One-loop effective action of the IKKT model for cosmological backgrounds

We study cosmological solutions of the IKKT model with $k=-1$ FLWR geometry, taking into account one-loop corrections. A previously discussed covariant quantum spacetime is found to be stabilized through one-loop effects at early times, without adding a mass term to the model. At late times, this background is modified and goes through a period of acceleration, before settling down to a coasting FLRW geometry with scale parameter $a(t) \sim t$. This is qualitatively close to observation without any fine-tuning, irrespective of the detailed matter content of the universe.

The IKKT matrix model [20] is defined by the action SrT, Ψs " 1 g 2 Tr `rT A , T B srT A , T B s `ΨΓ A rT A , Ψs ˘, where T A pA " 0, ..., 9q are hermitian matrices and Ψ Grassmann-matrix-valued Majorana-Weyl spinors of SOp9, 1q.The model is uniquely determined by maximal supersymmetry, which is essential to obtain a sufficiently local quantum effective action on 3+1 dimensional background branes.
Due to its special properties, the IKKT model shares the special status of superstring string theory as an approach to incorporate gravity into a consistent quantum theory.Remarkably, it offers a novel mechanism to obtain gravity in 3+1 dimensions, which is distinct from the standard mechanism considered in string theory leading to 9+1 dimensional gravity.This new mechanism arises in the weak coupling regime on 3+1 dimensional branes at one loop, where the fluctuation modes propagate on the brane, and do not escape into the bulk.More specifically, it was recently shown [39,40] that the Einstein-Hilbert action arises in the oneloop effective action on suitable backgrounds.The mechanism requires 3+1 dimensional spacetime branes with a certain product structure, without compactifying target space.Therefore the issue of the string theory landscape does not arise.However, it remains to be shown that this "emergent" gravity arising on the 3+1 dimensional spacetime brane can be (near-) realistic.
In the present paper, we address this question for the particular case of the cosmological FLRW solution given in Ref. [23], and study its properties at one loop.We will study the one-loop dynamics of backgrounds T A which can be interpreted in the semiclassical regime as FLRW cosmological spacetime.These backgrounds can then be described as symplectic manifolds M embedded in target space via the matrices We will restrict ourselves to the semiclassical regime of the geometry in this paper, where all matrices are replaced by functions, and commutators by Poisson brackets, i.e., r., .s" it.,.u.Moreover, we consider only 3+1 dimensional spacetime branes M 3,1 embedded along the first 4 matrix directions labeled by a " 0, .., 3. However these will be accompanied by fuzzy compact branes K in transversal directions i " 4, ..., 9, which is essential to obtain the Einstein-Hilbert action at one loop.An introduction and motivation for this framework can be found in Refs.[41,42], see also e.g.Refs.[22,23,26,39,[43][44][45] for related work in this context, as well as Refs.[46][47][48] for efforts towards establishing the emergence of 3+1 large dimensions in numerical simulations.

Geometrical structures
In this section, we recall some geometric structures relevant to the present framework, and study properties of classical divergence-free frames, ignoring possible higher spin (hs) contributions for now.These arise generically on the class of backgrounds due to a hidden internal S 2 [23], and will be taken into account in section 3.
The fundamental geometrical object in the matrix model is the frame [49] which arises as Hamiltonian vector fields generated by the matrix background T a .The metric is related to the frame through where the dilaton ρ relates the symplectic density ρ M on M3,1 to the Riemannian density a |G| via (with G :" det G µν and γ :" det γ µν ) In Cartesian coordinates, the symplectic volume form is given explicitly by Since symplectic manifolds are rigid, this can also be used for deformed backgrounds.
Considering that the frame arises in the present framework as symplectic vector field, it must satisfy the following divergence constraint [50]: in local coordinates on M.
Weitzenböck connection and torsion.In the framework of matrix models, the Weitzenböck connection associated to that frame (3) turns out to be useful.This connection is defined by ∇E a " 0, with trivial curvature but nontrivial torsion.This is natural because the torsion with two frame indices satisfies [23] T where is the "field strength" of the background T a .Recasting T abµ as a covariant tensor using the frame, the following contraction identities hold: where ´T σ ν µ ˘are the torsion and contorsion tensors, respectively.

