A dynamical systems perspective on the thermodynamics of late-time cosmology

Abstract

A thermodynamic description of cosmological spacetimes may provide insights into the fundamentals of the cosmic evolution that remain otherwise obscure, similar to `black hole thermodynamics'. We investigate the thermodynamic properties of late-time cosmological evolution using the dynamical systems approach, focusing on $Λ${}CDM model and scalar field models with exponential potentials. Thermodynamic quantities obtained through the Hayward-Kodama formalism are mapped onto the phase-space of these models. Specifically, we express the thermodynamic quantities as functions of the phase-space variables, allowing us to study the thermodynamic behavior across the phase space, particularly at the critical points. We focus on thermodynamic stability and phase transitions, analyzed in an initial condition-independent manner. In these models, the universe inevitably undergoes a thermodynamic phase transition, marked by diverging specific heats, irrespective of its initial configuration. We further demonstrate that the thermodynamic stability can occur only during an accelerating phase of the universe. For $Λ$CDM and quintessence models, the necessary stability conditions are never satisfied anywhere in the phase space, rendering both models thermodynamically unstable within the Hayward-Kodama framework and the canonical ensemble based stability criteria. Interestingly, the phantom models, although dynamically unstable, allow for the universe to attain thermodynamic stability in its asymptotic future. This can indicate the limitations of applying canonical ensemble based thermodynamic stability criteria to cosmological horizons. Through these archetypal descriptions of late-time cosmology, we show that the dynamical system approach is a robust framework to probe the thermodynamic aspects of cosmological evolution.

Figures (5)

• Figure 1: Phase space trajectories of the $\Lambda$CDM universe. The triangle bounded by $0 \leq x,y \leq 1$ and $x + y \leq 1$ shows phase space with critical points $R$ (radiation domination), $M$ (matter domination), and $D$ (dark energy domination). Light gray arrows represent the slope field for all viable orbits. The dashed gray line marks the transition from deceleration to acceleration ($q=0$), with the gray shaded region indicating accelerated expansion. The red dashed line denotes the thermodynamic phase transition ($\sigma=0$). The green and blue regions highlight where $C_V>0$ and $C_P>0$, respectively. The black orbit traces the evolution of the $\Lambda$CDM concordance model ($x(\tau=0) = \Omega_{M0} = 0.3153$, $y(\tau=0) = \Omega_{R0} = 10^{-4}$), and the blue orbit shows a case with reduced matter and enhanced radiation ($x(\tau=0) = \Omega_{M0}/2$, $y(\tau=0) = 10^2\Omega_{R0}$).
• Figure 2: Physical phase space of the $\Lambda$CDM model mapped onto the specific heat ($H^2 C_P - H^2 C_V$) plane. The physical bounds ($0 \le x, y \le 1$ and $x+y \le 1$) map to two disjoint shaded patches bounded by the dashed gray curves. The dark shaded region marks accelerated expansion. The physical regions never intersect the first quadrant ($C_P > 0, C_V > 0$), showing that the necessary conditions for thermodynamic stability are never simultaneously satisfied. The solid blue and black curves represent the same cosmic orbits shown in Fig. \ref{['fig:LCDMR']}. The trajectories emerge from the unstable radiation node $R$ (located at infinity in the second quadrant), approach the matter-dominated saddle point $M$ and encounter a thermodynamic phase transition where the specific heats diverge. The trajectories consequently diverge to infinity, re-entering from the third quadrant, before finally converging at the stable, dark energy-dominated future attractor $D$.
• Figure 3: Phase space of the quintessence model with exponential potential, $\lambda =1$. The half-circle bounded by $x^2 + y^2 \leq 1$ and $y>0$ shows the phase space and thermodynamic regions in panels (a) and (b), respectively. Both panels show the critical points $K_{\pm}$ (Kinetic domination), $M$ (matter domination), and $\Phi$ (scalar field–domination). (a) Phase space: The light gray arrows show the slope field of the orbits. The dashed gray line marks the transition from deceleration to acceleration ($q=0$), with the gray shaded region indicating accelerated expansion. The green dashed line marks $q = 1$. The red dashed line denotes the thermodynamic phase transition ($\sigma=0$). Yellow region indicates the overlap of $C_V>0$ and $C_P>0$. The blue orbit shows cosmic evolution with boundary condition $x(\tau = 0)=0.2124$, $y(\tau = 0)=0.7888$ which yields $\Omega_\phi(\tau = 0) \approx 0.667$ at present. (b) Thermodynamic regions: The panel shows regions for three thermodynamic stability conditions, with forward and backward hatching representing $C_V>0$ and $C_P>0$, respectively. The gray shaded region shows $C_P - C_V > 0$.
• Figure 4: Phase space of the quintessence model with exponential potential, $\lambda =2$. Format and convention follow Fig. \ref{['fig:Q1']}, apart from the additional critical point $S$ (scaling solution) appearing in the case. The blue orbit in panel (a) shows cosmic evolution with boundary condition $x(\tau = 0)=y(\tau = 0)=0.6123$ which yields $\Omega_\phi(\tau = 0) \approx 0.75$ at present.
• Figure 5: Phase space of the phantom model with exponential potential, $\lambda =1$. The region bounded by hyperbola $-x^2 + y^2 \leq 1$ and $y>0$ shows the phase space and thermodynamic regions in panels (a) and (b), respectively. Both panels show the critical points, $M$ (matter domination), and $\Phi$ (scalar field–domination) while the critical points for Kinetic domination are at the infinities in the $(x,y)$ plane. (a) Phase space: The light gray arrows show the slope field of the orbits. The gray shaded region indicate the accelerated expansion: light gray corresponds to the standard accelerated expansion $-1<w_{\rm eff}<-1/3$, while the dark gray region corresponds to the phantom regime $w_{\rm eff}<-1$. The red dashed curve denotes the thermodynamic phase transition ($\sigma=0$). The yellow region indicates the overlap of $C_V>0$, $C_P>0$, and $C_P-C_V>0$, i.e., region of thermodynamic stability. The blue orbit shows cosmic evolution with a boundary condition $x(\tau = 0)=-0.212$, $y(\tau = 0)=0.788$, which yields $\Omega_\phi(\tau = 0) \approx 0.576$ at present. (b) Thermodynamic regions: The panel shows regions for three thermodynamic stability conditions, with forward and backward hatching representing $C_V>0$ and $C_P>0$, respectively. The gray shaded region shows $C_P - C_V > 0$.