On de Sitter vacua in O(d,d) invariant cosmology
Yang Liu Antonio Padilla Paul M. Saffin Robert G. C. Smith

Deep multi-prototype capsule networks
    [
    Received September 15, 1996; accepted March 16, 1997
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§ INTRODUCTION

Cosmological observations <cit.> indicate that our universe has undergone two phases of accelerated expansion, at early <cit.> and at late times <cit.>.  At early times we have inflation, consistent with the evolution of a scalar field in slow roll.  At late times we have dark energy, which may correspond to our universe approaching a de Sitter vacuum, or else quintessence, which is a low scale version of early universe inflation. Data recently released by the DESI collaboration shows some preference for dark energy with an evolving equation of state <cit.>, consistent with  quintessence. Whether it takes the form of quintessence or de Sitter, obtaining an accelerated expansion from string theory is an ongoing challenge.  Although we can construct stable de Sitter vacua using background flux and brane uplifts (see eg. <cit.> and for reviews <cit.>) the consistency of these constructions is a matter of considerable debate (see eg. <cit.>).
This has led to the so-called de Sitter conjecture which implies that fully consistent and stabilised de Sitter vacua are absent from the spectrum of weakly coupled string theory at or near the boundary of moduli space <cit.>. If the conjecture is correct and these vacua are indeed absent, then quintessence is also expected to be ruled out <cit.>. 

String theory is a consistent quantum theory of gravity, with a well defined low energy description in terms of higher dimensional supergravities. For this to be relevant to our universe, it must be able to admit a compactification down to four dimensions, where gravity is well described by General Relativity and matter by a suitable extension of the Standard Model of Particle Physics. It should also be able to accommodate an accelerated cosmological expansion. Of course, it is still possible that the concerns surrounding de Sitter vacua in perturbative string theory will go away and we will be able to find consistent accelerated cosmological solutions that everyone is happy with. However, if this is not the case, we will need a non-perturbative understanding of the theory to probe the full vacuum structure and establish the existence of de Sitter vacua far from the boundary of moduli space. This will likely require a third string revolution - a step change in our current approach to string theory/M theory. We can also consider the possibility that de Sitter is a resonant or excited state of string theory <cit.>. 

We can try and get some hints into what can happen non-perturbatively by focussing on the effective field theory for classical strings in the homogeneous limit. The Neveu-Schwarz sector describing the metric, the Kalb-Ramond field and the dilaton, are known  to possess an  symmetry to leading order <cit.>. This symmetry is expected to be preserved to all orders in the slope parameter α', motivating corrections to the leading order Neveu-Schwarz theory using higher order  invariant operators <cit.>. Note that  invariance can be made manifest using double field theory, albeit at the expense of manifest diffeomorphism invariance <cit.>. Duality invariant cosmology has been used to tackle the problem of the initial singularity (see eg.  <cit.>). 

Remarkably, for homogeneous fields, Hohm and Zwiebach have been able to provide a complete classicfication of all higher order terms  invariant under   <cit.>.  With field redefinitions, the system is reduced to first order time derivatives allowing them to write down the most general duality invariant theory to all orders in α' for the metric, Kalb-Ramond field and dilaton. To leading order in the expansion in α' the  theory
coincides with the Neveu-Schwarz sector of the supergravity theories describing the effective theory of classical strings, in the homogeneous limit. In the limit of vanishing Kalb-Ramond field and an isotropic spatially flat metric, the theory exhibits scale factor duality a→ 1/a as a remnant of the underlying  symmetry.  The full theory is now tractable at all orders in α', enabling the authors of <cit.> to identify string frame de Sitter vacua that are non-perturbative in α'. The dynamics of this new class of theories has been studied in a series of subsequent papers <cit.>. In particular, one can extend the system to include  invariant matter fields <cit.>, or find theories that admit de Sitter solutions even in Einstein frame <cit.>. Instabilities for homogeneous and isotropic perturbations were studied in detail in <cit.> with subsequent extensions to anisotropic perturbations <cit.>. Bianchi I universes were studied in <cit.>, along with two-dimensional black hole solutions <cit.> and bouncing cosmologies <cit.>.
 
In this paper, we revisit the dynamics of  invariant cosmologies, starting with isotropic and spatially flat metrics before moving on to the anisotropic case. Our focus is on the existence of de Sitter vacua in both the string frame and the Einstein frame. Of course, the isotropic case is simpler and much more extensively studied. Within the isotropic framework, string frame de Sitter vacua are known to exist but have been shown to be unstable <cit.>, something we are able to verify using slightly different methods. de Sitter solutions in the Einstein frame have been obtained in the presence of a source <cit.> and in vacuum <cit.>. The latter only exist when the theory takes on a very bespoke form at all orders in α'. As we will show, these Einstein frame de Sitter solutions are the only solutions to the vacuum field equations for these particular theories for a finite time period,  ensuring their local stability under homogeneous fluctuations.  However, the solutions are not eternal in both past and future. This is because the dilaton evolves. After a finite time (towards either the past or future), the solution is kicked into a new dynamical phase where Einstein frame de Sitter vacua are no longer possible.  This resonates (!) with the idea of de Sitter as a resonance <cit.>. 