The undeformed FLRW background M 3,1
We start with the undeformed background considered in Ref. [23], which is defined by the matrix configuration considered as a function on the underlying 6-dimensional bundle space CP 1,2 , which is a coadjoint orbit of SOp4, 2q with generators M ab (here a, b " 0, ..., 5).Locally, this bundle has the structure CP 1,2 -M 3,1 ˆS2 , where the spacetime manifold M 3,1 is described by Cartesian coordinates x µ , and the internal sphere by the t ν .These satisfy the following relations [23]: They generate the algebra of functions on CP 1,2 , which can accordingly be viewed as algebra of higher-spin hs valued functions on M 3,1 , which decomposes into the spin s sectors C s spanned by (irreducible) polynomials of order s in t µ .
We also recall the following identity which relates M 3,1 with a hyperboloid thereby defining x 4 .This entails For more details we refer the reader to Ref. [23].

SOp3q-invariant time-dependent frames
In order to describe more general SOp3q-invariant time-dependent frames, we need to generalize some results of Ref. [50].Using Cartesian coordinates x µ and introducing the notation (here the time parameter t should not be confused with the t µ generators in Eq. ( 11) ff.; moreover, Latin indices i, j, ¨¨¨" 1, 2, 3), the most general spherically symmetric frame E a µ can be written as in Cartesian coordinates.It is simple to show that D and F can be made to vanish via a change of coordinates [50]; moreover, the totally antisymmetric part of the torsion tensor, which is related to the contribution of the axion, is zero if S " 0. With these assumptions, the inverse frame E µ a is Bearing in mind Eqs. ( 16) and ( 17), the relation (10) governing the behaviour of the dilaton ρ yields The spacelike component µ " k of the above equations leads to whose solution is f `t ˘being an arbitrary function of t.On the other hand, the timelike component µ " 0 of Eq. ( 18) gives where we have exploited Eq. ( 20), and the dot stands for differentiation with respect to t.The solution of Eq. ( 21) can be easily obtained if we assume the simplifying hypothesis E " 0. Indeed, in this case we get where g prq is an arbitrary function of r.It follows from Eqs. (20) and ( 22) that the divergence constraint (see Eq. ( 7)) is automatically fulfilled independently of the form assumed by f `t ˘and g prq.This means that we can consistently put f `t ˘" 1 " g prq, which owing to Eq. ( 22) implies A " B. Then the effective metric G µν becomes finally (cf.Eq. ( 4)) which provides the SOp3, 1q-invariant geometry considered in the following.More generally, we can set g prq " 1 but leave f `t ˘arbitrary.Then we get where

