Stability of the de-Sitter spacetime. The anisotropic case

We elaborate on the cosmological implications of the recently established non-perturbative $O(D,D$)-symmetric approach. Following the $\alpha'$-complete Friedmann equations, we provide a qualitative description of the dynamics of anisotropy perturbations. We find that the spacetime stability depends on the sign of the generalized dilaton time derivative. We consider isotropic limits and find possible mechanism for pre-big bang cosmology, governed by the value of cosmological constant.


I. INTRODUCTION
The discovery of the positive cosmological constant has put the existence of de-Sitter solutions and their stability as one of the central research questions within the string theory community.The so-called de-Sitter conjecture states that the de-Sitter vacua are unstable within string theory [1][2][3].
On the other hand a suitable O(D, D) formalism is at hand to study the de-Sitter solutions.As found by K.A. Meissner and G. Veneziano [4,5], in D + 1 dimensions the low-energy stringy effective action once reduced to time-dependent backgrounds possess continuous O(D, D) symmetry, see also [6,7].The symmetry emerges in the massless sector of (super) string theories: scalar dilaton φ(x), the spin-2 graviton G µν and anticommuting Kalb-Ramond 2-form field B µν .The low-energy action takes the form where D is the number of spatial dimensions, R is the Ricci scalar curvature of the metric G µν and H µνρ is a field strength of B µν , defined as H µνρ := ∂ µ B νρ + ∂ ν B ρµ + ∂ ρ B µν .For the time-dependent backgrounds G µν =: g µν (t), B µν =: b µν (t), φ =: φ(t) the effective action (1) may be expressed in the manifestly O(D, D) invariant form: We call Φ = 2φ − ln √ det g the generalized (shifted) dilaton field and g is the spatial metric tensor.Following the formalism of [4], 2d-dimensional matrix S is defined: The global O(D, D) group acts as As shown by [8] also for the α corrected actions the fields can be transformed in such a way that O(D, D) symmetry is manifest.Using this reasoning, in [9], the full, nonperturbative form of the α corrected effective action on cosmological backgrounds has been derived +multitrace .
A generic multitrace term at order α k−1 is of the following form: i Tr Ṡ2ni (7) with i n i = k and n i > 1.Note that also a cosmological constant term may be added in the O(D, D) manner, [10].The higher order potential terms in Φ will be O(D, D) symmetric as well, but will cause the instability in the equations [11].This general form of the string theory effective action and the associated equations revealed a new possibility of a non-perturbative de-Sitter solution once summing all the possible α corrections in the O(D, D) formalism [9], see also other developments here [10,[12][13][14][15][16][17][18].Finally, in our previous article, we studied the isotropic case for constant shifted dilaton Φ = 0 [11] as well as in the presence of cosmological constant.
In this work, we extend our study towards the anisotropic case and the stability of the de-Sitter solution across different spatial dimensions.As it will turn out, the O(D, D) framework will not be suitable to study the anisotropies, thus here we have adopted the perturbative approach including the multitrace effects.We also comment on possible Pre-Bing Bang scenarios.

