Quantized open FRW cosmology from Yang-Mills matrix models

We present simple solutions of IKKT-type matrix models describing a quantized homogeneous and isotropic cosmology with $k=-1$, finite density of microstates and a resolved Big Bang. At late times, a linear coasting cosmology $a(t) \propto t$ is obtained, which is remarkably close to observation. The solution consists of two sheets with opposite intrinsic chiralities, which are connected in a Euclidean pre-big bang era.


Introduction
Quantum field theory and general relativity (GR) provide the basis of our present understanding of fundamental forces and matter. However, GR is a classical theory, and its incorporation into a consistent quantum theory poses fundamental challenges. In particular, general arguments suggest a "foam-like" quantum structure at the Planck scale 10 −33 cm. One possible framework for such a description is provided by matrix models. Here we will focus on the IKKT model [1] which is singled out by maximal supersymmetry, and is related to IIB string theory. In this approach, we should recover cosmological space-time as a solution, and the known physics should emerge from fluctuations on this background.
In this letter, we will give such a cosmological solution, which could be considered as near-realistic. It realizes a quantum structure of space-time which is exactly homogeneous and isotropic with k = −1, with a finite density of microstates. A Big Bang (BB) arises through an appealing mechanism as in the k = 1 solutions 1 [2], and the late-time evolution could be considered as near-realistic, corresponding to a coasting universe [4]. This has been discussed as a possible alternative to the CDM model [5,6], however we refrain from claiming that the cosmology is fully realistic. The background also leads to spin 2 fluctuations, which could be the basis for emergent gravity, while avoiding the issues (notably Wick rotation) which arose in a previous semi-classical approach [7]. Quantum corrections are assumed to be small, which seems reasonable for cosmological considerations.
E-mail address: harold.steinacker@univie.ac.at. 1 Fuzzy cosmological solutions were also given in [3], but they are not fully homogeneous and isotropic in 3 + 1 dimensions.
We will consider solutions of the following IKKT -type matrix model Here η ab = diag(−1, 1, ..., 1) is interpreted as Minkowski metric of the target space R 1,D−1 . This captures the bosonic part of the IKKT model [1], supplemented by a mass term. This leads to the classical equations of motion plays the role of the d'Alembertian. These are also the effective eom for the IKKT model put forward in [8], which arise after taking into account an IR cutoff and integrating out the scale factor in the matrix path integral dropping the fermions for simplicity. The mass term introduces a scale to the model, and it arises naturally from an IR regularization as discussed in [8].

Euclidean fuzzy hyperboloids
To define the fuzzy 4-hyperboloid H 4 n , let M ab be hermitian generators of so(4, 2) ∼ = su (2,2) and η ab = diag(−1, 1, 1, 1, 1, −1) be the invariant metric. We choose a particular type of (massless discrete series) positiveenergy unitary irreps H n known as "minireps" or doubletons [9][10][11], which have the remarkable property that they remain irreducible under S O (4, 1) ⊂ S O (4,2). They have positive discrete where the eigenspace with lowest eigenvalue of M 05 is an n + 1-dimensional irreducible representation of either SU (2) L or SU (2) R . Then the hermitian generators with R 2 = r 2 (n 2 − 4) [12]. Since X 0 = rM 05 > 0 has positive spectrum, this describes a one-sided Euclidean hyperboloid in R 1,4 , denoted as H 4 n . However the full semi-classical geometry underlying fuzzy H 4 n is CP 1,2 [12]. This is a coadjoint orbit SU (2, 2)/SU (1, 2) × U (1), which is an S O (1, 4)-equivariant bundle over the 4-dimensional hyperboloid H 4 with fiber given by S 2 . Thus X a can be viewed as quantization 2 of the Hopf map where ψ ∈ C 4 transforms in the (4) of su(2, 2) through ab . We work mostly in this semi-classical limit, where H 4 n has the same local structure as the fuzzy 4-sphere S 4 n , with a Poisson tensor M μν ∼ θ μν (x, ξ) transforming as a selfdual 2-form under the local stabilizer 3 S O (4) x of any point x ∈ H 4 . This realizes the local fiber ξ ∈ S 2 , which in the fuzzy case is fuzzy S 2 n . Then the averaging over this S 2 can be achieved using the same local formula (16) as for S 4 N , which will be important below. In particular, H 4 n has a finite density of microstates, since the number of states in H n between two given X 0 -eigenvalues is finite. Note that n is not required to be large here, in contrast to fuzzy S 4 N .