SOp3, 1q-invariant frames
To obtain a FLRW cosmology with k " ´1 geometry, it is natural to require that not only the metric but also the frame is invariant under the SOp3, 1q isometry group.This isometry should naturally act on the frame indices as in Cartesian coordinates.Then SOp3, 1q invariance implies that or equivalently Here we define η through (cf.( 12)) which is a measure for the FLRW time, labeling the spacelike hypberboloids H 3 .Again, we can remove the F term using a suitable change of variables xµ " f pηqx µ , so that (hereafter, a prime indicates the derivative with respect to x 4 ) with This holds if the following two conditions are satisfied f Ãpηq " Apηq , which determines f through We will accordingly assume that F " 0 in the adapted coordinates.Then the divergence constraint (7) implies where c is an arbitrary constant that will be set equal to one.However, ρ M " ρ M pηq should then be considered as undetermined.The dilaton ρ is determined by the contraction identity (10), which similarly as before (cf.Eq. ( 20)) yields Then the effective metric is This provides the most general FLRW metric with k " ´1.That metric is simply obtained by setting F " 0 in the above ansatz (27), which will be assumed henceforth.
It follows from Eq. ( 36) and the hyperbolic parametrization [24] that the effective metric can be written as where is the invariant length element on the spacelike hyperboloids H 3 .We can bring Eq. ( 38) to the standard FLRW form via the relations This leads to a coasting late-time evolution [23] with The Weitzenböck connection associated to the frame (31) reads as where we have exploited the relation [23] Therefore, the torsion tensor is formally However, it should be kept in mind that in the presence of nontrivial Apηq, the torsion typically acquires hs valued components which may not be negligible.
3 Generalized k " ´1 FLRW matrix background So far, we considered the cosmological frames which may arise in the matrix model setting.
However, these frames need to be implemented through generators T a via Poisson brackets E aµ :" tT a , x µ u (cf.Eq. ( 3)).For the undeformed solution M 3,1 [23] of the IKKT model with mass term, this background is given by leading to the frame Requiring SOp3, 1q invariance strongly suggests to consider the following generalized SOp3, 1q-covariant matrix background which will lead to homogeneous and isotropic cosmological backgrounds corresponding to a FLRW geometry with k " ´1.As shown in appendix A, it is always possible to use a SOp3, 1q-invariant gauge transformation δ Λ " tΛpx 4 q, .u to reduce this background to the form denoted as standard FLRW gauge.Note that all SOp3, 1q-invariant functions on the underlying background are functions of To compute the resulting frame, we note that Poisson brackets of η " ηpx 4 q can be evaluated as follows Then the frame is found to be As shown in appendix B, these hs components proportional to t a t µ can be eliminated locally in generalized normal coordinates, at the expense of manifest SOp3, 1q invariance.The resulting framework of "higher-spin geometry" is still somewhat obscure, and we shall compute what appears to be the most reasonable effective metric in section 6 and Appendix B. However as a first and more conservative step, we accept the presence of small C 2 components of the frame in this section, and demand that these are negligible in some "linearized" or "weak-gravity" regime.

Linearized regime
The C 0 and C 2 components of the frame can be separated by projecting or averaging over the S 2 fiber, which is achieved using [23] rt a t µ s 0 " Then the classical and C 2 -valued components of the effective frame are The above projection r.s 0 to C 0 is justified in the weak-gravity regime, which means that the C 2 components of the frame are much smaller than the C 0 components: This boils down to i.e.
Here the parameter ε measures the quality of the classical approximation (neglecting the hs components), denoted as "slow-rolling" approximation henceforth.However, ε ! 1 would imply which would imply that the cosmology is close to the unperturbed one with α " const.To see this, consider the cosmic scale factor aptq obtained in the late-time regime η " 1 using Eq. ( 42) as which gives (hereafter a dot denotes the time derivative with respect to t, e.g., 9 a " d dt aptq) where (cf.Eq. ( 59)) and we have used Eq. ( 41).This can be rewritten using where Eq. ( 41) has been employed again.At late times η " 1, this simplifies as while Eq. ( 61) gives the Hubble rate This means that the condition ε ! 1 (cf.Eq. ( 58)) is valid if and only if where are the Hubble rate and cosmic scale parameter for the "slow-rolling" approximation α " const.More explicitly, then the slow-rolling condition becomes hence which means that the cosmic evolution is close to the undeformed background geometry M 3,1 with aptq " 3 2 t.Therefore, to describe significantly different cosmic time evolutions, one needs to keep the hs components.This will be done in the following.

hs-valued inverse tetrad and metric
In order to evaluate the metric G µν , we first need to work out the tetrad frame components E a µ in the presence of hs components.Starting from Eq. ( 53) and exploiting the relation E a µ E bµ " η ab , we find which yields (cf.Eq. ( 4)) where (cf.Eq. ( 58)) Using Eq. ( 72), we can also derive the dilaton.To do this, we need the modulus |γ| of the determinant of the metric γ µν .By employing the Jacobi formula, we obtain in Cartesian coordinates, which in view of Eqs. ( 5) and ( 6) leads to giving in the late-time regime the expansion It follows from Eqs. ( 53) and (75) that the inverse effective metric is We observe that both the frame and the metric are hs valued.The following relation will also be useful in the following (in Cartesian coordinates), which can be easily derived by means of Eqs. ( 72) and (74).