These solutions enjoy a natural  extension to anisotropic universes thanks to the  symmetry. Here we consider a Bianchi I universe, with a number of different scale factors with corresponding Hubble parameters.  In a notable departure from the  isotropic framework, the  symmetry now implies an invariance under the exchange of any two of those Hubble parameters, placing important restrictions on the structure of the underlying theory.  Although isotropy is no longer assumed at the level of the theory, we are interested in isotropic solutions: namely, the de Sitter vacua in both string frame and Einstein frame. In string frame, de Sitter solutions are possible, provided the anisotropic theory obeys properties analogous to the isotropic case. These solutions are once again unstable. In the Einstein frame, we find conditions for the de Sitter vacuum to exist. This mirrors the isotropic case discussed in the previous paragraph, with additional conditions pointing along the anisotropic directions. Remarkably, these extra conditions hold automatically thanks to the  symmetry.

The rest of this paper is organized as follows. In section <ref> we give a brief review of O(d,d)-invariant cosmology. In section <ref>, we review the structure of the underlying theory for an isotropic and spatially flat universe, including sources. In section <ref> we review the derivation and instability of de Sitter vacua in the string frame, providing some added insight into the source of the instability. In section <ref> we demonstrate how de Sitter solutions can also be found in the Einstein frame. As discussed above, these solutions exist for a finite time period, during which time they are stable against homogeneous fluctuations. However, the solutions are not external, and will eventually give way to a new cosmological phase. In section <ref>, we change gears and extend our analysis to include Bianchi I universes. In section <ref> we establish the appropriate conditions for string-frame de Sitter vacua, commenting on the stability. In section <ref> we  do the same for de Sitter vacua in the Einstein frame, noting the crucical role played by the  symmetry. In section <ref>, we summarise our conclusions, and extend the results to non-critical dimensions. 



§ REVIEW OF O(D,D)-INVARIANT COSMOLOGY
 
We begin with a review of all orders in α' duality invariant cosmology,  as originally proposed by Hohm and Zwiebach <cit.> building on earlier pioneering work <cit.>.  Consider the leading order theory describing the universal massless sector of closed strings in d+1 dimensions. This corresponds to the well known Neveu-Schwarz action depending on the string frame metric, 𝕘_μν, the antisymmetric Kalb-Ramond field, 𝕓_μν, and the scalar dilaton ϕ,

    I_0=1/2ł_s^d-1∫^1+dx√(-𝕘)e^-2ϕ[ R+4(ϕ)^2 -1/12H^μνρH_μνρ-2Λ] ,

where the string length l_s=√(α'), H_μνρ = 3∂_[μ𝕓_νρ] is the Kalb-Ramond field strength and R is the Ricci scalar for the string frame metric. The cosmological term, Λ, is present for non-critical strings and scales as l_s^-2. For critical strings it vanishes, so we shall neglect it in the following. We make a cosmological ansatz for the metric, b-field and dilaton, 

    𝕘_μν   =
    [ -n^2(t)       0;       0 g_ij(t) ],
    𝕓_μν   =
    [       0       0;       0 b_ij(t) ],
    ϕ   =ϕ(t),

where n(t) is the lapse and g_ij(t) are the components of the spatial metric. This breaks  diffeomorphisms down to the subgroup t → f(t), x^i→ x^i, with ϕ,  g_ij and b_ij transforming as scalars, and the lapse as a vector, n → n ḟ.  In this limit, the corresponding field equations also exhibit an  symmetry. Indeed, the full system can be written in terms of  covariant objects: a generalised dilaton, 

    Φ=2ϕ- ln√( g),

and a generalised metric 

    ℋ=
    [     g^-1   -g^-1b;    bg^-1 g-bg^-1b ].

The latter transforms as a tensor, ℋ→Ω^TℋΩ, under Ω∈ 
where 

    Ω^T ηΩ = η,     η=
    [ 0 1; 1 0 ].
 
In contrast, the generalised dilaton and the lapse function are  scalars,  transforming trivially as Φ→Φ and n→ n.

The spatial contribution to the action can now be integrated  out, resulting in a one-dimensional, two-derivative action that can be written in a form that is manifestly invariant under the residual diffeomorphisms as well as the underlying  symmetry <cit.>,

    I_0=κ^2 ∫ t  n e^-Φ[-(_nΦ)^2 - 1/8[(_n S)^2] ].

where κ^2>0 is some dimensionful constant that plays no role in what follows. Note that we have introduced the following notation for the covariant time derivative

    _n = 1/n(t)/ t,

and further defined

    S= ηℋ
    =
    [    bg^-1 g-bg^-1b;     g^-1   -g^-1b ].

Note also that both η and  S  are involutory matrices, squaring to the identity, S^2=η^2=1.  The action  is now seen to be manifestly invariant under  since S transforms as[To see this, note that S=ηℋ→ S'=ηΩ^T ℋΩ=ηΩ^T η S Ω, where we have used the fact that η^2=1. It then follows from (<ref>) that ηΩ^T η=Ω^-1 and so S'=Ω^-1 S Ω, as stated in (<ref>).  In <cit.>, these transformations are expressed in terms of h=Ω^-1  where (<ref>) along with the fact that η^2=1 now implies that h η h^T=η. ]

    S → S'= Ω^-1 S Ω,

We now consider the residual diffeomorphism and  invariant α' corrections to I_0. These terms are constructed out of  scalars such as Φ and its covariant  derivatives, _n Φ,^2_n Φ, … along with traces of  covariant terms such as _nS and higher covariant  derivatives _n^2 S, _n^3 S, …. Through a judicious use of equations of motion, field redefinitions, integration by parts, and useful relations of the form

    (S)=(_n S)=(_n^2S)=...   =0 ,
    ((_n S)^2k+1)   =0     for     k=0,1,...,
    (S(_n S)^k)   =0     for     k=0,1,...,

we can eliminate all higher order derivatives in the system. Indeed, Holm and Zwiebach <cit.> showed that the most general action at all order in α' must take the following form

    I=κ^2 ∫ dt  n e^-Φ[-(_n Φ)^2+∑_k=1^∞ c_k α'^k-1[(_n S)^2k] +multitrace]

where c_1=-1/8 to match the leading order Neveu-Schwarz action (<ref>). For any particular string theory, there ought to be a tower of dimensionless coefficients of higher order operators given by c_k, along with a tower of coefficients for the multitrace operators.

