Stamps of state on structure: Probing the state of ultralight dark matter via its density fluctuations

TL;DR

Ultralight dark matter may reside in a coherent-state ensemble or in a squeezed state produced by inflation via a Bogoliubov transformation. The authors show that, even in the large-particle-number limit and without DM self-interactions, the DM state leaves distinct imprints on density perturbations: the squeezed state can double the two-point density correlator and, under in-phase Bogoliubov coefficients, quadruple the three-point correlator relative to a coherent ensemble. These differences translate into detectable non-Gaussianity in the matter bispectrum with a local-type signature, enabling cosmological observations to diagnose the current DM state and its inflationary origin. The work motivates further numerical studies of how cosmological dynamics might modify the initial squeezed state and its density fluctuations, and suggests higher-point statistics as potential discriminants when phase coherence is imperfect.

Abstract

Dark matter (DM) candidates with very small masses, and correspondingly large number densities, have gained significant interest in recent years. These DM candidates are typically said to behave "classically". More specifically, they are often assumed to reside in an ensemble of coherent states. One notable exception to this scenario is when isocurvature fluctuations of the DM are produced during inflation (or more generally by any Bogoliubov transformation). In such contexts, the ultralight DM instead resides in a squeezed state. In this work, we demonstrate that these two scenarios can be distinguished via the statistics of the DM density fluctuations, such as the matter power spectrum and bispectrum. This provides a probe of the DM state which persists in the limit of large particle number and does not rely on any non-gravitational interactions of the DM. Importantly, the statistics of these two states differ when the modes of the squeezed state are all in-phase, as is the case at the end of inflation. Later cosmological dynamics may affect this configuration. Our work motivates future numerical studies of how cosmological dynamics may impact the initial squeezed state and the statistics of its density fluctuations.

Figures (3)

• Figure 1: Schematic diagram of an ensemble of coherent states (left) and a squeezed state (right) in the phase space parametrized by $\hat{X}$ and $\hat{Y}$. Each circle/ellipse represents the $2\sigma$-contour of the probability density of a state. On the left, each colored circle represents a single coherent state, with the brown state being the vacuum. The center of each circle lies at $(\hat{X},\hat{Y})=(\mathop{\mathrm{\text{Re}}}\nolimits[\gamma],\mathop{\mathrm{\text{Im}}}\nolimits[\gamma])$. A Gaussian coherent state ensemble is a classical ensemble of such states, with $\gamma$ distributed as in Eq. (\ref{['eq:probability']}). On the right, the grey ellipse represents a state which is squeezed along the $\hat{X}$-direction. Time evolution in these phase space diagrams corresponds to clockwise rotation. Note that the coherent state ensemble is time-invariant, while the squeezed state is not. In light grey, we show several contours of fixed $\hat{N}=\hat{X}^2+\hat{Y}^2-\frac{1}{2}$. It is apparent that these two states exhibit different statistics for $\hat{N}$, that is if both are scaled to have the same expectation of $\hat{N}$, the squeezed state will have a larger variance than the coherent state ensemble. In the case of a quantum field, the roles of $\hat{X}$ and $\hat{Y}$ are played by (the momentum modes of) $\hat{\phi}$ and $\partial_t\hat{\phi}$, respectively. Importantly, inflation produces a squeezed state where all modes are squeezed along the $\hat{\phi}$-direction [see discussion following Eq. (\ref{['eq:2point_squeezed']})].
• Figure 2: Contractions which contribute to the three-point function of the density perturbations [see Eqs. (\ref{['eq:3point_coherent']}) and (\ref{['eq:3point_squeezed']})]. Each diagram consists of six nodes representing the three creation and three annihilation operators in Eq. (\ref{['eq:3point_op']}) [with their corresponding momenta labeled in the top-left diagram], while lines indicate operator contractions. As $\mathbf p_1,\mathbf p_2,\mathbf p_3\neq0$, horizontal contractions are forbidden. There are two valid bipartite contractions, shown in black, which contribute to the result for the coherent state ensemble in Eq. (\ref{['eq:3point_coherent']}). Meanwhile, there are six other contractions, shown in blue, which contribute to the squeezed state result in Eq. (\ref{['eq:3point_squeezed']}). All diagrams in the top row yield the first term in Eq. (\ref{['eq:3point_coherent']}), while the diagrams in the bottom row yield the second term.
• Figure 3: Contractions which contribute to the four-point function of the density perturbations. As in Fig. \ref{['fig:3point']}, each diagram consists of nodes corresponding to creation/annihilation operators, with contractions shown by solid lines. In this figure, we set $\mathbf p=\mathbf p_1=-\mathbf p_2$ and $\mathbf q=\mathbf p_3=-\mathbf p_4$. In red and black, we show all bipartite contractions, which contribute to the four-point function of the coherent state ensemble. The red contractions are disconnected, and so do not contribute to $K_\Psi(\mathbf p,\mathbf q)$, as in Eq. (\ref{['eq:4point_def']}). [Only the leftmost is nonzero when $\mathbf p\neq\pm\mathbf q$.] The blue diagrams show non-bipartite contractions which are phase-independent, and so contribute to $K_\Omega(\mathbf p,\mathbf q)$ for the squeezed state. The dashed lines represent pairs of nodes which have not been contracted, but whose momenta are equal because we have fixed $\mathbf p_1=-\mathbf p_2$ or $\mathbf p_3=-\mathbf p_4$. As a result, the non-bipartite contractions in the blue diagrams depend on the same momenta. This leads to a cancellation of the Bogoliubov phases, as in Eq. (\ref{['eq:blue_diagram']}). The bottom two rows of diagrams are arranged so that diagrams in the same column yield the same term in Eq. (\ref{['eq:4point_squeezed']}).