α ′ -Cosmology: solutions and stability analysis
αα -Cosmology: solutions and stability analysis Heliudson Bernardo ∗ and Guilherme Franzmann † Physics Department, McGill University, Montreal, QC, H3A 2T8, Canada The Oskar Klein Centre, Department of Physics, Stockholm University, AlbaNova, SE-10691Stockholm, Sweden Nordita and KTH Royal Institute of Technology, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden
Abstract
We review O( d, d ) Covariant String Cosmology to all orders in α in the presence ofmatter and study its solutions. We show that the perturbative analysis for a constantdilaton in the absence of a dilatonic charge does not lead to a time-independet equationof state. Meanwhile, the non-perturbative equations of motion allow de Sitter solutions inthe String frame parametrized by the equation of state and the dilatonic charge. Amongthis set of solutions, we show that a cosmological constant equation of state implies a deSitter solution both in String and Einstein frames while a winding equation of state impliesa de Sitter solution in the former and a static phase in the latter. We also consider thestability of these solutions under homogeneous linear perturbations and show that they arenot unstable, therefore defining viable cosmological scenarios. The very early universe remains the most conceivable laboratory for a theory of Quantum Grav-ity (QG). Assuming a smooth geometric phase, its dynamics can be divided into a backgroundcosmology and perturbations on top of it. Typically, it is the latter that is encoded in obser-vations like the Cosmic Microwave Background (CMB), which is a map of the fluctuations ofthe matter content that were present at the time of recombination. Given current observationalwindows, what the CMB tells us about the very early universe is encoded into two parameters:the amplitude of these fluctuations, A , and the deviation from scale invariance in their powerspectrum, P S , parametrized by the spectral index n s . Neither of them provides us with a directaccess to the underlying background dynamics that was evolving these fluctuations before theCMB was emitted [1, 2, 3].Moreover, the fluctuations imprinted in the CMB are deep in the infrared in relation to QG’sscale. That means that their dynamics can be treated approximately by the linear perturbationtheory in General Relativity (GR). Within this paradigm, the P S of these fluctuations is fullydetermined by the cosmological background evolution, parametrized by the Hubble parameter H ( t ), and their initial conditions (IC). QG may play a direct role into both of them, for instanceby deforming [4, 5] or explaining the IC (e.g. [6]) or by altering the background dynamics, i.e.,modifying Friedmann equations ([7, 8, 9] and references therein). ∗ Email: [email protected] † Email: [email protected] a r X i v : . [ h e p - t h ] M a y ithin the space of models to account for the observed values of {A , n s } , inflationary cosmol-ogy remains the most successful proposal [10, 11, 12, 13, 14]. It considers the IC to be quantummechanical while its background evolution is quasi-de Sitter, ˙ H ( t ) ≈
0, which suffices to providethe necessary ingredients for an almost scale-invariant P S . In the scope of inflation, QG maychange the IC for ultraviolet (UV) modes [15] or even constrain how much it lasts [16]. All inall, inflation has largely motivated to seek for de Sitter (dS) like solutions in UV completions ofgravity.Among the different QG coexisting approaches, String Theory remains the most successfulproposal to this date. As expected, most of the early universe physics considered in its frameworkhave been dS oriented [17, 18, 19]. However, alternative models can also be constructed. StringGas Cosmology (SGC) [20], for instance, is a toy model that considers the fluctuations to bethermal in their origin, and their stringy nature guarantees them to have a holographic scaling[21, 22], while the background evolution is almost quasi-static, H ( t ) ≈