 

§ ISOTROPIC COSMOLOGIES
 
In the case of isotropic and spatially flat cosmology, g_ij=a^2(t)δ_ij, with a vanishing b field, the system simplifies considerably and the action (<ref>)  can be written in the following compact form <cit.>:

    I = κ^2∫ t  n e^-Φ[ - Φ^2_n + 1/α'f(√(α')H) ],

where we have introduced the shorthand Φ_n=_n Φ, Φ_nn=_n^2 Φ, … and define the string frame Hubble factor as H=_n ln a. 
Matching with (<ref>) we note that the function f is analytic in a neigbourhood of the origin where it takes the following form

    f(√(α' ) H)    = ∑^∞_k=1 d_k α'^k  H^2k,      d_1= d.

and the dimensionless coefficients  d_k can, in principle,  be determined from the c_k and the coefficients of the multitrace operators. Note that f is a manifestly even function in H. This follows from the  symmetry which takes the form of a simple scale factor duality, a → 1/a and so H → -H, in the homogeneous and isotropic limit. 

In the remainder of the paper, we choose units where α'=1. It will also be convenient to represent the scale factor through its logarithm,  q=ln a.   The equations of motion for the full system  can be obtained from varying the action with respect to the lapse, the dilaton and the scale factor. This yields the equations E_a=0 where a=n, Φ, q and 

    E_n   =δ I/δ n =κ^2  e^-Φ[Φ^2_n - L_f(H)],
    
    E_Φ   =δ I/δΦ =κ^2 n e^-Φ[2Φ_nn - Φ^2_n- f(H) ],
    
    E_q   =δ I/δ q =-κ^2 / t[e^-Φ f'(H)].

Here, L_f(H) =Hf'(H)-f(H)
denotes the Legendre transform of f and f'(H)= f/ H. Thanks to the residual diffeomorphisms, these equations are not independent and satisfy the following identity

    / t E_n=Φ_n E_Φ+H E_q.

We can include matter in an  invariant way by including a matter action of the form S_m[Φ, 𝒮, χ_I], where χ_I(t) are  invariant matter fields <cit.>. For a homogeneous and isotropic cosmology, this is equivalent to S_m[Φ, q, χ_I]. The vacuum equations of motion are now sourced as follows

    E_n=ρ̅,    E_q=-d n p̅,      E_Φ=1/2 n σ̅

where the  covariant energy density, pressure and dilaton charge are respectively given by 

    ρ̅=- δ S_m/δ n|_Φ, q,  p̅=1/dnδ S_m/δ q|_n, Φ,   σ̅=-2/nδ S_m/δΦ|_n, q.

Note that the  covariant sources differ from their diffeomorphism invariant counterparts by a factor of √(-g)=a^d, namely, ρ=ρ̅/a^d,  p=p̅/a^d,  σ=σ̅/a^d. For the pressure, we should also note that p is now defined by holding Φ=2ϕ-d q fixed when we vary the matter action. This differs from the usual definition of pressure given by  
    p_standard=1/dn a^dδ S_m/δ q|_n, ϕ,
  where ϕ is held fixed instead. It is easy enough to show that  the two pressures can be related using the dilaton charge, p_standard= p+1/2σ <cit.>. Note also that diffeomorphism invariance in the matter sector implies that 

    _n ρ+d H (ρ+p)=1/2Φ_n σ.

Although we have included the 
matter sources for completeness, in the remainder of this paper we shall mostly neglect them, since our focus will be on finding consistent solutions to the vacuum field equation. In particular, we will seek conditions for consistent de Sitter vacua in  both the string and the Einstein frame.  We  refer the reader to <cit.> for more details on matter couplings. 



 §.§ dS solutions in the string frame
 
It was already shown in <cit.> that string frame de Sitter solutions are possible in this class of α' complete cosmologies   for suitable choices of the function f(H). These solutions are necessarily non-perturbative in α', and could not have been found using standard techniques in perturbative string theory.  However, it has also been  shown that these solutions are unstable under vacuum fluctuations <cit.>. We will review  the derivation of these solutions and add some additional insight into the nature of the instability. 

We know that in a neighbourhood of the origin, the function f(H) is even and  admits the following Taylor expansion in H^2

    f(H)=dH^2 +d_2H^4 +…

Although we cannot assume that the function remains analytic for all values of H, we shall assume that it is twice differentiable in order to have well defined field equations. We shall also assume it to be an even function in order to preserve the  symmetry.

Here we are interested in the properties of the function in a neighbourhood of a string frame de Sitter solution with H=H_c>0, for some constant H_c.  Whilst it is tempting to  motivate this with the current phase of accelerated expansion, it is important to note that we  are working in d+1 dimensions.  For the critical dimension (d+1=10, 26), we must perform  compactification   down to four dimensions before making contact with observation. Therefore, we remain agnostic as to the value of H_c.

