Universal accelerating cosmologies from 10d supergravity Paul Marconnet andDimitrios Tsimpis

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Universal accelerating cosmologies

from 10d supergravity

Paul Marconnet and Dimitrios Tsimpis

Institut de Physique des Deux Inﬁnis de Lyon

Université de Lyon, UCBL, UMR 5822, CNRS/IN2P3

4 rue Enrico Fermi, 69622 Villeurbanne Cedex, France

marconnet@ipnl.in2p3.fr,tsimpis@ipnl.in2p3.fr

Abstract

We study 4d Friedmann-Lemaître-Robertson-Walker cosmologies obtained from

time-dependent compactiﬁcations of Type IIA 10d supergravity on various classes

of 6d manifolds (Calabi-Yau, Einstein, Einstein-Kähler). 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, corre-

spond 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 in-

ﬁnity, 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.

arXiv:2210.10813v4 [hep-th] 6 Aug 2023

Contents

1 Introduction and summary 2

2 The general setup 11

3 1d consistent truncation 13

4 Cosmological 4d consistent truncation 15

5 Analytic solutions 16

5.1 Minimal (zero-ﬂux) solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.2 Single-ﬂuxsolutions ................................ 18

6 Two-ﬂux solutions: dynamical system analysis 21

6.1 Case study I: λ, k .................................. 26

6.2 Case study II: k, m ................................. 31

6.3 Case study III: λ, m ................................ 33

6.4 Case study IV: φ, χ ................................. 35

7 Conclusions 37

A Type IIA supergravity 39

B Analytic solutions 39

B.1 Compactiﬁcation on Calabi-Yau manifolds . . . . . . . . . . . . . . . . . . . . 39

B.2 Compactiﬁcation on Einstein manifolds . . . . . . . . . . . . . . . . . . . . . . 54

B.3 Compactiﬁcation on Einstein-Kähler manifolds . . . . . . . . . . . . . . . . . 62

C Two-ﬂux dynamical systems 65

C.1 Compactiﬁcation on Calabi-Yau manifolds . . . . . . . . . . . . . . . . . . . . 65

C.2 Compactiﬁcation on Einstein manifolds . . . . . . . . . . . . . . . . . . . . . . 69

C.3 Compactiﬁcation on Einstein-Kähler manifolds . . . . . . . . . . . . . . . . . 73

1 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 R00 ≥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 no-

go and can lead to 4d Einstein-frame accelerated expansion for some period of time [5–10].

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 [1–3]: 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 con-

structed. Indeed, the eternal accelerating cosmologies of [16,17] are such that the accel-

eration 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)∼T1

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 ef-

fective theory with multiple scalar ﬁelds were studied in [20–22]. 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 [25–37].

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

1The no-go was also extended in [3] to the case of massive IIA supergravity, a theory which does not obey

the SEC.

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 equa-

tions: 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 de-

pend on a time coordinate. In other words, there is a 1d consistent truncation of the theory,

S1d. The ﬁelds in question are the dilaton ϕand 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 S1d, cf. Table 2.

Moreover we show that a two-scalar 4d cosmological consistent truncation, S4d, of the

10d theory to ϕ,Ais possible in certain special cases. In other words, every solution of cos-

mological Friedmann-Lemaître-Robertson-Walker (FLRW) type of S4duplifts to a solution

of the ten-dimensional equations of motion. Again, the diﬀerent ﬂuxes show up as constant

coeﬃcients in the potential of S4d. At least for a certain range of the parameters in the

potential, we expect our S4dto 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,2we are able to provide analytic cos-

mological 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 param-

eters. 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. [4,12,39–44] in particular are closely related

to the strategy employed here. One of the novelties of the present paper is to give an ex-

haustive 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 cor-

responding 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

2Here we use an extended notion of “ﬂux” that also includes non-trivial curvature, but excludes the ﬁelds

A,B,ϕ(the warp factors and the dilaton).

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)∼Ta, 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 trajec-

tory 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.

β1

β20 4 6

0∅ ∅

4∅ ∅

Table 1: Diﬀerent types of universal two-ﬂux compactiﬁcations and their corresponding

acceleration regions in the phase space. The values of β1,2– 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 ∅.

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Universalacceleratingcosmologiesfrom10dsupergravityPaulMarconnetandDimitriosTsimpisInstitutdePhysiquedesDeuxInfinisdeLyonUniversitédeLyon,UCBL,UMR5822,CNRS/IN2P34rueEnricoFermi,69622VilleurbanneCedex,Francemarconnet@ipnl.in2p3.fr,tsimpis@ipnl.in2p3.frAbstractWestudy4dFriedmann-Lemaître-Robertson-Wal...

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