Universal accelerating cosmologies from 10d supergravity

We study 4d Friedmann-Lema\^{i}tre-Robertson-Walker cosmologies obtained from time-dependent compactifications of Type IIA 10d supergravity on various classes of 6d manifolds (Calabi-Yau, Einstein, Einstein-K\"{a}hler). The cosmologies we present are universal in that they do not depend on the detailed features of the compactification manifold, but only on the properties which are common to all the manifolds belonging to that class. Once the equations of motion are rewritten as an appropriate dynamical system, the existence of solutions featuring a phase of accelerated expansion is made manifest. The fixed points of this dynamical system, as well as the trajectories on the boundary of the phase space, correspond to analytic solutions which we determine explicitly. Furthermore, some of the resulting cosmologies exhibit eternal or semi-eternal acceleration, whereas others allow for a parametric control on the number of e-foldings. At future infinity, one can achieve both large volume and weak string coupling. Moreover, we find several smooth accelerating cosmologies without Big Bang singularities: the universe is contracting in the cosmological past ($T<0$), expanding in the future ($T>0$), while in the vicinity of $T=0$ it becomes de Sitter in hyperbolic slicing. We also obtain several cosmologies featuring an infinite number of cycles of alternating periods of accelerated and decelerated expansions.


Introduction and summary
Obtaining realistic 4d cosmologies from the ten-dimensional supergravities that capture the low-energy limit of superstring theory has proven notoriously difficult. At the turn of the century, it was thought that accelerating cosmologies were as difficult to achieve as de Sitter space itself, being subject to a famous no-go theorem first discovered by Gibbons [1,2] and later rediscovered by Maldacena-Nuñez [3] in a string-theory context. More precisely, a matter whose stress-energy tensor in the higher-dimensional theory implies R 00 ≥ 0, as a consequence of the Einstein equations, is said to satisfy the strong energy condition (SEC). The SEC implies that time-independent compactifications of the higher-dimensional theory can never lead to 4d cosmologies with accelerated expansion (which includes de Sitter space as a special case). 1 However, as was first pointed out in [4], time-dependent compactifications evade the nogo and can lead to 4d Einstein-frame accelerated expansion for some period of time [5][6][7][8][9][10]. Such transient acceleration is in fact generic in flux compactifications, see [11] for a review, although de Sitter space is still ruled out by the SEC, if the time-independence of the 4d Newton's constant is obeyed in a conventional way [12]. If instead the time-independence of the 4d Newton's constant is obeyed in an averaged way, even de Sitter space is not ruled out by the SEC, although the so-called dominant energy condition does rule out non-singular de Sitter compactifications [13] (see also [14,15] for a recent discussion on energy conditions).
In [12], Russo and Townsend greatly refined the no-go of [1][2][3]: one of their conclusions is that, even if one imposes the SEC and the time-independence of the 4d Newton's constant in a conventional way, late-time accelerating cosmologies are not ruled out. However, no late-time accelerating cosmologies from compactification of the ten-or eleven-dimensional supergravities arising as low-energy effective actions of string theory have ever been constructed. Indeed, the eternal accelerating cosmologies of [16,17] are such that the acceleration of the scale factor tends to zero at future asymptotic infinity, so that there is no cosmological horizon.
Ref. [10] reinterpreted the accelerating solutions of [4] from the point of view of a 4d theory with a scalar potential. It was found that there is always a big-bang singularity near which the scale factor behaves as a power law: S(T ) ∼ T 1 3 , which does not lead to enough e-foldings for inflation [16,18]. The transient acceleration of the solutions could, however, be used to describe the current cosmological epoch [9]. The general characteristics of one-field inflation with an exponential potential were studied in [19], while cosmologies from an effective theory with multiple scalar fields were studied in [20][21][22]. In particular, the analysis of [23] could be relevant for the swampland conjecture [24]. Cosmological solutions of gauged supergravities and F-theory have been studied in [25][26][27][28][29][30][31][32][33][34][35][36][37].
In the present paper we re-examine some of the previous statements in the context of universal cosmologies, obtained by compactification to 4d of 10d Type IIA supergravity (with or without Romans mass) on 6d Einstein, Einstein-Kähler, or Calabi-Yau (CY) manifolds. In this context, the term "universal" means that the ansätze that we consider do not depend on the detailed features of the manifold on which we compactify, but only on the properties which are common to all the manifolds belonging to that class. For example, our ansatz for the CY compactification exploits the existence of a holomorphic three-form and a Kähler form, but does not assume the existence of any additional harmonic forms on the manifold.
Our universal cosmologies are obtained by solving the ten-dimensional supergravity equations: we do not start from a 4d effective action. Nevertheless, it turns out that all of the resulting 10d equations of motion are obtainable from a 1d action where all fields only depend on a time coordinate. In other words, there is a 1d consistent truncation of the theory, S 1d . The fields in question are the dilaton ϕ and two warp factors A, B (one for the internal and one for the external space), while, by virtue of our ansatz, all fluxes manifest themselves as constant coefficients in the potential of S 1d , cf. Table 2.
Moreover we show that a two-scalar 4d cosmological consistent truncation, S 4d , of the 10d theory to ϕ, A is possible in certain special cases. In other words, every solution of cosmological Friedmann-Lemaître-Robertson-Walker (FLRW) type of S 4d uplifts to a solution of the ten-dimensional equations of motion. Again, the different fluxes show up as constant coefficients in the potential of S 4d . At least for a certain range of the parameters in the potential, we expect our S 4d to coincide with the universal sector of the effective 4d theory of the compactification, see [38] for a recent discussion on consistent truncations vs effective actions. Much of the literature on string-theory cosmological models uses a 4d effective action and a 4d potential. We make contact with the potential description in Section 4. Whenever a single "species" of flux is turned on, 2 we are able to provide analytic cosmological solutions to the equations of motion. These are described in detail in Section 5. Whenever multiple fluxes are simultaneously turned on, one cannot give an analytic solution in general, although this can still be possible for certain special values of the flux parameters. Whenever exactly two different species of flux are turned on, although an analytic solution is not possible in general, a powerful tool becomes available: as we show in Section 6, the equations of motion can be cast in the form of an autonomous dynamical system of three first-order equations and one constraint. The use of dynamical-system techniques in general relativity is of course not new: refs. [4,12,[39][40][41][42][43][44] in particular are closely related to the strategy employed here. One of the novelties of the present paper is to give an exhaustive analysis of universal cosmological solutions from compactification on the previously mentioned classes of manifolds. The resulting dynamical system description is rather intuitive and captures several generic features of cosmological solutions coming from models of flux compactification. Each solution is represented by a trajectory in a three-dimensional phase space. The constraint forces the trajectories to lie in the interior of a sphere, with expanding cosmologies corresponding to trajectories in the northern hemisphere. Whenever the system of equations admits fixed points, these correspond to analytic solutions.
The boundary of the allowed phase space corresponding to expanding cosmologies, i.e. the equatorial disc and the surface of the northern hemisphere, are both invariant surfaces. This implies that trajectories that do not lie entirely on these surfaces can only approach them asymptotically. Trajectories that do lie entirely on either of these surfaces also correspond to analytic solutions (namely analytic solutions with a single species of flux turned on: indeed restricting the trajectory to either the surface of the sphere or the equatorial disc, corresponds to turning off one of the two species of flux). Moreover, the intersection of these two invariant surfaces -the equator -constitutes a circle of fixed points, and corresponds to a family of analytic solutions with all flux turned off, described in Section 5.1. The dynamical system generally possesses a third invariant surface, an invariant plane P, whose disposition depends on the fluxes of the solution.
The analytic solutions corresponding to fixed points are all cosmologies with a power-law scale factor, S(T ) ∼ T a , with 1 3 ≤ a ≤ 1. The fixed points at the equator correspond to power-law (scaling) cosmologies with a = 1 3 , while a = 1 corresponds to either a fixed point at the origin of the sphere (whenever the dynamical system admits such a fixed point), or a fixed point in the bulk of the sphere and on the boundary of the acceleration region. In the former case the corresponding cosmology is that of a regular Milne universe, whereas in the latter case it is a Milne universe with angular defect. In addition, we find fixed points corresponding to cosmologies with a = 3 4 , 19 25 , or 9 11 . The list of analytic solutions is given in Table 3.
Trajectories interpolating between two different fixed points asymptote the respective scaling solutions in the past and future infinity. Note in particular that we find no fixed points with a > 1 (which would correspond to eternally accelerating scaling cosmologies).
The question of acceleration becomes particularly transparent in the dynamical system description: the acceleration period of the solution corresponds to the portion of the trajectory that lies in a certain region in phase space. This "acceleration region" is entirely fixed by the type of flux that is turned on, with the different cases summarized in Table 1.  Table 1: Different types of universal two-flux compactifications and their corresponding acceleration regions in the phase space. The values of β 1,2 -corresponding to the two species of flux that are turned on in the solution -depend on the type of flux and its number of legs along the external-space, internal-space and time directions, cf. (35). The absence of an acceleration region is denoted by the empty set ∅.