Field strength and torsion
In this section, we work in Cartesian coordinates x µ .We first compute the field strength for the background (50): which takes values in the spin 1 sector C 1 of hs-valued functions on M 3,1 .This will be used below to evaluate the torsion tensor for late times η Ñ 8.

Torsion on the unperturbed background
The Weitzenböck torsion of the unperturbed background with α " 1 can be written as [49] T where τ " τ µ B µ is the cosmic timelike vector field, given by τ µ " x µ in Cartesian coordinates, and τ µ " G µν τ ν .
The contraction of the torsion can be computed in general using which for the unperturbed cosmic background gives As a check, we also find using the frame formalism which is consistent with Eq. (82) due to Eq. (81).

hs-valued torsion on the deformed background T a " αt a
The torsion of the background (50) is obtained using Eqs.( 8) and (79) as Using the relation [49] valid in the late-time regime, Eq. (85) yields where we have exploited Eqs. ( 58) and (68), and denoted the higher-spin components of the torsion with T abµ p2q .We can estimate their size in the late-time regime as follows: Notice that here we consider α 2 α to be of the same order as `α1 α ˘2.At this stage, we are ready to evaluate the term T σ ρ µ T ρ σν G µν which will occur in the oneloop effective action (see Eq. ( 117), below).To this end, let us first calculate T abµ T ν ab .Bearing in mind Eqs. ( 83) and (87), we obtain after a lengthy calculation where we have employed Eq. ( 12) and we have defined Owing to Eqs. ( 72) and (76), Eq. ( 89) leads to for late times, and together with Eq. (81) the desired term T σ ρ µ T ρ σν G µν is obtained as in the late-time regime.As a consistency check we reconsider the ansatz without hs components, where the divergence constraint (7) led to Eq. (34), which in turn implies that α " const.Therefore, it follows from Eqs. ( 36) and ( 46) that for late times This is consistent both with the unperturbed background calculation (82), and with the leading-order term of Eq. (92).

Evaluation of the classical action
From now on, we consider a background brane with product structure embedded in flat (uncompactified4 ) target space, as required for the present mechanism for gravity.Here M 3,1 plays the role of spacetime given by the generalized FLRW background (50) T a " αpηq t a (95) and K " K N are fuzzy extra dimensions which support only finitely many degrees of freedom, see section 4.2.This means that the total Hilbert space is given by H M b C N , where C N is the Hilbert space for the extra dimensions with N P N. The K factor is essential to obtain an induced Einstein-Hilbert term in the one-loop effective action, with effective Newton constant set by the Kaluza-Klein scale of K.
On such a background, the Yang-Mills (YM) action can be written in the semiclassical regime as dropping the fermions for now.Here the invariant (symplectic) volume form Ω " |G| on M 3,1 can be written as in hyperbolic coordinates.The first term in Eq. ( 96) is the bare action for the cosmological background M 3,1 , the second represents the contribution from the extra dimensions K, and the last "mixing" term denotes a kinetic term for time-dependent K.We will evaluate these contributions separately in the following.
4.1 Contributions from the FLRW spacetime M 3,1 In this section, we work in Cartesian coordinates x µ .Using the field strength (79), we can obtain the YM term for the background (95) as in the late time regime, using along with Eq. ( 12).The corresponding contribution to the YM action can be written as where is a spacelike volume factor, and N arises from the trace over the extra dimensions.This geometric YM contribution can be interpreted as brane tension of M 3,1 , which will be seen to be dominant at late times.Remarkably, this large tension is not in conflict with obtaining massless spin 2 excitations on the brane [23].