II. ANISOTROPIC PERTURBATIONS
The anisotropic solutions for the full α O(d, d) dynamics have been discussed in [15,19].Yet in both of the articles only a special class of the anisotropies have arXiv:2301.06616v1[hep-th] 16 Jan 2023 been covered, namely for the d − n static dimensions, i.e.H i = 0 for i ∈ (n, d).As we will shortly discuss, the full unperturbative, dynamical investigation of anisotropic perturbations within Hohm-Zwiebach setup is beyond the feasible scope of the theory.We are however able to recognise stability features present in the isotropic non-perturbative case.To investigate the stability of the d dimensional de-Sitter solution, let us now consider the anisotropic case: In that case, the matrix S equals: And its derivative is: Then it can be seen that the square of the derivative takes a diagonal form: where we define In [9], a simpler case of a 1 = a 2 = . . .= a d is considered.It is then shown that in the low-energy effective action the multitrace terms can be ignored, because they contribute exclusively to the value of the constants c k .It is because then (as one can see by setting all H i to be equal in (11)): Since the multitrace terms at order α k−1 consist of the traces of powers of Ṡ that sum to 2k, they will all have the form CH 2k , where C is constant.Therefore, they will all sum to a term proportional to H 2k and can be included by simply changing the constants c k .However, the anisotropic case is different.A generic trace appearing in the action is: Plugging this into (7), it is easily seen that the multitraces will introduce terms proportional to where some of the j i are not equal, into the action.The consequence of this is that the equations become hard to derive, and they would be a lot more complicated than in the isotropic case.
However, in [9], it has been shown that the first multitrace terms will appear at the third order in α .Therefore we will disregard the multitrace terms if we truncate the expressions to a sufficiently small order.Alternatively, the issue of existence of de-Sitter vacua can be investigated by assuming, that the matrices g, ġ, and ḃ commute (the so called commuting matrices ansatz [20]).That approach allows for some non-perturbative analysis, but results in very complicated equations which are not suitable for stability analysis.In our study, we take one another perspective: the perturbative approach quantifying multitrace effects affecting the stability of a general anisotropic solution.The equations derived in [9] by a direct variation of the action (without the multitrace terms) with the added contribution from the cosmological constant are: They follow from variation with respect to S, Φ, and the lapse function n(t), which has been set to 1.In the considered case, the equation ( 17) transforms into: the following relation between Φ2 and Φ holds: and (15) gives d equations of the form: A. The perturbative approach Now, we proceed to the calculation of the terms introduced to the equations by the first multitrace term.The part of the action that includes it is: In order to vary it with respect to S, we use an approach laid out in [9].First, we perform a variation as if S was an unconstrained variable: The expression multiplying δS defines F (m) S .The contribution to equation ( 15) is given by Using , it can be brought to the following form: Using (13), this can be shown to give the following corrections to equations ( 20): Additionally, the equations ( 16) and ( 17) will receive contributions from the multitrace term.For our purposes, only the change to the latter needs to be calculated.After restoring g 00 = −n(t), the multitrace term will be multiplied by 1 n 7 .Therefore, its contribution to the equation will be: Finally, the equations to the third order take the form: Using equation ( 28), Φ can be expressed in terms of H i : By plugging ( 29) into ( 27), we get d differential equations for the H i 's, where the only appearance of Φ is the sign of its derivative.They are linear equations for Ḣi s and can be used to express them in terms of H i and sgn( Φ).Time evolution of Φ is entirely determined by the evolution of anisotropic Hubble parameter.We note that the sign of the generalised dilaton time derivative, is governed by the equation of motion containing the second time derivative, following from (16).Moreover, we should consider such constants c n , so that the expression under the squere root is positively defined or find a dynamical mechanism restricting the flow into this area.We discuss such mechanism in section IV.

III. EXACT SOLUTION TO ZERO-ORDER ANISOTROPIC EQUATIONS
If we disregard the α terms in the system of equations ( 15), ( 16) and ( 17) for the case considered in section II with Λ = 0, they can be solved analytically.We consider the cases Λ = 0 and Λ = 0 separately.

A. Zero potential
In this case, ( 16) and ( 17) imply the following relation: The general solution for Φ is given by: with C 1 ∈ {0, 1}.For C 1 = 0 the Eq.( 15) implies that H i are also constant.For C 1 = 1, after plugging (31) into (15), one gets: They can be solved to give: where β i are constants.By plugging ( 33) and ( 31) into ( 17), a constraint on β i 's is: Now, since H i = ȧi ai , we solve for a i and obtain: And since φ = 1 2 (Φ + log( √ detg)) and √ detg = j a j , we have: Interestingly, for j β j = 1 the dilaton remains constant for time dependent g.The dilaton and Ricci scalar curvature are singular for t − → t 0 .This means that the action is infinite at this limit and in the spirit of Finite Action Principle [21][22][23][24][25][26] such solution will be dynamically excluded, as the exponential weight in the path integral will vanish.There is however, an exception to this.Notice, when β i = 1 the dilaton and curvature remain finite.Assuming all |β i | are equal, the only possible choice of 1  3 , } corresponding to six contracting dimensions and three expanding.Originally, this result was discovered by Meissner and Veneziano in [5].At higher orders of α , the exact solution is much harder to obtain, since terms without Φ in ( 16) and ( 17) differ and (31) doesn't hold.