0. In more generality, SGCfulfills the requisites demanded by the emergent scenario discussed in [3], which also providesthe necessary elements in order to recover an almost scale-invariant P S .Given that the fluctuations’ dynamics can be accounted by the linear perturbation theoryin GR, our interest is to develop further the background cosmology in the context of StringTheory. Our approach consists of looking to its universal NS-NS massless sector in a cosmologicalbackground including all classical string corrections, namely α -corrections, that are fixed by an O ( d, d ) symmetry which is intrinsic to such a background.Historically, this particular avenue of research goes back to the early 90’s after the realizationthat the cosmological background provided by these fields had a scale-factor duality [23, 24],which was later generalized to a global O ( d, d ) symmetry [25, 26]. These developments wereconsidered in the context of 0 th order in α -corrections and an explicitly O ( d, d ) covariant for-malism was developed, with a manifestly duality invariant action [27]. Then, it was showed thatthis symmetry remained even after considering the full tower of α -corrections [28] as long asthe fields remained spatially independent. Although the O ( d, d ) transformations also receivedcorrections, in [29] it was shown that the form of the transformations were preserved, at least to1 th order in α . Recently, assuming this to be true to all orders, the form of the α -correctionswere found, allowing a suitable classification of these corrections [30] . A natural extension oftheir results involved the inclusion of a matter sector [33], which finally allows us to considerrealistic cosmological solutions.Our goal in this paper is to briefly review O ( d, d ) String Cosmology including α -corrections[30, 33], then to present and discuss some of its cosmological solutions and finally to considertheir stability. This will open precedent to consider realistic cosmological scenarios in futureworks and to investigate the feasibility of the models discussed above and beyond.The paper is organized as follows. In Section 2 we review the formalism and write down theequations of motion. Then, in Section 3 we introduce cosmological solutions for which a stabilityanalysis will be considered in Section 4. Finally, in Section 5 we conclude. O ( d, d ) String Cosmology including α - corrections The massless NS-NS sector of all superstring theories includes the metric, G µν ( x ), the Kalb-Rammond 2-form field, B µν ( x ), and the dilaton field, φ ( x ). For a cosmological background, weconsider the fields given as G = − n ( t ), G i = 0, G ij = g ij ( t ), B = 0, B i = 0, B ij = b ij ( t )and φ = φ ( t ). Then, the action for these fields coupled to matter including α -corrections to all Cosmological applications in the absence of matter were considered in [31, 32]. d, d ) symmetry, has been shown to have the form [33] S = 12 κ Z d d xdtne − Φ (cid:2) − ( D Φ) + X ( DS ) (cid:3) + S m [Φ , n, S , χ ] (1)where D ≡ /n∂ t is the covariant temporal derivative, Φ ≡ φ − ln √ det g is the shifted dilaton, χ represents the matter sector and X ( DS ) depends only on the first derivative of the 2 d × d matrix S = η H = (cid:18) bg − g − bg − bg − − g − b (cid:19) , (2)where η = (cid:18) (cid:19) , H = (cid:18) g − − g − bbg − g − bg − b (cid:19) , (3)with η being an O ( d, d ) metric and H ∈ O ( d, d ), such that S = 1, while Φ is a scalar under O ( d, d ) transformations. Considering only single trace contributions to X ( DS ), that was shownin [30] to be enough for applications in flat FRLW cosmologies, the action can be written as S = 12 κ Z d d xdtne − Φ " − ( D Φ) + ∞ X k =1 α k − c k tr( DS ) k + S m [Φ , n, S , χ ] , c = −
18 (4)which is manifestly invariant under the global O ( d, d ) transformations and time reparameteriza-tions. The values of the coefficients { c k } for k > c = 1 /
64 for heterotic strings and it vanishes for type IIstrings, for instance).The equations of motion (EOM) are given by2 D Φ − ( D Φ) − ∞ X k =1 α k − c k tr( DS ) k = κ e Φ ¯ σ, (5a)( D Φ) − ∞ X k =1 α k − (2 k − c k tr( DS ) k = 2 κ ¯ ρe Φ , (5b) D e − Φ ∞ X k =1 α k − kc k S ( DS ) k − ! = − κ η ¯ T , (5c)where we defined a dilatonic charge [34] σ ≡ − √− G δS m δ Φ , (6)the bar denotes multiplication by √ g , ¯ σ ≡ √ gσ , and we defined the O( d, d ) covariant energy-momentum tensor ¯ T ≡ n (cid:18) η δS m δ S S − η S δS m δ S (cid:19) , (7)while the energy density is defined as usual, ρ = ¯ ρ √ g = T n , T µν = − √− G δS m δg µν . (8)3rom the EOM, we can also write down a continuity equation, D ¯ ρ + 14 tr( S ( DS ) η ¯ T ) −