Contributions from fuzzy extra dimensions K
We recall that the IKKT model comprises 10 matrices T A .Among these, the first 3+1 are used to describe space-time.The fuzzy extra dimensions are realized by the remaining 6 transversal matrices T i , which acquire a nontrivial vacuum expectation value interpreted as (fuzzy) embedding function The precise structure of K is not relevant for our discussion except for the finite, discrete (positive) spectrum, labelled by Λ, of its Laplacian 2 K arising form the splitting 2 " 2 M 1,3 2K : associated with eigenmodes λ Λ P EndpH K q.Here µ Λ is dimensionless, while m 2 K sets the scale of K. Then the second term in the YM action (96) contributes to the potential for m 2 where F 2 K is a discrete number depending on the structure of K. Therefore ż

Mixed term or kinetic term for m K
The last (mixed) contribution in Eq. ( 96) amounts to a kinetic term for m K , which can be evaluated as where is a discrete number depending on the structure of K. Let us evaluate this term explicitly for late times.First of all, by using Eq. ( 77) we find, up to corrections Opε 3 q, which upon employing Eq. ( 45) jointly with the identities (12a) and (12c) yields Therefore, using Eqs.( 76) and (78) and neglecting Opε 3 q corrections we finally obtain Notice that Eq. ( 109) is consistent with an evaluation using the classical metric without the hs components, which for SOp3, 1q-invariant frames gives exploiting Eq. (38).That term clearly suppresses any variations of m K .
Scalar matter contribution.Even though we do not consider matter in this paper, it is not hard to see that matter will typically not significantly affect the late-time cosmic evolution, in contrast to general relativity.To see this, it suffices to consider the contribution from nonabelian scalar fields, which would have an action similarly as in Eq. ( 106), with T i " ϕ viewed as (nonabelian) scalar fields.Then their contribution to the action is much smaller than the contribution of the YM brane tension of M 3,1 tT α , T i utT α , T i u " ρ 2 G µν B µ ϕB ν ϕ !tT α , T β utT α , T β u " α 4 cosh 2 η (112) for large η; note that the energy density " G µν B µ ϕB ν ϕ of matter will decay like 1 aptq 3 " 1 ρ 3 with the cosmic expansion, assuming α " 1.The inclusion of quantum corrections through αpηq will not change this conclusion.

Combined classical YM action
Summing up all the above terms, the YM action (96) becomes in the late-time regime.

One-loop effective action
We want to understand the dynamical evolution of the above FLRW spacetime (50), realized as brane solution of the IKKT model, taking into account quantum effects at one loop.
The one-loop effective action on covariant spacetime branes was computed in Ref. [40].
As pointed out before, we need to assume that the background has a product structure M 3,1 ˆK as in Eq. ( 94) to obtain the Einstein-Hilbert action at one loop.Then the combined geometric one-loop effective action on the spacetime brane M 3,1 takes the form Here is the (bosonic part of the) bare action of the IKKT model with "field strength" (which includes contributions from K computed above), while the induced gravitational action at one loop has the form where the second term indicates higher-order contributions from the one-loop effective action.This determines the effective Newton constant as [40] G N " where c 2 K is a (large) constant depending on the structure of K.Note that the coupling to matter -including fermions 5 , but also the nonabelian bosonic sector arising from K -is contained in S YM .Finally, subsumes the remaining contributions of the induced vacuum energy which are independent of the geometry of M 3,1 , where are the contributions from K, from S 2 n , and mixed contributions from S 2 n ´K, respectively.Here C 1 , C 2 and C 3 are (large) constants determined by the structure of K [40].
Gravitational action without hs components.We first evaluate the above gravitational action using the geometrical results in section 2.3, assuming that the hs components of the frame and torsion can be neglected.However, this is justified only in the linearized regime, as long as the hs components are negligible.Bearing in mind Eqs. ( 29), (36), and ( 46), the gravitational action (117) then becomes where in the last line, which is valid for η Ñ `8, we have used Eqs.( 35) and ( 62) assuming that α is constant.This will be refined below by taking into account the hs contributions.