This clarifies why (transient) accelerated expansion is essentially generic in flux compactifications: there is always a trajectory that passes by any given point in phase space (corresponding to some particular choice of initial conditions). By making sure that a trajectory passes by some point in the acceleration region of the northern hemisphere, we can thus obtain a cosmological solution that will necessarily feature a period of accelerated expansion. Moreover, the only non-trivial requirement, namely the existence of an acceleration region in phase space, is not difficult to satisfy, as can be seen from Table 1. Figure 1: Trajectories in phase space corresponding to two-flux cosmological solutions with accelerated expansion. The corresponding dynamical systems are given in Sections 6.2, 6.3, 6.4. The pair of fluxes that are turned on in each case is indicated below each subfigure. The trajectories (blue lines) interpolate between a fixed point on the equator in the past infinity, corresponding to a cosmology with a power-law scale factor S ∼ T 1 3 , and another fixed point at future infinity. For cases 1c and 1d the second fixed point is also on the equator, and thus the solution asymptotes the same scaling cosmology in the past and future infinities. Cases 1a and 1b asymptote at future infinity a fixed point in the interior of the sphere which corresponds to a cosmology with a power-law scale factor S ∼ T and S ∼ T 19 25 respectively. Cases 1c and 1d describe the same system, but employ different parametrizations which interchange the role of the two fluxes that are turned on. The transient accelerated expansion corresponds to the portion of the trajectory within the acceleration region depicted in green. In the dynamical system description, the calculation of the number of e-foldings, N , also becomes particularly transparent, since the flow parameter is simply the logarithm of the scale factor. Most instances of transient acceleration are such that the number of efoldings is of order one, N ∼ O(1), as was mentioned earlier. The exceptions that we find all correspond to dynamical systems with fixed points on the boundary of the accelerated region, and they all give rise to hyperbolic FLRW cosmologies (k < 0). In addition, there is a flux turned on corresponding to one of the following being non-zero: λ, m, c f , c 0 , or c φ ; see Table 2 for the 10d origin of these coefficients.
The first of these solutions is examined in Section 6.1 and is obtained from compactification on a 6d Einstein manifold with negative curvature (λ < 0). The second solution occurs in IIA supergravity with non-vanishing Romans mass (m ̸ = 0) compactified on a Ricci-flat space, and is described in Section 6.2. Despite having vanishing Romans mass, the three remaining solutions (k < 0 and c f , c 0 , or c φ ̸ = 0) are all qualitatively very similar to the one of Section 6.2.
The k, λ < 0 solution has been studied before in [16] (see also [17]) with somewhat different methods. It features a conical accelerating region, with a fixed point p 0 at the origin of the cone (which coincides with the origin of the sphere), and a second fixed point p 1 on the surface of the cone. Both p 0 , p 1 lie on the invariant plane P. The invariant manifolds of p 0 are the equatorial disc (in which p 0 is an attractor) and the vertical z-axis (in which p 0 is an unstable node). There are generic trajectories that asymptote some point on the equator at past infinity, cross into the acceleration region (the interior of the cone) at some point p ′ , then exit the acceleration region again at some point p ′′ , and asymptote p 1 at future infinity, see Figure 2.
These trajectories correspond to accelerating cosmologies with N → ∞ as p ′′ → p 1 . In other words, we can continuously increase the number of e-foldings and make it as large as desired, by choosing the trajectory such that the exit point from the acceleration region is sufficiently close to p 1 , see Figure 3.
Note also the presence of semi-eternal trajectories that enter the acceleration region at some point p ′ , and reach the fixed point p 1 at future infinity, without ever exiting the cone. The limiting case, p ′ → p 0 , is the unique trajectory which reaches p 0 in the past and p 1 at future infinity: it corresponds to eternal acceleration, with acceleration vanishing asymptotically at future infinity. In the vicinity of p 0 the universe becomes de Sitter in hyperbolic slicing. This is not asymptotic de Sitter, however, as p 0 is reached at finite proper time in the past. The solution can be geodesically completed beyond p 0 in the past, by gluing together its mirror trajectory in the southern hemisphere. Figure 2: Three trajectories lying on the invariant plane P of the dynamical system obtained from the compactification with k, λ ̸ = 0. The point p 0 corresponding to a Milne universe is depicted in green. The point p 1 (and its mirror in the southern hemisphere) is drawn in blue and corresponds to a Milne universe with angular defect. The equator fixed points coincide with a scaling cosmology with S ∼ T 1 3 and are illustrated in purple. The blue trajectory features transient acceleration with tunable number of e-foldings, whereas the red one corresponds to semi-eternal acceleration. Depicted in purple is the unique (finetuned) trajectory corresponding to eternal acceleration. The solution becomes de Sitter in the vicinity of the origin, which is reached at finite proper time. The solution can be geodesically completed in the past beyond the point at the origin, by gluing together its mirror trajectory in the southern hemisphere.   Figure 4: Plot log-log of the scale factor S (in blue) as a function of the cosmological time T . It corresponds to the (λ, k)-compactification (with l = 2). The green dot corresponds to the beginning of inflation; the red one to its end. The red dashed line coincides with the Milne fixed point S ∝ T and asymptotes the curve at future infinity.
From a given trajectory in phase space, it is then straightforward to reconstruct the scale factor S(T ) as a function of the cosmological time, as depicted in Figure 4. This gives access to all the cosmological data (Hubble parameter H, slow-roll parameters ϵ and η, the spectral tilt n s , or the tensor-to-scalar ratio r) needed for phenomenology.
To our knowledge, the k < 0, m ̸ = 0 solution has not appeared before in the literature. Like the previous solution, it features a conical accelerating region, with a fixed point p 0 at the origin, and a second fixed point p 1 on the surface of the cone. Both p 0 , p 1 lie on the invariant plane P. However, unlike the previous solution, the fixed point p 1 is now a stable focus on P, and an attractor in the direction perpendicular to P. This allows for generic trajectories that asymptote some point on the equator at past infinity, cross into the acceleration region (the interior of the cone) at some point p ′ , exit the acceleration region at some point p ′′ , then enter again at some point p ′′′ , and so on, as they spiral into p 1 asymptotically at future infinity, see Figure 5. This asymptotic spiraling corresponds to an infinite number of periodic cycles of alternating periods of accelerated and decelerated expansion, each cycle contributing a finite number of e-foldings, so that N → ∞. As in the previous system, there is here a unique (fine-tuned) trajectory which reaches p 0 at finite proper time in the past and asymptotes p 1 at future infinity. In the vicinity of p 0 , the universe becomes de Sitter in hyperbolic slicing. The solution can be geodesically completed beyond p 0 in the past, by gluing together its mirror trajectory in the southern hemisphere. Figure 6: Two trajectories lying on the invariant plane P of the dynamical system obtained from the compactification with m, k ̸ = 0. The point p 0 corresponding to a Milne universe is depicted in green. The point p 1 (and its mirror in the southern hemisphere) is drawn in blue and corresponds to a Milne universe with angular defect. The equator fixed points coincide with a scaling cosmology with S ∼ T 1 3 and are illustrated in purple. The red trajectory enters and exits the acceleration region an infinite number of times, spiraling around p 1 , see Figure 5 for a zoom around this region. Depicted in blue is the unique (fine-tuned) trajectory: the solution becomes de Sitter in the vicinity of the origin, which is reached at finite proper time. The solution can be geodesically completed in the past beyond the point at the origin, by gluing together its mirror trajectory in the southern hemisphere.
To our knowledge, none of the remaining systems, exhibiting a parametric control on the number of e-foldings, corresponding to k < 0 and non-vanishing c φ (351), c 0 (408) or c f (425), has appeared before in the literature. These are all captured by the 4d consistent truncation (22) with potential given in (25). While they all correspond to solutions with vanishing Romans mass, they are all qualitatively very similar to the (m, k) case analyzed above: they all feature a hyperbolic FLRW universe (k < 0) with one type of non-vanishing flux, and they all admit solutions with an infinite number of cycles of alternating periods of accelerated and decelerated expansion. In the "rollercoaster cosmology" scenario of [45] it was argued that an oscillatory behavior of this type could be relevant for inflation.
The rest of the paper is organized as follows. In Section 2 we explain the general setup and ansatz of our 10d solutions, and make contact with the 4d cosmological parameters, such as the FLRW scale factor, etc. In Sections 3, 4 we obtain the 1d, 4d consistent truncations respectively. Section 5 discusses the analytic solutions presented in the paper. Section 6 explains the dynamical system techniques used here, illustrated with some notable cases in subsections 6.1 -6.4. We conclude with some discussion and future directions in Section 7. Further details of all analytical solutions and all two-flux dynamical systems studied here, can be found in the appendices.

The general setup
Our ten-dimensional metric is a warped product of a four-dimensional FLRW cosmological factor and a six-dimensional compact internal space. The ansatz for the ten-dimensional metric in 10d Einstein frame reads ds 2 10 = e 2A(t) e 2B(t) g µν dx µ dx ν + g mn dy m dy n , where the scalars A, B only depend on the conformal time coordinate, while y m are coordinates of the internal six-dimensional space. The unwarped 4d metric is assumed to be of the form where t is the conformal time, and the spatial 3d part of the metric is locally isometric to a maximally-symmetric three-dimensional space of scalar curvature 6k. Explicitly, with i, j = 1, 2, 3, and R ij is the Ricci tensor of the metric γ ij . The 3d metric is thus locally a sphere (k > 0) or a hyperbolic space (k < 0); the case k = 0 corresponds to flat space.