Setting E_a=0 for a=n, Φ, q, we immediately see  that H=H_c>0, for some constant H_c, if and only if 

    Φ^2_n=f(H_c)=L_f(H_c),      f(H_c) f'(H_c)=0,

which is equivalent to Φ_n=f(H_c)=f'(H_c)=0. This means that f(H) must have a double zero at H=H_c. Thanks to the  symmetry, it must also have a double zero at H=-H_c. This suggests  the following general form

    f(H)= g(H)/H_c^4 (H^2-H_c^2)^2

for some even function g(H) that admits a Taylor expansion in H^2-H_c^2

    g(H)=g_0+g_1 (H^2-H_c^2)+ ….

For the form of f(H) given in (<ref>) to overlap with the form  close to the origin  (<ref>), we obtain the following constraints

    g(0)=g'(0)=0,      g”(0)=2 d.

It is also important to establish the stability of these de Sitter vacua. Of course, a full analysis of cosmological perturbations requires knowledge of the underlying theory beyond the homogeneous limit under consideration. Although there has been some interesting recent work in this direction <cit.>, it is well known that  invariance in d+1 dimensions is at odds (!) with manifest diffeomorphism invariance, as evidenced through double field theory <cit.> (see also <cit.> for a related duality observed in cosmological perturbations).  Faced with these limitations, we can only consider the stability of solutions under homogeneous perturbations captured by the general action (<ref>). Instabilities for homogeneous and isotropic perturbations of string frame de Sitter vacua were identified in <cit.> with subsequent extensions to anisotropic perturbations <cit.>.  Here we take a subtlely different approach to those works but draw the same conclusions.

We consider only homogeneous and isotropic perturbations, so that the system of equations E_i=0 remains unchanged. Assuming f”(H) ≠ 0, these can be written in the following form

    f'(H)=c  e^Φ,     _n H=s √(L_f(H)) f'(H)/f”(H) ,

where c is a constant of integration. Here s=±1 corresponds to the sign of Φ̇=dΦ/dt, appearing when we take the square root in (<ref>). This ability to take either sign was identified as a potential source of error in the stability analysis, as discussed in <cit.>. Next we will show how the choice of sign plays a crucial role in opening up the unstable channel.

We  begin by making a choice of the time co-ordinate by setting the lapse function to unity, n=1, and solve for the Hubble parameter in a neighbourhood of the vacuum, giving 

    H =H_c+δ H

where δ H satisfies

    δ̇ ̇Ḣ≈ s  √( H_c f”(H_c))δ H^3/2.

This gives a consistent solution

    δ H ≈4 /H_c f”(H_c)(t-t_c)^2

with s=-sgn(t-t_c). We immediately see the unstable channel if and only if the initial time t_i <t_c, or equivalently s=+1. This means the  instability is excited if and only if Φ̇(t_i)>0. 



 §.§ dS solutions in Einstein frame
 
We now turn our attention to the Einstein frame. Einstein frame de Sitter solutions with constant Φ̇ were obtained in <cit.> in the presence of a source with p̅=-ρ̅. Here we shall present Einstein frame de Sitter vacua as solutions to the vacuum equations of motion for bespoke choices of the function f(H).   Although the solutions are stable, we will also show that the required form of f(H) cannot be smoothly connected to the leading order Neveu-Schwarz theory in a neighbourhood of H=0. This agrees with the conclusions drawn in <cit.>.  

The string frame and the Einstein frame are related by a conformal transformation.  Labelling the Einstein frame with the letter “E", we write 𝕘_μν =𝒜^2 𝕘_μν^E  and so n = 𝒜 n_E, a_S = 𝒜 a_E, where 𝒜 is the conformal factor to be derived presently. Focussing on the leading order curvature term in the action, we note that

    ∫^1+dx√(-𝕘)e^-2ϕ R=∫^1+dx√(-𝕘_E)e^-2ϕ𝒜^d-1[R_E +…]

where the ellipsis include derivatives of 𝒜 and R_E is the Ricci scalar for the transformed metric. For the latter to coincide with the Einstein frame metric, we must choose 𝒜=e^2ϕ/d-1.

Setting  E_i=0 for i=n, Φ, q, we see that the vacuum field equations are equivalent to

    f'(H)=c e^Φ,     Φ_n=s√(L_f(H))

where we recall that c is a constant of integration and s=± 1. We wish to recast these equations in terms of the Hubble parameter defined in the Einstein frame

    H_E=_n_E q_E

where _n_E=1/n_Ed/dt and  q_E=ln a_E. From the definition of the generalised dilaton 
Φ=2ϕ -dq and the conformal mapping  between frames q-q_E=ln(n/n_E)=2ϕ/d-1, we can derive the following useful relations

    Φ=-q-(d-1) q_E  Φ_n=-H-(d-1)n_E/n H_E

and

    ϕ=-(d-1)/2 (Φ+dq_E) n/n_E=e^-Φ/

where =a_E^d is volume scale factor in Einstein frame. After some manipulations,  the system of equations (<ref>), (<ref>) and (<ref>) can be expressed as follows

    e^Φ=f'(H)/c,     n/n_E=c /f'(H),     ∂_n_E=c ℱ(H),     ∂_n_E H=c 𝒢(H)/

where 

    ℱ(H)=d/1-d[H+s √(L)/f'(H)],     𝒢(H)=s √(L)/f”(H)

and we have assumed f'(H)f”(H)≠ 0. To explore the possibility of a de Sitter vacuum in the Einstein frame, it is convenient to construct 

    _n_E H_E =dH_E^2 [_n_E^2 /(_n_E)^2 -1]=dH_E^2[ℱ'(H) 𝒢(H)/ℱ(H)^2-1 ],     c ℱ(H)≠ 0.