FLRW background and solution at one-loop level
In this section, we will evaluate the one-loop effective action (114) for the deformed FLRW background (50), taking into account also the hs contributions.This will be used to derive the equations of motion for the cosmic scale parameter aptq.
Let us analyze the various terms occurring in the one-loop effective action S 1loop separately.Due to the presence of higher spin components, it is safer to use Eq. ( 117) for the one-loop gravitational action, rather than its Einstein-Hilbert form.This can be evaluated using Eqs.
(76), (78), and (92), which give in the late-time regime.Transforming the derivatives with respect to x 4 to derivatives with respect to η (cf.Eq. ( 51)) and using Eq. ( 97) , S grav can be written as which after partial integration, assumes the first-order form For the YM piece, we employ the calculation performed before, see Eq. ( 113).Finally, the one-loop vacuum term (119) gives using Eqs.( 97) and (75).

Lagrangian and equations of motion
In of Eqs. ( 113), (124), and (125), the action S " ş dηL can be written in the late-time regime in terms of an effective Lagrangian having the form where (hereafter we will omit the Opε 3 q symbol) Here L m is the matter Lagrangian for some generic matter field ψ, and we have adopted the compact notation for the geometric degrees of freedom of interest.Then the Euler-Lagrange equations yield -after a lengthy calculation -the following equations of motion for α and m K : in the late-time regime.Notice that we have dropped the contributions coming from L m for brevity.

Stabilization of m K and equation of motion for α
Now consider the effective potential V pα, m K q , which can be read off from the Lagrangian (127) written in the form L " T ´V : The plots of V pα, m K q both in the early-time and the late-time regimes are given in Figs. 1 and 2, respectively.We observe that the potential attains a stable minimum for small η, but not for large η.
Stabilization of K and α.At early times i.e. for sufficiently small η, we can assume that the vacuum energy is dominant with C 1,2 " 0, and dominates the contribution from the Einstein-Hilbert term " m 2 K α 2 as well as the YM term.Dropping the Opεq terms, this potential has a stable minimum for m K at This means that K is indeed stabilized by quantum effects, which is an important result justifying the present scenario for emergent gravity, cf.Ref. [40].Moreover looking at Fig. Figure 2: The effective potential (135) with all constants set to unity in the late-time regime (η " 10).
1, it is manifest that the potential V pα, m K q has a stable local minimum for both variables, leading at early times.Therefore the undeformed M 3,1 background ( 11) is consistent with quantum effects for early times.
Assuming m K " const more generally (which holds for sufficiently large C 1 {F 2 K " 1), the Lagrangian functions (128)-(130) simplify as where recall that the first term in L YM arises from K, and the second term from the M 3,1 background.We can already recognize that at late times η " 1, the YM contribution from the M 3,1 background will be dominant.Then α will no longer be constant, which means that the late time evolution is modified by quantum effects.We will now examine this regime.
Late-time regime and accelerated expansion.Using Eqs. ( 138)-( 140), the Euler-Lagrange equation for α becomes where which can be derived also from Eq. (133) assuming m K constant.
Let us consider the leading late-time terms occurring in Eq. ( 141).This means that we retain in Eq. ( 142) only the terms proportional to cosh η " 1, and we obtain the equation This is consistent with the classical solution6 derived in Appendix C for m 2 " 0, see Eq. (C.7).Notice that the exponential factor in Eq. ( 146) leads to ε " 1 at late times.
We have thus found the one-loop corrected FLRW background of the model.Identifying the effective metric for this background with time-dependent α is not trivial, because the hs components of the frame cannot be neglected, as discussed in section 3.1.The effective metric can be obtained by going to adapted local normal coordinates, as discussed in appendix B. This leads to the effective metric (B.17) which at late times with ε " 1 can be written as using Eq. ( 38).This is recognized as a k " ´1 FLRW metric by comparing with the standard FLRW metric (40): at late times.Together with α " e η this leads to t " ce 5η{2 , and the cosmic scale parameter is obtained as at late times, corresponding to an asymptotically coasting FLRW cosmology.This should be compared with the undeformed metric (36) with α " const, which leads to aptq " 3 2 t [23].We conclude that the present one-loop corrected background features an early phase where T a " t a is stabilized by the one-loop effective potential, and a late-time phase where T a " αpηqt a is driven by the kinetic term in L YM , see Eq. ( 139).The cross-over between these two regimes implies a period of accelerated expansion at the transition between the classical and the one-loop regime, jumping from aptq " 3 2 t to aptq " 5 2 t.Although a quantitative assessment of the model would be premature, that qualitative feature is certainly reminiscent of the late-time acceleration observed in our universe, which is usually attributed to dark energy or a small cosmological constant.
Naive projection of the metric.It is interesting to compare the above result with the metric which is obtained by naively projecting out the hs components by averaging over the internal S 2 .The result would be significantly different, leading to the effective metric using Eq. ( 72), where we have neglected Opε 2 q corrections.Here the hs contributions t µ t ν dx µ dx ν are evaluated by replacing t µ t ν with its spin-zero projection (54), and we have exploited Eq. ( 38) to compute η µν dx µ dx ν and Eq. ( 13) to work out the the term x µ x ν dx µ dx ν .A comparison with the standard FLRW metric (40) yields at late times.Bearing in mind the solution (146), Eq. (152a) gives again t " ce 5η{2 , which would now lead to the scale parameter This still amounts to an asymptotically coasting FLRW cosmology, but with a slower expansion rate and in conflict with the result (150).Since the above analysis using local normal coordinates seems more trustworthy, we shall discard the naive treatment by projecting out the hs components.
Some technical comments are in order.First of all, we note that the parameter ε is of order one owing to Eq. (146) (cf.Eq. ( 58)).This is consistent with the above observation that the effective metric (147) differs significantly from the naive projection (151).
Furthermore, one might object that we have only considered the variations w.r.t. the cosmic scale parameter in deriving the equations of motion, and dropped all other possible variations of the background.However this is sufficient for our purpose, because we are considering the most general SOp3, 1q-invariant background7 , and SOp3, 1q is an exact symmetry of the quantum effective action.The general equations of motion at one loop for generic geometries will be derived elsewhere [51].

