical conclusions that may be derived from it.
• The First Swampland Conjecture (FSC) [2] 1 In some supercritical string cosmologies [11], the Universe approaches at large cosmic times a linearlyexpanding universe [14] that is characterised by a logarithmic dependence of the dilaton on cosmic time, in which case the perturbative S-matrix can be well-defined asymptotically [13].
This conjecture restricts the self-interactions of scalar field(s) in an EFT stemming from string theory via the gradient of the scalar potential V in field space, so that the EFT is valid: where c 2 is a positive (dimensionless) constant of O(1). The gradient in field space is in the multicomponent space of scalar fields φ i , i = 1, . . . N that the EFT contains.
The constraint (2) can be derived from the DFSC (1), using entropy in de Sitter space [3], by using the Bousso bound [15], which is in this case saturated by the Gibbons-Hawking entropy [16]. Specifically, let us consider an EFT of a quintessence field φ(t) in which, say, the latter increases with cosmic time, i.e., φ(t) hasφ > 0 2 . As the time evolves, if the field lies in regions in which (2) is violated, towers of string states with masses m ∼ exp(−a|∆φ|), where a > 0 is a constant with mass dimension −1 that depends on the details of the microscopic string theory, become lighter and lighter and descend from the UV-complete theory to contaminate the EFT as the 'distance' ∆φ from the initial point increases.
The number of these effective light degrees of freedom N (φ) depends on the value of the field φ at any moment in time, and can be parametrised as [2]: where n(φ) indicates the number of states in the tower that are becoming light, and b, like a, is another positive constant that depends on the mass gap and other details of the underlying string theory. Since, according to the distance conjecture, more states become light as φ grows, one has to take the (positive) function n(φ) > 0 to be monotonically increasing with φ, i.e. dn(φ) dφ > 0. In an accelerating Universe, characterised by a Hubble parameter H, which for (near-)de Sitter-times is (approximately) constant, the entropy of this tower of string states increases with the Hubble horizon 1/H as where γ, δ > 0. On the basis of string examples, the authors of [3] argued that 0 < δ ≤ 2. If the tower of string states that are becoming light behave effectively as point particles, then δ = 0, but this is not necessarily the case, and the value of δ depends on the underlying microscopic string theory.
Since the expanding Universe is characterised by the presence of the Hubble horizon of area A = 4πH −2 , the entropy is bounded by Bousso's covariant entropy constraint applied to a cosmological background [15]: S string states (N, H −1 ) ≤ 1 4 A, which is nothing other than the Gibbons-Hawking entropy [16]: from which we derive If the potential energy of the scalar field dominates the energy density of the Universe, and one assumes the standard Friedmann equation i.e., the kinetic energy of the scalar field responsible for inflation is ignored, compared to the potential energy, then we can use (7) to rewrite (6) as After some straightforward manipulations, and taking into account the fact that the derivative of the potential with respect to the field φ is negative 3 , we find that where the prime denotes the derivative with respect to the field φ. Using (3), one then where the last inequality follows from the fact that n > 0. Comparing (9) with (2), we obtain the FSC with c 2 = 2 b γ 2−δ > 0, provided that the parameters b , γ and δ are such that c 2 is of order O(1). 4 3 Here we adopt the conventions of, e.g., the work in [17], as we are interested in the flow of the field from large values, where the de Sitter phase occurs, to small values, where the exit from inflation takes place. 4 As already remarked, δ = 0 if only point-like states are included in the tower of states N (φ).
In fact, it was argued in [3,4] that the FSC does not necessarily hold, but that consistency of the EFT requires either the constraint (2), or : • The Second Swampland Conjecture (SSC) [3,4] According to the SSC, the minimum eigenvalue of the Hessian in theory space min(∇ i ∇ j V ), with the notation ∇ i denoting the gradient of the potential V with respect to the scalar field φ i , should satisfy the following constraint near the local maximum of the potential V , provided there is one: where c 3 > 0 is an appropriate positive (dimensionless) constant that is O(1).
We note that, for single-field inflationary models, the condition (11) would also be incompatible with the smallness in magnitude of the second of the slow-roll parameters η, as required for conventional slow-roll inflationary models.
