[Paper Review] Large-scale computation of the exponentially expanding universe in a simplified Lorentzian type IIB matrix model

Yuta Ito, Jun Nishimura+1 more|arXiv (Cornell University)|Dec 7, 2015

Black Holes and Theoretical Physics6 references4 citations

TL;DR

This study investigates the emergence of an exponentially expanding universe in a simplified Lorentzian type IIB matrix model using large-scale Monte Carlo simulations up to N=256. By introducing a tunable infrared (IR) cutoff parameter p, the authors find that results become universal for p > 1, suggesting IR cutoff effects vanish in the infinite volume limit, supporting the robustness of the observed 3+1-dimensional spacetime dynamics.

ABSTRACT

The type IIB matrix model is a conjectured nonperturbative formulation of superstring theory. Recent studies on the Lorentzian version of the model have shown that only three out of nine spatial directions start to expand after some critical time. On the other hand, due to the unbounded action of the Lorentzian model, one has to introduce infrared (IR) cutoffs in order to make the partition function finite. In this work we investigate whether the effects of the IR cutoffs disappear in the infinite volume limit. For that purpose, we study a simplified model with large matrix size up to $N=256$ by Monte Carlo simulation. First we confirm the exponentially expanding behavior of the "universe". Then we generalize the form of the IR cutoffs by one parameter, and find that the results become universal in some region of the parameter. It is suggested that the effects of IR cutoffs disappear in this region, which is confirmed also from the studies of Schwinger-Dyson equations.

Motivation & Objective

• To assess whether infrared (IR) cutoffs used to regularize the unbounded action of the Lorentzian type IIB matrix model affect the emergence of spacetime in the infinite volume limit.
• To investigate whether the observed exponential expansion of three spatial dimensions and spontaneous SO(9) to SO(3) symmetry breaking are robust against IR regularization.
• To determine the dependence of simulation results on the form of IR cutoffs by introducing a tunable parameter p.
• To validate the universality of results through both Monte Carlo simulations and Schwinger-Dyson equation analysis.

Proposed method

• A simplified Lorentzian IIB matrix model is employed, omitting the A_i-dependent term in the fermionic action to enable large-N simulations.
• Monte Carlo simulations are performed for matrix sizes up to N=256 with IR cutoffs parameterized by p in the form of (A_μ)^2 > p for μ=0,i.
• The spatial extent R²(t) is computed as a function of time to analyze the expansion dynamics.
• The IR cutoff dependence is studied by varying p across 1.0, 1.3, and 1.5, with results compared for consistency and universality.
• Schwinger-Dyson equations are analyzed to probe the scaling behavior of IR cutoff contributions with increasing N.
• Block averaging with different block sizes (n=6,10) is used to improve statistical accuracy in the time evolution analysis.

Experimental results

Research questions

• RQ1 Does the exponential expansion of three spatial dimensions persist when IR cutoffs are systematically varied via the parameter p in the simplified model?
• RQ2 For which values of p do the simulation results become independent of the IR cutoff form, indicating the disappearance of cutoff effects in the infinite volume limit?
• RQ3 How do the Schwinger-Dyson equations reflect the scaling behavior of IR cutoff contributions as N increases?
• RQ4 Is the observed spontaneous SO(9) to SO(3) symmetry breaking robust under different IR regularization schemes?
• RQ5 Can the results for p > 1 be considered universal, suggesting that the physical dynamics are independent of the specific IR cutoff choice?

Key findings

• For p = 1.3 and p = 1.5, the spatial extent R²(t) shows nearly identical exponential expansion behavior, with only minor differences near the peak region.
• The result for p = 1.0 deviates significantly from the others across the entire time range, indicating strong IR cutoff dependence.
• The exponential expansion is well-fitted by R²(t)/R²(t_c) = a + (1−a)exp(bx) for p = 1.3 and 1.5, confirming the inflationary-like dynamics.
• The IR cutoff effects diminish for larger p, as the cutoffs affect only large eigenvalues of (A_μ)², leading to universal behavior in the p > 1 regime.
• Schwinger-Dyson analysis confirms that the IR cutoff-induced terms scale down with increasing N when p > 1, supporting the disappearance of cutoff effects in the large-N limit.
• The results suggest that the observed 3+1-dimensional expanding universe is robust and not an artifact of the regularization, particularly for p > 1.