On the emergence of an expanding universe from a Lorentzian matrix model

We present evidence that recent numerical results from the reduced classical equations of a Lorentzian matrix model can be interpreted as corresponding to the emergence of an expanding universe. In addition, we propose an effective metric to describe the emerging (3+1)-dimensional spacetime. This metric gives, at all times, finite values for the Ricci and Kretschmann curvature scalars. With these results, we are able to give a heuristic discussion of the origin of the Universe in the context of the matrix model.


I. INTRODUCTION
The IKKT large-N matrix model [1] has been proposed as a nonperturbative definition of (9+1)-dimensional type-IIB superstring theory.
Numerical investigations [2,3] of the Lorentzian version of the IKKT matrix model have found indications that three spatial dimensions emerge which behave differently than the remaining six spatial dimensions and that the "late-time" behavior of these three spatial dimensions may be controlled by a classical configuration of the matrices. Recently, further numerical results have been presented [4] from the matrix-model classical equations with a particular Ansatz for the ten bosonic matrices, where the Ansatz implements the split of an (3+1)-dimensional "spacetime" and a 6-dimensional "internal space." The goal, here, is to present a simple check that Figs. 3-5 in Ref. [4] indeed correspond to a flat 3-space expanding with cosmic time |t|. In addition, we suggest an effective metric for the emerging (3+1)-dimensional spacetime and give a heuristic discussion of the origin of the Universe, assuming that the matrix model describes a new phase replacing the big bang singularity [5][6][7]. We have also obtained exact solutions of the reduced classical equations of the Lorentzian matrix model, which are presented and discussed in App. A.

II. PROCEDURE
The basic idea is to calculate certain observables from a simple model describing random points inside a box embedded in Euclidean 3-space and to compare with the observables obtained from the reduced classical equations of the Lorentzian matrix model. The procedure consists of six steps: 1. Consider a cube of volume (2 L) 3 in R 3 with a constant flat metric, g ij (x) = δ ij for spatial indices i, j ∈ {1, 2, 3}.
2. Generate n points in the cube with random (uniform-distribution) values for the Cartesian coordinates x i in the range [−L, L] ⊂ R.
3. Make diagonal n × n matrices X 1 , X 2 , and X 3 (the matrix X 1 , for example, has the x 1 coordinates of the n random points on the diagonal and zeros elsewhere). 4. Define the 3 × 3 matrix T as in Eq. (3.5) of Ref. [4] and calculate its eigenvalues λ i . 5. Define the n × n matrix Q as in Eq. (3.7) of Ref. [4] and calculate its eigenvalues q r , for r = 1, . . . , n.
6. Define the number R 2 as in Eq. (3.8) of Ref. [4] and calculate its value.
For definiteness, we give the explicit definitions of T , Q, and R 2 : where the central dot on the right-hand sides of (2.1a) and (2.1b) stands for matrix multiplication. The flat metric δ ij in (2.1b) effectively traces back to the coupling constants η KL in the Lorentzian version of the IKKT matrix model [1,2], using the notation of Ref. [4] with (9+1)-dimensional indices K and L running over {0, 1, 2, 3, . . . , 9}. The observables obtained from the random-point model carry a hat, in order to distinguish them from the observables obtained from the classical equations of the matrix model.

III. NUMERICAL RESULTS
The dimensionless box scale L used in the procedure of Sec. II can be considered to be a function of the dimensionless cosmic time t, L = L |t| . (3.1) The n random points from Sec. II are then generated inside a cube of volume For the moment, t is considered to be just an arbitrary cosmic time coordinate, but t has, in fact, been found to arise from the matrix-model calculation of Ref. [4] (see Sec. V for further discussion). We start by taking n = 10 random points (in order to allow for a direct comparison with the results of Ref. [4]; see below) and assume the following numerical values for the box scale L(|t|): where the actual L values are chosen to obtain the desired values for R 2 . In fact, the values for R 2 obtained in a typical run are shown in the top-left panel of Fig. 1. The corresponding eigenvalues λ i are shown in the middle-left panel of Fig. 1 and the corresponding eigenvalues q r in the bottom-left panel of Fig. 1. Two remarks are in order. First, with the small number of points (n = 10), the scatter of λ i values at |t| = 0.75 is significant. Second, in the bottom-left panel of Fig. 1, the apparently increasing gap near zero for larger and larger values of |t| just appears because a larger and larger interval is covered by a fixed number (n) of points. Both issues will be discussed further below.
For a first comparison, we put n = 10 random points in a flattened box of volume with L(0) = L(0.02) from (3.3). The corresponding numerical results are shown in the left panels of Fig. 2. The middle-left panel of Fig. 2 shows that two eigenvalues grow as |t| increases, whereas one eigenvalue remains more or less constant. The behavior in the middle-left panel of Fig. 2 is fundamentally different from that in the middle-left panel of Fig. 1. For a second comparison, we put n = 10 random points in a ball of radius L(t) = (6/π) 1/3 L(t) and volume The corresponding numerical results are shown in the left panels of Fig. 3, which are qualitatively the same as those in the left panels of Fig. 1 (but there are differences, as will become clear later on).