We also remark that in case of models for which the SSC applies but not the FSC, the entropy-bound-based derivation of FSC still holds, but for a range of fields further away from the regime for which the local maximum of the potential occurs (as required for the implementation of the SSC). The critical value (range) of the field magnitude for the entropy-bound implementation of the FSC depends on the underlying microscopic model, upon which the parameters γ, b in the bound (10) also depend.

III. SUPERCRITICAL STRING THEORY AND THE SWAMPLAND
The central point of our discussion is the modification of the First Swampland Conjecture (FSC) (2) in supercritical string theory, in which the right-hand-side (r.h.s.) of the Friedmann equation of motion (7) receives additional contributions due to non-critical string degrees of freedom [8,9], which arise from identifying time as the Liouville mode of the supercritical string [10]. To see this, we consider a concrete but representative example of a supercritical string cosmology where the Liouville mode is identified with cosmic time, and the dilaton plays the role of a quintessence field. The scenario is discussed in detail in [9], and here we simply outline the main results relevant for our purposes.

A. The Supercritical String Cosmology Approach
Before going into calculational details, it is important to recall briefly the underlying philosophy of the non-critical approach to string cosmology of [8]. This necessarily involves infinite towers of stringy states that fail to decouple from the light degrees of freedom.
In the approach of [8] these string states provide a non-trivial environment for the lowenergy effective field theory (EFT), whose description incorporates only local field theory modes. When the string is non-critical, the world-sheet Weyl-anomaly coefficients, which appear as the renormalization-group β-functions of the couplings/target-space fields g i of the σ-model that describes string propagation, are no longer zero as in a conformal field theory corresponding to a critical string model. Instead, they are determined by a standard Liouville-dressing procedure: 5 where the g i are Liouville-dressed couplings/background fields in the σ-model, which has d + 1 coordinates, of which d are space-like coordinates in target space, and one is time-like and interpreted as the target time. The prime in (12) denotes differentiation with respect to the world-sheet zero-mode ρ of the Liouville field ρ(σ), where σ denotes collectively the dependence on world-sheet coordinates, which is identified with the target time in the approach of [8]. The Liouville mode itself is viewed as a dynamical renormalization-group scale, which is local on the world-sheet and promoted to a fully-fledged quantum field on the two-dimensional world-sheet geometry. The right-hand side of (12) vanishes in critical string theories, and this absence provides the standard target-space equations of motion of local fields in a low-energy string EFT. The identification with the target time of this extra coordinate in the target space of the string, which in supercritical strings has a time-like signature, imposes a constraint on the (d + 2)-dimensional target manifold, such that there is a single physical time. 5 For notation, conventions and more details we refer the reader to [8] and references therein.
A prototype that illustrates how towers of string states may affect the Weyl-invariance conditions and leading to Liouville dressing is the cigar-type black-hole background in twodimensional string theory [19], which may be embedded in higher dimensions [20]. As observed in [7], such a stringy background leads to the mixing with light modes of infinite towers of topological modes of (non-propagating) massive string states with discrete momenta. If they are not taken into account, the truncated σ-model that includes only backgrounds from the massless string states (the graviton and dilaton in this example) is no longer conformally invariant, rendering the string non-critical. Liouville dressing is then required to restore world-sheet conformal invariance, and thus independence of the targetspace physics from world-sheet dynamics. 6 Other examples of such mixing between massive and massless states occur in D-brane theories [17,21], when one takes into account the space-time distortions induced by the recoil of such extended objects during scattering with string states representing low-energy matter. Such an approach can also lead to a time-dependent vacuum energy, with a dark energy-like equation of state that exhibits temporal dependence (relaxation) [22] 7 . This cosmological situation is similar in spirit to the effective field theory model of a running vacuum [24], but involves brany objects whose recoil induces the temporal dependence via non-criticality of the underlying string theory. 8 .
When applied to specific non-critical string models [8,9,11], the Eqs. (12) result in a coupled system of equations that constitute an extension of the cosmological equations of standard critical string cosmologies. The simplest and most representative systems we deal with here, which suffice for our purposes, contain dilaton and metric backgrounds only. In more realistic string models [11] one considers other fields, such as fluxes, string moduli, antisymmetric tensors, etc., which provide additional equations of motion and constraints among the relevant parameters of the models, but do not affect our basic conclusions on the 6 Studies of such stringy black holes, and the mixing of the environment of higher string modes, prompted the authors of [7] to discuss quantum coherence issues and modifications of naive quantum mechanics within such an EFT framework, identifying the Liouville mode with time and mapping the problem to a suitable supercritical string. 7 For phenomenological consequences of relaxation models for cosmological dark energy in the context of the non-critical string cosmologies of [14], see [23]. 8 An embedding of the running vacuum model in critical string theory and its cosmological implications have been discussed in [25].
fate of the First Swampland Conjecture (2) in the non-critical cosmology framework.
