Dynamical friction in fuzzy dark matter: Circular orbits

Dynamical friction (DF) is the gravitational force experienced by a body moving in a medium as a result of its density wake. In this work, we investigate the DF acting on circularly moving perturbers in fuzzy dark matter (FDM) backgrounds. After condensation in the early Universe, FDM is described by a single wave function satisfying a Schrödinger-Poisson equation. An equivalent, hydrodynamic formulation can be obtained through the Madelung transform. Here, we consider both descriptions and restrict our analysis to linear response theory. We take advantage of the hydrodynamic formulation to derive a fully analytic solution to the DF in steady state and for a finite time perturbation (corresponding to a perturber turned on at $t=0$). We compare our prediction to a numerical implementation of the wave approach that includes a nonvanishing FDM velocity dispersion $\sigma$. Our solution is valid for both a single and a binary perturber in circular motion as long as $\sigma$ does not significantly exceed the orbital speed $v_{\mathrm{circ}}$. While the short-distance Coulomb divergence of the (supersonic) gaseous DF is no longer present, DF in the FDM case exhibits an infrared divergence which stems from the (also) diffusive nature of the Schrödinger equation. Our analysis of the finite time perturbation case reveals that the density wake produced by perturbers diffuses through the FDM medium until it reaches its outer boundary. Once this transient diffusive regime is over, both the radial and tangential DF oscillate about the steady-state solution with a decaying envelope. Steady state is never achieved. We discuss two astrophysical applications of our results: we revisit the DF decay timescales of the five Fornax globular clusters and point out that the inspiral of compact binary may stall because the DF torque about the binary center of mass sometimes flips sign to become a thrust rather than a drag.

Figure 1

The orbital radius $r_0$ for which ${\lambda }_{\mathrm{NL}}=0.1r_0$ [i.e., ${\alpha }_0$ defined in Eq. (25) is ${\alpha }_0=0.1$] is shown as a function of the total binary mass $M$ and the axion mass $m_{18}$. The nonlinear scale ${\lambda }_{\mathrm{NL}}$ is calculated assuming circular Keplerian motion. The contour levels indicate the value of $r_0$ above which higher-order contributions to the linear response approach considered here become larger than $\gtrsim 10\%$.

Figure 2

The retarded Green’s function Eq. (23) assuming $R_{\mathrm{\Omega }}=4$. For convenience, $G_{\mathrm{ret}}(\tilde{\mathbf{r}},)$ is normalized to $\frac{R_{\mathrm{\Omega }}}{4\pi \mathrm{\Omega }{r}_0^3}$ and both $\tilde{r}$ and $\tilde{t}$ axes are logarithmic. The white contours indicate the locus for which $G_{\mathrm{ret}}$ vanishes.

Figure 3

Density wake in the single-perturber case computed after 1.25 rotations. The Born approximation to the Lippmann-Schwinger equation (left panel) and the linearized hydrodynamic approach (middle panel) are used. The corresponding overdensities ${\alpha }^{\mathrm{LS}}$ and ${\alpha }^{\mathrm{M}}$ are normalized by ${\alpha }_0$ (see text for details). The right panel shows the difference ${\alpha }^{\mathrm{LS}}-{\alpha }^{\mathrm{M}}$, which is at the few percent level. The parameter values $R_{\mathrm{\Omega }}=4$ and $R_{\sigma }=0.1$ adopted here match our single-perturber configuration.

Figure 4

Similar to Fig. 3 but for our fiducial compact binary with mass ratios $q_1=q_2=0.5$, $R_{\mathrm{\Omega }}=0.0177$ and $R_{\sigma }=0.1$. The predicted wavelike and fluidlike overdensities ${\alpha }^{\mathrm{LS}}$ and ${\alpha }^{\mathrm{M}}$ are normalized to ${\alpha }_0$ (see text for details). The right panel shows the difference ${\alpha }^{\mathrm{LS}}-{\alpha }^{\mathrm{M}}$. The current position of the binary components are indicated by white filled symbols, while the binary orbit is shown as a thin white circle.

Figure 5

The real and imaginary parts of the function $I(R_{\mathrm{\Omega }})$ (solid curves) in FDM backgrounds calculated from Eq. (45) with (48)–(50) with ${\tilde{k}}_{\min }=0.3$. For comparison, we also show the corresponding gaseous $I({\mathcal{M}}_g)$ (dashed curves) computed in [41]. Although the quantity $R_{\mathrm{\Omega }}$ plays a role similar to the gas Mach number ${\mathcal{M}}_g$, we show $I(R_{\mathrm{\Omega }})$ as a function of $\sqrt{R_{\mathrm{\Omega }}}$ for convenience. Furthermore, we have rescaled the real and imaginary parts of $I$ such that they all asymptote to a constant in the limit of large $R_{\mathrm{\Omega }}$ or ${\mathcal{M}}_g$. For the imaginary part of $I(R_{\mathrm{\Omega }})$, this rescaling leads to an artificial divergence at $R_{\mathrm{\Omega }}=0$. The multipoles have been summed up to $l_{\max }=100$ in both the gas and FDM cases.

Figure 6

The imaginary part $\mathrm{Im}(I)$ of the complex friction $I(R_{\mathrm{\Omega }})$ is shown for different values or $R_{\mathrm{\Omega }}$ as a function of the upper limit $l_{\max }$ of the (truncated) multipole expansion Eq. (45). Unlike the gaseous case, $\mathrm{Im}(I)$ converges to any desired accuracy for finite values of $l_{\max }$.