Four-dimensional interpretation
The correct frame in which the predictions of our model should be compared to the cosmological data is the four-dimensional Einstein frame: this is the frame in which the effective four-dimensional Newton constant becomes time-independent. The 4d Einstein-frame metric reads where the scale factor is given by and we have introduced of a new time variable τ defined via dt dτ = e 8A+2B .
Note that τ is neither the conformal nor the cosmological time. It is simply the coordinate with respect to which the equations of motion are simplest to solve.
In terms of the cosmological time coordinate T , the metric takes the standard FLRW form, Suppose we have an explicit solution of our set of equations for all fields, so that in particular we can construct the explicit expression of scale factor. We may then always engineer a 4d perfect fluid of density and pressure ρ and p respectively, such that the scale factor in question can be seen as resulting from the 4d Einstein equations sourced by the stress-energy tensor of that fluid, where the dot refers to differentiation with respect to T . Solving the above system for ρ, p we obtain In particular, it follows that for k > −Ṡ 2 (and in particular for k = 0), the condition for acceleration is equivalent to w < − 1 3 . Of course, unless w is constant, the first equality in (10) is simply a definition, rather than an equation of state. Nevertheless, even for non-constant w, this quantity is useful insofar as it allows us to compare to the equations of state of the different cosmological eras.
The conditions for expansion and acceleration read The number of e-foldings, N , is defined as where the limits of the integral should be taken at the beginning and the end of the acceleration period.

1d consistent truncation
In all of the compactifications presented here, the internal 6d components of the Einstein equations and the dilaton equation reduce to 3 where U is a function (potential) that depends on the compactification and on the flux that is turned on, given in (16) below. The external 4d Einstein equations reduce to the following two equations, The second line above is a constraint, consistently propagated by the remaining equations of motion (i.e. the τ -derivative of this equation is automatically satisfied by virtue of the remaining equations). The important point to note here is that, as we shall see, the remaining (flux) equations of motion can be solved without imposing any additional conditions. The above equations of motion "know" about the flux content of the background which enters via U in the form of constant parameters, cf. Table 2. The bottom line is that we are left with three second-order equations for three unknowns (the two warp factors A, B and the dilaton ϕ), together with a constraint, which only needs to be imposed once at some fixed initial time, and amounts to imposing an algebraic condition on the constants parameterizing the fluxes.
Moreover, equations (13), (14) can be derived from a one-dimensional action given by where N is a non-dynamical Lagrange multiplier which should be set to N = 1 at the end of the calculation; it can be thought of as originating from the lapse function. Variation of the action (15) with respect to N imposes the constraint (second line of (14)), while variation with respect to the fields A, B, ϕ is equivalent to their respective equations of motion in (13), (14). The potential U is in general a function of all three scalars A, B, ϕ. Explicitly, it is given by In the above we have indicated the type of compactification in which the potentials appear (E stands for Einstein, EK for Einstein-Kähler, CY for Calabi-Yau), as well as numbers of the respective equations of motion.
Note that the potential encodes all the information about the flux (which is generically non-vanishing from the ten-dimensional point of view), as well as the external and internal curvature contributions. Indeed, this information is encoded in U via the different constants appearing in (16), whose 10d origin is summarized in Table 2. Note also that, in terms of the potential, the derivatives of the scale factor with respect to cosmological time, cf. equations (11), take the form Moreover, the quantity that appears in (10), is written in terms of the k-independent part of the potential, In particular we see that

Cosmological 4d consistent truncation
In a subset of the cases we present here, the equations of motion (13), (14) can be written in terms of a potential V (A, ϕ) which only depends on A and ϕ, As can be verified, these equations can then be "integrated" into a four-dimensional action, Indeed, in terms of the scale factor (5) and the cosmological time coordinate (7), taking suitable linear combinations thereof, equations (21) can be written equivalently as On the other hand, the 4d equations of motion resulting from (22) read Inserting the Einstein metric (4) into the above, and assuming that A, B, ϕ only depend on the time coordinate, results precisely in (23). Therefore, S 4d of (22) is a two-scalar consistent truncation of IIA for cosmological solutions: all cosmological solutions (i.e. all solutions with a metric of FLRW type and scalar fields that only depend on time) of S 4d lift to ten-dimensional solutions of IIA supergravity.
For the different compactifications admitting a consistent cosmological truncation of the form (22), the potential reads, Of course terms with k (the external 4d spatial curvature) cannot appear at the level of the 4d action, but rather they potentially arise as part of its solutions.
It is known [46,47] that there exists a consistent 4d truncation in the case of the universal CY sector, i.e. a consistent truncation to the the gravity multiplet, one vectormultiplet and one hypermultiplet. Remarkably, the action S 4d of (22) is a sub-truncation thereof to the graviton and two scalars, such that all information about the flux (which is generically non-vanishing from the ten-dimensional point of view) is carried by the potential V via the constants m, b 0 , c 0 , c φ . The latter correspond respectively to non-vanishing zero-form flux (Romans mass), internal three-form flux, internal four-form flux, and external four-form flux; λ is the scalar curvature of the internal Einstein manifold, cf. (260).
Note that: (i) not all compactifications considered in the present paper admit the subtruncation (22), (25) to gravity plus two scalars; (ii) the Einstein and Einstein-Kähler cases in (25) reduce to CY in the λ → 0 limit.

Analytic solutions
In the present paper we find three different types of analytic solutions: • Critical (scaling) solutions. Their metrics are of power-law form, in terms of the cosmological time T , with a = 1 3 , 3 4 , 19 25 , 9 11 , 1. They admit an interpretation as critical points in an appropriately defined phase space, as described in Section 6.
• Type I solutions. They interpolate between two asymptotically power-law metrics, as τ → −∞, τ → +∞ respectively. In terms of the dynamical system description, they correspond to trajectories interpolating between two fixed points on the equator.
The corresponding coordinate patches are parameterized by coordinates T ± ∝ e b ± τ , for certain parameters b ± which depend on the particular solution. If b − is positive, τ → −∞ corresponds to T − → 0. It follows that the solution reaches a singularity at finite proper time in the past. If b − is negative, τ → −∞ corresponds to T − → ∞, and is therefore reached at infinite proper time in the past. An identical analysis holds for the patch parameterized by T + , corresponding to τ → +∞. All four different possibilities are realized in the analytic solutions presented here: τ → ±∞ corresponding to T ± → 0, or T ± → ∞.
• Type II solutions. They interpolate between an asymptotically power-law metric, at either τ → −∞ or τ → +∞, with a ± = 1 3 , and a critical solution at τ = 0 (which is reached at infinite proper time) with a = 1 or a = 3 4 . In terms of the dynamical system description, they correspond to trajectories interpolating between one fixed point on the equator and an interior fixed point. Table 3 below summarizes the different types of analytic solutions constructed here, the corresponding type of compactification, and the section in which the explicit details of the solution can be found.

Critical
Type I Type II Table 3: All analytic solutions presented in this paper, besides the zero-flux, flat 4d space solution of Section 5.1. We list the corresponding compactification manifold, the type of non-vanishing flux, together with a reference to the explicit details of the solution. Critical solutions correspond to fixed points in the dynamical-system description. All CY critical points correspond to regular Milne universes except the one with φ ̸ = 0, which is a Milne universe with angular defect. The λ < 0 critical point corresponds to a scaling cosmology with a = 3 4 . The m ̸ = 0, λ < 0 critical point corresponds to a scaling cosmology with a = 19 25 . The k, λ, m, φ ̸ = 0 solution corresponds to AdS 4 in hyperbolic slicing. The h ̸ = 0, λ < 0 critical point corresponds to a scaling cosmology with a = 9 11 . All the other critical points correspond to Milne universes with angular defects. Type I solutions always correspond to trajectories interpolating between two points of the equator. Type II solutions interpolate between a fixed point on the equator and an interior fixed point. For all type II CY solutions, the internal space warp factor e A and the dilaton ϕ tend to a constant at future infinity, except for the φ ̸ = 0 solution for which e A , e ϕ → ∞. For all type II Einstein solutions, e A → ∞, ϕ → const. at asymptotic infinity, except for the (m, k) solution for which e ϕ → 0.

Minimal (zero-flux) solution
The simplest solution to the form equations is given by vanishing flux. Let us consider the remaining equations of motion. The internal Einstein and dilaton equations give, cf. (13), for some constants c A , d A , c ϕ , d ϕ . The external Einstein equations give, cf. (14), where c B , d B are constants. The second equation above is the constraint, which implies in particular that the ratio r defined by must satisfy r ≤ − 1 2 or r ≥ − 1 6 . The points where the constraint is saturated correspond to constant dilaton (c ϕ = 0).
This solution thus implies a power-law form for the 4d part of the Einstein-frame metric, where we have appropriately rescaled the τ , ⃗ x coordinates to absorb d A , d B . In terms of a coordinate T ∝ e (12c A +3c B )τ we have a power-law expansion, The time coordinate T ranges from T = 0, where we have a singularity, to T = ∞. Depending on the values of c A , c B and c ϕ , we can have solutions with constant dilaton such that the warp factor collapses (e A → 0), or decompactifies (e A → ∞) as T → 0 or ∞ respectively. The opposite behavior is also possible, i.e. e A → ∞, 0 as T → 0, ∞ respectively. We can also have solutions with constant internal space warp factor, such that either ϕ → ±∞ or ϕ → ∓∞, as T → 0 or ∞.