We conclude this summary by commenting on the relation between the approach adopted here and the linear-dilaton string cosmology of [14]. The latter is regarded as a critical, "equilibrium" string configuration, which is described by a conformal field theory on the world-sheet, even though the internal central charge is non-critical. In the approach of [8], the presence of the Liouville mode and its identification with target time provides a non-equilibrium "transient" period for the associated string universe, which may in some instances tend asymptotically [9,11] to the equilibrium critical string model of [14].

Conjectures
Considering metric and dilaton cosmological backgrounds, along with some generic matter and radiation (denoted by the suffix m), one finds the following cosmological equations in the Einstein frame, upon identifying the (supercritical) Liouville mode with the target time: 9 where˜ m andp m are the conventional matter density and pressure, φ and p φ are the dilaton energy density and pressure, and: 10 with Q the central-charge surplus, a the cosmic scale factor and q the cosmic decceleration, and we note that only two of the equations (13) are independent. We have used in the above a dilaton field with a canonically-normalised kinetic term, following the notation in [17]. In particular, the dilaton φ used in [17] and appearing in (13) is related to the standard dilaton Φ used in [14] and [8,9] by In our conventions, the slope of the potential is positive, as the potential increases for increasing φ, until it assumes an approximately constant value for large values of φ.
In the convention (15), the string coupling is given by and, as we discuss below, in all our non-critical cosmology examples [8,9,14,17], the string coupling is weak, including during the inflationary era, in the sense that the dilaton takes on large (trans-Planckian) positive values. Indeed it is an increasing function of the cosmic time. One can take g (0) s < 1 to guarantee string-loop perturbation theory in all the phenomenologically relevant epochs of the Universe evolution, from early cosmology to the current era.
The dots above a quantity in (13) denote derivatives with respect to the Einstein-frame cosmic time, which is the standard time used in cosmology to compare models with observations [18]. The r.h.s. of these equations are provided by the terms that are associated with the non-critical string behaviour. The time derivatives appearing on the r.h.s. are essentially derivatives with respect to the Liouville mode, which is the zero mode of the world-sheet renormalisation scale and has been constrained to be identical to the cosmic time [8]. Such terms are absent in critical string cosmologies, including the asymptotic situation met in [14].
We ignore for our purposes here the matter/radiation contributions denoted by the suffix m, as we are only interested in early (inflationary) epochs, in particular the validity of the Swampland Conjectures for de Sitter space time and the EFT in the context of non-critical string cosmologies, and hence set˜ m =p m = 0.
The reader should notice the dissipative terms proportional to Q(t)φ in (14), where Q(t) is the central charge surplus, that depends in general on the cosmic time [9]. There are many explicit ways to generate such as a surplus, and we shall come back to explicit examples discussed below. In general, the variation of the central charge deficit Q(t) with the cosmic time is provided [9,11] by the Curci -Paffuti equation [26], which expresses the renormalisability of the world-sheet theory. To leading order in an α expansion, to which we restrict ourselves in this work, this equation is given by in the Einstein frame.
V all of [17]. In other non-critical string scenarios [11], the dilaton evolves in such a way that the potential exhibits a decaying behaviour, V all ∼ e −c 1 φ withc 1 > 0, at very large times (indicated by the doubly-dotted-dashed red curve), and reaches asymptotically a logarithmic behaviour corresponding to the linearly-expanding Universe of [14], which constitutes an asymptotic "equilibrium" situation for non-critical string cosmology [8,9,11].
In Eq. (14) above, q denotes the deceleration parameter of the non-critical string Universe q ≡ −äa/ȧ 2 as function of the time. We shall be interested in (almost) de Sitter space-times for which The potential appearing in (13) above is defined by where c 0 is a constant to be determined by consistency with the modified swampland criteria, and for generality we have allowed for a potential term in the string action − √ −G V 0 in addition to that dependent on the central charge deficit term. We expect that V 0 might be generated by, e.g., string-loop effects, or else by interaction with D-particle defects [17], which themselves provide a source of non-criticality. In our convention, inflation is obtained for large positive values of the field φ, which correspond to early cosmic times.
