Lattice QCD for Cosmology

TL;DR

This work delivers a comprehensive first-principles determination of the QCD equation of state (EoS) across 2+1+1 and 2+1+1+1 flavors, incorporating charm and bottom thresholds, and couples it with electroweak inputs to produce a full Standard Model cosmological EoS. It introduces advanced lattice techniques—fixed sector integrals, eigenvalue reweighting, and overlap fermions—to control topological observables, enabling a precise determination of the temperature-dependent topological susceptibility $\\chi(T)$ to guide axion cosmology. The results constrain the axion mass in the post-inflationary scenario to $m_A \,\in\, [50,1500]\,\mu\mathrm{eV}$ and provide a robust link between $\,\chi(T)$ and dark matter abundance, with practical implications for current and next-generation axion experiments. Together, the nonperturbative QCD insights and the SM-EoS parametrization offer a rigorously grounded framework for early-ununiverse thermodynamics and axion phenomenology across temperatures from MeV to hundreds of GeV.

Abstract

We present a full result for the equation of state (EoS) in 2+1+1 (up/down, strange and charm quarks are present) flavour lattice QCD. We extend this analysis and give the equation of state in 2+1+1+1 flavour QCD. In order to describe the evolution of the universe from temperatures several hundreds of GeV to several tens of MeV we also include the known effects of the electroweak theory and give the effective degree of freedoms. As another application of lattice QCD we calculate the topological susceptibility (chi) up to the few GeV temperature region. These two results, EoS and chi, can be used to predict the dark matter axion's mass in the post-inflation scenario and/or give the relationship between the axion's mass and the universal axionic angle, which acts as a initial condition of our universe.

Figures (34)

• Figure 1: The effective degrees of freedom for the energy density ($g_\rho$) and for the entropy density ($g_s$). The line width is chosen to be the same as our error bars at the vicinity of the QCD transition where we have the largest uncertainties. At temperatures $T<1$ MeV the equilibrium equation of state becomes irrelevant for cosmology, because of neutrino decoupling. The EoS comes from our calculation up to $T=100$ GeV. At higher temperatures the electroweak transition becomes relevant and we use the results of Ref. Laine:2015kra. Note that for temperatures around the QCD scale non-perturbative QCD effects reduce $g_\rho$ and $g_s$ by 10-15% compared to the ideal gas limit, an approximation which is often used in cosmology. For useful parametrizations for the QCD regime or for the whole temperature range see SI.
• Figure 2: Continuum limit of $\chi(T)$. The insert shows the behaviour around the transition temperature. The width of the line represents the combined statistical and systematic errors. The dilute instanton gas approximation (DIGA) predicts a power behaviour of $T^{-b}$ with $b$=8.16, which is confirmed by the lattice result for temperatures above $\sim 1$ GeV.
• Figure 3: Relation between the axion's mass and the initial angle $\theta_0$ in the pre-inflation scenario. The post-inflation scenario corresponds to $\theta_0=2.155$ with a strict lower bound on the axion's mass of $m_A$=28(2)$\mu eV$. The thick red line shows our result on the axion's mass for the post-inflation case. E.g. $m_A$=50(4)$\mu eV$ if one assumes that axions from the misalignment mechanism contributes 50% to dark matter. Our final estimate is $m_A$=50-1500$\mu eV$ (the upper bound assumes that only 1% is the contribution of the misalignment mechanism the rest comes from other sources e.g. topological defects). For an experimental setup to detect post-inflationary axions see SI. The slight bend around $m_A\sim 10^{-5}$$\mu$eV corresponds to an oscillation temperature at the QCD transition Aoki:2009scBorsanyi:2010bp.
• Figure S1: Lattice spacing dependence of the zero temperature topological susceptibility. The grey squares are obtained with the standard approach, the red circles after dividing by the taste singlet pion mass squared. The line is a linear fit. The blue cross corresponds to leading order chiral perturbation theory. The plot shows $n_f=2+1+1$ flavor staggered simulations at zero temperature.
• Figure S2: Continuum extrapolations of the $w_0$-scale in lattice units (up) and $m_\pi w_0$ (down). The plots show $n_f=3+1$ flavor staggered simulations at zero temperature.