[Paper Review] Causality, Stability and Sound Speed in Scalar Field Models

Axel de la Macorra, Hector Vuccetich|arXiv (Cornell University)|Dec 12, 2002

Fluid Dynamics and Turbulent Flows3 citations

TL;DR

This paper demonstrates that scalar field models with $ w < -1 $, while seemingly violating causality due to superluminal sound speeds, remain causal when $ w $ is time- or space-dependent and dispersion is present. The key insight is that the effective sound speed in such models is always subluminal, ensuring causality is preserved even when the equation of state parameter $ w $ exceeds $-1$ in magnitude.

ABSTRACT

The result from the SN1A projects suggest that the dark energy can be represented by a fluid with $w&lt;-1$. However, it is commonly argued that a fluid with $|w|&gt;1$ contradicts causality. Here, we will show that a fluid with $|w|&gt;1$ does not contradict causality if $w$ is not constant. Scalar field are the most promising candidates for describing the dark energy and they do not have a constant equation of state parameter $w$. For typical scalar potentials there are regions where $w $ is larger or smaller than one and even regions where $d p/d\ ho$ diverges. We study the evolution of scalar field perturbations and we show that the "sound speed" is always smaller than the speed of light independently of the value of $w=p/\ ho$ or $d p/d\ ho$. In general, it is neither the phase velocity nor the group velocity that gives the "sound speed". Our results can be applied to all fluids and allows for a fluid with $w&lt;-1$ without contradicting causality as long as there is dispersion.

Motivation & Objective

• To resolve the apparent conflict between dark energy models with $ w < -1 $ and the principle of causality.
• To investigate whether scalar field models with time- or space-dependent $ w $ can maintain causality despite $ |w| > 1 $.
• To clarify the physical meaning of 'sound speed' in non-ideal fluids and scalar field theories with variable $ w $.
• To establish conditions under which $ w < -1 $ fluids remain causal, particularly in the presence of dispersion.

Proposed method

• Analysis of scalar field perturbations in time- and space-dependent potentials to derive the effective sound speed.
• Derivation of the effective sound speed $ c_s^2 = dp/d\rho $, showing it remains bounded below $ c^2 $ even when $ w = p/\rho $ or $ dp/d\rho $ diverges.
• Use of dispersion relations to show that phase and group velocities do not determine causality; instead, the effective sound speed governs signal propagation.
• Generalization of results to all fluids with variable $ w $, particularly those with $ |w| > 1 $, by analyzing the full perturbation spectrum.
• Application of relativistic fluid dynamics and scalar field theory to model dark energy with $ w < -1 $ without violating microcausality.

Experimental results

Research questions

• RQ1 Can a fluid with $ w < -1 $ be causal if its equation of state is not constant?
• RQ2 What determines the effective sound speed in scalar field models with time- or space-dependent $ w $?
• RQ3 Why does the standard argument linking $ |w| > 1 $ to superluminal propagation fail in non-constant $ w $ models?
• RQ4 How does dispersion affect the propagation of perturbations in dark energy models with $ w < -1 $?

Key findings

• The effective sound speed in scalar field models is always less than or equal to the speed of light, even when $ w = p/\rho $ or $ dp/d\rho $ exceeds $ c^2 $.
• Divergences in $ dp/d\rho $ do not imply superluminal signaling, as long as the full dispersion relation is considered.
• The phase and group velocities are not the correct measures of signal propagation speed; the effective sound speed $ c_s^2 = dp/d\rho $ governs causality.
• Causality is preserved in $ w < -1 $ models as long as there is sufficient dispersion, which stabilizes the effective sound speed.