Single-flux solutions
In the case where a single type of flux is turned on, it is always possible to solve the equations analytically. Let us suppose a potential of the form for some real constants c, α, β, γ, whose precise values depend on the type of flux turned on. More specifically, let n t , n s , n i be the number of legs the form has along the time, 3d space, and internal directions respectively. Then, for an RR form we have For an NS-NS three-form the constants α, β are as above, but γ = −(−1) nt .
In the following, it will be useful to set Case q ̸ = 0: Let us set where c E , d E are arbitrary constants. The A, B, ϕ-equations of motion are then solved by where c A , d A , c B , d B , c ϕ , d ϕ are arbitrary constants subject to the conditions Moreover, the constraint (14) reduces to For τ → ±∞, the scale factor ln S grows linearly in τ . This results in a scaling solution with S(T ) ∼ T 1 3 . For q > 0, S goes to a constant in the τ → 0 limit. In the case q < 0 we have instead, in the τ → 0 limit, where we took (40) into account. This results in a scaling solution with Taking (35) into account, we see that there are no scaling solutions with a > 1 (accelerated expansion). However, there is one case which gives a = 1 (scaling solution with vanishing acceleration): it involves negative external curvature and is described in Section 6.1.
Case q = 0: For special values of n t , n s , n i it is possible to have q = 0. It can be seen that this occurs only if the flux is anisotropic in the 3d spatial dimensions. The A, B, ϕ-equations of motion are then solved by where

arbitrary constants and
Moreover, the constraint (14) reduces to The scale factor is given by The conditions for accelerated expansion read where we have set For concreteness, let us assume c E > 0. The conditions (47) can then be written as (47). Since B ± ≥ 0 for allc 1 ≤ 1 8 , one needsc 2 > 0 for the second condition to be satisfied, so we restrict to that case in the following. Furthermore, it holds for t 1 < t < t 2 with, If 0 <c 1 < 1 8 : the first condition in (47) is always satisfied and the number of e-folds is thus given by For the valuec 1 = 1 8 , B − = B + and there is no accelerated expansion.
Note that for allc 1 < 0, t 1 < t 0 < t 2 . In this case, the number of e-folds is given by Plugging (50) into (51) and (53), we observe that N only depends onc 1 , and reads explicitly Forc 1 < 0, N is monotonously increasing from 0 to 1 2 . For 0 <c 1 < 1 8 , N admits a maximum N max = 0.59980 atc 1 = 0.038148, see Figure 7. It is the maximal number of e-folds one can obtain in such one-flux solutions. 6 Two-flux solutions: dynamical system analysis Let us now consider the case where the potential consists of two terms, where c i , α i , β i , γ i are real constants. We shall assume that the potential is not everywhere non-positive, so that we can take c 1 > 0, while c 2 is unconstrained.
Let us define the following phase-space variables, In terms of these, the equations of motion become an autonomous dynamical system, where dσ := e E 1 /2 dτ , and we have used the constraint to eliminate the terms with c 2 from the equations of motion. Moreover, the constraint reads To better analyse the behavior of the flow at infinity, it is useful to compactify the phase space as in [42,44]. We introduce the new variables Note that in these variables the condition of expansion is equivalent to z > 0. The system (58) becomes where f ′ = d ω f and dω := √ c 1 √ 6z dσ, while the constraint (59) now reads Let us also note that as follows from the previous definitions. The equations of motion (61) imply so that the unit sphere, is an invariant surface. This implies that trajectories which include some interior (resp. exterior) point of the three-dimensional unit ball will remain there; such trajectories correspond to c 1 , c 2 having the same (resp. opposite) sign. Similarly, trajectories that include a point on the sphere S must lie entirely on S. As can be seen from the constraint (62), trajectories on S must have c 2 = 0.
Moreover, it follows immediately from the third line in (61) that the plane z = 0 is another invariant surface, so trajectories which include some point in the upper (resp. lower) half of the three-dimensional space z > 0 (resp. z < 0) must lie there entirely. Similarly, trajectories which include a point of the (x, y)-plane must lie entirely on that plane. The latter trajectories must have c 1 = 0, as follows from (62).
The intersection of the two invariant surfaces above, the unit circle C in the z = 0 plane, is also an invariant surface. In fact C is a circle of fixed points, Each point p C corresponds to a trajectory (solution) with c 1 , c 2 = 0, i.e. the minimal solution of Section 5.1. In particular, the ratio r defined in (31) is related to the polar angle via, Another consequence of the system (61) is that the plane is an invariant surface, where the constants a, b, c are obtained as solutions of the system of equations 4 Indeed, in this case, (61) implies Allowed trajectories must therefore either lie entirely on P, or be limited on either side of it.
As follows from (61), the flow equations are invariant under (z, ω) → −(z, ω), so that each trajectory in the z > 0 region is paired to a "mirror" trajectory in the z < 0 region. As we are ultimately interested in expanding cosmologies, we will restrict our attention to the z ≥ 0 region. On the other hand, taking (17), (62) into account, the condition for acceleration is written as 5 which defines an acceleration region in the phase space. Depending on the values taken by β 1 , β 2 , it can be a cone, a cylinder, a ball (regular or deformed) or the region above a horizontal plane, see Table 1 and Figure 8 below. Here, the acceleration region is a cone, depicted in green. The generic equator fixed points are illustrated in purple. This example corresponds to the case λ, k ̸ = 0, studied extensively in Section 6.1, and coincides with the northern hemisphere of Figure 2.
Note that different pairs of β 1,2 lead to acceleration regions of different shape. Although the solutions are invariant under β 1 ↔ β 2 , this leads to a reparametrization of the phase space of the corresponding dynamical system. Put differently, although the solutions are invariant under reflections along the diagonal of Table 1, the shapes of the acceleration regions are not.
Moreover, using (10), (18) and (62) we have Note that in the presence of external curvature (with the convention c 2 = −6k), the above expression is to be replaced by Eqs. (71), (72) are consistent with the fact that the acceleration condition is equivalent to It is also possible to obtain an expression for the number of e-foldings N directly in terms of phase space variables. Indeed, solving (60) for for v, u, w, and taking (57) into account, we obtain In particular this implies d ω (4A + B) = 1, which, taking (5) into account, is equivalent to This means that the flow parameter of the system (61) is simply ω = ln S S 0 , so that S = S 0 at ω = 0. It follows that where the limits of the integral should be taken as the points of entry and exit of the trajectory into and out of the acceleration region (71), respectively.

The scale factor
Given a solution (x(ω), y(ω), z(ω)) of the above dynamical system, it becomes possible to reconstruct the corresponding expression for the scale factor S(T ), which is the quantity we are ultimately interested in, in order to construct all the observables related to the cosmological model.
One can integrate (63) to obtain where T = 0 corresponds to the lower bound ω → −∞ and S = e ω = 0. Note that since we restrict to z > 0, T is ensured to be positive. Here, A, B, ϕ are completely determined by the solution (and the data of initial conditions) via (74), and can be computed by numerical integration. In practice, we solve the system over a finite range [ω min , ω max ], which gives the bounds to be used in the integrals. One can then numerically invert (77) to obtain ω(T ) = log S.
Alternatively, we can compute the parametric curve (log T (ω), ω) with parameter ω which corresponds to the log-log plot (log T, log S), as shown in Figure 4. In such a plot, the freedom in the parameter c 1 can be thought as a freedom to move the curve left and right. 6 Furthermore, since the dynamical system is autonomous, meaning it does not depend on ω explicitly, every shift of a solution is also a solution, viz.
This means that the log-log plot can also be shifted up or down at will. The only freedom left is the choice in the initial conditions (x 0 , y 0 , z 0 ).
From (77), one can expressṠ andS as functions of ω, where t(ω) is the integrand in (77), and t ′ = d ω t. From this, one may compute the Hubble parameter H =Ṡ/S and its derivatives as functions of ω, as well as other quantities such as the tensor-to-scalar ratio r or the scalar spectral index n s . In principle, these could allow to further assess and restrict the viability of the models obtained in this way (beyond the mere number of e-foldings N ), but this goes beyond the scope of the present paper, although it would be interesting to explore these constraints in future work.
In the following four subsections we will study in depth four dynamical systems. The first two correspond to the two open FLRW cosmologies described in the Introduction. The one with negative internal curvature is studied in Section 6.1, whereas the one with nonvanishing Romans mass in Section 6.2. Both admit solutions with infinite or parametrically controlled number of e-foldings. The remaining two systems, studied in Sections 6.3 and 6.4, were chosen for the richness of their fixed-point structure, allowing for different interpolating solutions. All the other possible two-flux dynamical systems can be found in Appendix C, and are summarized in Table 4 below. Table 4: All possible two-flux dynamical systems. A ∅ indicates that the corresponding pair of fluxes does not relate to a possible subcase of (16).
Besides S and the invariant plane z = 0, the plane y = 0 is also an invariant surface. Since eqs. (80) are invariant under y → −y, we may restrict our attention to trajectories lying in the y, z ≥ 0 quadrant.
The condition for acceleration (71) reduces tö so that accelerated expansion occurs in the portion of the trajectory that lies in the upper half of the cone defined in (82). Moreover, from (72) we have so that w = −1 whenever the trajectory passes by the z-axis. Equations (82), (83) imply that the acceleration condition is equivalent to w < − 1 3 .