12 ¯ σ D Φ = 0 . (9)Restricting ourselves to a flat FRLW background, meaning we consider a vanishing two-form, n ( t ) = 1 and g = a ( t ) I , and taking the matter sector to be given by a perfect fluid defined as T = ρ , T i = 0 and T ij = pg ij , the equations reduce to the α -corrected Friedmann equations˙Φ + HF ( H ) − F ( H ) = 2 κ e Φ ¯ ρ (10a)˙ HF ( H ) − ˙Φ F ( H ) = − dκ e Φ ¯ p (10b)2 ¨Φ − ˙Φ + F ( H ) = κ e Φ ¯ σ, (10c)where p is the pressure, H ( t ) is the Hubble parameter, denotes derivatives w.r.t. H and thefunction F ( H ) is defined as F ( H ) = 2 d ∞ X k =1 ( − α ) k − c k k H k . (11)As expected, these equations are invariant under the scale factor duality transformation a → /a ,since under this transformation we have H → − H, Φ → Φ , f ( H ) → − f ( H ) , g ( H ) → g ( H ) , ¯ ρ → ¯ ρ, ¯ p → − ¯ p, ¯ σ → ¯ σ, (12)which relies on the fact that the matter action is duality invariant, which is the case if oneconsiders it to be given by a gas of free strings [27, 24]. This is a remnant of the O( d, d )symmetry in a FLRW background.The continuity equation (9) reduces to˙¯ ρ ¯ ρ + dHw − λ , (13)where we have introduced a barotropic equation of state (EOS), w ≡ p/ρ , and the density ratio(DR), λ ≡ σ/ρ , which measures how strongly the matter is coupled to the dilaton in relationto the metric [35]. Throughout this paper, we will assume { w, λ } to be constants, thus we cansolve the continuity equation for the energy in terms of the scale factor and shifted dilaton,¯ ρ = ¯ ρ (cid:16) a a (cid:17) dw e λ (Φ − Φ ) , (14)where ¯ ρ , a and Φ are constants.In the next sections we will study closely and systematically cosmological solutions to theseequations. Since they are so far only written in the String frame (S-frame), it is also usefulto write the Hubble parameter in the Einstein frame (E-frame), H E , in terms of the S-framevariables. In [30], H E as a function of the cosmic time in the E-frame, t E , was shown to be H E ( t E ) = − ( a ( t )) dd − e Φ d − d − (cid:0) ˙Φ + H (cid:1) , dt = dt E e φd − (15)and its evolution is given by dH E ( t E ) dt E = − e φd − d − (cid:20) ¨Φ + ˙ H + 1 d − dH + ˙Φ)( H + ˙Φ) (cid:21) . (16)4 Cosmological Solutions
The solutions to the α -Cosmological equations (10) started to be considered in [33]. Before weconsider their stability, we will briefly review them and also introduce some new solutions thatprovide appealing cosmological scenarios. It is known that at the lowest order in α , i.e., considering c k = 0 for k > w = 1 /d [24]. In[33] it was shown that upon turning on the α -corrections, it is not possible to obtain a solutionwith a constant EOS after imposing ˙ φ = 0 = σ . However, a perturbative solution was found, H ( t ) = H t + α H t + α H t + . . . , (17) w ( t ) = 1 d − dc w α H + 128 dc w α H − dc w α H + . . . , (18)where the EOS is time dependent instead and the coefficients w i and H i are completely fixedby the c k and the spacetime dimensionality. Note that, indeed, the first term corresponds tothe radiation solution expected from the 0 th order equations. The above solution is potentiallyrelevant for late-time cosmology since a rolling dilaton can lead to violations of the weak equiv-alence principle [34], which is very constrained (see for instance [36]), and its running may leadto a breakdown of the classical regime, given the dilaton modulates the strength of the stringcoupling. If the Hubble parameter is zero, then F (0) = F (0) = 0 and F (0) = 16 c d . Thus, (10c) givesas solution for the shifted dilaton,˙Φ( t ) = − t (cid:0) λ (cid:1) + C , Φ( t ) = C −
21 + λ ln (cid:20) t (cid:18) λ (cid:19) + C (cid:21) , (19)where C i are constants and λ = −
