Lattice simulations of SU(N) Scalar+Gauge theories for holographic cosmology

Abstract

In the work by Skenderis and McFadden, the power spectrum of the Cosmic Microwave Background (CMB) in a 4d gravitational theory is mapped to the two-point function of the Energy-Momentum Tensor (EMT) of a 3d QFT with generalised conformal structure. Perturbative analysis of this QFT and subsequent comparison with data from the Planck and WMAP surveys suggest that the most general form for it is a theory of SU(N) scalar fields in the adjoint representation minimally coupled to gauge fields. In order to access the nonperturbative regime of this perturbatively IR-divergent, super-renormalisable theory, lattice simulations are necessary.

We start from the simplest candidate, a φ⁴ theory of pure SU(N) adjoint scalars at its critical point, and show through finite-size scaling that it is indeed IR-finite in the infinite volume limit for N=2 and N=4. With IR finiteness guaranteed, we must obtain the continuum limit of the EMT two-point function in order to compare it with CMB data. Due to the breaking of translational symmetry on the lattice description of the theory and thereby the need for a way to renormalise the EMT, a novel position-space method is explored that filters out divergent contributions. This method can successfully renormalise the EMT operator, although its two-point function suffers from high statistical noise.

The theory is then extended to its more general case: a φ⁴ theory of scalars minimally coupled to a gauge field. In this extension, the phase diagram is explored for N=2 and its parameter space is examined, with the intent of finding a line of second-order phase transitions where the theory becomes massless, and a comparison with perturbative expectations for the critical mass is given.

All of these lattice analyses were made possible through the development and optimisation of a Heatbath-Overrelaxation (HBOR) algorithm written using the Grid library, which is shown to achieve statistical decorrelation between configurations significantly faster than with the Hybrid Monte-Carlo (HMC) method. The HBOR code was initially written for CPU runs but then ported and optimised for GPU machines, where it exhibits good scaling properties and is more performant than its CPU counterpart by about an order of magnitude. The Scalar and Scalar+Gauge codes for arbitrary N and d have been made publicly available.