With the simple random-point model of Sec. II, it is easy to push the numerics further, for example, by taking n = 10 2 random points (results from n = 10 3 random points are similar and a selection will be given later). The numerical results from n = 10 2 random points are given by the right panels of Figs. 1-3. The top-right and middle-right panels of Figs. 1 and 3 show a more or less stable behavior of R 2 (t) and λ i (t) for n → ∞ at fixed t, the eigenvalues λ i approach each other and λ i ∼ R 2 /3 , while the bottom-right panels of Figs. 1 and 3 show a dense distribution of q r over a range increasing with |t|. For n = 10 2 random points in the flattened box, the middle-right panel of Fig. 2 gives results similar to those for n = 10 in the middle-left panel, with two eigenvalues λ 2,3 (t) growing with increasing |t| and a single eigenvalue λ 1 (t) staying more or less constant (the same behavior is seen with n = 10 3 random points in the flattened box).
We have already mentioned the significant spread of λ i values if the number of points is relatively small (n = 10) and the gaps occurring in the q r distribution. Figure 4 shows, for the case of a ball, the behavior with increasing values of n. Recall that the quantity R 2 is given by the sum of the three eigenvalues λ i or, alternatively, by the average of the eigenvalues q r , according to (2.1c). As to the detailed distribution of q r eigenvalues, Figs. 1 and 3 show some differences between the case of the cube and the case of the ball. Figure 5 gives the corresponding distributions of q r values for n = 10 3 random points and cosmic time |t| = 0.75.
To summarize, we have found that the left panels of Fig. 1 and 3 from the random-pointmodel calculations directly match Figs. 3-5 from the matrix-model calculation [4]. The conclusion is that the classical matrix-model equations (2.6) of Ref. [4] with the appropriate Ansatz (2.7) of Ref. [4] indeed seem to generate a cosmic time t and a flat 3-space which expands with |t|.

IV. EFFECTIVE METRIC
Taking the classical results from Ref. [4] at face value and relying on the spacetime interpretation from Sec. III, we introduce the following effective metric [5,6]: where relativistic units are used (c = 1) and where b can be interpreted as the length scale of a spacetime defect [8]. The square of the cosmic scale factor a(t) in this metric can be identified with the quantity R 2 (t) obtained from the large-N matrix model [4]. Specifically, we take so that a(0) ≈ 1 from the R 2 (t) results of Fig. 3 in Ref. [4]. The actual numerical value of b in (4.1) is not really fixed by the matrix model (a model with only dimensionless numbers), but may perhaps be estimated [6] as b ∼ √ α ′ in terms of the Regge slope α ′ from the inverse string tension of the corresponding type-IIB superstring theory [1]. Here, we have switched to natural units with = 1 and c = 1.
With the metric (4.1a), the Ricci curvature scalar R(x) ≡ g νσ (x) g µρ (x) R µνρσ (x) and the Kretschmann curvature scalar K(x) ≡ R µνρσ (x) R µνρσ (x) are given by [5] R a(t) = 6 K a(t) = 12 (4.3b) Both curvature scalars are perfectly regular at t = 0 for the series a(t) = 1 + a 2 t 2 /b 2 + a 4 t 4 /b 4 + . . . ; see Ref. [6] for further discussion. Scalar metric perturbations of the background (4.1) have been studied in Ref. [7]. If nonrelativistic matter appears later (as argued in Sec. V), the matter density perturbations are found to grow with |t| and it is possible to define a "thermodynamic" time T (t) = |t|. But, here, we only have an emerging spacetime.
There is, however, an important caveat. It is, namely, not at all clear that the genuine Lorentzian large-N matrix model [1][2][3], and not just the set of corresponding classical equations [4], also gives a nonsingular bouncing cosmology as suggested by Fig. 3 of Ref. [4] and the top-row panels of Figs. 1 and 3 here. If a cosmic bounce does not result from the matrix model, then the effective spacetime metric (4.1a) only holds for t > 0, perhaps with b = t emergence ∼ √ α ′ . The metric (4.1a) is then equivalent [5,6] to the spatially flat Robertson-Walker metric with cosmic time coordinate τ The authors of Ref. [4] have obtained numerical solutions of the classical equations for the ten bosonic N × N traceless Hermitian matrices A K of the Lorentzian IKKT matrix model [1]. With a particular Ansatz to implement the 3 + 6 split of spatial dimensions found previously [2,3], the classical results of Ref. [4] indicate the emergence of a cosmic time coordinate t and a flat 3-space expanding with |t|. The results of the present article support this interpretation if we compare the observables from Ref. [4] with those obtained from having random points in an expanding 3-dimensional cube or ball embedded in R 3 (Secs. II and III). In addition, we have proposed an effective metric to describe this emerging spacetime (Sec. IV).
