Accelerated Universes from type IIA Compactifications

We study slow-roll accelerating cosmologies arising from geometric compactifications of type IIA string theory on $T^{6}/(\mathbb{Z}_{2}\,\times\,\mathbb{Z}_{2})$. With the aid of a genetic algorithm, we are able to find quasi-de Sitter backgrounds with both slow-roll parameters of order $0.1$. Furthermore, we study their evolution by numerically solving the corresponding time-dependent equations of motion, and we show that they actually display a few e-folds of accelerated expansion. Finally, we comment on their perturbative reliability.


Introduction
Since the end of last century, various cosmological observations have provided evidence for the existence of dark energy. Combined measurements coming from supernovae [1], the Cosmic Microwave Background (CMB) radiation [2] and the Baryonic Acoustic Oscillations (BAO) [3] have lead to the conclusion that we live in a universe with positive and small cosmological constant. The energy/matter content giving the best fit gave rise to the socalled concordance model of cosmology.
However, despite the increasing precision of the cosmological data released in the past decade by WMAP [4], and most recently by Planck [5], we are still unable to exclude that such dark energy is described by a quintessence rather than a cosmological constant. Indeed, the current best fit coming from Planck + WMAP9 + eCMB + BAO + SNe for the w ≡ p/ρ parameter is [6] w(a) = w 0 + w a (1 − a) , with In parallel to this development, it has turned out to be remarkably difficult to construct rigorous models in string theory with perturbatively stable de Sitter (dS) vacua. To find dS solutions one typically needs to introduce ingredients that are not well under control. This includes KKLT [7], where non-perturbative effects play an important role. It is the aim of the present paper to study what happens if you relax the requirement of time independence of the cosmological constant, and instead investigate quintessence scenarios with quasi dS solutions, i.e. accelerated expanding solutions which are slowly decaying in time. Apart from serving the purpose of describing the late-time accelerated phase of the universe, this class of models has also been considered in the attempt of describing inflation within string theory [8][9][10] and supergravity [11][12][13][14][15][16].
Focusing on flux compactifications of type IIA string theory, one of the first examples of moduli stabilisation at large volume and small string coupling at a classical level can be found in ref. [17], where it was achieved by means of a Calabi-Yau compactification with NS-NS 3-form flux, R-R fluxes, D6-branes and O6-planes. Unfortunately, the corresponding vacua exhibit negative values of the cosmological constant.
When concentrating on dS solutions in the context of type IIA flux compactifications on a six-torus with D6-branes and O6-planes, the above setup turns out to fall into a no-go theorem [18] providing a lower bound of O(1) for the first slow-roll parameter, thus ruling out dS vacua and quasi-dS solutions. A possible way to circumvent this result would be to replace the flat six-torus by a negatively curved internal manifold, i.e. by adding metric flux.
A lot of work has been done in this context by making use of the underlying minimal supergravity description in four dimensions [19][20][21][22][23][24][25][26][27]. However, many indications were found that not only are there no stable dS vacua in this setup, but also no quasi-dS solutions, at least not in the isotropic sector of the theory. Here, the best one can achieve is a family of unstable dS solutions with second slow-roll parameter of O(−1), which were studied in refs [28][29][30][31][32][33][34] Very recently solutions that display cosmic acceleration were presented in [36] where noncritical strings were considered, and in [37] where discrete Wilson lines among other objects were used.
In the present work, we would like to stress that, by exploring the non-isotropic sectors of the N = 1 supergravity theories arising from geometric type IIA compactifications with O6/D6 on T 6 /(Z 2 × Z 2 ), it is actually possible to find flux backgrounds describing quasi-dS cosmologies with slow-roll parameters both of O(.1). 2 These solutions are hence about 2σ away from the current cosmological data presented in (1.1). The method used to find these solutions were by the means of a genetic algorithm which is very similar to the one designed and presented in ref. [35].
The paper is organised as follows. In section 2, we review the details of type IIA flux compactifications T 6 /(Z 2 × Z 2 ) with D6-branes and O6-planes, and give a description thereof in terms of superpotential deformations of minimal supergravity in four dimensions. In 1 In [35] perturbatively stable dS solutions were found provided that non-geometric fluxes were introduced. 2 This distinguishes our approach from [38] where it is described how acceleration can be achieved beyond a slow-roll regime. section 3, we introduce the study of the dynamics of a set of scalar fields coupled to gravity in a FRW-like background, and discuss the corresponding equations of motion. In section 4, we first present the explicit flux backgrounds realising slow-roll cosmologies and subsequently, after solving the equations of motions, we analyse their time-dependent evolution and show that they exhibit a few e-folds of accelerated expansion. In section 5, we discuss some related issues such as perturbative control and scale separation. Finally, we add our conclusions in section 6.
2 Geometric type IIA compactifications with O6/D6 Reductions of type IIA string theory on a twisted T 6 with fluxes, and one single O6-plane, have been extensively studied in the literature. Such orientifold planes split the space-time coordinates into transverse and parallel directions as follows and can be located at the fixed points of the following Z 2 involution σ : (y 1 , y 2 , y 3 , y 4 , y 5 , y 6 ) −→ (y 1 , y 2 , y 3 , −y 4 , −y 5 , −y 6 ) , where {y i } denote the six-dimensional compact coordinates.
According to (2.1), all the fields and fluxes arising from such a compactification undergo a discrete truncation only retaining parity-allowed objects. This gives rise to an effective fourdimensional supergravity theory with 16 supercharges. In what follows we will see how to use the orbifold symmetry in order to reduce the amount of supersymmetry of the resulting supergravity down to N = 1 in four dimensions.