The critical points
For z > 0 there are two critical points of the system (80), given by Both p 1,2 lie on the invariant y = 0 plane. Moreover, p 1 lies on the boundary of the acceleration cone and in the interior of S, while p 2 lies outside the cone and on the boundary of S.
On the z = 0 plane, the origin p 0 = (0, 0, 0) is an isolated fixed point. In addition, we have an invariant circle of fixed points p C : the equator of the sphere S. These points require k, λ = 0 and correspond to the minimal solutions of Section 5.1.
The linearized system at p 1 has eigenvalues −1 (double) and −2. The corresponding eigenvectors are along the x and y directions, respectively. The linearized system at p 0 has eigenvalues −2 (double) and 1. The corresponding eigenvectors are along the x, y and z directions, respectively.
The critical points of the dynamical system correspond to solutions that can be given analytically: p 1 corresponds to the singular Milne universe given in (283); p 2 requires k = 0 (which is consistent with the fact that it lies on S) and corresponds to the critical solution with λ < 0, which is a decelerating power-law expansion. The origin p 0 requires λ = 0 (which is consistent with the fact that it lies on the z = 0 plane) and corresponds to the regular Milne universe of (181).

The invariant surface S
Restricting to trajectories on S, the system (80) implies It follows that the projections of all trajectories on S to the z = 0 plane are of the form i.e. straight lines passing by the point (x, y) = 2 3 , 0 , which is the projection of p 2 onto the z = 0 plane. These trajectories correspond precisely to the solutions of (268), and require k = 0. More specifically, the slope of the line (86) is related to the constants in (268) via The z = 0 invariant plane Restricting to trajectories on the z = 0 plane, the system (80) reduces to It follows that all trajectories on the z = 0 plane are of the form i.e. straight lines passing by the point (x, y) = (0, 0). These straight lines correspond precisely to the solutions of (176), and require λ = 0. More specifically, the slope of the line (89) is related to the constants in (176), (177) via The invariant plane y = 0 On the plane y = 0, the system (80) reduces to All trajectories of the reduced system are attracted by the stable node p 1 , except for the two trajectories on S which start at the two antipodal points (x, y, z) = ±(1, 0, 0) ∈ C, and end on either side of the unstable node p 2 .
Using a perturbative analysis, it is possible to obtain an analytic description of the trajectory connecting p 0 and p 1 near the critical endpoints. In the neighborhood of p 0 the system (91) admits the solution so that the (x, z) trajectory attains p 0 in the ω → −∞ limit, tangentially along the vertical direction (z-axis). Moreover the constraint (81) imposes It can be seen from (83), (92) that w → −1 as the trajectory tends to p 0 . However it would be incorrect to conclude that the solution becomes asymptotic de Sitter, since the p 0 point is reached at finite cosmological time in the past. Indeed this can be seen explicitly by reconstructing the metric corresponding to the solution (92), (93): taking into account the relation between dT and dω, cf. (63), we obtain Moreover from (74) and (92) we obtain the perturbative expression for A and z. Plugging into (94) and integrating we obtain 7 where we imposed T → 0 as ω → −∞. Taking (93) into account gives where we have set S 0 := e 4A 0 +B 0 , so that S = S 0 e ω , cf. below (75). Finally, inverting the perturbative series (97) we obtain It then follows from (10) that w = −1 + O(T 2 ), in agreement with our previous result. Indeed up to and including terms of order O(T 4 ) the spacetime metric becomes that of de Sitter space in hyperbolic slicing, where Λ is related to the scalar curvature R of de Sitter via Λ 2 = 12/R. This is not asymptotic de Sitter, however, as T = 0 is reached at finite proper time in the past, where the space becomes a regular Milne universe. The solution can thus be geodesically completed in the past to T < 0, by gluing together its mirror trajectory in the z < 0 region, cf. the comment in the paragraph preceding (71).
A similar analysis can be performed in the neighborhood of p 1 . In that case we obtain S → 2 |k| T , so that w → − 1 3 , as the trajectory approaches p 1 .

Comparison with the analytical solutions (280)
It follows from the system (80) that the ellipse, is

Potential and kinetic energies
The acceleration period can in fact be understood as a competition of the kinetic and potential energies of the system, as we would like to illustrate here.
Let us recall that, by comparing the energy-momentum tensor of a perfect fluid and that of a homogeneous scalar field φ, one can assign to the latter the following pressure and energy density, The acceleration condition w = p φ /ρ φ < −1/3 then translates tȯ i.e. there is acceleration whenever the potential energy dominates (twice) the kinetic energy.
In our models, the 2-field 4d potential has the shape of an exponential "wall", and the system can be thought in field space as thrown against that wall: initially the potential energy V is exponentially small and the kinetic energy K dominates. Eventually the system reaches the wall and starts climbing it, K decreases while V increases; when 2K = V , acceleration begins, which starts to significantly dissipates energy (the Hubble term in the EOM of φ acts a friction term). When the potential energy becomes too important, the trajectory undergo a turnaround and goes back down the wall; the system starts to accelerate again and inflation stops whenever 2K = V . Such a trajectory is depicted in Figure 9. More quantitatively, in that case study the 4d potential is given by To compute the kinetic energy, one first has to canonically normalize the fields, Then, using g τ τ = S(τ ) −3 = exp (−13A − 3B), the kinetic energy reads We further use that dω/dτ = √ −λ/z e 8A+3B , cf. (63), and to obtain The condition for accelerationS > 0 ⇔ V > 2K then precisely recovers the condition (111), which defines the boundaries of the acceleration region.

Case study II: k, m
We now turn to the case where k, m ̸ = 0. Here, the system of equations reduces to while the constraint reads The invariant plane P is given by the equation The acceleration condition reads The critical points are Both p 0 and p 1 lie on the boundary of the acceleration region and on the invariant plane. The point p 0 requires m = 0, and corresponds to the critical solution of (181); the point p 1 corresponds to the critical solution of (293).
The behavior close to the fixed point p 0 is similar to that of the (λ, k) system analyzed in the previous section. As the trajectory approaches p 0 , we have w → −1, and the solution becomes de Sitter-like. This however is not an asymptotic de Sitter, as p 0 is reached at finite proper time in the past. At p 0 spacetime becomes a regular Milne universe, and the solution can be geodesically completed in the past by gluing its mirror trajectory in the z < 0 region.
Close to the fixed point p 1 we may linearize and solve (108) analytically. The solution reads, up to terms of order O(e −2ω ), where c is an integration constant, and we have defined In addition, the constraint imposes It can be seen that, up to and including linear terms in ω, the solution (114) corresponds to the critical solution of (293), with τ ∼ e −2ω . The acceleration can also be calculated analytically, cf. (71) and Footnote 5, where to lowest order, ln T ∼ ω + const. The fixed point is reached as T → ∞.
The oscillations of the system can be captured for instance by the equation of state parameter w. Since these are exponentially damped, we rather consider the following quantity, plotted in Figure 10, which is positive whenever there is acceleration, and negative otherwise. As ω increases, the duration of these accelerated periods tends to 7 17 π in the variable ω.

Case study III: λ, m
We proceed with the case where λ, m ̸ = 0. Here, the dynamical system is given by together with the constraint For λ < 0, this forces the trajectories to lie within a unit ball in three-dimensional phase space. The invariant plane is given by The acceleration condition reads with, The critical points There is a unique critical point of the system (121) away from the z = 0 plane, given by The critical point p 1 lies on the invariant plane P. The linearized system at p 1 has one real and two complex conjugate eigenvalues: 16 19 The eigenvectors corresponding to the real and complex eigenvalues are orthogonal and parallel to the invariant plane P, respectively. It follows that for trajectories in P, p 1 is a stable focus; for trajectories orthogonal to P, p 1 is a stable node.
In addition we have a circle of fixed points on the equator C of the sphere S, and an isolated fixed point, which lies on the x-axis. The linearized system at p 2 has eigenvalues 2 3 (−5, −5, 1). The eigenvectors corresponding to the negative eigenvalue are along the x and y directions, whereas the eigenvector corresponding to the positive eigenvalue is along the z direction. It follows that for trajectories in the z = 0 plane, p 2 is a stable singular node; for trajectories orthogonal to the z = 0 plane, p 2 is an unstable node.
The critical points of the dynamical system correspond to solutions that can be given analytically: the point p 1 corresponds to the solution (286). The point p 2 corresponds to the critical solution with λ < 0, and requires m = 0. The critical points p C correspond to the minimal solution of Section 5.1, and require m, λ = 0.