The form of the potential (19) is indicated in Fig. 1. The Q-dependent part of the potential is typical of non-critical string cosmologies, and is associated with the central-charge surplus of the underlying world-sheet Liouville theory [8,14]. Indeed, for constant Q (as, e.g., in the case of [14]) and V 0 = 0 the potential exhibits a standard quintessence-like behaviour, which is compatible with the First Swampland Conjecture (FSC) (2), provided c 0 is of order one. For instance, the case of a linearly-expanding universe with four uncompactified target-space dimensions studied in [14] has c 0 = 1, and thus falls in this category.
The problems with the FSC start when one considers a large-field de Sitter phase (no matter how brief) within the EFT framework, for which the dilaton φ > 1/κ, V > 0 but also, as we shall discuss [9], Q(t) is no longer constant. In fact, the purpose of this paper is to discuss the swampland criteria for non-critical string potentials of the form depicted in Fig. 1.
Depending on the details of the V 0 part of the potential, the Universe may enter a (relatively brief) de Sitter era, before exiting into either (i) an era in which the dilaton is stabilised at a constant value φ 0 , as in dilaton-induced Starobinsky-like inflation in the context of the non-critical string/brane scenario of [17] (with the value of the potential V all (φ 0 ) at its minimum being either zero, or small and positive so as to match the current cosmological constant value), or (ii) an era [11] in which the dilaton potential exhibits another non-critical string quintessence form, V all ∼ e −c 1 κφ , withc 1 > 0 some positive constant of O(1), and the dilaton evolves with the cosmic time as where a is the scale factor of the Universe. This solution is asymptotic for very large cosmic times to the solution of [14], in which the dilaton scales logarithmically with the Einsteinframe cosmic time, and the Universe is linearly expanding in this frame. 11 .
The dilaton energy density and pressure are We observe in (21) that in a de Sitter background of the type we are interested in, under the assumption that the potential dominates, one can ignore theφ 2 terms in front of V all , so the field φ acts as a quintessence field inducing an (almost) de Sitter space-time, with The first of Eqs. (13) is actually the analogue of the Friedmann equation for non-critical string cosmologies. Using (14) and˜ m =p m = 0, we may write it in the following form in the de Sitter phase: During the de Sitter phase, we may make the reasonable approximation that the behaviour (20) characterises the dilaton field. In particular, we assume the proportionality constant in (20) to be that in the solution of [14], which actually describes the asymptotic exit phase from the non-critical string inflation model of [9]. In such a solution one has: which, on account of the second of the equations (13), implieṡ From this we easily obtain, using (24): 11 It should be noted that, if the expansion is measured in string rods, the linear expansion is not really an expansion of the Universe, but being an asymptotic equilibrium point in the non-critical string evolution [11], this is not an issue.
where the prime denotes differentiation with respect to φ, and H I is an inflationary scale to be determined phenomenologically.
We now observe that, because of (24) and (25), the Curci-Paffuti equation (17) implies a constant of mass dimension [+2], during the de Sitter phase, which is realised for large values of φ (see Fig. (1)). Using (14) and (26), this yields the following evolution with respect to the cosmic time, or equivalently φ(t), of the central charge deficit Q(t) [9]: We now remark that, although the supercriticality of the string corresponds to Q 2 (t) > 0, we may without loss of generality assume that Q(t) > 0, which allows us to take As we see later, this is a self-consistent choice when formulating the modified swampland conditions for supercritical strings.
Then, we obtain from (23) Using the Bousso bound (5), or equivalently (6), which is valid independently of the criticality or otherwise of the string, being a geometric bound on the entropy of states, one may obtain from (24) and (30): where as can be inferred from (28) and (29). It is clear that the presence of the function D φ (φ) which depends on the dilaton may drastically modify the swampland bound, depending on the details of the non-critical string model, in particular the values of the parameters c 0 and Indeed, deep in the de Sitter phase, the field φ takes on trans-Planckian values, for which the distance conjecture (1) is violated. We may also assume that most of the states that are becoming light behave either in a point-like way, corresponding to δ 0, or are string-like, with δ = 1. 12 Hence, without loss of generality we may assume 0 < δ < 2 as a physically relevant range of this parameter for our non-critical string cosmologies, which is also in agreement with a plethora of critical string models [3].
