[Paper Review] Monte Carlo studies on the expanding behavior of the early universe in the Lorentzian type IIB matrix model

Yuta Ito, Sang‐Woo Kim+2 more|arXiv (Cornell University)|Nov 21, 2013

Black Holes and Theoretical Physics9 references4 citations

TL;DR

This paper investigates the cosmological evolution of the early universe in the Lorentzian type IIB matrix model via Monte Carlo simulations. It finds that three spatial directions undergo exponential expansion—indicating inflation—followed by a transition to power-law expansion ($t^{1/2}$), consistent with the radiation-dominated Friedmann-Robertson-Walker universe, suggesting a dynamical emergence of spacetime from nonperturbative string theory.

ABSTRACT

The type IIB matrix model is a conjectured nonperturbative formulation of superstring theory. Recently the Lorentzian version of the model has been studied by Monte Carlo simulation, and it has been shown that only three out of nine spatial directions start to expand after a critical time. We extend this work by investigating the expanding behavior for much longer time. We find that the 3d space expands exponentially for some period of time, which may be interpreted as inflation. We also simulate a simplified model, which is expected to capture some qualitative features of the original model at much later times. We observe that the exponential expansion eventually changes into a power-law t^(1/2) behavior, which agrees with the expanding behavior of the Friedmann-Robertson-Walker (FRW) universe in the radiation dominated era.

Motivation & Objective

• To investigate the long-time dynamical evolution of spacetime in the Lorentzian type IIB matrix model, a nonperturbative candidate for M-theory.
• To determine whether the observed early-time expansion in previous studies evolves into a behavior consistent with known cosmological models at later times.
• To clarify the role of fermionic degrees of freedom in driving initial exponential expansion and their diminishing influence as the universe evolves.
• To test whether the transition from exponential to power-law expansion can be captured in simplified models that isolate key physical mechanisms.

Proposed method

• Monte Carlo simulations are performed on the Lorentzian type IIB matrix model with $N \times N$ traceless Hermitian matrices $A_\mu$ and $\Psi_\alpha$, using a Wick-rotated action to enable numerical evaluation.
• A simplified model is introduced by retaining only the term proportional to $A_0$ in the fermionic action, emphasizing the repulsive force between eigenvalues of the temporal matrix.
• A quenched model is constructed by omitting all fermionic matrices entirely, to study late-time behavior where fermionic effects are expected to be negligible.
• The extent of spatial expansion is measured via the eigenvalue distribution of the temporal matrix $A_0$, with $R^2(t) = \frac{1}{N} \mathrm{Tr} \left( A_i(t)^2 \right)$ used as a proxy for spatial size.
• Symmetry breaking from SO(5) to SO(3) is monitored as a signature of three-dimensional space formation.
• Fits to the time evolution of $R^2(t)$ are used to identify exponential ($e^{bt}$) and power-law ($t^{1/2}$) behaviors.

Experimental results

Research questions

• RQ1 Does the three-dimensional spatial expansion observed in earlier simulations persist as exponential growth over longer timescales?
• RQ2 What is the origin of the exponential expansion in the early universe within the Lorentzian IIB matrix model, and which terms in the action drive it?
• RQ3 Does the expansion behavior transition from exponential to power-law at late times, and if so, does it match the $t^{1/2}$ scaling of the radiation-dominated FRW universe?
• RQ4 Can simplified models—particularly the quenched model—accurately reproduce the late-time transition to power-law expansion?
• RQ5 Is the E-folding number of inflation dynamically determined in this model, and can it be extracted from the simulation data?

Key findings

• The three spatial directions exhibit exponential expansion for a finite period after a critical time $t_c$, which may be interpreted as a dynamical realization of cosmic inflation.
• This exponential behavior is confirmed in a simplified model that isolates the fermionic term proportional to $A_0$, indicating that the repulsive force between eigenvalues of $A_0$ is essential for early-time inflation.
• In the quenched model—where fermionic matrices are omitted—the expansion transitions from exponential to a power-law behavior $R(t) \sim t^{1/2}$, matching the radiation-dominated era of the FRW universe.
• The transition to $t^{1/2}$ scaling is observed in simulations with $N \geq 16$, and the fit parameters for late-time behavior are $c = 17.0(1)$ and $d = -23.3(3)$ in the linear fit $y = c x + d$ to $R^2(t)/R^2(t_c)$.
• Spontaneous symmetry breaking from SO(5) to SO(3) is observed after $t_c$ in both the original and simplified models, signaling the emergence of three-dimensional space.
• The results suggest that the Lorentzian IIB matrix model can dynamically generate a spacetime consistent with early-universe cosmology, including inflation and radiation domination, from first principles.