The invariant surface S
Restricting to trajectories on S, the system (121) implies It follows that the projections of all trajectories on S to the z = 0 plane are of the form i.e. straight lines passing by the point (x, y) = 1 2 , − 5 2 √ 3 . These trajectories correspond precisely to the solutions of (275), and require λ = 0. More specifically, the slope of the line (130) is related to the constants in (275) via The z = 0 invariant plane Restricting to trajectories on the z = 0 plane, the system (121) reduces to It follows that all trajectories on the z = 0 plane are of the form i.e. straight lines passing by the point (x, y) = 2 3 , 0 . These straight lines correspond precisely to the solutions of (268), and require m = 0. More specifically, the slope of the line (133) is related to the constants in (268) via The invariant plane P On P, the system (121) reduces to All trajectories of the reduced system spiral into the stable focus p 1 .

Case study IV: φ, χ
Let us finally consider the case where φ, χ ̸ = 0. The system of equations is given by together with the constraint, The invariant plane is defined by and the acceleration condition is simply given by so that a trajectory undergoes acceleration whenever it passes above the z = This system admits no fixed points, apart from those lying on the equator C. Nevertheless, it is worth being discussed, in view of its connection with analytic solutions.
As argued previously, the particular case φ = 0 and χ = 0 corresponds to one of the fixed points on C, and coincide with the minimal solution (33). When only φ is turned on, the solutions are restricted to the boundary S of the phase space (and their projection on the z = 0 plane are straight lines); if both fluxes φ and χ are turned on, trajectories live generically in the bulk and connect two fixed points of C. In the former case, the trajectory that maximizes the number of e-foldings N is the one passing by the North pole: it obviously maximizes its length inside the acceleration region, but also turns out to maximize its "time" spent inside the region (which is not necessarily the same). Numerically, this maximal number of e-foldings can be determined to be N max = 0.30408. Now, let us turn to the corresponding analytic solutions (184) found for φ ̸ = 0, χ = 0. There exists a family of solutions parameterized by a real number 8 r ≤ − 11 16 , with the following scale factor, Having the explicit scale factor S at hand allows us to compute the derivativesṠ,S and determine the values of τ for which accelerated expansion starts and stop (or in other words determine the values of τ for which one can satisfy bothṠ > 0 andS > 0), see Figure 11. We can then readily compute the number of e-foldings N (r) and extremize it with respect to r, see Figure 12. The maximal value is reached for r max = − 9 8 , which precisely gives the same N max = 0.30408 as above, and is of course in line with the bound N max ≤ 0.59980 found in Section 5.2 for one-flux compactifications.

Conclusions
It has been known for some time that transient accelerated expansion is not difficult to achieve in flux compactifications arising from string-theory effective 10d supergravities. What our work is suggesting is that cosmologies featuring eternal or semi-eternal acceleration, or a parametric control on the number of e-foldings are also generic! The necessary ingredients in all instances thereof seem to be a negative spatial 4d curvature (open universe), and a fixed point on the boundary of the acceleration region in the interior of the phase space.
To our knowledge, this is also the first time where examples of spiraling cosmologies with an infinite number of cycles alternating between accelerated and decelerated expansion have been shown to arise from compactification of string-theory effective 10d supergravities. This so-called "rollercoaster cosmology" has been argued to be a potentially viable alternative for inflation [45].
In all examples exhibiting a parametric control on the number of e-foldings, the acceleration vanishes asymptotically at future infinity, where spacetime approaches a Milne universe with angle defect. Moreover, these are all captured by the 4d consistent truncation (22) with potential given in (25), and require turning on one of the parameters λ, m, c φ , c 0 or c f . Smooth accellerating cosmologies, without Big Bang singularities, are also possible, and they correspond to unique fine-tuned (therefore unstable) trajectories in phase space.
Instead of a singularity at T = 0, the spacetime approaches de Sitter space (in hyperbolic slicing). This, however, is not an asymptotic de Sitter, as T = 0 is reached at finite proper time. These solutions can be geodesically completed to T < 0 in the past, as explained previously.
As we have shown, the techniques of the present paper allow us to straightforwardly translate the trajectories in phase space to the explicit form of the scale factor S(T ) of the corresponding FLRW solution, as a function of cosmological time. From this, the other cosmological observables can all be readily computed.
Our approach has been to work with the 10d equations of motion, not with a 4d effective potential. Nevertheless, in certain cases a graviton plus two-scalar (dilaton and warp factor) consistent truncation to a 4d theory S 4d is possible, such that all cosmological solutions of S 4d lift to ten-dimensional solutions of IIA supergravity. This does not mean that solutions of S 4d uplift to 10d solutions with only dilaton and warp factor: all information about the flux (which is generically non-vanishing from the ten-dimensional point of view) enters the 4d potential of S 4d via certain constants.
The action S 4d is in fact a consistent sub-truncation of the universal CY consistent truncation of [46,47]. Indeed, it was shown in those references that the 4d effective theory of the universal sector of CY type II compactifications is also a consistent truncation of 10d type II supergravity. We thus expect the consistent truncation S 4d of (22) to be part of the 4d effective action, and thus subject to e.g. the analysis and constraints presented recently in [37].
A general stability analysis of the cosmological solutions presented here would require considering (small) perturbations in the space of all 10d fields. We shall leave this important point for future work.
The dynamical system techniques as applied in the present paper, limit us to solutions with a maximum of two species of flux. It would be desirable to overcome this limitation and explore richer solutions with all possible fluxes turned on, in each universal compactification class.
A lot of work has been done recently on classical de Sitter solutions using smeared orientifolds [48][49][50][51][52][53][54]. Setting aside the still unresolved conceptual issues associated with the latter, including smeared orientifolds in our analysis would certainly enrich the structure of the phase space of the dynamical systems presented here, potentially leading to the appearance of fixed points corresponding to de Sitter solutions. It would be interesting to explore this possibility further.

A Type IIA supergravity
The ten-dimensional IIA action with Romans mass m reads where S CS is the Chern-Simons term. The resulting equations of motion (EOM) read as follows: • Einstein EOM's, where it is understood that Φ 2 ..Mp , for any p-form Φ.
• Dilaton EOM, • Forms EOM's, Additionally the forms obey the Bianchi identities, B Analytic solutions

B.1 Compactification on Calabi-Yau manifolds
We will now search for analytic solutions of ten-dimensional IIA supergravity. We follow the ansatz of the consistent truncation of [46], but we work directly with the 10d equations of motion. We will furthermore restrict to the case without fermion condensates.
The ansatz for the ten-dimensional two-form F , three-form H and four-form G reads where φ, χ, ξ, ξ ′ are 4d scalars, α, γ are 4d one-forms, and β is a 4d two-form. We shall also introduce the 4d three-form h := dβ. The ansatz for the ten-dimensional metric in 10d Einstein frame reads ds 2 10 = e 2A(x) e 2B(x) g µν dx µ dx ν + g mn dy m dy n , where the scalars A, B only depend on the four-dimensional coordinates x µ , while y m are the CY coordinates. The equations of motion are as follows, coming from the internal (m, n)-components of the ten-dimensional Einstein equations. The external (µ, ν)-components give rise to while the mixed (µ, m)-components are automatically satisfied. The dilaton equation reduces to The F -form equation of motion reduces to the condition The H-form equation gives rise to two equations, and, The G-form equation reduces to together with which can be integrated to for some constant c φ .

Cosmological ansatz
We will now make a cosmological ansatz for all fields, i.e. one that is (in general) only invariant under the isometries of the 3d spatial part of the metric. We assume that all scalars are functions of the time coordinate t alone. The one-forms are assumed to be of the form for some scalars α, γ which may be time-dependent in general. With some abuse of notation, we have denoted by the same letter the one-forms and the corresponding scalars. In the following all equations of motion will be expressed exclusively in terms of the scalars, so hopefully no confusion will occur. The three-form h is assumed to be of the form where c h is a constant. The unwarped 4d metric (1) is assumed to be of the form of eq. (2), while the 4d Einstein frame metric is thus given by (4).
Substituting the ansatz above into the form equations of motion (152)-(157), we obtain the following system: eq. (152) reduces to Eq. (153) reduces to where c χ , d χ are constants. Eq. (154) is automatically satisfied. Eq. (155) reduces to where d t denotes the derivative with respect to t. Note that if eq. (162) is satisfied for some ξ, ξ ′ ̸ = 0, this implies for some arbitrary real constant c ξξ ′ .
The system (162) can be integrated to give for some constants c ξ , c ξ ′ . Let us define a new time variable ν by If c h ̸ = 0, the solution to (164) reads where d ξ is an arbitrary constant. If c h = 0, the solution to (164) reads instead where d ξ , d ξ ′ are are arbitrary constants. This then concludes the solution of all equations of motion for the forms.

Adapted time coordinate
In the following we will use the time variable τ defined in (6). The internal Einstein and dilaton equations, (149), (151) reduce to The external Einstein equations (150) reduce to The second line above is a constraint, consistently propagated by the remaining equations of motion. Indeed the τ -derivative of this equation is automatically satisfied by virtue of (168) and the first equation in (169).