2. Then, it is easy to see from (10b) that the EOS vanishes.Thus, a static solution is compatible with a pressureless equation of state. Note that for H = 0all the α -corrections vanish since they enter into the equations in powers of H . Therefore, this isa solution to the lowest order equations that remains valid even after including all α -correctionsdespite its energy scale.This solution in the E-frame gives rise to H E ( t E ) = 2(2 + ( d − λ/ t E − t E, ) , a E ( t E ) ∝ t d − λ/ . (20)Note that for λ = 0 it corresponds to a radiation dominated solution. It has been shown [33] that after fixing H = H to be constant with σ = 0 and constant EOS,the shifted dilaton satisfies ˙Φ = dwH . It was also shown that if the condition for a constant w is relaxed, then there is a dynamical solution with varying Φ and w that reduces to the former5ase in the asymptotic limits t → ±∞ , with a different w in each limit. In fact, as discussedin the next section, dS solutions of the form ˙Φ = − βH with constant β are fixed points of thedynamical equations. This is true even for a non-zero DR, for which β depends on λ as well. Aswe see below, for β < λ = 0. Thefirst one comes about after analyzing the dilaton’s velocity, which satisfies˙ φ = dH w ) . (21)Thus, for w = −
1, corresponding to a cosmological constant EOS, we get a dS solution in bothframes since the dilaton is constant. The second special case can be found by looking to (15) andnoticing that a static solution in the E-frame can be obtained by considering a winding EOS,namely w = − /d . Thus, we have a time dependent dilaton solution with Minkowski metric inthe E-frame. This opens precedent for considering the quasi-static phase required by String GasCosmology [20] for the background cosmology and more generally for implementing the emergentscenario advocated in [3] as an alternative to inflation in the context of string cosmology. Bothspecial cases are only possible due to the entire tower of α -corrections, therefore characterizingnon-perturbative solutions. The covariant String Cosmology equations of motion with α -corrections are coupled and non-linear. To proceed with the stability analysis, we treat the equations as a dynamical system andconsider linear perturbations around a given solution, in a similar fashion to what was done in[37] to handle first order α -corrections. In order to develop some intuition first, let’s look to thedynamical analysis of the Friedmann equations in GR. For D = 1 + 3 in the absence of spatial curvature, the dynamical equation after taking intoaccount the constraint equation is given by˙ H = −
32 (1 + w ) H , (22)assuming a barotropic EOS. We see that a fixed point, namely ˙ H = 0, is given by w = −
1, a dSsolution as expected, or vacuum, H = 0.In order to look at the stability of the dS solution, let us write the dynamical equation as˙ H = C ( H ) and look at fluctuations around the fixed point solution, H . In fact, this results into˙ δH = C ( H ) δH. (23)It is easy to see that C ( H ) = 0, so any homogeneous perturbation in the dS solution will shiftthe value of the Hubble parameter. That does not mean that the dS solution is unstable though,it just means that a perturbation in the energy density has happened, as one can easily checkafter looking to the continuity equation. Both the energy density and the Hubble parametershift by a constant amount . Another insightful exercise is to consider a power-law solution, H = n/t . Then, solving (23) shows that acrude phantom EOS, w < −
1, leads to instabilities. .2 Dynamical Analysis of α -Cosmology Following the same prescription, we need first to rewrite (10) as a system of first order ordinarydifferential equations (ODE). Given we are interested in a relatively short, single phase evolutiondominated by a certain type of matter, we consider a constant EOS and a constant DR. Thus,(10) reduce to ˙ H ≡ C ( H, y ) = 1 F (cid:2) yF − dw ( y + HF − F ) (cid:3) (24a)˙ y ≡ C ( H, y ) = y − F λ y + HF − F ) , (24b)where ˙Φ ≡ y and we have assumed F ( H ) = 0, together with the constraint equation nowwritten as y + HF − F = 2 κ e Φ ¯ ρ. (25) The fixed points, also called equilibria, X , given by solutions of the ODEs where ˙ H = ˙ y = 0, aredefined by C i ( X ) = 0, which implies y F − dw ( y − F ) − dwH F = 0 (26a) (cid:18) λ (cid:19) ( y − F ) + λ H F = 0 . (26b)where H and y are constants and F ( n )0 ≡ F ( n ) ( H ). They can be combined into F (cid:20) H − y dw (cid:18) λ (cid:19)(cid:21) = 0 , (27)where we assumed w = 0 (the case of a vanishing EOS will be considered later). The fixed pointscorresponds to dS solutions in the S-frame with a constant shifted dilaton’s velocity. In order to analyse stability around the fixed points, we need to linearize the system of ODEsand then look at the eigenvalues of the matrix defined by A ≡ ∂ { H,y } C i ( X ) at the equilibria.The matrix is given by A = y − dH w F ( F − dwy ) − F + λ H F y (cid:0) λ (cid:1) ! . (28)The cases in which we are mostly interested in this paper are all of the form y = − βH , assummarized in Section 3.3. Then, the eigenvalues of A are given by, α ± = − H β − H (cid:18) β + βλ dw (cid:19) ± vuut H " λF H + (cid:18) dw + βλ (cid:19) − F F ( F + 2 dwβH ) . (29)Unstable modes are characterized by positive eigenvalues, while decaying ones are associatedto negative eigenvalues. When we have vanishing eigenvalues further analysis is required. Inparticular, that was the case for the dS solution in GR studied above in Section 4.1, where theperturbations only cause a shift on the background parameters and do not lead to exponentialinstabilities. We will also find that among our solutions.7 .3 Stability of α -cosmological solutions We are finally ready to consider the stability of the solutions introduced in Section 3.3. FromSection 4.2, it is clear that they are fixed points solutions with ˙Φ = − βH = constant, where β = 1 gives the static phase in the E-frame and β = d gives the dS solution in both frames. Theequations of motion imply (for w = 0) H F dw (cid:20) dw + β (cid:18) λ (cid:19)(cid:21) = 0 (30) β H − F = 2 κ e Φ ¯ ρ (cid:18) dwβ (cid:19) , (31)where we have used the fact that for constant ˙Φ, w and λ we have e Φ( t ) ¯ ρ ( t ) = e Φ ¯ ρ . (32)The condition for fixed points (27) can now be written as dw + β (cid:18) λ (cid:19) = 0 , (33)where we have assumed F = 0 . The values of F , F and H can be fixed by the EOM (for1 − λ/ = 0 and λ = − F = 2 d w H λ , (34) F = 1 + λ λ H F , (35) H = −
11 + λ κ e Φ ¯ ρ dwβ . (36)From the last equation, we see that either w > λ < − w < λ > −
2. Our mainfocus is on the latter case. Note that the last equation shows that the Hubble parameter’s scaleis defined by the matter content, which does not have to be necessarily close to the string scale,potentially allowing for realistic phenomenological values.To check the solutions’ stability at fixed points, let us look at the sign of the eigenvalues atequilibria after considering (33)-(35), α ± = − β H (cid:18) ∓ (cid:12)(cid:12)(cid:12)(cid:12) dwβ (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) . (37)For λ = 0, then β + dw = 0 and the eigenvalues are { , − βH } : one mode is decaying and oneis constant. The constant mode is not worrying, since it corresponds to exactly what happenswhen one considers perturbations around dS solutions in GR, now with a constant shift not onlyon the Hubble parameter but also in the shifted dilaton’s velocity.More generally, what we need to guarantee is that the eigenvalues are never positive. Thus, β has to be positive and the term in the square root implies w (cid:18) dwβ (cid:19) ≤ , (38) The solution with F = 0 was considered in [33]. In order to recover these relations, one can follow the prescription already outlined in [33] in Section 6.2.,where the case λ = 0 was considered. w ∈ [ − β/d, w = 0, λ = {− , − } . In order to check them, one can makedirect use of (26) and (28). For w = 0 and λ = −
2, which also implies y = 0 (since we are notinterested in vacuum solutions), we find H F = 2 κ e Φ ¯ ρ (cid:18) λ (cid:19) , F = λκ e Φ ¯ ρ (39) α ± = ± s − F F + λH F , (40)so that the eigenvalues have opposite signs and this fixed point will be a saddle point as long asthe term in the square roots is positive, otherwise the two eigenvalues will be imaginary and theperturbations are periodic, having a fixed amplitude (the fixed point is a neutrally stable center[38]). While for w = 0 and λ = − y = ˙Φ < F . Our findings aresummarized in Figure 1. stability region w = - β d λ = - λ = - w λ { λ , w } - plane Figure 1: The shaded region indicates where the solutions are stable despite the properties of thefunction F . For w = 0 or λ = −
