Power-law expansion of the Universe from the bosonic Lorentzian type IIB matrix model

Abstract

Recent studies on the Lorentzian version of the type IIB matrix model show that (3+1)D expanding universe emerges dynamically from (9+1)D space-time predicted by superstring theory. Here we study a bosonic matrix model obtained by omitting the fermionic matrices. With the adopted simplification and the usage of a large-scale parallel computer, we are able to perform Monte Carlo calculations with matrix size up to N = 512, which is twenty times larger than that used previously for the studies of the original model. When the matrix size is larger than some critical value N c 110, we find that (3+1)D expanding universe emerges dynamically with a clear large-N scaling property. Furthermore, the observed increase of the spatial extent with time t at sufficiently late times is consistent with a power-law behavior t 1/2 , which is reminiscent of the expanding behavior of the Friedmann-Robertson-Walker universe in the radiation dominated era. We discuss possible implications of this result on the original supersymmetric model including fermionic matrices.

Exponential and power-law expansion of the Universe from the type IIB matrix model

“…In the VDM model, we observe an exponential expansion [6], which is reminiscent of the inflation. On the other hand, in the bosonic model, we observe a power-law expansion [5], which is reminiscent of the expansion in the radiation dominant era of the Universe.…”

“…3(Left), we plot the expectation values of the nine eigenvalues of T i j (t) against t for N = 512 with the block size n = 32 and observe that the SO(9) symmetry is spontaneously broken down to SO(3) after the critical time t c [5]. In Fig.…”

“…In Fig. 3(Right), we plot the extent of space R 2 (t) normalized by R 2 (t c ) against x = (t − t c ) /R (t c ) for N = 128, 256, 384 and 512 [5]. We observe scaling behavior for different N. We see that the behavior of R 2 (t) at t > t c can be fitted to an exponential function only for a finite range and that at later times it can be fitted to a linear function, which corresponds to the power-law expansion…”

“…One is called the VDM model [4], and the other is called the bosonic model [5]. The former is expected to effectively describe the expanding behavior at early times, while the latter that at late times.…”

Complex Langevin analysis of the space-time structure in the Lorentzian type IIB matrix model

“…In ref. [2] and our subsequent work [15][16][17][18] on the Lorentzian type IIB matrix model and its simplified models, the sign problem was avoided by integrating out the scale factor of the bosonic matrices by hand, which yields a function of the bosonic action S b sharply peaked at the origin. Approximating this function by a sharply peaked Gaussian function, we can perform Monte Carlo simulations without the sign problem.…”