At this point, it may be of interest to clarify the meaning of the emerging "time" in the large-N matrix model [4]. From the Ansatz (2.7) of Ref. [4] with N = N X + N Y , there are four N X × N X Hermitian matrices X µ , for µ ∈ {0, 1, 2, 3}. Next, the particular matrix X 0 , singled out by the coupling constants η KL , is diagonalized and its real eigenvalues α p , for p ∈ {1, 2, . . . , N X }, are ordered as follows: This essentially gives a dimensionless "time" variable t, which runs parallel to the index p of the eigenvalues α p , as will become clear shortly.
In fact, the definition (3.3) in Ref. [4] can be written as follows: with n a fixed number (1 ≤ n < N X ) used for averaging the eigenvalues α p and k an index running over {1, 2, . . . , N X − n}. In Ref. [4] and the present article, we simply write "t" for this emerging time, The results of Fig. 1 in Ref. [4] show that the distance between neighboring eigenvalues α p is more or less uniform. Write ∆α for this average distance, so that ∆α = O(1/N X ). Then, we have approximately where the zero of t has been defined to occur "halfway," at k = N X /2 − n/2. With fixed values of n and N X , the increase of this dimensionless variable t just follows the index k on the α eigenvalues in (5.5).
The crucial insight, now, is that "time" is really a tool for ordering events. And precisely the X 0 eigenvalues α p were ordered, according to (5.1). Hence, the notion of "time" in t k of (5.2) is carried by the index k, which corresponds to the index of the ordered eigenvalues from Fig. 1 in Ref. [4].
We, thus, have an understanding of how the cosmic time t emerges from the classical solutions of the large-N matrix model. In addition, the particular classical solutions of Ref. [4] have an emerging flat 3-space, which has been found to expand with |t|. The obvious question then is: what drives this expansion? The answer appears to be: algebra. How precisely the algebra produces the expansion is not yet completely clear, but we can sketch the beginning of an explanation.
The results of Fig. 2 in Ref. [4] show that the spatial matrices X i of the classical solution are nearly diagonal, in the basis which makes X 0 perfectly diagonal (needed to define "t" as explained above). The effective size R 2 (t) of the universe is defined by Eq. (3.4) in Ref. [4] and shown by Fig. 3 in that reference. This R 2 (t) behavior is more or less consistent with the behavior on the diagonal i = j of Fig. 2 in Ref. [4]. Let us look at how the classical solutions appear.
The reduced classical equations are given by Eqs. (2.16) and (2.17) of Ref. [4]. For simplicity, we can take a trivial solution Y a = 0 for the six "internal-space" N Y × N Y Hermitian matrices labeled by a ∈ {4, 5, . . . , 9} and having dimension N Y = N − N X . (The fermionic matrices Ψ α , for α ∈ {1, 2, . . . , 16}, are also set to zero.) We then focus on the four "spacetime" N X × N X Hermitian matrices X µ labeled by µ ∈ {0, 1, 2, 3}, for which the classical "equations of motion" are with Lagrange multipliers ξ = 0 and ζ = 0. We are only interested in nontrivial solutions X µ = 0. Fixing the matrix size N X = 64, for definiteness, we can start with a diagonal block-type solution at i ∼ j ∼ 32 and then work out towards i ∼ j ∼ 0 and i ∼ j ∼ 64, finding that the amplitudes of the matrices X i increase (cf. Fig. 2 in Ref. [4]). As the distance along the diagonal i = j can be interpreted as the cosmic time t [roughly t ∝ (i − 32)], we have that the X i amplitudes grow with |t|, i.e., the 3-space expands (cf. Fig. 3 in Ref. [4] and the top-row panels of Fig. 3 in the present article). Further discussion is relegated to App. A, which presents exact solutions of the reduced classical equations from the matrix model and makes a conjecture as to the algebraic origin of the expansion.