Taking into account the fact that in this large-dilaton regime there is a large number of states (3) that are light, since n(φ) 1, with b γ = O(1) > 0, our analysis indicates that the term D φ (φ), proportional to Q(t), dominates the N -dependent entropy term on the r.h.s of the inequality (31). Therefore, using (32) we may approximate the r.h.s. of the inequality in the second line of (31) by where we stress once again that the (dimensionful) constant c 4 is connected to the supercritical-string central-charge surplus Q, and vanishes for critical strings: c 4 = 0.
The potential (19) may be written as For consistency with the (quintessence-like) exponential behaviour of the potential depicted in Fig. 1, which we take as our prototype of supercritical string cosmology, it suffices to consider potentials (34) for which either (i) all of the exponents of the exponentials are negative, or (ii) the largest of them vanishes, and the other two are negative, i.e., • Case (i): All the exponents in the exponential are negative For large positive φ, the leading term in the potential is that with coefficient c 2 4 . We then see from (19) and (31) that for consistency with inflationary phenomenology, a sufficient condition for a de Sitter phase at large φ is In this way satisfying the Second Swampland Criterion becomes almost trivial. Indeed, taking the logarithms on both sides of (33), differentiating with respect to the field φ, and recalling that in our case V all < 0 during and before the inflationary phase (see Fig. (1)), we which provides a much weaker constraint on the potential than (2)  We now check the conditions on the parameters of the potential (34) under which it satisfies the condition (37) in the large-φ (de Sitter) regime (see Fig. 1). To this end, we compute the quantity: where the . . . denote subleading terms in the large-field de Sitter regime φ 1 with |c 0 | > 5/2. Comparing (38) with (37), we observe that (37) is satisfied provided: which, since V 0 ∼ H 2 I (36), implies: where we have reinstated the factors of κ units, for clarity. On the other hand, from the Bousso bound on the potential ((36) or equivalently (33)), we must have for κ φ 1. Taking into account the fact that phenomenologically H I κ ∼ 10 −4 1 [18], we find that a stronger lower bound on |c 4 | comes from (40), which is the result of the modified swampland condition (37) in the supercritical string model.

• Case (ii): One exponent vanishes
Let us now consider the case in which one exponent vanishes, and the other two are negative In this case, we see from (19) and (31) that for consistency with inflationary phenomenology, a sufficient condition for obtaining a de Sitter phase at large φ is where we have again reinstated the factors of κ, for clarity. Performing a similar analysis as before, i.e., by computing V all /V all from (34) at large φ 1, but now under the condition (42), we reach the conclusion that in order for V all to satisfy (37) and (43) simultaneously one must have and The stronger bound on |c 4 | is again that coming from the modified swampland constraint (37) on the slope of the dilaton potential.
The condition (37) is necessary and sufficient for the validity of the EFT during the de Sitter phase of supercritical string cosmology. The latter contains a plethora of stringy states, which result in non-criticality, but collectively their presence is taken into account by the presence of the EFT terms on the r.h.s. of (13), stemming from the identification of the Liouville mode with the cosmic time [8]. The modified criterion (33) or equivalently (37) is compatible with slow-roll inflation, in contrast to the First Swampland Conjecture (2). The latter is recovered in the critical string limit Q → 0, since the N -dependent term on the r.h.s. of the inequality in (31) is the only term present in that case.
C. The Special Case of a Dilaton-Induced Starobinsky-Like Potential [17] In light of the above discussion, we infer that the dilaton/D-particle-induced Starobinskylike potential of [17] constitutes a consistent non-critical string cosmology, as it also satisfies (37). However, for completeness we should mention that this model is constructed somewhat differently, in that its dilaton central charge deficit is assumed to be subcritical, as the brane world lives in d = 4 dimensions, and the non-criticality arises from the interaction with ensembles of D-particles living on the brane world, which enter the latter from the bulk as the brane Universe propagates in cosmic time.
