A renormalization group method for studying the early universe in the Lorentzian IIB matrix model

Abstract

We propose a new method for studying the early universe in the Lorentzian version of the IIB matrix model, which is considered to be a nonperturbative formulation of superstring theory. This method is based on the idea of renormalization group, and it enables us to study the time-evolution of the universe for much longer time than in the previous work, which showed that the SO(9) rotational symmetry is spontaneously broken down to SO(3) after a "critical time". We demonstrate how this method works in a simplified model, which is expected to capture the behaviors of the original model when the space is not so large. In particular, we present clear evidence that the three-dimensional space expands exponentially after the critical time in this simplified model.

On the structure of the emergent 3D expanding space in the Lorentzian type IIB matrix model

“…In this way, it was found that only three out of nine directions start to expand at some critical time, which implies that the model predicts the emergence of a (3+1)d expanding space-time from superstring theory in (9+1)d. The expanding behavior for a longer time was investigated by simulating simplified models. The obtained results suggested a scenario for the full model that the expansion is exponential at early times [9], which is reminiscent of the inflation, and that it turns into a power law [10] at later times, which is reminiscent of the Friedmann-Robertson-Walker universe in the radiation dominated era. See also refs.…”

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

“…The expanding behavior for a longer time was investigated by simulating the simplified models. The obtained results suggested a scenario for the full model that the expansion is exponential at early times [15], which is reminiscent of the inflation, and that it turns into a power law [16] at later times, which is reminiscent of the Friedmann-Robertson-Walker universe in the radiation dominated era. See also refs.…”

“…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.…”

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

“…where we put L = 1 without loss of generality since the action is homogeneous in A µ and ψ (for details, see [4]), and we take the κ → ∞ limit as N → ∞. We can fix SU(N) symmetry A µ → UA µ U † where U ∈ SU(N).…”

“…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.…”