Phase transition between shock formation and stability in cosmological fluids

Abstract We demonstrate a novel phase transition from stable to unstable fluid behaviour for fluid-filled cosmological spacetimes undergoing decelerated expansion. This transition occurs when the fluid speed of sound <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>c</mml:mi> <mml:mtext>S</mml:mtext> </mml:msub> </mml:mrow> </mml:math> exceeds a critical value relative to the expansion rate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow> <mml:mi>α</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> of spacetime. We present an explicit relationship between α and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>c</mml:mi> <mml:mtext>S</mml:mtext> </mml:msub> </mml:mrow> </mml:math> , which subdivides the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>c</mml:mi> <mml:mtext>S</mml:mtext> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> -parameter space into two regions. Using rigorous techniques, we establish stability of quiet fluid solutions in the stable region. Numerical experiments reveal that the complement of the stable region consists of unstable solutions, implying sharpness of our stability result. We provide a definitive analytical bound and high-precision numerical evidence for the exact location of the critical line separating the stable from the unstable region.