The corresponding effective action in the Einstein-frame is given by [17] S staro where the four-dimensional gravitational constant is given by κ 2 = 8πα V −1 C , and α = M −2 s is the Regge slope where M s is the string mass scale, and V C is the volume of the extra compact dimensions (or the volume of the portion of the bulk space between the brane worlds, depending on the construction). The potential V NC−Staro is given by [17] V The quantity s < 1, depends on the (three-)density of D-particles on the brane world n, which is assumed to be approximately constant, with its dilution due to cosmic expansion being compensated by an influx of D-particles from the bulk. The density n and masses of the D-particles can be adjusted so that the potential vanishes at φ = 0. The magnitude of the quantity A provides the inflationary scale in the model.
We now make some important remarks concerning the structure of such a potential.
Notice that in the conventions of [17], the potential increases for increasing positive φ, which is a crucial difference from the non-critical string potentials (34). On the other hand, as in the supercritical-string case, the inflationary phase occurs for large positive κ φ 1.
Nonetheless, because now V NC−Staro > 0, some crucial modifications need to be made in the non-critical string analysis presented above that led to the swampland conditions.
The potential (46) corresponds to (19), but now Q 2 < 0 as in sub-critical strings. Formally, the non-critical string analysis can be performed in such a case by analytically continuing to Euclidean cosmic time maintaining otherwise the structure of (23). Using the analytic continuations of the expressions (25) and (26) where we maintain the same sign in front of the constant c 4 as in the supercritical string case (33), as required by the positivity of the potential (the reader can check from (27) that we have the freedom to do so). However the sign of the H 2 I -dependent term is now the opposite, in view of (47).
Taking the logarithm of (48) and differentiating with respect to φ, we now have From this we see that the condition (37) is now modified by changing the sign of the inequality, since now the slope of the (positive) potential is also positive V NC−Staro > 0.
It is thus straightforward to see that, during the trans-Planckian inflationary de Sitter phase with κ φ 1, the Starobinsky-like potential (46) satisfies the modified Swampland conjecture (49) quite comfortably. Indeed, we see that Thus, the condition (49) is naturally satisfied and an EFT description is valid.
We stress once more that, in our considerations in this work, and more generally in our non-critical cosmological string models [8,9,11], the string coupling remains weak: g s < 1 during the entire cosmic evolution from the origin of cosmic time, which corresponds to φ = 0, until the present era, including the inflationary epoch (see Fig. 1). Hence the D-particles remain heavy in the scenario of [17], since their masses are proportional to and Nonetheless, as the dilaton takes on large values, other string states (e.g., scalar moduli or stringy extended objects) can descend from the UV to contaminate the EFT in the way described above. However, in the context of our supercritical strings this is taken into account by representing collectively the "environment" of such states via extra contributions to the EFT cosmological equations, incorporated via the appropriate Liouville dressing and identification of the (time-like) Liouville mode with the cosmic time (see the r.h.s. of (13) in the concrete example discussed in the current work). Thus, we always retain an EFT description, which is reflected in the triviality of the modified swampland condition (37) in supercritical string cosmologies. There are, however, some restrictions on the parameters of the respective dilaton potentials, see, e.g., (35) and (36), needed to match the relative phenomenology.

IV. DISCUSSION: ISSUES WITH THE WEAK GRAVITY CONJECTURE
Before closing, we discuss the consistency of our results with the Weak Gravity Conjecture (WGC) [27] and its extension to include scalar fields [28]. The WGC states essentially that gravity is the weakest of the fundamental forces in four dimensions. According to the WGC, if a theory contains a U(1) gauge symmetry with coupling g, then there must exist a state of charge q and mass m, such that [27] g qM P = m where M P = √ 8π M Pl is the four dimensional Planck scale. Its scalar field extension [28] implies that if m(φ i ) is the mass gap of some scalar field ϕ (not the dilaton) interacting with the light scalar moduli fields φ i (which include the dilaton) within some four space-time dimensional EFT, then gravity remains the weakest force, which requires that interactions involving exchanges of the light scalars be such that the mass of the scalar field ϕ satisfies whereg ij is the metric in the space of the moduli fields φ i . As we do not deal explicitly with charged matter in this work, it is this scalar extension (53) that is of interest to us here.
