Dynamical and Gravitational Instability of Oscillating-Field Dark Energy and Dark Matter

TL;DR

Coherent oscillations of a scalar field can mimic fluids with a potential-determined equation of state and may drive inflation or dark energy or behave as dark matter. The authors map perturbations to two coupled anharmonic oscillators and derive a simple stability criterion: negative $w$ oscillations are dynamically unstable on large scales, while nearly harmonic potentials allow controlled analysis and a gravity-inclusive Jeans analysis; for axion dark matter, the resulting small-scale cutoff is $M_J \approx 1.8\times10^{-13}(m_a/10^{-5}\mathrm{eV})^{-3/2}(1+z)^{-3/4} M_\oplus$, implying a cutoff near $10^{-15} M_\oplus$ in the primordial spectrum. Gravity modifies but does not remove the dynamical instability, and the overall result is that oscillating-field models for acceleration face generic large-scale instabilities, with only temporary or narrow-stability windows. These findings place strong constraints on oscillating-field proposals for dark energy and inflation and provide a precise, physically transparent criterion that connects microphysical potential properties to cosmological stability.

Abstract

Coherent oscillations of a scalar field can mimic the behavior of a perfect fluid with an equation-of-state parameter determined by the properties of the potential, possibly driving accelerated expansion in the early Universe (inflation) and/or in the Universe today (dark energy) or behaving as dark matter. We consider the growth of inhomogeneities in such a field, mapping the problem to that of two coupled anharmonic oscillators. We provide a simple physical argument that oscillating fields with a negative equation-of-state parameter possess a large-scale dynamical instability to growth of inhomogeneities. This instability renders these models unsuitable for explaining cosmic acceleration. We then consider the gravitational instability of oscillating fields in potentials that are close to, but not precisely, harmonic. We use these results to show that if axions make up the dark matter, then the small-scale cutoff in the matter power spectrum is around $10^{-15} M_\oplus$.

Figures (5)

• Figure 1: Examples of potentials. The red dashed potential produce an equation of state corresponding to accelerated expansion ($w < -1/3$). The average intercept for some amplitude is shown as the red dot at $V>0$. The harmonic potential, which produces an equation-of-state parameter $w=0$, is the blue dot-dashed curve, and its average intercept the blue dot at $V<0$. Oscillations in the solid green potential produce an equation-of-state parameter $w > 0$, with an average intercept at the green dot at $V<0$.
• Figure 2: Two examples of a potential that can produce accelerated expansion and have stability on small scales---but only with oscillation amplitudes near those indicated by the dots.
• Figure 3: The first seven bands of instability for oscillations in the presence of a power-law potential.
• Figure 4: The first six bands of instability for oscillations in the presence of the potential shown in the left cell of Fig. \ref{['fig-noinstab']}. The solid line represents an amplitude corresponding to $w = -0.4$, dashed line $w=-0.6$ and dot-dashed line $w=-0.8$. In this example, $w$ remains less than one, but as the amplitude of oscillations decays a large-scale band of instability develops.
• Figure 5: The instability band for oscillations in the double-well potential shown in the right cell of Fig. \ref{['fig-noinstab']}. There is just one band of instability until $k$ becomes very large. Also plotted are lines that indicate the amplitudes at which $w=0.4$, $w=0.2$, and $w=0$ (from top to bottom). Note that $w\rightarrow -1$ as $x\rightarrow \sqrt{2}\simeq 1.41$, and so this model features large-scale stability with $-1<w<0$. However, this period of negative pressure is short-lived as the Universe expands.