If we choose the function f(H) such that the right hand of this equation vanishes identically, the Hubble parameter in Einstein frame is guaranteed to be constant, H_E=H_⋆, for a solution to the vacuum field equations.  Thus, the existence of the de Sitter vacuum in Einstein frame requires that

    (1/ℱ)'+1/𝒢=0,

or in other words, that f(H) satisfies the following differential equation, 

    d+1/d-1 s H √(H f'-f) f” + 1/d-1(H f'-f)f” + d/d-1 H^2 f” + s √(H f'-f) f' + 1/2H f' f”=0.

 Note further that we can use the third equation in  (<ref>)  to show that the Hubble parameters in the Einstein and string frame are related according to the following formula 

    H_E = c ℱ(H)/d.

For a suitable choice of f(H) satisfying (<ref>), the corresponding constant curvature solution is stable under vacuum perturbations, as it is the only solution to (<ref>). 

Finding a suitable choice of f(H) is not easy as the differential equation (<ref>) is challenging to solve in full generality.  However, if we assume that H>0, it does admit the following analytic solution when s=-1, 

    f̅(H) = (H + H_m)^2 - H^2_m,

where H_m is a constant. 

With  f=f̅ and taking the negative (s=-1) root  we find that ℱ=0 and 𝒢=-H/2.  The vanishing of ℱ suggests this solution lies outside of the original regime of validility for which (<ref>) was derived.  Nevertheless, it is interesting to explore the solutions to the system of equations (<ref>) in this particular case. It turns out that they imply a constant but vanishing Hubble rate in Einstein frame

    H_⋆=0,

consistent with (<ref>), along with the following solutions for the string frame Hubble parameter, the dilaton and the lapse, 

    H(t)    =    H_c exp(-ct/2a_0^d), 
    Φ(t)    =   ln[2/c(H_c exp(-ct/2a_0^d)+H_m) ],  
    
    n(t)    =   c/2 a_0^d[H_cexp(-ct/2a_0^d)+H_m]^-1,

where H_c is a dimensionful constant of integration and we have assumed d ≠ 1 and c≠ 0. Clearly the solution runs into a singularity as H → -H_m  although this can be trivially avoided by taking H_m>0.

We have not been able to find an exact  solution to (<ref>) that gives non-vanishing ℱ and, correspondingly,  H_⋆>0. However, numerical solutions are presented in fig:f_H, with the corresponding value of H_E shown in fig:HE, where we recall that we have set units such that α'=1.



We can also seek a perturbative solution to (<ref>). To this end, we consider  

    f(H) = f̅(H) [1+ ϵ (H)],

where we assume |ϵ|≪1. Plugging this into (<ref>) (with s=-1) and keeping terms up to linear order in ϵ, we arrive at the linear differential equation

    (H_m-H)(2H_m-H)H ϵ”(H)+(2H_m^2-3H H_m+2 H^2) ϵ'(H)-H_m ϵ (H)=0

with the following solution 

    ϵ (H) =  C_1 H_m + C_2 H + h_m C_2 ln|H/H_m|/H+2h_m,

where C_1 and C_2 are constants of integration. Note that C_1 can be removed by redefining H_m → H_m(1-C_1/2) and working to linear order in C_1. Thus we can set C_1=0 without loss of generality and express (<ref>) as a simple  function of the ratio δ=H/H_m, 

    ϵ (H)=C_2[δ+ln |δ|/δ+2 ].

Recall that this expression is valid as long as |ϵ(H)| ≪ 1. If we assume  |C_2|≪ 1 then this condition is violated if and only if δ  strays too close to the singularities at δ=0, -2. More precisely, we have ϵ(H) ≪ 1 whenever |C_2|≪ 1, provided

    |δ|  ≫ e^-2/|C_2| ,   | δ+2 | ≫ |C_2| (2 -ln 2 ).

We now revisit the field equations (<ref>) and (<ref>) using the perturbed form of f given by (<ref>). To leading order in |C_2| ≪ 1 (<ref>) gives the following expression for Hubble rate in Einstein frame

    H_E ≈C_2 c/4(d-1).

If we assume H_E = H_⋆ for some constant H_⋆ and  set n_E=1, it follows that that we have =a_0^d e^d H_⋆ t. We can then solve (<ref>) explicitly, giving

    H_⋆≈1/dt W(dC_2 ct/4(d-1)a_0^d) ≈C_2 c/4(d-1)a_0^d

where we have used the fact that the Lambert W function W(x) ≈ x for small x.  We see that we have a constant and non-vanishing  Hubble rate in the Einstein frame, with the approximation valid as long as |t| ≪ 4(d-1)a_0^d/d|c||C_2|. Since H_⋆ is required to be positive in an expanding universe, we assume that C_2 c>0. The remaining fields are dominated by their non-vanishing zeroth order (in ϵ) solutions (<ref>)-(<ref>). 
Recall that in a neighbourhood of the origin, perturbative string theory suggests that the function f(H) should admit a Taylor expansion of the form (<ref>).  We will now show that any solution to the differential equation (<ref>) valid for H>0 cannot be analytically continued towards a solution of the form (<ref>) in a neighbourhood of H=0.