(2.2)
At the fixed points of {σθ 1 , σθ 2 , σθ 1 θ 2 } one can locate a triplet of O6-planes placed as Dividing out by this extra discrete orbifold symmetry further truncates the theory to a minimal supergravity in four dimensions.
Orbifold compactifications of type IIA string theory on T 6 /(Z 2 × Z 2 ), with O6-planes (and duals thereof) and generalised fluxes, can all be placed within the same framework as the four-dimensional supergravity models that are known as ST U -models. These theories enjoy N = 1 supersymmetry and SL(2) 7 global bosonic symmetry. The action of such a global symmetry on the fields and couplings can be interpreted as the effect of string dualities.
The scalar sector contains seven complex fields spanning the coset space (SL(2)/SO(2)) 7 , which we denote by Φ α ≡ (S, T i , U i ) with i = 1, 2, 3. The kinetic Lagrangian follows from the Kähler potential This yields The presence of fluxes induces a scalar potential V for the moduli fields, which is given in terms of the above Kähler potential and a holomorphic superpotential W by where K αβ is the inverse Kähler metric and D α denotes the Kähler-covariant derivative.
The general form of a superpotential induced by geometric fluxes in type IIA with O6planes is given by where P 1 , P 2 and P (k) 3 are (up to) cubic polynomials in the complex structure moduli given by The IIA flux interpretation of the above superpotential couplings is summarised in table 1, from which one can see that the total parameter space of geometric fluxes in this duality frame consists of 24 superpotential couplings. However, in order to have a twisted torus interpretation, one needs to impose the following Jacobi constraints where A, B, . . . are fundamental SL(6) indices, which are split into {a, m} by the orientifold involution. This restricts the number of independent metric flux parameters from 3 + 9 = 12 down to 3 6.
couplings Table 1: Mapping between fluxes and couplings in the superpotential in type IIA with O6planes. The six internal directions of T 6 are split into " − " labelled by a = 1, 2, 3, and " | " labelled by m = 4, 5, 6 ( i.e. parallel and transverse to O6 || respectively). Note that the orbifold involution forces i, j, k to be all different any time they appear as indices of fields of the same type (T or U ).

Analysis of the scalar dynamics
In this section, we will analyse the dynamics of the fourteen scalars generically obtained from the class of type IIA backgrounds described in section 2 coupled to gravity. This will lead us to the derivation of a system of coupled differential equations, which generalises what is normally obtained in the case of single-field inflation.
The sector of the four-dimensional supergravity theory representing the coupling of scalars with gravity is described by the following Lagrangian where I = 1, · · · , 14 and K IJ is derived from the Kähler metric K αβ when rewriting the seven complex fields introduced in section 2 in terms of their real degrees of freedom according to On an FRW-like background with the metric defined as and after choosing a time-dependent profile for all the scalars, one can rewrite the Lagrangian The corresponding Euler-Lagrange equations read where we have introduced the Hubble parameter H ≡˙a a .
In order to write down the Einstein equations describing the dynamics of the metric we need to introduce the energy density ρ and the pressure p parametrising the stress-energy tensor in the rest frame In terms of the above quantities, one can write the Einstein equations in the form of the

7)
4 Note that in order to be consistent with the supergravity potential given in (2.5) we will have to set the reduced Planck mass equal to 1, i.e. M 2 where the first equation can be essentially interpreted as a Hamiltonian constraint while the second actually describes the dynamics of the scale factor a(t).
Moreover, the stress-energy tensor defined in (3.6) should obey a conservation law which can be written in the form of the continuity equatioṅ Such a condition happens to be already implied by the equations of motion for the scalars Finally this means that the full set of equations describing the dynamics of the 14 scalars coupled to FRW gravity is given by which consists of 15 coupled differential equations in the 15 unknown functions a(t), φ I (t) , appearing up to first and second order respectively.

Slow-roll accelerated expansion
As we have briefly discussed in the introduction, both inflation and dark energy require a (quasi-)dS phase describing a universe in accelerated expansion. The preferred regime in which one can construct such a cosmological model is the so-called slow-roll approximation.
This essentially corresponds to a situation where the kinetic energy inside (3.6) is negligible w.r.t. the potential term V (φ). This already intuitively results in scalar fields which vary quite slowly in time while the scale factor expands exponentially.
Explicitely, the validity of the slow-roll approximation is encoded in the first and second slow-roll parameters in the following way: 1 and |η| 1. Generically these slow-roll conditions can be translated into properties of flatness of the scalar potential through However, whenever such an accelerated expanding phase is driven by more scalars, the conditions (3.11) are not sufficient for guaranteeing the existence of such a phase. The typical problem that one can run into in this case is an V parameter mainly generated by a scalar with a big and positive mass combined with the presence of a nearly-flat tachyon driving η V . In order to overcome this generic problem, one needs to require that all the directions of the scalar manifold significantly contributing to V have a nearly-flat normalised mass. In particular, if one defines the direction of rolling as 12) and the projection of the mass matrix along such a direction as one would also desire |η proj. | to be as small as possible.
In the next section we will present some particularly interesting type IIA string theory backgrounds which have values for V , η V and η proj. all about O(.1) or slightly smaller. After discussing some issues related to the choice of suitable initial conditions, we will then solve the system of differential equations given in (3.9) with the aid of numerical methods and show that they actually provide some e-folds of accelerated expansion. Thus, the only thing that needs to be fixed carefully is the first time-derivative of the scalars at the initial time.

The explicit accelerated models
Since we are looking for a slow-roll accelerated dynamics, we have chosen these velocities to be exactly those of slow-roll. This is achieved by imposing the conditions in (3.9) at t = 0 in the absence of the terms withφ andφ 2 , i.e.  Table 2: Values of the normalised energy, first and second slow-roll parameters, and the projection of the mass matrix along the direction of rolling for the four best solutions which were found. Here we adopt the definitionγ ≡ |DW | 2 3|W | 2 given in ref. [39]. The values in the last column represent the averaged sGoldstino mass as defined in ref. [14]. In none of the cases the supersymmetry breaking scalars seem to significantly contribute to the dynamical process of accelerated expansion.

This yieldsφ
Clearly, this choice of initial condition is only possible when V is positive in the origin.

Time evolution and plots
By performing the aforementioned choice of initial conditions, one puts the system in a slowroll regime at t = 0 and lets the system evolve in time to see how long it actually stays in this phase of accelerated expansion before decaying. We have solved the coupled differential equations (3.9) for our system in all the cases presented in tables 4-5 by means of numerical methods. The corresponding resulting plots are collected in figures 1 and 2.
As can be seen from figure 1, all of the cases show an accelerated expansion phase and the corresponding amount of e-folds for each solution are given in table 3. Figure

Discussion
All the cosmologies presented in the previous section are obtained as valid four-dimensional supergravity solutions arising from geometric compactifications of type IIA string theory.
However, one would like to directly address the issue of their reliability as honest solutions of perturbative string theory together with the question of scale separation in order to meet other physical requirements. All of this could be in principle be achieved in a regime in which the following conditions are met (where R = ρ 1/2 denotes the compactification radius): • Large volume approximation, i.e. √ α R