IIB solution
Although the main focus of the present paper is on IIA supergravity, let us note that the minimal solution can be easily generalized to include a non-trivial axion C. The tendimensional Lagrangian, together with the ansatz (1) for the metric, now lead to the following equations of motion: for the axio-dilaton we have for some real constant c a . The Einstein equations reduce to for some real constants c A , c B , together with the constraint The solution for the axio-dilaton reads for some constants d a , d ϕ . Taking the above into account, (173) reduces to the constraint (30). It readily follows that, apart from the axio-dilaton, the IIB solution is identical to the minimal solution.
k ̸ = 0 As before, we will take  168), for some constants d A , d ϕ , exactly as for the minimal solution. From the first equation in (169), we obtain the solution for B, where, for some constants c B , d B . Moreover, the second equation in (169) (the constraint) reduces to an algebraic condition, for k of either sign. In particular this implies, r ≤ | 1 4 |, cf. (31). In the following we consider the two cases in (178) in more detail.

Type I solution: closed universe
Let us set k > 0. The 4d Einstein-frame metric reads where we have set d A , d B = 0 for simplicity. Since the sign of c B can be absorbed in the definition of τ , we may suppose c B ≥ 0 without loss of generality; the inequality is saturated in case c A , c B , c ϕ all vanish. In terms of coordinates T ± ∝ e ∓ 3 2 c B τ the metric asymptotically takes the form of (27), where a ± = 1 3 . For c B ≥ 0, τ → ±∞ corresponds to T ± → 0, where a singularity is reached at finite proper time.

Type II solution: open universe
Let us now set k < 0. The 4d Einstein-frame metric reads where we have set d A , d B = 0 for simplicity. Since the sign of c B can be absorbed in the definition of τ , we may suppose c B ≥ 0 without loss of generality.
The τ → ±∞ asymptotics of the warp factors are exactly the same as for k > 0. For τ → 0, on the other hand, the function f tends to f → − ln(4|k|τ 2 ), while the constraint imposes c A , c ϕ = 0. Hence in the τ → 0 limit the solution reads Note that comoving geodesics reach τ = 0 at infinite proper time. The metric becomes asymptotically that of a regular Milne universe, 9 where we have defined T = 2(c 3 B τ ) − 1 2 . The warp factor of the internal space and the dilaton are constant.
φ ̸ = 0 Let us now take while (169) reduce to the following two equations, To obtain an analytic solution we proceed as follows. We define f := −ϕ/2 + 6A + 6B, which by virtue of the equations of motion (185),(186) obeys, d 2 τ f = −2c 2 φ e f . The solution to this equation reads 9 Recall that the spatial 3d part of the metric is locally isometric to a hyperbolic space of scalar curvature 6k, cf. (3). An explicit parametrization is given by, with dσ 2 the line element of the two-sphere.
for some constants c ϕ , d ϕ . Plugging back into (185), and the first line of (186), we obtain the solution for some constants c A , c B , d A , d B , and f as given in (187). Moreover, the second line of (186) imposes the constraint which implies r ≤ − 11 16 . The 4d Einstein-frame metric reads where we have set d A , d B , d ϕ = 0 for simplicity.
Let us assume c ϕ ≥ 0, without loss of generality, since the sign of c ϕ can be absorbed in the definition of τ . In terms of coordinates T ± ∝ e (12c A +3c B ± 3 4 c ϕ )τ , the metric asymptotically takes the form of (27), where a ± = 1 3 . If c B > 0, τ = −∞ corresponds to T − = ∞ while τ = ∞ corresponds to T + = 0, where a singularity is reached at finite proper time. The situation is reversed for c B < 0: τ = −∞ corresponds to T − = 0, where a singularity is reached at finite proper time, while τ = ∞ corresponds to T + = ∞. For a certain range of parameters, this model exhibits transient accelerated expansion, whereṠ(T ),S(T ) > 0, cf. Figure 13. Critical solution: φ ̸ = 0, k < 0 Let us now consider k ̸ = 0. The equations of motion (168), (169) admit the following solution, for an arbitrary constant d A , and k, d ϕ subject to the conditions The 4d metric is that of a singular Milne universe with angular defect, cf. Footnote 9, where we set T ∝ |τ | − 1 2 and dΩ 2 k is the line element of a locally hyperbolic three-space. The warp factor of the internal space and the dilaton scale as e A ∝ T 3/28 , e ϕ ∝ T 1/7 respectively. Type II solution: φ ̸ = 0, k < 0 We will now construct a solution interpolating between future or past infinity τ → ±∞, and the above critical solution in the τ → 0 limit. Let us use an ansatz of the form for some (non-linear) function f and constants c, d, e. Moreover we will require that the equations of motion (168) and the first line of (169) reduce to a single differential equation for f . These requirements impose the following condition on the spatial curvature, (open universe) and, after some redefinitions of the constants, lead to the solution, where g = 1 112 ln The second line of the constraint then reduces to The 4d Einstein metric reads where we have set d A , d ϕ = 0 for simplicity. The metric asymptotically takes the form (27), where a ± = 1 3 . On the other hand, for τ → 0 we have In this limit the solution (196) thus takes the form given in eqs. (191), (192). In other words, for τ → 0 the solution asymptotes the critical solution. Comoving geodesics reach τ = 0 at infinite proper time.
Let us now take which still solves the form equations (152)-(157) in the case of χ ̸ = 0. Let us consider the remaining equations of motion. Equations (168) and the first line of (169) are solved by with f := ln while the second line of (169) reduces to The constraint imposes in particular, r ≤ − . The 4d Einstein metric reads where we have set d A , d B , and d ϕ = 0 for simplicity. In terms of the coordinate T ∝ e (12c A +3c B )τ , T ∈ [0, ∞), the metric takes the form (27), with a ± = 1 3 .
Type I solution : k > 0 Let us now consider k ̸ = 0. Equations (168) can be solved to give for some constants c A , d A , where Taking the above into account, the first of (169) can be solved for B, for some constants c B , d B . Plugging the solution into the second line of (169) we obtain the constraint for either sign of k.
Let us first consider the case of a closed universe (k > 0). The 4d metric reads, with d A , d B , d ϕ = 0 for simplicity, Type II and critical solutions: k < 0 Let us now set k < 0. The 4d Einstein metric reads ξ, ξ ′ ̸ = 0 Let us assume that eq. (163) is satisfied for some arbitrary real constant c ξξ ′ , and let us take This ansatz thus solves the form equations (152)-(157) in the case of ξ, ξ ′ ̸ = 0. Let us consider the remaining equations of motion. Equations (168) and the first of (169) are solved by with f := ln while the second line of (169) reduces to which implies in particular, r ≥ − 1 8 . The 4d Einstein metric reads Type I solution: k > 0 Let us now take k ̸ = 0. Equations (168) can be solved to give for some constants c A , d A , where g := ln Moreover the first of (169) can be solved for B, for some constants c B , d B . Plugging the solution into the second line of (169) we obtain the constraint for either sign of k, which imposes |r| ≤ 1 8 . Let us first consider the case of closed universe (k>0). The 4d Einstein metric reads, where we have set d A , d B , d ϕ = 0 for simplicity.
Type II and critical solutions: k < 0 Let us now set k < 0. The 4d part of the Einstein metric reads χ, ξ, ξ ′ ̸ = 0 Let us now take, This ansatz thus solves the form equations (152)-(157) in the case of χ, ξ, ξ ′ ̸ = 0. Let us consider the remaining equations of motion. Equations (168) and the first of (169) are solved by with f := ln while the second line of (169) reduces to which imposes |r| ≤ √ 3. The 4d Einstein metric reads where we have set d A , d B , and d ϕ = 0 for simplicity.

Type I solution: k > 0
Let us now assume that k > 0. Equations (168) are solved by with, Moreover the first equation of (168) can be combined with the first of (169) to solve for B, where for some constants c B , d B . Plugging the solution into the second line of (169) we obtain the constraint which imposes |r| ≤ √ 3 2 . Let us first consider the case of a closed universe (k > 0). The 4d Einstein metric reads where we have set d A , d B , and d ϕ = 0 for simplicity.
Type II and critical solutions: k < 0 Let us now set k < 0. The 4d Einstein metric reads h, χ, ξ, ξ ′ ̸ = 0 Let us take This ansatz thus solves the form equations (152)-(157) in the case of h, χ, ξ, ξ ′ ̸ = 0. Let us consider the remaining equations of motion. Equations (168) are solved by with Moreover the first equation of (168) can be combined with the first of (169) to solve for B, where, for some constants c B , d B . Plugging the solution into the second line of (169) we obtain the constraint for either sign of k. Let us first consider the case of a closed universe (k > 0). The 4d Einstein metric reads where we have set d A , d B , d ϕ = 0 for simplicity.
Type II and critical solutions: k < 0 Let us now set k < 0. The 4d Einstein metric reads

Compactification with background flux
The cosmological ansatz can be easily modified to accommodate a non-vanishing background flux for the three-and four-forms, as in [47], 10 with background charges b 0 , c 0 ∈ R, where the covariant derivative is defined as: Dξ ′ = dξ ′ + b 0 α. The internal (m, n)-components of the Einstein equations now read 10 We have redefined: ξ → c0 + 4ωξ with respect to [47].
The external (µ, ν)-components read while the mixed (µ, m)-components are automatically satisfied. The dilaton equation reads The F -form equation of motion reduces to the condition The H-form equation reduces to The G-form equation of motion reduces to together with the constraint The latter can be integrated to solve for φ in terms of the other fields, where C is an arbitrary constant.
The second line above is a constraint, consistently propagated by the remaining equations of motion. Indeed the τ -derivative of this equation is automatically satisfied by virtue of (254).
Equations (254) and the first of (255) are solved by with, while the second line of (255) reduces to which imposes |r| ≤