To see this, we begin by assuming that f(H) admits a Taylor expansions of the form (<ref>) in a neighbourhood of the origin 

    f(H)= d H^2+𝒪(H^4) .

In this neighbourhood, it immediately follows that

    ℱ'(H) 𝒢(H)/ℱ(H)^2=𝒪(H^2).

In order to satisfy (<ref>), the right hand side of this expression should be equal to unity, which is clearly not possible in a neighbourhood of H=0, regardless of the sign of s=± 1.  Thus, it is not possible to smoothly connect a solution to (<ref>) to the desired leading order Neveu-Schwarz theory at low curvatures. However, perhaps this is too demanding. As noted earlier, for the field equations to be well defined, we only need the form of f(H) to be twice differentiable. For any finite N, it is certainly possible to match the first N derivatives of a solution to (<ref>) to a function of the form (<ref>), at some  value of H=H_match>0.

It is, perhaps, optimistic to expect such a matching  to emerge from a non-perturbative understanding of string theory. Nevertheless, we could imagine a discontinuity in the higher order derivatives of f(H) to be a remnant  of the truncation to the massless sector of the theory. Indeed, if it emerges from string oscillators with string scale masses, we might expect H_match∼ 1 (in units of α'=1).

Even if this desired form of f(H) can be justified, it is clear that we can only trust our de Sitter vacuum in the regime where H>H_match. However, recall that H is the Hubble parameter in the string frame and this will change over time. If the system evolves into a  regime where H<H_match, the form of f(H) will no longer admit de Sitter solutions and the Einstein frame de Sitter vacuum is destabilised.  

To avoid this instability, we need to stop H evolving into the unstable region with H<H_match in either time direction.  In other words, we need H to have a turning point in its evolution at some H_turn>H_match. At such a turning point, the last equation in (<ref>) tells us that 𝒢(H_turn)=0. Since H_turn>H_match by assumption, the  differential equation (<ref>) ought to be satisfied at H=H_turn, further implying that ℱ(H_turn)=0 must have a zero  at the same point.  Since we assume that f(H) is twice differentiable, ℱ(H_turn)=𝒢(H_turn)=0 if and only if H_turn=0. This contradicts the underlying assumption that H_turn>H_match>0. It follows that  our de Sitter vacua in Einstein frame cannot be eternal. 



§ ANISOTROPIC COSMOLOGIES
 
We now turn our attention to anisotropic cosmologies in d+1  spacetime and consider a string frame line element of the form

    ds^2=-n(t)^2 dt^2 +∑_i=1^d a_i(t)^2 dx_i^2,

where a_i(t) is the scale factor along the ith spatial direction. A detailed derivation  of the  invariant action requires a careful computation of all the terms in (<ref>), including the multitrace operators. Instead of computing that explicitly, we note that the action will take the form 

    I = κ^2∫ t  n e^-Φ[ - Φ^2_n + f(H_i) ],

where we have assumed units with α'=1. The Hubble parameter along the ith direction is given by H_i=_n q_i for q_i=ln a_i, while the generalised dilaton is given in terms of the standard dilaton via the following relation

    Φ=2ϕ-∑_i q_i.

To match with the leading order Neveu-Schwarz action (<ref>) at low curvatures, we expect that f admits a Taylor expansion near the origin such that

    f(H_i) =∑_i H_i^2 + 𝒪(H^4).

The  symmetry now includes scale factor duality along each direction, a_i → 1/a_i and scale factor exchange a_i ↔ a_j for each i and j. This suggests that the function f(H_i) can also be expressed as

    f(H_i)=F(ê^d_i(H_j^2)) ,

where we define the normalised elementary symmetric polynomials 

    ê_i^d(X_j)=1/di∑_1 ≤ j_1 < … j_i ≤ d X_j_1… X_j_i

for i=1, … d.

Varying the action (<ref>) with respect to the lapse, the dilaton and each of the scale factors, we obtain E_a=0 for a=n, Φ, q_i, where

    E_n   =δ I/δ n =κ^2  e^-Φ[Φ^2_n - L_f(H_i)],
    
    E_Φ   =δ I/δΦ =κ^2 n e^-Φ[2Φ_nn - Φ^2_n- f(H_i) ],
    
    E_q_i   =δ I/δ q_i =-κ^2 / t[e^-Φ f_i],

where we denote partial derivatives f_i= f/ H_i and the Legendre transform L_f(H_i) =∑_i H_i f_i-f(H_i). It is also convenient to define the average Hubble parameter in string frame as H=1/d∑_i H_i.



 §.§ dS solutions in the string frame
 
We now seek solutions to the anisotropic field equations corresponding to the string frame de Sitter vacua. 
Using similar arguments to the isotropic case in section <ref>, we see that  there is an isotropic solution to (<ref>) to (<ref>) with H_i=H_c>0 for all i and some constant H_c if and only if

    Φ_n=f(H_c, … , H_c)=f_i(H_c, … , H_c)=0.

Given the form of f dictated by the  symmetry (<ref>), this suggests that

    f(H_i)=∑_j g_j(H_l)/H_c^4j(ê_j^d(H_l^2)-H_c^2j)^2

where  the functions g_j(H_k) take the  symmetric form (<ref>)  and admit a Taylor expansion in each of ê_j^d(H_l^2)-H_c^2j,

    g_j(H_k)=g_j0+∑_k g_jk(ê_k^d(H_l^2)-H_c^2k)+….

