Structure formation in a non-canonical scalar field model of clustering dark energy

TL;DR

We study a non-canonical scalar field (k-essence) with Lagrangian $L(X,\phi)=X\left(\frac{X}{M^4}\right)^{\alpha-1}-V(\phi)$ and an exponential potential as clustering dark energy in a spatially flat FLRW universe. Using dynamical-system analysis for the background, linear perturbation theory in the pseudo-Newtonian framework, and the spherical collapse model for non-linear growth, we identify five critical points (C1–C5) and quantify the growth of structure via the growth factor $D(z)$, the observable $f(z)\sigma_8(z)$, and halo statistics, including mass-function corrections for DE clustering. The model yields a late-time acceleration close to $\Lambda$CDM while permitting measurable deviations in growth at higher redshift and in the high-mass end of the halo population, controlled by $\alpha$, $\lambda$, and $M$. These results offer observational pathways to test clustering dark energy and distinguish non-canonical scalar-field cosmologies from standard scenarios.

Abstract

This paper examines the growth of dark matter and dark energy perturbations within a non-canonical scalar field model characterized by an exponential potential. Through dynamical system analysis, we identify critical points and track the background evolution of a spatially flat FLRW universe dominated by dark energy and pressureless dark matter. We systematically derive key cosmological quantities, including the Hubble parameter, deceleration parameter, density parameters, and the scalar field's equation of state, and explore their dependence on model parameters. Within the linear perturbation framework, employing the pseudo-Newtonian formalism, we compute the growth factor of matter density perturbations. To investigate the non-linear regime of structure formation, we employ the spherical collapse model and derive its key parameters. Building on these findings, we compute the function $f(z)σ_8(z)$ and the relative number density of halo objects exceeding a given mass threshold. Our results indicate that non-canonical scalar field models can effectively account for both background cosmic evolution and the growth of structure, offering potential insights into observational constraints and large-scale dynamics.

Figures (12)

• Figure 1: Projection of the evolution of phase space trajectories near the critical point $C_5$ along $x$-axis (left panel), $y$-axis (middle panel), and $\zeta$-axis (right panel) for $\Gamma=1$ and $\alpha=3$.
• Figure 2: Projection of the phase space trajectories for $\Gamma=1$ and $\alpha=3$ along the $x-y$ plane (left panel) and $x-\zeta$ plane (right panel). Initial conditions are set as $x_0=0.001$, $\zeta_0=0.005$ with different values of $\zeta_0\in(0,1)$. The evolution directions are illustrated by arrows. The black dots denote the critical points.
• Figure 3: Evolutions of $\Omega_m$ (black solid line), $\Omega_\phi$ (red solid line), $\omega_\phi$ (blue dashed line) and $\omega_{\rm eff}$ (purple dashed dotted line) for $\alpha=3$. Initial conditions are set as $x_0=0.001$, $y_0=0.005$ and $\zeta_0=0.3$.
• Figure 4: Evolution of the background quantities for $\lambda=0.5$ and $\tilde{M}=1$ in three different cases of $\alpha=1$, $\tilde{V}_0=2.222$ (blue dashed line), $\alpha=3$, $\tilde{V}_0=2.688$ (red dashed-dotted line) and $\alpha=10$, $\tilde{V}_0=3.059$ (purple dotted line). (a): Relative difference of normalized Hubble parameter with that in $\Lambda$CDM model $\Delta E$, (b): The deceleration parameter $q$, (c): the density parameter of scalar field $\Omega_\phi$, (d): the density parameter of matter $\Omega_m$, (e): the EoS parameter of scalar field $\omega_\phi$, (f): the effective EoS parameter $\omega_{\rm eff}$, (g): the normalized scalar field $\tilde{\phi}$, (h): the normalized potential of the scalar field $V/V_0$. The solid lines in panels (c), (d), and (f) denote the results of $\Lambda$CDM.
• Figure 5: Evolution of the growth function $D\equiv \delta_m/\delta_{m_0}$ normalized by the scale factor, $a$ (the growth function of a pure matter model). The top, middle, and bottom panels display the variation of the function with respect to the parameters $\alpha$, $\lambda$, and $M$, respectively. In each panel, the solid curve represents the results of $\Lambda$CDM.