Perturbative control and separation of scales
In the following we will use a ten dimensional string metric of the form where τ and ρ are the so called universal moduli. Compactifying down to four-dimensions gives rise to a four dimensional Planck mass given by M 2 Pl = R 6 τ −2 g 2 s (α ) 4 . If we want to end up in Einstein frame with M Pl = 1, we need to pick τ such that where vol 6 is the internal volume, φ is the ten-dimensional dilaton. The various flux-induced terms in the scalar potential scale as [18] where V Fp represents the various contribution to the vacuum energy coming from the p-form field strength fluxes in type IIA, p = 0, 2, 4, 6 and h 3 , ω and f p denote the corresponding flux quanta.
In particular, by scaling the universal moduli according to ρ ∼ N and τ ∼ N δ and the flux quanta as where N is a very large number and α and δ are suitable positive numbers, one finds where |V | ∼ H 2 M 2 Pl . In the case of vanishing metric flux this regime was achieved in ref. [17] 5 . In our more general setup, one can still go into a perturbative regime, but then an arbitrarily small internal curvature (i.e. metric flux) would be needed in order to achieve separation of scales at the same time. Whether this is possible still remains to be seen 6 .
Moreover, we have so far neglected the issue of tadpole cancellation. Whenever O6-planes and D6-branes are added to these compactifications, they induce the following cross-term in the scalar potential where we must make sure that the net charge N 6 is cancelled by fluxes in the following way The quantity N 6 is quadratic in the flux quanta, and will, for that reason, generically be very large in the supergravity limit unless one performs some careful finetuning. It is useful to distinguish bewteen three different cases: • N 6 > 0 : This is unproblematic. High flux quanta is accompanied by large N 6 and a large number of D6-branes.
• N 6 = 0 : All flux quanta can be kept very high as long as they satisfy f 0 h 3 + f 2 ω = 0.
No sources at all need to be added in this case and the corresponding contribution inside the scalar potential is just absent.
• N 6 < 0 : This corresponds to a net orientifold configuration. In this case one cannot make |N 6 | arbitrarily large since it is bounded by the number of fixed points of the corresponding orientifold involution, which is typically a small number, say O(1).
This last case turns out to be a bit troublesome, not only because it would require an enormous finetuning to combine very large flux quanta into N 6 ∼ O(1), but also because such a contribution to the energy would tend to disappear in the supergravity limit. This effect would be due to the scaling behaviour for large M Pl , under which the local source term would go to zero while all the others stay finite. This somehow suggests that orientifold planes are intrinsically flux-quantisation sensitive objects and cannot readily be seen by supergravity.
All our solutions fall in this third category and hence suffer from the same problem. In the next section, we will show that in fact any slow-roll accelerated quasi-dS background requires N 6 < 0, and hence the presence of O-planes. 6 We thank Thomas Van Riet for pointing this out to us.

Slow-roll accelerated solutions need O-planes
The fact that dS solutions imply the presence of O-planes is well-known in the literature (see e.g. how the technique introduced in ref. [18] was used to show this in four dimensions [29,40] and higher [41]). This can be seen very easily by writing V as In the case of quasi-dS backgrounds, the term in (5.8) proportional to the equation of motion for τ no longer needs to be zero. Thus, in order to extend this result to this more general case, we need to relate this term to the first slow-roll parameter V introduced in (3.11). One finds 1 4 This implies and hence still N 6 < 0 whenever V > 0 and V < 1.

Summarising
By switching to the string frame 7 , one can relate the scaling behaviours presented in equation (5.3) to the following dependences on the KK radius R, α and g s (5.11) The above scaling behaviours show that it is possible to achieve the large volume approximation and small string coupling at the same time and, in this regime, all terms in the scalar potential become very small compared to the Planck scale. However, in order to keep them finite, one needs to choose very high flux quanta. This is what one would correctly expect in the supergravity description, where one should not be able to see flux quantisation. 7 Please note that the string frame can be obtained by exchanging τ for g s via τ = 1 gs ρ 3/2 . In this frame, the string length s ∼ √ α rather than the Planck scale is kept constant. Accordingly, the Planck length Pl ∼ 1/M Pl will now scale dynamically.
It is reassuring to see that one can achieve scale separation from (5.4) according to in particular, it would also be much smaller than m 3/2 .
The issue of quantisation, and the lower limit on N 6 , is something that possibly could put the supergravity approach into doubt. Note, though, that the usual loop corrections are of order g 2 s V [42], and thus under control.

Conclusions
We have studied the problem of cosmic accelerated expansion in type IIA backgrounds with metric and gauge fluxes in the context of Z 2 × Z 2 orbifold compactifications. With the aid of a genetic algorithm, we were able to find backgrounds with both slow-roll parameters within order of 10%. Subsequently, by explicitly studying the corresponding time-dependent dynamics in these cases, we have shown that there are some backgrounds developing an accelerated expansion phase lasting for a few e-folds.
When discussing the possibility of achieving perturbative control and separation of scales, we found that there are different cases to be analysed according to the possible different choices for the exponents α and δ scaling the fluxes in (5.4). The most promising case for achieving scale separation without having metric flux going to zero, seems to be when δ = α.
In order to avoid the tadpole N 6 to grow arbitrarily high (as N α−1 ), one would need some finetuning for cancelling this leading divergent contribution between the two different terms appearing in (5.7). This same finetuning will also make the cosmological constant small compared to the Planck scale and simultaneously separate all the other scales. In addition, loop-corrections are estimated to be under control in such a situation, since their size is roughly given by g 2 s V . To summarise, we find it extremely encouraging that the relaxation of the requirement of time independence, opens up new possibilities to look for cosmologically interesting solutions.
It is intriguing to note that the quasi-dS solutions we have found exhibit a sufficient number of e-foldings to be relevant for late time dark energy. Furthermore, through an appropriate choice of scaling we can tune down quantities such as the gravitino mass and the cosmological constant to quite small values, and achieve scale separation with respect to other important scales. With finetuning one should be able to separate these two scales and make the