Close to the origin, the form of f should coincide with (<ref>) to recover the leading order Neveu-Schwarz theory at small curvatures. This implies that

    ∑_j g_j(0, …, 0)=∑_j _H_k g_j(0, …, 0)=0,     ∑_j _H_k_H_l g_j(0, …, 0)=2δ_kl.

On the isotropic line where  H_i=H for all i,  the form of f is compatible with the isotropic solutions (<ref>), as expected

    f(H_i)|_iso=∑_j g_j(H,…, H)/H_c^4j(H^2j-H_c^2j)^2.

It immediately follows that the corresponding de Sitter vacua will once again be unstable to isotropic fluctuations. 



 §.§ dS solutions in the Einstein frame


We now switch our attention to the Einstein frame to see if the family of de Sitter solutions found in the isotropic framework <cit.> carries over to the anisotropic framework. As usual, we perform a conformal transformation to switch to Einstein frame, with  n=𝒜 n_E and a_i=𝒜 a_i^E  where 𝒜=e^2 ϕ/d-1. We therefore  have ln (n/n_E)=q_i-q_i^E=2ϕ/d-1 for q_i^E=ln a_i^E. This allows us to extract the following useful relation

    ϕ=-d-1/2(Φ+∑_i q_i^E),

implying 

    n/n_E=e^-Φ/,      q_i-q_i^E=-Φ-ln

where =∏_i (a_i^E)^k_i is the Einstein frame volume scale factor. 

In the Einstein frame, we define Hubble parameters H_i^E=_n_E q_i^E and the corresponding average H^E=1/d∑_i H_i^E. These are easily related to their string frame counterparts via the relation

    H_i=n_E/n(H_i^E-dH^E)-Φ_n.


From the field equations (<ref>) to (<ref>) we can derive expressions for the generalised velocities in Einstein frame

    _n_EΦ= e^-Φ/s√(L_f) , 
           H_i^E-H^E= e^-Φ/(H_i-H),
           H^E= e^-Φ/Σ/dℱ

  where 

    ℱ=d/1-d(H+s√(L_f)/Σ),     Σ=√(∑_j f_j^2)

and s=± 1, depending on the sign of Φ_n. 
  From equation (<ref>), we have that 
 
    e^-Φ f_i=c_i  e^-ΦΣ=|c|≡√(∑_j c_j^2)
 
for some constants c_i.  We can now derive the following constraints on the dynamics of the system, analogous to the last two equations in (<ref>)
  
    _n_E=| c|ℱ,     _n_E H_i=| c|𝒢_i/
 
where 

    𝒢_i=s√(L_f)∑_j f^ij f_j/Σ

 and f^ij denotes the inverse of the Hessian of the function f, assumed to be well defined.   We proceed in a similar way to the isotropic case, and compute the rate of change of the Einstein frame Hubble parameters. We find that

    _n_E H_i^E=d (H^E)^2[ (1+ξ_i) (𝒪lnℱ-1 ) +𝒪ξ_i ],

where 

    𝒪   =1/ℱ∑_j 𝒢_j _H_j,
    ξ_i   =d (H_i-H)/Σℱ.

Since ∑_j ξ_j=0, it follows that the rate of change of the average Hubble factor is given by a somewhat simpler formula

    _n_E H^E=d (H^E)^2 (𝒪lnℱ-1 ),

which is strongly reminiscent of the corresponding isotropic equation (<ref>), as expected.
We may alternatively express (<ref>) as

    _n_E H_i^E-(1+ξ_i) _n_E H^E =d (H^E)^2𝒪ξ_i.

Let us now evaluate our system on the isotropic line H_i=H, along which we have ξ_i=0 from (<ref>). For consistency, it must follow that 

    (_n_E H^E)|_iso=d ( H^E)^2 (𝒪lnℱ-1 )|_iso,      (𝒪ξ_i)|_iso=0

where |_iso indicates evaluation on the isotropic line.  Requiring the vacuum to be de Sitter in the Einstein frame on the isotropic line, i.e. H_i^E=H^E=const, now yields the following constraints

    𝒪 (lnℱ)|_iso=1,     (𝒪ξ_i)|_iso=0 .

Defining   𝒢=∑_i𝒢_i/d, we note that 

    𝒪(lnℱ)=-∑_i 𝒢_i _H_i(1/ℱ), 
          𝒪(ξ_i)=1/Σℱ^2(𝒢_i-𝒢)-ξ_i𝒪ln (Σℱ),

and impose (<ref>), giving

    [∑_i _H_i(1/ℱ)+1/𝒢]_iso=0,     𝒢_i|_iso=𝒢|_iso≠ 0.

To explore the implications of these results, consider the N dimensional  Euclidean space spanned by the Hubble parameters, (H_1, …, H_d). It is convenient to introduce a new set of coordinates (H, v_α), adapted to the isotropic line, defined in terms of the average  Hubble parameter H=∑_i  H_i/d and d-1  orthogonal directions

    v_α   =∑_i(H_i-H)n_i^α,

so that 

    H_i=H+∑_α n^α_iv_α.

Here we have introduced an orthonormal set of vectors in Hubble space, n^α_i,  orthogonal to the  tangent vector, u_i=1/d, on the isotropic line 

    ∑_i u_i n_i^α= 1/d∑_i n_i^α=0,     ∑_i n_i^α n_i^β=δ^αβ.

