Oscillatons in Scalar-Field Dark Matter from a Full Fourier Expansion of an Exponential Potential

TL;DR

Oscillating, self-gravitating real scalar fields (oscillatons) are studied as scalar-field dark matter candidates with an exponential self-interaction. The authors develop a full Fourier–Bessel (Jacobi–Anger) framework that resums the time dependence of both the metric and the potential, yielding a dimensionless boundary-value problem for radial profiles solved by truncation of Bessel series. They show the geometry contains only even harmonics, energy density and other observables have a DC part plus even harmonics, and the radial energy flux is dominated by the second harmonic; they also verify the small-amplitude limit reproduces the quadratic oscillaton. The framework provides a reproducible baseline for stability analyses and observational constraints, linking microphysical coupling to macroscopic core properties through a mass-radius map and harmonic content.

Abstract

Real, time-dependent scalar fields can form oscillating, self-gravitating configurations-oscillatonsthat are viable candidates for scalar-field dark matter (SFDM). We revisit oscillatons with an exponential self-interaction and develop a full Fourier (Jacobi{Anger) treatment that resums the time dependence of both the metric and the potential, thereby unifying quadratic, quartic, and higher-order interactions within a single framework. After fixing the small-amplitude normalization V0 = m2 Φ=(λ2k0), we derive a closed, dimensionless boundary-value problem for the radial profiles and solve it numerically via Bessel-series truncation with controlled convergence. We compute time-resolved and time-averaged observables energy density, radial energy flux, radial/tangential pressures, and total mass and map their dependence on the coupling λ and central amplitude. The geometry exhibits only even harmonics of the fundamental frequency, while composite observables inherit a DC part plus even harmonics; the radial flux oscillates predominantly at 2!. Apparent negative instantaneous pressures arise from coherent oscillations and are assessed consistently through classical energy-condition diagnostics (WEC/NEC/SEC). Our formulation provides a reproducible and extensible baseline for stability analyses and observational constraints on SFDM oscillatons

Figures (7)

• Figure 1: Radial profiles for the time-periodic oscillaton with central amplitude $\sigma(0)=\frac{1.2}{\sqrt{2}}$ and coupling $\lambda=1$ (gauge: $\nu_0(0)=-\mu_0(0)=0.7855$; $\Omega$ tuned to enforce asymptotic flatness). Left: time-averaged metric exponents $\nu_0(x),\mu_0(x)$ and leading even-harmonic amplitudes $\nu_1(x),\mu_1(x)$. Right: scalar radial amplitude $\sigma(x)$. Regularity imposes $\sigma'(0)=0$ and $\nu_1(0)=\mu_1(0)=1.2141$, while the constraint $\nu_1+\mu_1=x\,\sigma\sigma'$ is satisfied to solver tolerance.
• Figure 2: Metric coefficients at $t=0$ and their leading harmonic content from Eqs. \ref{['equ20']}-\ref{['equ21']} for $\sigma(0)=\frac{1.2}{\sqrt{2}}$, $\lambda=1$. Left:$-g_{tt}(0,x)-1$ (green) together with $-g_{tt,0}(x)-1$ (blue) and $-g_{tt,2}(x)$ ($2\omega$–harmonic). Right:$g_{rr}(0,x)-1$ (green) with $g_{rr,0}(x)-1$ and $g_{rr,2}(x)$ (blue). The $2\omega$ mode dominates the time dependence; higher even harmonics are subleading once the Bessel truncation converges.
• Figure 3: Dependence of the geometry on the self-interaction strength $\sigma$ with other boundary data fixed. We compare $-g_{tt}(0,x)-1$ (left) and $g_{rr}(0,x)-1$ (right) for $\sigma=\{0.851,709,567\}$. Increasing $\sigma$ enhances $I_n(2\lambda\sigma)$, leading to larger departures from Minkowski and steeper core gradients, a trend robust under truncation/mesh refinement.
• Figure 4: Time dependence of the scalar and the exponential potential over one period for $\sigma(0)=\frac{1.2}{\sqrt{2}}$, $\lambda=1$ and $x\in(10^{-4},1.3]$. Left:$\sqrt{k_0}\Phi(t,x)=2\,\sigma(x)\cos(\omega t)$ shows the fundamental $\omega$ oscillation modulated by the radial envelope. Right:$V(\Phi)=V_0 e^{-\lambda\sqrt{k_0}\Phi}$ exhibits the full harmonic content implied by Eq. \ref{['eq:Vexp-Fourier']}; the small amplitude limit is consistent with $V_0=m_\Phi^2/(\lambda^2 k_0)$ from Eq. \ref{['eq:V0-consistency']}.
• Figure 5: Energy density and radial energy flux for $\sigma(0)=\frac{1.2}{\sqrt{2}}$, $\lambda=1$. Left:$\rho_\Phi(t,x)=\tfrac{1}{2}\![e^{-(\nu-\mu)}\dot{\Phi}^{2}+e^{-(\nu+\mu)}\Phi'^{2}+2V(\Phi)]$ has a dominant DC part plus even harmonics, peaking near the core and decaying rapidly with $x$. Right:$j_r(t,x)=e^{-(\nu+\mu)}\dot{\Phi}\,\Phi'$ oscillates predominantly at $2\omega$ with the expected phase shift; it averages to zero over one period at fixed $x$, indicating no net mass transport in the stationary oscillaton.