Discussion and conclusions
Let us discuss the above results.The first important observation is that the quantum corrections corresponding to vacuum energy and the induced Einstein-Hilbert term significantly affect the cosmic evolution only in the early stage of the universe, where the vacuum energy leads via the potential V pα, m K q to a stabilization of the classical M 3,1 ˆK background.
This justifies the use of M 3,1 as a consistent background, without having to introduce a mass term to the model by hand8 as in earlier works [23,24].
However at sufficiently late times η " 1, the semiclassical YM action dominates over these quantum effects, at least at one loop.Then the lack of a stabilization by a mass term leads to an instability of M 3,1 , which is thus accelerated, leading to an exponential pre-factor αpηq in the generalized background (50).We have explicitly determined the resulting modified time evolution of the scale parameter aptq, which turns out to accelerate during some finite period of time, and settle down to a somewhat more rapid but still uniform expansion corresponding to a coasting universe.
In particular, the transition between the early vacuum dominated and late-time epoch of the universe entails a novel mechanism for a period of "acceleration" of the universe.Although the present understanding is not yet mature enough to provide explicit physical results, that mechanism is clearly very interesting in view of the observed late-time acceleration of the universe, which is usually attributed to some unexplained form of dark energy.This certainly deserves to be studied in more detail elsewhere.
To obtain a more complete understanding of the physics on these cosmological backgrounds, one must include matter and study the local perturbations.We have argued that matter is not expected to significantly affect the late-time evolution of the universe in the present framework, in stark contrast to general relativity.However, a more complete understanding can only be obtained once a practical form of modified Einstein-like equations is available for the present framework; this will be addressed in future work.
for a " 0, ..., 3 i.e. it rotates the x a and t a components.This can be done independently for any time η, hence we can use such time-dependent SOp3, 1q gauge transformations to rotate the background into the form T a " αpηqt a (A.3) denoted as standard FLRW gauge.