Cosmology from squashed fuzzy hyperboloid H 4 n
It was shown in [2] that in the presence of S O (4, 1)-breaking mass terms, the matrix model has solutions based on H 4 n which describe fuzzy FRW space-times with Minkowski signature. Here we insist on invariant mass terms, and look for solutions of the Note that the generator X 1 is dropped. The indices i indicate Euclidean directions, and 0 is the time-like direction. Geometrically (i.e. in the semi-classical limit), this corresponds to a brane M which is embedded in R 1,3 via a projection along x 1 , The appropriate su (2,2) representations in the fuzzy case are obtained from (8) if the ψ satisfy a bosonic oscillator algebra, see e.g. [10]. 3 Note that the induced metric on the hyperboloid H 4 ⊂ R 1,4 is Euclidean, despite of the S O (4, 1) isometry; this is obvious at the point x = (R, 0, 0, 0, 0).
Therefore the Y μ for μ = 0, 2, 3, 4 are a solution of (2) with m 2 = t (one for each sheet except for t 0 ), whose effective metric depends on the time parameter t. The two sheets M + ∪ t 0 M − are connected at t = t 0 , indicated by ∪ t 0 , as indicated in Fig. 1. We obtain a double-covered FRW space-time with hyperbolic (k = −1) spatial geometries.
One may worry about possible instabilities due to the negative mass term. However, this mass is extremely small, set by the cosmological scale. Moreover as shown in the case of S 4 N [13], even a positive bare mass term can lead to a stabilization at one loop. Thus one may hope that quantum effects stabilize the present solution without a bare negative masses. The computation of the effective metric below would then essentially go through.
We can write this as a S O (3, 1)-invariant FRW metric with k = −1, for t ≥ 0. This gives the Milne metric: Here d 2 is the S O (3, 1)-invariant metric on a space-like H 3 with k = −1, and d 2 is the S O (3)-invariant metric on the unit sphere S 2 .
We need P 00 c 0 (y 0 ) (17) where the time parameters y 0 or η will be used appropriately. Therefore While the first component is non-negative, there is a signature change at c(η) = 0 i.e. cosh 2 (η 0 ) = 3 (19) which marks the Big Bang. The effective metric is Euclidean for η < η 0 , and Minkowskian for η > η 0 . In contrast, the induced metric always has Minkowski signature.
Scale factor Now we rewrite the above metric from local Cartesian coordinates y μ into FRW coordinates (τ , χ , θ, ϕ) (13). At the reference point p with where d 2 is the length element on a spatial standard 3-hyperboloid H 3 . Therefore at the reference point ds 2 Thus the FRW form of the effective metric is obtained as where the scale parameter a(t) is determined by , a(t) = |c(y 0 )| dropping R for simplicity. We thus read off and therefore a(t) = y This describes a linear coasting universe [4,17], which provides a remarkably good fit with observation [4,5]. It has been considered [5] and disputed [18] as a possible alternative to the CDM model. While such a cosmology seems artificial within GR, the present framework provides a good theoretical basis, and should motivate a careful re-assessment of this scenario. In particular, the age of the Universe is correctly reproduced as ≈ 13.9 × 10 9 years from the present Hubble parameter, If ȧ were 1, we would recover the Milne universe (14). However (23) gives a larger spatial curvature radius, which is large compared with the scale of the visible universe. This modification might have some effect e.g. on the CMB, and could have an impact on the detailed assessment of the model. To address this in a reliable way however requires a more detailed analysis of the physics on the present background. Now consider the Big Bang singularity. Shortly after the Big Bang η η 0 (19), we can write dt = |c(y 0 )| 3 4 c 0 (y 0 ) and therefore dropping some constants. Hence we recover the same initial a(t) ∼ t 1/7 expansion as in [2]. The physical implications of this singular but non-exponential initial expansion remain to be understood. Recall that the BB arises only through the effective metric as in [2], while there is no singular expansion and no signature change in the induced metric. Together with a Euclidean pre-Big Bang era, this should have interesting implications on the early cosmology including the CMB, and resolve the horizon problem even in the absence of standard inflation. Note that all propagating modes on M, including the gravitational modes, are governed by the effective metric, as discussed in the next section.

Further perspectives
The effective physics on this background M should be elab- As explained in [15,19], this amounts to a decomposition into higher spin modes, whose propagation is governed by 2 Y (3), hence by the effective metric G μν . The maximal spin in (25) is n, determined by the maximal mode on the internal fuzzy sphere S 2 n . In particular, the tangential fluctuations Y μ + A μ include the following modes where P μ ∈ so(4, 1) is the local generator of translations, cf. [15].
The symmetric part of h μν could naturally play the role of the spin 2 graviton. Whether or not this leads to physically viable gravity is a non-trivial question, which should be examined in detail elsewhere, cf. [15,19]. Here we only note that the issues encountered on S 4 N may not arise here since n can be small. If this emergent gravity works out correctly below the cosmological scale, this could provide an elegant solution of the cosmological constant or dark energy problem.
Once matter and fields on M are taken into account, the energy density near the BB would formally diverge in the semiclassical picture. However, the inherent discreteness of y 0 (5) leads via (24) to a natural discretization of t, hence to a UV cutoff. At this point any continuous evolution equation will break down, and must be replaced by some (yet to be determined) boundary condition relating the Euclidean and Minkowski regimes. Nevertheless, the high energy density near the BB will certainly have a significant effect on the geometry, which remains to be clarified.
Another interesting aspect of the present solution is the twosheeted structure of M. Since the bundle of Poisson tensors θ μν on H 4 n is self-dual on one sheet and anti-selfdual on the other, this could provide the seed for a chiral gauge theory, e.g. through leftright symmetric models. This may also provide a dynamical justification for the present solution over e.g. a Euclidean H 4 solution where all five X a are embedded, since branes with opposite chirality tend to form a bound state, cf. [13,20]. The additional structure required for particle physics might arise from the remaining 6 dimensions in the matrix model, perhaps along the lines of [21,22]. It is thus quite conceivable that all fundamental interactions could arise from a suitable refinement of the present background.