B.2 Compactification on Einstein manifolds
We will now consider (massive) IIA backgrounds for which the internal 6d manifold is Einstein, where R mn is the Ricci tensor associated to g mn , and λ ∈ R. The 10d metric is as in (1). We assume the following form ansatz, where φ is a 4d scalar.
The resulting equations of motion are as follows. The internal (m, n)-components of the Einstein equations read where m is the Romans' mass. The external (µ, ν)-components read while the mixed (µ, m)-components are automatically satisfied. The dilaton equation reads The F , H-form equations are automatically satisfied, while the G-form equation of motion reduces to φ = c φ e −2A+4B−ϕ/2 , where c φ is an arbitrary constant.
As before, we will assume that the unwarped 4d metric is of the form (2) and moreover that A, B, ϕ only depend on time. Eq. (262) and the dilaton equation (264) reduce to The external Einstein equations (263) reduce to the following two equations, The second line above is the constraint, consistently propagated by the remaining equations of motion.
For m, k, φ = 0, the system of equations in (266) and the first of (267) can be solved to give , for some constants c B , d B . Plugging the solution into the second line of (267) we obtain the constraint for either sign of λ. The constraint imposes |r| ≤ 3 10 .
Type I solution: λ > 0 Let us first consider an internal space of positive curvature (λ > 0). The 4d Einstein metric (1) reads where we have set d A , and d ϕ = 0 for simplicity. To examine the asymptotic behavior of the metric it suffices to consider c B ≥ 0 (the other cases are obtained by inverting the sign of τ ). The metric asymptotically takes the form of (27), where a ± = 1 3 .
Type II and critical solutions: λ < 0 Let us now set λ < 0. The 4d Einstein metric reads The τ → ∞ asymptotics of the warp factors are the same as for λ > 0. For τ → 0, on the other hand, the function g in (269) tends to g → − ln(5|λ|τ 2 ). Moreover, the constraint imposes c A , c ϕ = 0. Hence, in the τ → 0 limit the solution reads Asymptotically, at infinite proper time, the metric reaches the form where we have defined T ∝ τ − 4 5 . The warp factor of the internal space scales as e A ∝ T 1/4 , while the dilaton is constant. This is also an exact solution of the theory in its own right. For a certain range of parameters, this model exhibits transient accelerated expansion, wherė S(T ),S(T ) > 0, cf. Figure 15. Type I solution: m ̸ = 0 Setting k, λ, φ = 0, the system of equations in (266) and the first of (267) can be solved to give for some constants c A , d A , c B , and d B where for some constant d ϕ . Plugging the solution into the second line of (267) we obtain the constraint 25 12 which imposes r ≤ − 13 24 or r ≥ − 1 8 . The 4d Einstein metric (1) reads where we have set d A , d B and d ϕ = 0 for simplicity. To examine the asymptotic behavior of the metric it suffices to consider c ϕ ≥ 0. The metric asymptotically takes the form (27), where a ± = 1 3 . For a certain range of parameters, this model exhibits transient accelerated expansion, whereṠ(T ),S(T ) > 0, cf. Figure 16. λ, k ̸ = 0 Assuming B is constant (we may set B = 0 for simplicity) and setting the system of equations in (266) and the first of (267) can be solved to give B = 0, ϕ = c ϕ τ + d ϕ , for some constants c ϕ , d ϕ , and for some constants c A , d A . Plugging the solution into the second line of (267) we obtain the constraint for either sign of λ.
Type I solution: λ > 0 Let us first consider the case λ > 0. The 4d Einstein metric reads where we have set d A , and d ϕ = 0 for simplicity, and we took (279) into account. For τ → ±∞ the metric asymptotically takes the form of a power-law expansion (27) for a ± = 1 3 .
Type II and critical solutions: λ < 0 Let us now set λ < 0. The 4d Einstein metric reads where we took (279) into account. The τ → ∞ asymptotics of the warp factors are the same as for λ > 0. For τ → 0, on the other hand, the function A in (280) tends to A → − 1 16 ln(8|λ|τ 2 ). Moreover, the constraint imposes c ϕ = 0. Hence, in the τ → 0 limit, which is reached at infinite proper time, the solution reads The metric asymptotes a singular Milne universe with angular defect, where we have defined T ∝ τ − 1 2 . The warp factor of the internal space scales as e A ∝ T 1/4 , while the dilaton is constant. This is also an exact solution of the theory in its own right.
Critical solution: λ, m ̸ = 0 Setting k, c φ = 0, the system of equations in (266), (267) admits the solution for arbitrary constants d A , d B , d ϕ subject to the conditions The 4d Einstein metric takes the form where T ∝ τ − 25 32 . The warp factor of the internal space and the dilaton scale as e A ∝ T 1/4 , e ϕ ∝ T −1/5 respectively.
Type I solution: φ, m ̸ = 0 Setting c 2 φ = m 2 , the system of equations in (266) and the first of (267) can be solved to give for some constants c A , d A , c B , and d B where for some constant d ϕ . Plugging the solution into the second line of (267) we obtain the constraint The 4d Einstein metric reads where we have set d A , and d ϕ = 0 for simplicity. The metric asymptotically takes the form (27), with a ± = 1 3 . For a certain range of parameters, this model exhibits transient accelerated expansion, whereṠ(T ),S(T ) > 0, cf. Figure 17. Type II and critical solution: k, m ̸ = 0 Setting k = − 3 4 m 2 , the system of equations in (266) and the first of (267) can be solved to give for some constants c A , d A , c ϕ and d ϕ . Plugging the solution into the second line of (267) we obtain the constraint The scale factor reads In the τ → 0 limit, which is reached at infinite proper time, the solution reads The metric asymptotes a singular Milne universe with angular defect, where we have defined T ∝ τ − 1 2 . This is also an exact solution of the theory in its own right.
We thus obtain a singular Milne universe, for A 0 , B 0 , ϕ 0 arbitrary real constants and We thus obtain a singular Milne universe, AdS solution: k, λ, m, φ ̸ = 0 A different analytic solution is obtained by setting A, ϕ = 0, and λ = m 2 , c φ = ± √ 5m. Equations (266), (267) then reduce to These can be integrated to give for some real constants c B , d B . The 4d Einstein metric reads where we set d B = 0 for simplicity. This is an AdS 4 metric in conformal coordinates. To see this, we may define a new coordinate T , such that cos T = sech(c B σ), in terms of which the metric takes the well-known form, ds 2 4E = 2 m 2 (−dT 2 + |k| cos 2 T dΩ 2 k ) .
λ, φ ̸ = 0 Setting k, m = 0, the system of equations in (266) and the first of (267) can be solved to give for some constants c A , d A , where for some constants d B , d ϕ . Plugging the solution into the second line of (267) we obtain the constraint for either sign of λ. The constraint allows any r ∈ R.
Type I solution: λ > 0 Let us first consider the case λ > 0. The 4d Einstein metric (1) reads where we have set d A , d ϕ and d B = 0 for simplicity. To examine the asymptotic behavior of the metric it suffices to consider c φ ≥ 0. The metric asymptotically takes the form (27), with a ± = 1 3 .
The metric takes the form ds 2 → −dT 2 + T where we have defined T ∝ τ − 4 5 . This is not an exact solution of the theory, unless c φ = 0. In this case we recover precisely the critical solution with λ < 0. Hence the model interpolates between two metrics of the form (27), one with a = 1 3 for |τ | → ∞, and one with a = 3 for τ → 0. For a certain range of parameters, we can have transient accelerated expansion, whereṠ(T ),S(T ) > 0, cf. Figure 18.

B.3 Compactification on Einstein-Kähler manifolds
We will now consider (massive) IIA backgrounds for which the internal 6d manifold is Einstein-Kähler, R mn = λg mn , where R mn is the Ricci tensor associated to g mn , and λ ∈ R. In addition there is a real, closed Kähler two-form J, dJ = 0.
Let us first consider the massless limit, and we take the form ansatz, The external (µ, ν)-components read The F -form equation of motion reduces to the condition The H-form equation reduces to d e −ϕ+4A+2B ⋆ 4 dχ = c 0 φ vol 4 .
The G-form equation of motion reduces to together with the constraint The latter can be integrated to solve for φ in terms of the other fields, where C is an arbitrary constant.
for A 0 , B 0 , ϕ 0 arbitrary real constants and We thus obtain a power-law, flat universe expansion, where we have set T ∝ |τ | − 11 16 . The warp factor of the internal space and the dilaton scale as e A ∝ T 1/4 , e ϕ ∝ T 1/11 respectively.
(339) c f ̸ = 0 A different ansatz with non-vanishing two-form flux is also possible, where c f is a constant. This automatically satisfies the Bianchi identities and the form equations of motion provided φ = c φ e −2A+4B−ϕ/2 , for some constant c φ . The remaining equations of motion are as in (13), (14), with potential given by φ, k ̸ = 0 Acceleration condition, 1 3 > x 2 + y 2 .
The point p 1 lies on the invariant plane, outside the acceleration region.