B Local normal coordinates and FLRW metric
Consider again the background (50), i.e., T a " αpηqt a , leading to the hs-valued frame (53), which we write here again for convenience E aµ " tT a , x µ u " sinh η αη aµ `R tT 0 , xi u| ξ " ϕ i j E 0j | ξ " 0, tT i , x 0 u| ξ " E i0 | ξ " 0, tT 0 , x 0 u| ξ " E 00 | ξ " ´sinh η α. (B.4) Only the first equation is nontrivial, and it is solved by the ansatz ϕ i j " δ i j `bt i t j .(B.5) Then the first equation gives `δi j `bt i t j ˘´sinh η αδ aj `R We can thus use the xµ as local Cartesian coordinates to describe the local physics near ξ for any x 0 .However, the relation (34) arising from the divergence constraint cannot be assumed in local normal coordinates, since the hs components of the torsion are not negligible.We should therefore determine the dilaton directly using Eq.(5) ρ2 " ρ´1 M a |G| " ρM det Ẽaµ , (B.12) where ρM is the reduced symplectic density on M 3,1 in xµ coordinates.Note that the symplectic volume form Ω " ρ M d 4 xΩ t " ρM d 4 xΩ t on CP 1,2 (here Ω t is the normalized volume form on the internal S 2 ) is rigid and not affected by the deformation of the frame.We can thus compute ρM using det ˆBx µ Bx ν ˙" 1 `bt µ t µ " 1 `b R´2 cosh 2 η " Recalling that ρ M " 1 sinh η in Cartesian coordinates x µ , the symplectic density on M 3,1 is therefore given by Then the effective metric at ξ is Gµν " ρ´2 η ab Ẽaµ Ẽbν " psinh ηq ´1 p1 `ε coth ηq ´1α ´2η µν , (B.17) in the above local normal coordinates.For α " e η we have ε " 1, so that this metric behaves like Gµν " e 3η p1 `εqη µν , (B.18) at late times; recall that this metric and the underlying normal coordinates apply near x i " 0 for any (late) times.Since the relation xµ xµ " ´R2 cosh 2 η given in Eq. (B.11) still holds, this looks exactly like a k " ´1 FLWR geometry (31) for such a local comoving observer, so that we can identify the scale parameter aptq as (150).
C Classical solution for m " 0 In this section, we will discuss a k " ´1 FLRW solution of the form (49) T a " αpx 4 qt a (C.1) of the classical IKKT model without mass term, with manifest SOp3, 1q symmetry.Since the classical action dominates at late times, this is expected to be an approximate solution of the one-loop effective action at late times, dropping K for simplicity; recall that the simple M 3,1 background (11) assumes an explicit mass term, which was added to the model by hand for simplicity in Ref. [23].To compute 2 T T b , we can use (79) ´Fab " tαt a , αt b u " ´1 R2 R 2 α 2 θ ab `1 R αα 1 px a t b ´ta x b q (C.For late times x 4 " R, this has the asymptotic solution αpx 4 q " x 4 " R sinh η, corresponding to a background T a " x 4 t a , (C.7) consistent with Eq. ( 146).This leads to the frame (53) with ε " Op1q, hence with significant non-negligible higher-spin components.As discussed in Appendix B, these hs components can be removed at any given (comoving) point in suitable local normal coordinates, where the effective geometry is recognized as k " ´1 FLRW cosmology with aptq " 5 2 t.

Figure 1 :
Figure 1: The effective potential (135) with all constants set to unity in the early-time regime (η " 0.1).

2
Rα 1 t a t µ .(B.1)It is not possible to find Cartesian normal coordinates xµ which are manifestly covariant under the spacelike isometry SOp3, 1q such that the hs components of the frame vanish.This appears to be a rather generic feature of geometries with curvature in the present framework.