[Paper Review] QCD Cosmology from the Lattice Equation of State

Michael McGuigan, Wolfgang Söldner|ArXiv.org|Oct 1, 2008

Particle physics theoretical and experimental studies43 references3 citations

TL;DR

This paper computes the time evolution of the cosmic scale factor $ a(t) $ using the lattice QCD equation of state at zero chemical potential, showing that early-time expansion follows a radiation-like $ t^{1/2} $ behavior, while late-time dynamics exhibit a complex, matter-like $ t^{0.666} $-like scaling. It compares lattice results with the hadronic resonance gas, Hagedorn, and AdS/CFT models, finding that only the dilaton-deformed AdS/CFT model qualitatively reproduces the lattice's 'hockey stick' behavior in log-log plots of $ a(t) $.

ABSTRACT

We numerically determine the time dependence of the scale factor from the lattice QCD equation of state, which can be used to define a QCD driven cosmology. We compare a lattice approach to QCD cosmology at late times with other models of the low temperature equation of state including the hadronic resonance gas model, Hagedorn model and AdS/CFT.

Motivation & Objective

• To compute the time evolution of the cosmic scale factor $ a(t) $ using nonperturbative lattice QCD results for the equation of state.
• To compare lattice QCD results with alternative low-temperature QCD models: hadronic resonance gas (HRG), Hagedorn, and AdS/CFT.
• To assess whether the AdS/CFT correspondence, with a dilaton deformation, can qualitatively reproduce the lattice QCD behavior in the late-time regime.
• To identify the cosmological implications of nonperturbative QCD dynamics, particularly the transition from early radiation-like to late-time complex expansion.

Proposed method

• Numerically integrate the Friedmann equation using the lattice QCD equation of state at $ N_t = 6 $, with zero chemical potential and no bulk viscosity.
• Derive $ T(t) $ from the energy density and pressure via $ dt/dT = [3( ho + p) ho^{1/2}/(3M_P)^{1/2}] / (d ho/dT) $, then invert to get $ T(t) $.
• Compute $ a(t) $ via $ a(t) = \exp\left( \frac{1}{\sqrt{3}M_P} \int_{t_0}^t dt \, \sqrt{\varepsilon(T(t))} \right) $, using the time-dependent temperature.
• Compare results with the HRG model using resonance masses from the PDG, the Hagedorn model with a limiting temperature, and the AdS/CFT model with a dilaton field.
• Introduce a deformation parameter $ \sigma $ in the AdS/CFT action to tune the dilaton boundary condition and match lattice-like behavior.
• Use numerical solutions of the black hole background in AdS with the dilaton to compute entropy, energy density, and ultimately $ a(t) $.

Experimental results

Research questions

• RQ1 Does the lattice QCD equation of state lead to a scale factor $ a(t) $ that deviates from the standard radiation-dominated $ t^{1/2} $ behavior at late times?
• RQ2 How does the hadronic resonance gas model compare quantitatively to lattice QCD in predicting $ a(t) $ over the same temperature range?
• RQ3 Can the AdS/CFT correspondence, with a dilaton field, reproduce the nontrivial late-time scaling of $ a(t) $ observed in lattice QCD?
• RQ4 What is the role of the Hagedorn model in describing late-time cosmology, and how does it differ from lattice QCD in its prediction of energy density and scale factor?
• RQ5 Does the inclusion of a dilaton in the AdS/CFT model lead to a qualitatively similar 'hockey stick' behavior in $ \log a(t) $ vs. $ \log t $ plots as seen in lattice QCD?

Key findings

• The scale factor $ a(t) $ follows $ t^{1/2} $ at early times, consistent with radiation domination, as expected from the high-temperature lattice QCD equation of state.
• At late times, the lattice QCD result shows a complex time dependence that is closer to $ t^{2/3} $, indicating a matter-like expansion, and displays an upward-bending 'hockey stick' shape in log-log plots.
• The hadronic resonance gas (HRG) model predicts a scale factor that lies above the lattice QCD data in the log-log plot, with a slope between 0.5 (radiation) and 0.666 (matter), indicating intermediate behavior.
• The Hagedorn model predicts a finite limiting temperature and diverging energy density at that point, which contrasts with the finite, well-behaved lattice QCD data.
• Without the dilaton, the AdS/CFT model yields a $ t^{1/2} $ expansion over the lattice temperature range, failing to reproduce the lattice's late-time behavior.
• With a tuned dilaton deformation parameter $ \sigma = 0.01 $, the AdS/CFT model produces a scale factor evolution that qualitatively matches the lattice QCD 'hockey stick' behavior, suggesting the dilaton captures essential nonperturbative dynamics.