Heuristically, we now have the following picture of the origin of the Universe, within the context of the IKKT matrix model [1]. The fundamental matrix model has no spacetime and no gravity. The classical solutions from Ref. [4] show an emerging spacetime, with a flat 3-space expanding with |t| due to the algebra of the reduced equations of motion (5.6). Gravitation and matter are expected to appear dynamically (all that information resides in the infinite matrices of the matrix model, as N → ∞). For example, with ponderable matter present, the gravitational force will be mediated by the appropriate massless modes (virtual gravitons) of the emerging superstrings. An interesting phenomenon will be the cross-over from matrix-model expansion driven by the algebra to Friedmann-type expansion driven by gravity and matter [the latter type of expansion is, strictly speaking, also due to (complicated) algebra, but that expansion can be described by Einstein's general relativity as an effective theory]. However, at this moment, the urgent task is to show unequivocally that the matrix model [1] indeed gives rise to the type of classical solutions found in Ref. [4] and discussed in the present article.
so that the set of four equations reads explicitly This set of algebraic equations is surprisingly subtle. We have two preliminary remarks. First, none of the matrices X µ can be proportional to the identity matrix (excluding the null matrix), as the commutator terms in (A2) then vanish but the single-matrix terms not. Second, the equations are nonlinear and allow only for a trivial sign rescaling, X µ → s µ X µ , without sum over µ and with s µ = (±1, ±1, ±1, ±1). We start by considering 2 × 2 real symmetric matrices X µ , which later will be inserted on the diagonals of the respective X µ matrices. An exact solution of the reduced classical equations (A2), restricted to 2 × 2 matrices, is given by: with Hence, we have a solution X µ with four moduli which lie in a particular domain D, so that the other three coefficients are real: We now establish that D is nonempty.
As to the actual shape of the 4-dimensional manifold D from (A4), we have three simple remarks. First, we see from the root in (A3e) that the moduli (b 1 , b 3 ) lie on a disk of radius 1/(2 √ 2). Second, we see from the numerator of the root in (A3c) that the point (a 1 , a 3 ) = (0, 0) is excluded and that |a 1 | and |a 3 | must lie above a boundary value determined by (b 1 , b 3 ). Third, we also see from the numerator of the root in (A3c) that the moduli a 1 and a 3 can run off to ±∞ and we conclude that D is a noncompact manifold.
With the X µ solution (A3) in hand, it is straightforward to construct N X ×N X Hermitian matrix solutions X µ of the algebraic equations (A2). Taking we have the following block-diagonal solutions: where the suffixes (1), (2), . . . , (k) remind us that the individual blocks may have different values of the moduli, as long as these moduli remain in the domain D from (A4). At this moment, it may be of interest to compare our exact solutions (A8b) with the earlier exact solutions of Refs. [9,10]. Apart from one exception (mentioned in App. C of Ref. [10]), these earlier solutions have the property that all spatial matrices commute between themselves. That space-space commutativity property does not hold for the solutions presented here. For the record, we can mention that other space-space noncommutative solutions have been considered before; see, e.g., Ref. [11] and references therein.

Special results
We, next, investigate whether or not we can qualitatively reproduce the numerical results from Ref. [4] with the exact solutions of App. A 1.
Implementing the last two ingredients listed above, the diagonal X 0 matrix from (A12) with moduli (A13) is reordered by use of the transformation (A10) and the X i matrices are transformed simultaneously. We then get the following values for the diagonal matrix X 0 and the matrix ∆ from (A11): which are also shown in Fig. 6. The growth along the diagonal of ∆ pq is significant (by a factor of approximately 3, going from the middle of the diagonal towards the edges) and there is a modest jump at τ = 0 in the X 0 eigenvalues, but the diagonal band structure of ∆ pq is somewhat disturbed by small entries straddling the anti-diagonal. It is, now, straightforward to generalize the N X = 6 construction to larger values of N X . With l > 1 defined by (A9), the block-diagonal solutions are given by X µ = diag X µ (τ =−1) , . . . , X µ (τ =−2/l) , X µ (τ =−1/l) , X µ (τ =0) , X µ (τ =1/l) , X µ (τ =2/l) , . . . , X µ (τ =1) .
We also make a slight change in the moduli functions (the reason will become clear shortly), a 1 (τ ) = a 3 (τ ) = 1/4 + τ 2 /2 , with τ ∈ [−1, 1]. From these moduli, the three other coefficients entering the 2 × 2 matrices (A3) are: with a 0 (0) = 0, as desired. Reordering the diagonal X 0 matrix from (A16) with moduli (A17) and transforming the X i simultaneously, we get the diagonal matrix X 0 and the matrix ∆ from (A11). Results for N X = 18 are shown in Fig. 7. Generalization to even larger values of N X is trivial and the results will look like those of Fig. 7. The exact results from Fig. 7 are qualitatively similar to the numerical results of Figs. 1 and 2 in Ref. [4]. Admittedly, the right panel of our Fig. 7 has some "noise" along the anti-diagonal, but the main feature of ∆ pq is the growth on the diagonal (by a factor 9, going from the middle of the diagonal towards the edges).
It still needs to be explained how the particular moduli functions (A17) for the blockdiagonal solution (A16) come about. One conjecture is that there would be an "entropy effect" operative in the numerical procedure of Ref. [4]: the entropy for embedded blockdiagonal solutions with τ -dependent moduli would be larger than the entropy for embedded block-diagonal solutions with constant (τ -independent) moduli.