1, in general the stability is contingent on the sign of F ( H ),represented by the dashed line on these boundaries, while the case { w, λ } = { , − } is stable aslong as ˙Φ < − βH for β > The dS solutions with constant ˙Φ can be recovered from the asymptotic limit of the generalsolution with evolving ˙Φ and EOS. This was already shown in [33] for the case λ = 0. Forconstant λ > −
2, we may combine equations (10a) and (10c) to write the following equation for9( t ) ¨Φ − (cid:18) λ (cid:19) ˙Φ + 12 (cid:18) λ (cid:19) F − λ H F = 0 , (41)which is solved byΦ( t ) = Φ −
42 + λ ln (cid:20) cosh (cid:18) q (2 + λ ) | F (2 + λ ) − λH F | ( t − t ) (cid:19)(cid:21) . (42)Taking the asymptotic limitslim t →±∞ ˙Φ = p | F H λ − F ( λ + 2) |√ λ + 2 = ± dw λ H , (43)where we used equations (34) and (35) to write the last equality for w <
0. We see that werecovered the condition (33) for a fixed point solution, also serving as a consistency check of ouranalysis. For t → ∞ , β > t → −∞ we have β < We have reviewed the action and equations of motion encompassing O( d, d ) Covariant StringCosmology to all orders in α . Since the formalism has been introduced in the String frame,we have also written down the evolution of the Hubble parameter in the Einstein frame so thattypical cosmological scenarios could be considered and discussed. In order to check the viabilityof such scenarios, we have also gone through a linear stability analysis. The summary of theresults obtained is as follows.The perturbative solution for a constant dilaton in the absence of a dilatonic charge impliesa time-dependent equation of state for barotropic matter. Even though perturbatively the EOSis still completely fixed and it reproduces a radiation EOS at the 0 th order as it was expectedfrom the lowest order EOM, higher α -corrections imply that a constant EOS is not a solution.We have considered de Sitter solutions in the String frame and showed that they lead to aconstant velocity for the shifted dilaton asymptotically, which depends on the asymptotic valueof the EOS; in more generality, it also depends on the density ratio λ . Among the different valuesthe EOS can take, w = − w = − /d stand out when λ = 0.For a cosmological constant EOS, we have seen that the dilaton is constant, which contrastswith having a radiation EOS that was otherwise expected from the lowest order equations asdiscussed above. This solution implies an equivalence between String and Einstein frames, there-fore implying a de Sitter solution also in the latter. Moreover, we have shown that the stabilityof this solution is on the same grounds as the typical de Sitter solution encountered in GR.Our most promising result occurs for a winding equation of state, w = − /d . We have seenthat there is a de Sitter solution in the String frame with this EOS that implies a static solutionin the Einstein frame. Since first proposed, String Gas Cosmology relied on a quasi-static phasein the Einstein frame in order to recover an almost scale-invariant power spectrum, despite thefact that such a phase could not be realized through the evolution of the stringy EOS associatedwith a gas of closed strings after using the lowest order EOM in α . On the other hand, in α -Cosmology this phase is naturally realized by a winding EOS and that corresponds preciselyto the initial phase of the stringy EOS. Given that this solution is also shown to not be unstableunder a linear stability analysis, this can be seen as a proof of principle that the dynamicsproposed in SGC, and more generally advocated in the emergent scenario, may be realized inString Cosmology after considering all α -corrections [39].10 Acknowledgements
The authors thank Jerome Quintin and Robert Brandenberger for reading the manuscript andrelevant discussions. H. B. is thankful to Nordita for the hospitality while this work was devel-oped. The research at McGill is supported by funds from NSERC.
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