First we notice that in models involving (scalar) D-particles, as in the Starobinsky inflation model of [17], the masses of the D-particle states themselves satisfy (53), since that the D-particle mass is inversely proportional to the string coupling (51). In terms of the standard dilaton Φ (15) that is used in the canonical expression for the D-particle mass, the string coupling is given by g s = e Φ , so we see that condition (53) is satisfied, in fact it is saturated with a constant 'metric' in theory space that is equal to the identity. However, having said that, the conjecture is formulated at present for local field theory scalars, and its extension to stringy/brany objects is not yet formulated to a level of precision and understanding that allows us to apply the above result (53) with rigour.
Moreover, in our non-critical string cosmologies, the considerations around the scalar WGC pertain to masses of scalars other than the dilaton, which is one of the string moduli.
As such, it is not relevant for the non-critical string dilaton potentials (34) discussed in this work. One needs a formulation of the scalar WGC that incorporates the potential of the (self-interacting) dilaton. 13 In this connection, an improved version of the scalar WGC has been suggested in [29]: which is expressed in terms of the scalar potential V , for canonically-normalised fields. This improved WGC resolves the axion puzzle, leading to a plausible constraint on the axion decay constant, f a ≤ M P , as expected. In principle, (54) should be applicable to our case, where we deal with a canonically-normalised dilaton φ (15).
As discussed in Section II, in both the cases (42) and (39) that are compatible with the non-critical string modification of the First Swampland Conjecure (37), we are dealing with large dilaton field values κ φ 1 and exponential potentials in a de Sitter phase of the Universe: where γ is a negative dimensionless constant that takes the value in both cases. For such potentials, the condition (54) is not satisfied.
We remark, though, that there is no rigorous diagrammatic derivation of the improved scalar WGC (54). In particular, the coefficients of the various scalar self-interaction terms appearing on the r.h.s. of the inequality could in principle be modified. For this reason, a generalised form of the scalar potential WGC (54) has been proposed in [30]: where c 1 and c 2 > 0 are positive coefficients of order O(1). With appropriate values of these coefficients, the generalised form of WGC (57) can accommodate various scalar potentials of interest in the early Universe, as in Q-ball models, but also exponential scalar potentials of the form (55) for γ ≤ 1, such as our supercitical string case with a canonically-normalised dilaton (56). The condition would be saturated in that case for c 1 = c 2 = 2.
On the other hand, we note that the condition is satisfied, indeed saturated, if one uses the standard dilaton Φ (15) appearing in the definition of the string coupling, g s = e Φ , as the string loop-counting parameter. However, we note that in such a formalism the dilaton does not have canonically-normalised kinetic terms, but appears in the string-inspired low-energy gravitational action in a D-dimensional uncompactified target space in the form: where Q 2 is the central charge surplus of the supercritical string [8,14], and the . . . denote other fields, higher derivative terms, etc.. Thus, for four large uncompactified dimensions, D = 4, one obtains the result (15) for the canonically-normalised dilaton, φ, used in [17] and above.
The satisfaction of the improved WGC (54) by Φ but not φ is curious, but may be the correct form to be used in the case of non-critical strings with exponential scalar potentials.
We remark that in [29] the conjecture (54) was derived by looking at four-point scalar interactions, one with the scalar field as a mediator and the second a four-scalar contact interaction, which could be due to the exchange of a massive mediator. The four-point interaction is repulsive, and is associated with the term in the conjecture involving the fourth derivative of the field. Thus, subtracting the contact interaction from the the four-point amplitude with a scalar mediator (which is attractive, and corresponds to the term in the conjecture involving the square of the cubic derivatives of the potential with respect to the scalar field) should give a result at least as weak as the fourth-order interaction mediated by a four-dimensional graviton exchange, which is essentially the square of the second derivative of the potential with respect to the field (which defines the mass-squared of the scalar field). This is the philosophy behind (54). For non-critical strings with exponential potentials the situation needs to be revisited, which might lead to an explanation on the aforementioned discrepancy, and why the (non-canonically-normalised) form Φ of the dilaton should be used in the string effective action.
Finally, we note that, for the dilaton-induced Starobinsky potential (46), in the inflationary phase occurring for large positive φ, and with a canonical kinetic term as in (15), the condition (54) is trivially satisfied, as can be seen by direct computation. The exponent √ 2 is crucial for this result. In this model the potential (46) is of the following generic form for large κ φ 1: