Haloes of k-Essence

TL;DR

This paper investigates whether k-essence, a non-canonical scalar field, can form static, spherically symmetric haloes that mimic dark matter. By solving Einstein equations with L(X) theories and focusing on X = 1/2 ∇φ ∇φ, the authors show that static configurations yield anisotropic stress-energy, unlike the cosmological perfect-fluid form for timelike gradients. They analyze two Lagrangian families—barotropic and polytropic/Chaplygin-type—and derive halo solutions that produce flat rotation curves under certain parameter regimes, linking microscopic Lagrangians to macroscopic halo observables via the weak-field structure equations. The study also discusses stability (classical and quantum) and the initial-value formulation, and notes a de Sitter solution with spacelike gradients, offering a field-based alternative to particle dark matter and a route toward unifying dark matter and dark energy in some regimes.

Abstract

We study gravitationally bound static and spherically symmetric configurations of k-essence fields. In particular, we investigate whether these configurations can reproduce the properties of dark matter haloes. The classes of Lagrangians we consider lead to non-isotropic fluids with barotropic and polytropic equations of state. The latter include microscopic realizations of the often-considered Chaplygin gases, which we find can cluster into dark matter halo-like objects with flat rotation curves, while exhibiting a dark energy-like negative pressure on cosmological scales. We complement our studies with a series of formal general results about the stability and initial value formulation of non-canonical scalar field theories, and we also discuss a new class of de Sitter solutions with spacelike field gradients.

Figures (6)

• Figure 1: A plot of the Lagrangian (\ref{['eq:polL']}) for $p_*>0$ and $0<\gamma<1$ (case a). The zero occurs at $X=-M^{-4}(p*/\rho_*^{\gamma})^{2/(1-\gamma)}$, and the pole of $\rho(r_S)$ corresponds to the $X\to -\infty$ limit. Note that the Lagrangian is not defined for positive values of $X$.
• Figure 2: The density profile $\rho$ for a model with $p_*>0$ and $0<\gamma<1$ (case a). The parameters are $p_*/\rho_*=0.9$ and $\gamma=1/2$. Since $\gamma<2/3$, this halo has a finite mass.
• Figure 3: Figure showing the Lagrangian (\ref{['eq:polL']}) for $p_*>0$ and $\gamma>1$ (case b). The Lagrangian has a pole at $X=-M^{-4}(p_*/\rho_*^{\gamma})^{2/(1-\gamma)}$, and a zero at $X=0$. Note that the Lagrangian is defined for positive values of $X$ only if $\gamma$ is an odd integer, with the resulting plot being a mirror image of $X<0$.
• Figure 4: Figure showing the density profile for a model with $p_*>0$ and $\gamma>1$ (case b). The sample parameters are $p_*/\rho_*=3/4$ and $\gamma=3/2$. There is a point of zero density at finite $r$, and the solution can be joined to the vacuum at that point.
• Figure 5: Figure showing the Lagrangian (\ref{['eq:polL']}) for $p_*<0$ and $\gamma<0$ (case c). We plot both the original Lagrangian (full line) and the renormalized Lagrangian (dashed line). The asymptotic value of $\rho(r)$ corresponds to the energy density at $X=0$, which is a non-zero constant (full line). Note that the Lagrangian is defined for positive values of $X$ only if $\gamma$ is a negative odd number. If $(1-\gamma)/2$ is even, the plot of the Lagrangian for $X>0$ is a mirror image of the one for $X<0$. On the other hand, if $(\gamma-1)/2$ is odd, the Lagrangian is differentiable at $X=0$ and vanishes at a finite positive value of $X$.