We can now think of the isotropic line as being parametrized by the average Hubble parameter, H, with each of the remaining coordinates vanishing there, v_α|_iso=0.  At this point, we recall that the  symmetry implies that f(H_i) is given in terms of the normalised elementary symmetric polynomials, as in (<ref>). In a neighbourhood of the isotropic line, we can show that

    ê^d_i(H_j^2)=H^2i + i(d+1-2i)/d(d-1)H^2(i-1)∑_α, βδ^αβ v_α v_β+𝒪(v^3).

Correspondingly we find that the function f takes the form

    f(H_i)=f_0(H)+g_0(H)∑_α, βδ^αβ v_α v_β +𝒪(v^3),

where

    f_0(H)=F(H^2, …, H^2d),      g_0(H)=∑_i=1^d  i(d+1-2i)/d(d-1)H^2(i-1) F/ê^d_i |_ê^d_j =H^2j.

For the de Sitter vacuum to exist, we require the conditions (<ref>) to hold. The second of these is equivalent to 

    [ ∑  u_i 𝒢_i]_iso=𝒢_iso≠ 0,    [∑ n_i^α𝒢_i]_iso=0.

Making use of the chain rule

    / H=∑_i / H_i,     / v_α=∑_i n_i^(α)/ H_i,

and assuming g_0(H) ≠ 0, we immediately see that the condition (<ref>) holds automatically. Meanwhile, the first condition in (<ref>) yields

    [_H (1/ℱ)+1/𝒢]_iso=0,

which is equivalent  to  (<ref>) for f(H)=f_0(H).  It follows that the Einstein frame de Sitter vacua found in the isotropic framework extend naturally to the anisotropic framework. This is because the underlying   symmetry ensures that deviations from isotropy only enter the theory at 𝒪(v^2) ∼𝒪((H_i-H_j≠ i)^2). 




§ CONCLUSIONS
 

Can we obtain de Sitter solutions in string theory?  Although perturbative methods provide some rich phenomenology,   they have not been able to uncover de Sitter solutions that are universally accepted as consistent and fully under control. This motivates attempts to probe the non-perturbative structure of string theory. Of course, we do not have the tools to do this in full generality; but progress can be made by restricting access to the Neveu-Schwarz fields in the homogeneous limit. This is because they admit an  symmetry that can be extended to higher order operators, extending to all orders in α'. 

The underlying structure of these theories is easy to write down whenever we have a homogeneous and isotropic framework, where the  symmetry is equivalent to scale factor dulaity a → 1/a. For a suitable non-linear structure,  de Sitter solutions can be found in both the string frame and the Einstein frame. As shown previously  <cit.>, homogeneous and isotropic fluctuations about the string frame solutions  can trigger an instability, although we have now shown that this is excited if and only if the generalised dilaton is growing at the initial time Φ̇(t_i)>0.  The situation in the Einstein frame is even more delicate. de Sitter solutions can and do exist in the Einstein frame, for particular non-perturbative extensions of the Neveu-Schwarz theory. However, these extensions do not connect smoothly to the leading order Neveu-Schwarz theory. This means the Einstein frame de Sitter solutions cannot be eternal to both past and future. In other words, a de Sitter solution can survive for a finite period of time, or even a semi-infinite time, but at some point towards either the future or the past, the solution breaks down. The solution enters a new phase of the theory in a neighbourhood of small string frame curvature, where Einstein frame de Sitter solutions are no longer possible. Perhaps this should have been expected, as the breakdown occurs in the perturbative regime where de Sitter vacua are conjectured to be absent <cit.>. 

All of these results can be extended to include Bianchi I metrics.  The  symmetry now plays a more interesting role than in  the isotropic case, where scale factor duality is accompanied by scale factor exchange a_i ↔ a_j.  This is crucial, especially for the existence of de Sitter solutions in the Einstein frame where the additional constraints required for existence in an anisotropic setting are satisfied automatically thanks to  symmetry. 

Although we have focused our analysis on the case of classical strings in a critical dimension, for which the constant Λ=0 in (<ref>), the results are easily extended to non-critical dimensions.  This is because a non-vanishing Λ∝ D-D_c is easily absorbed into a redefinition of the function f that defines  theory to all orders in α', 

    f → f-2 Λ.

This changes the structure of the theory in neighbourhood of the origin, but little else. For the string frame de Sitter solutions in the isotropic framework, this will affect the boundary conditions (<ref>), so that we now have g(0)=-2Λ. Similarly, in the anisotropic case,  we take (<ref>) with ∑_i g_j(0, …, 0)=-2Λ. The stability analysis is unchanged.  For the Einstein frame, we can revisit the question of whether or not the bespoke choice of f(H) can be smoothly connected to the desired form in a neighbourhood of H=0. Focusing on the isotropic case, we set f(H)=-2Λ + dH^2+𝒪(H^4)  and compute

    ℱ'(H) 𝒢(H)/ℱ(H)^2=d-1/d+𝒪(H).

Clearly this cannot be set equal to unity, as required for Einstein frame de Sitter solutions. Thus, we conclude that the situation is qualitatively unchanged from the critical case, D=D_c, and one cannot have eternal de Sitter solutions in the Einstein frame. 




  
Acknowledgements 
We would like to thank Heliudson Bernardo and Guilherme Franzmann for useful discussions. AP and PMS were supported by STFC consolidated grant number ST/T000732/1 and YL by an STFC studentship. RGCS was supported by a Bell-Burnell studentship.  This paper was drafted during the period of the UCU marking and assessment boycott, demonstrating work carried out by AP and PMS as employees of the University of Nottingham. For the purpose of open access, the authors have applied a CC BY public copyright licence to any Author Accepted Manuscript version arising.

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