Figure 7

The imaginary part of the transient amplitude $S_{l,l-1}^{\mathrm{Tra}}$ normalized to its steady-state counterpart as a function of the (dimensionless) time $\tilde{t}$. Results are shown for a few choices of multipole $l$. The azimuthal number is fixed to $m=1$ and a parameter value $R_{\mathrm{\Omega }}=4$ is assumed. All the transient amplitudes shown here fall below the steady-state result after completing one rotation. Their envelope approximately decays as $S_{l,l-1}^{\mathrm{Tra}}\propto t^{-l+1/2}$.

Figure 8

The finite time perturbation amplitude $S_{l,l-1}^{\mathrm{Ftp}}$ in the special case $l=1$ and $m=0$. Results are shown as a function of the number of rotations $\tilde{t}/2\pi$ (in unit of $R_{\mathrm{\Omega }}$) for different lower cutoff wave numbers ${\tilde{k}}_{\min }$ as indicated on the figure. All the curves assume $R_{\mathrm{\Omega }}=4$, while the empty symbols represent the particular case $({\tilde{k}}_{\min },R_{\mathrm{\Omega }})=(0.3,1)$. The (also) diffusive nature of the Schrödinger equation leads to the square root behavior $S_{1,0}^{\mathrm{Ftp}}\propto \sqrt{\tilde{t}}$ at early time. Arrows mark the estimated timescale at which the initial ($t=0$) perturbation has diffused throughout the medium (see text). The horizontal lines indicate the steady-state limit which the finite time perturbation result converges to.

Figure 9

Contour levels of $\mathrm{Re}(I)/R_{\mathrm{\Omega }}^2$ (top panel) and $\mathrm{Im}(I)/R_{\mathrm{\Omega }}$ in the plane $(R_{\mathrm{\Omega }},q_1)$, where $q_1$ is the (normalized) mass of the first binary component. A value of ${\tilde{k}}_{\min }=0.3$ is assumed for the computation of $I(R_{\mathrm{\Omega }})$. The green contours indicate the zero level. The increase in the magnitude of $\mathrm{Re}(I)$ toward smaller $R_{\mathrm{\Omega }}$ is artificially caused by the normalization $\mathrm{Re}(I)/R_{\mathrm{\Omega }}^2$ adopted here. $\mathrm{Re}(I)$ linearly depends on $R_{\mathrm{\Omega }}$ for $R_{\mathrm{\Omega }}\lesssim 1$ but exhibits a quadratic dependence $\propto R_{\mathrm{\Omega }}^2$ for $R_{\mathrm{\Omega }}\gg 1$. Both $\mathrm{Re}(I)$ and $\mathrm{Im}(I)$ can be positive or negative depending on $(R_{\mathrm{\Omega }},q_1)$.

Figure 10

Top panel: radial ($\widehat{\mathbf{r}}$) and tangential ($\widehat{\mathit{\phi }}$) components of the DF force in the finite time perturbation case for our fiducial single perturber. The DF force is normalized to $\rho R_{\mathrm{\Omega }}{(\frac{GM}{\mathrm{\Omega }{r}_0})}^2$. The cross symbols are the outcome of a numerical implementation of the Lippmann-Schwinger approach on a three-dimensional grid, whereas the solid and dashed curves show the analytic predictions obtained from the Madelung approach (see text for details). We have adopted a cutoff wave number ${\tilde{k}}_{\min }=1/8$ which matches the size of our simulation boxes. The steady-state tangential DF is shown as the horizontal dashed line. In the radial direction, the steady-state DF is $\simeq -76.6$ for ${\tilde{k}}_{\min }=1/8$. Hence, it is not visible on the figure. Bottom panel: fractional difference between our wave and hydrodynamic predictions of the radial and tangential DF.

Figure 11

Stable fixed points $r_0^{*}$ of the one-dimensional dynamical system Eq. (66) as a function of $q_1$ ($q_2=1-q_1$). When $a_{\mathrm{GW}}=0$, there is one stable orbital radius for each $q_1$ shown as the solid (black) curve. When $a_{\mathrm{GW}}\ne 0$, there are no stable orbits for $q_1$ in the range $[q_{\min },0.5]$. Vertical lines mark $q_{\min }$ for the different values of $a_{\mathrm{GW}}$ indicated on the figure. For each $q_1$ in the range $[0,q_{\min }[$, there is, again, a unique stable orbit. Whenever they exist, the stable orbital radii $r_0^{*}$ change by at most 1% as $a_{\mathrm{GW}}$ is varied in the range $0<a_{\mathrm{GW}}<1$. Therefore, the solid (black) curve computed for $a_{\mathrm{GW}}=0$ accurately characterizes the fixed point also for $0<a_{\mathrm{GW}}<1$ (leftward of the corresponding vertical line).

Figure 12

Left panel: poles and semicircular contours $C_1$ and $C_2$ used for the calculation of the steady-state amplitude Eq. (47). The poles are located at ${\tilde{k}}_0=0$, ${\tilde{k}}_{1,2}=\pm \sqrt{m{R}_{\mathrm{\Omega }}+i\epsilon }$ and ${\tilde{k}}_{3,4}=\pm i\sqrt{m{R}_{\mathrm{\Omega }}+i\epsilon }$. We choose to shift the $\tilde{k}={\tilde{k}}_0$ pole into the upper half plane. The contour $C_1$ ($C_2$) extends from $-\infty$ to $\infty$ on the real axis and is completed in the upper (lower) half plane. Right panel: the contours $C_3$ and $C_4$ used for the calculation of the transient amplitude Eq. (c3). For $C_3$ ($C_4$), the circular arc subtends an angle $\pi /4$ and lies in the upper (lower) half plane. The two contours are closed by a diagonal line at an angle $\vartheta =\pm \frac{\pi }{4}$ which can be parametrized by $k=(1\pm i)\chi$ with $\chi$ in the range $[0,\infty [$.