B. Cosmological constant potential
Now, (16) together with (17) give us: This equation has two solutions: Then, similarly as before, we can obtain solutions for a i : (42) where a ni are solutions of equations with Φ n .Incidentally, (42) was obtained by a different method in [27].
Then, (17) gives us the following conditions for the coefficients: i For solutions ( 41) and ( 43), these conditions lead to a complex metric, and thus are not physical.
The answer to this question possibly lays already at the zeroth-order.Equations (37) at H = 0 , h = 0 give vanishing Φ so Φ = −Λ.Consider a case, where the evolution starts at a point on 5 and is being brought to the beginning of the coordinate system.After reaching it, we could possibly switch to the repulsive flow on 6. Value of the second derivative is determined by the sign of the cosmological constant.If the second derivative is positive (cosmological constant is negative), the gradients inverse, creating a "bouncing" universe at H = 0 , h = 0. Similar thing may happen on the red line for the perturbative approach, avoiding the forbidden region effectively.We have verified it is possible using the corrected equivalent of (37).

IV. SOLUTIONS AT HIGHER ORDER IN α
Now, we set H 1 , . . ., H d−1 = H and H d = H +h, where h is small.In this case, after using (29) and performing the time derivative in equations (27), they become linear equations for Ḣi and ḣ.We solve them to obtain Ḣ(H, h, sgn( Φ)) and ḣ(H, h, sgn( Φ)).This enables us to plot the phase diagram of this system.In table IV, the values of the necessary constants c i in different string theories given in [33] are displayed.The phase diagrams for both signs of Φ with d = 9, α = 1, c 1 = −  as in the bosonic theory are depicted in Figures 1 and  2, and ones with c i as in the HSZ theory are depicted in Figures 3 and 4. In heterotic and type II theories, the diagrams are similar to those in bosonic theory -conclusions drawn for the bosonic theory apply to them as well.The reason the diagram is in the HSZ theory is so different is that the coefficients at the H 8 terms under the square root in (29) vanish, leaving H 6 as the highest power.H 6 terms have negative coefficients, so apart from a region near zero the expression will be negative.Therefore most of the plane is forbidden by the requirement that Φ should be real.Further reasoning applies to theories other than HSZ.
Since h is assumed to be a small perturbation, mainly the parts of the phase diagrams with h ≈ 0 are of interest.One can see, that for Φ < 0, anisotropic perturbations are suppressed regardless of the value of H, but H tends to 0 (which corresponds to a static universe).For Φ > 0, anisotropic perturbations will grow for any value of H.For other theories, the diagrams are similar -the conclusions stay the same.The white regions on the top and bottom of both diagrams correspond to the values of H and h for which Φ2 < 0 and thus are prohibited.When Φ = 0, Ḣ = ḣ = 0 and (29) becomes an algebraic constraint.It is satisfied for H = h = 0 and on the curves marked in red on the diagrams.Change of sign of Φ is possible only there (as we expect Φ to be continuous).When Φ > 0, the arrows on the phase diagram point away from 0 and the curve, and when Φ < 0, they point towards 0 and the curve.Therefore, this change of sign is likely only possible in one direction -from negative to positive.
At the zeroth order in α , analogously derived equations are: Ḣ + ḣ = sgn( Φ) dH 2 + 2hH + h 2 + h); (46) which lead to phase diagrams depicted in Figures 5 and  6.Near the H axis, the diagrams at O(1) and O(α 3 ) are mostly identical, which means that for initial conditions in this regions the solutions will be similar to those described in the previous section.The most significant difference is the appearance of the forbidden regions and the possibility of change of sign of Φ.As one can see in (31), at the zeroth order the sign of Φ is determined by the choice of initial conditions.

V. CONCLUSIONS
Our investigation reveals that the de-Sitter solution (an isotropic one with constant H greater than 0), that to the third order in α , no stable solution of this kind is possible, because either the only stable point is H = 0 (for Φ < 0), or there are no stable points.This confirms the previous studies done at α 0 .However, no conclusions can be drawn about the stability of non-perturbative de-Sitter vacua, since higher-order terms can change the dynamics of the system.However, our analysis entails that for small values of H the phase diagrams at α 0 and α 3 do not differ significantly supporting the claim for the lack of de-Sitter solutions at the observed value of cosmological constant [1][2][3].Yet, for the non-perturbative, isotropic study see our previous work [11].Furthermore, we have discussed a possible Pre-Big Bang solution in a universe with negative cosmological constant.

TABLE I .
Values of constants ci in different string theories.