Classical field approximation of ultralight dark matter: Quantum break times, corrections, and decoherence

The classical field approximation is widely used to better understand the predictions of ultralight dark matter. Here, we use the truncated Wigner approximation method to test the classical field approximation of ultralight dark matter. This method approximates a quantum state as an ensemble of independently evolving realizations drawn from its Wigner function. The method is highly parallelizable and allows the direct simulation of quantum corrections and decoherence times in systems many times larger than have been previously studied in reference to ultralight dark matter. Our study involves simulation of systems in 1, 2, and 3 spatial dimensions. We simulate three systems, the condensation of a Gaussian random field in three spatial dimensions, a stable collapsed object in three spatial dimensions, and the merging of two stable objects in two spatial dimensions. We study the quantum corrections to the classical field theory in each case. We find that quantum corrections grow exponentially during nonlinear growth with the timescale being approximately equal to the system dynamical time. In stable systems the corrections grow quadratically. We also find that the primary effect of quantum corrections is to reduce the amplitude of fluctuations on the de Broglie scale in the spatial density. Finally, we find that the timescale associated with decoherence due to gravitational coupling to baryonic matter is at least as fast as the quantum corrections due to gravitational interactions. These results are consistent with the predictions of the classical field theory being accurate.

Figure 1

Gravitational collapse of a spatial over-density in a single spatial dimension. We plot the results for the classical field theory in red, and two quantum simulations of coherent states with $n_{\mathrm{tot}}\approx 6\times {10}^4$ and $n_{\mathrm{tot}}\approx 1\times {10}^6$ in black and cyan, respectively. Each column represents a different time, $t$. The top row shows the value of the each stream in the ensemble at $x=0$ and the bottom row shows the spatial density plotted such that each field has the same norm. Shell crossing occurs at $t=1$. During the collapse phase the field undergoes phase diffusion but the density remains well approximated by the MFT until after the collapse. Following the collapse, the density is smoothed out in proportion to the amount of phase diffusion achieved prior to the collapse, with high particle number simulations exhibiting larger fluctuations in the final number density. In these simulations $\tilde{\hslash }=2.5\times {10}^{-4}$ and $N_s=1024$, $M=256$, $M_{\mathrm{tot}}=L=1$. Plot taken from [47].

Figure 2

Here we show how the approximation scheme works diagrammatically for a single mode with a quartic nonlinearity. The left two plots show the Husimi distribution for the quantum distribution at the initial and final times. The red dots show the value of the many random streams sampling the Wigner function of the quantum distribution. We can see that the evolution of the ensemble of points approximates the evolution of the underlying quantum phase space. The right most plot shows the true value and ensembled approximation of $\mathrm{Var}[\widehat{q}]$. We can see that the evolution is well approximated by the ensemble. Here we use $N_s=1024$.

Figure 3

Here we plot the condensed object resulting from the collapse of a momentum space Gaussian density in two spatial dimensions, initial conditions described in Sec. 4b. The density shown here is the result of 1 Gyr of evolution. Each plot also shows the simulated value of the $Q$ parameter at this time. The left panel shows the classical field evolution, the condensed object in this case exhibiting the expected granular structure. The right panel shows the evolution for a simulation using the truncated Wigner approximation with $n_{\mathrm{tot}}={10}^6$ particles. Here, as in the one dimensional case, we see that the quantum corrections have removed most of the granular structure. In the middle panel we show the same simulation with $n_{\mathrm{tot}}={10}^9$ particles. The quantum corrections in this case are much smaller, and the resulting density is almost identical to the classical case. Here we set $M_{\mathrm{tot}}=6\times {10}^9{M}_{\odot }$, $\hslash /m=0.02{\mathrm{kpc}}^2/\mathrm{Myr}$, $L=60\mathrm{kpc}$, $M=512^2$, $2k_d^2=0.05{\mathrm{kpc}}^{-2}$, $T=1\mathrm{Gyr}$.

Figure 4

Here plot the prediction for the physical $Q_p$ for three different simulations all with different sampling parameters $n_{\mathrm{tot}}^s$. The simulated $Q_s(t)$ are then normalized according to Eq. (50). We can see that each simulation makes the same prediction for the growth of $Q_p$. The simulations are of the collapse of a momentum space Gaussian, described in Sec. 4b, in two spatial dimensions, with $M_{\mathrm{tot}}=6\times {10}^9{M}_{\odot }$, $\hslash /m=0.02{\mathrm{kpc}}^2/\mathrm{Myr}$, $L=60\mathrm{kpc}$, $M=512^2$, $2k_d^2=0.05{\mathrm{kpc}}^{-2}$.

Figure 5

Here we plot the simulated value of $Q$ for the gravitational collapse of an initial overdensity in a single spatial dimension at three different simulation resolutions. $M$ represents the number of grid cells, and $N_s$ the number of sampling streams. The evolution of $Q$ is the same in all three cases. This demonstrates that the evolution is independent of the specific simulation parameters. For these simulations $M_{\mathrm{tot}}={10}^8{M}_{\odot }$, $L=60\mathrm{kpc}$, and $\hslash /m=0.02{\mathrm{kpc}}^2/\mathrm{Myr}$.

Figure 6

The density of the initial conditions used to simulate collapsing objects in two spatial dimensions. The momentum density is Gaussian centered on $k=0$. The granular structure seen in the density is a result of the interference between different momentum modes. Here $M_{\mathrm{tot}}=6\times {10}^9{M}_{\odot }$, $L=60\mathrm{kpc}$, and $\hslash /m=0.02{\mathrm{kpc}}^2/\mathrm{Myr}$, $t_c\sim 2\mathrm{Gyr}$, $M=512^2$.

Figure 7

Here we plot the projected density evolution of the momentum space Gaussian initial conditions described in 4b in three spatial dimensions. Each column represents a different time, with initial conditions are shown on the left plot and the collapsed object on the right plot. We see the initially randomly distributed granules collapse into an object. Here we set $M_{\mathrm{tot}}=1\times {10}^{10}{M}_{\odot }$, $\hslash /m=0.02{\mathrm{kpc}}^2/\mathrm{Myr}$, $L=60\mathrm{kpc}$, $N=512^2$, $2k_d^2=0.05{\mathrm{kpc}}^{-2}$, $t_c\sim 2\mathrm{Gyr}$.

Figure 8

Here we plot the classical field evolution of the collapsed object resulting from the evolution of the Gaussian momentum distribution described in 4b in three spatial dimensions. Each column represents a different time, with initial conditions are shown on the left plot and the collapsed object on the right plot. We see the object is supported against further collapse with a continually evolving granular envelope. Here we set $M_{\mathrm{tot}}={10}^{10}{M}_{\odot }$, $\hslash /m=0.02{\mathrm{kpc}}^2/\mathrm{Myr}$, $L=60\mathrm{kpc}$, $N=512^3$.

Figure 9

Here we plot the classical field evolution of the merger of two collapsed object resulting from the evolution of the Gaussian momentum distribution described in 4b in two spatial dimensions. Each column represents a different time, with initial conditions are shown on the left plot and the collapsed object on the right plot. The collapse time is $t_c\sim 600\mathrm{Myr}$. Here we set $M_{\mathrm{tot}}=1.2\times {10}^{10}{M}_{\odot }$, $\hslash /m=0.02{\mathrm{kpc}}^2/\mathrm{Myr}$, $L=120\mathrm{kpc}$, $N=1024^2$.

Figure 10

Here we plot the evolution of the physical $Q$ parameter for the three test problems described in Sec. 4b. From left to right the plots show $Q(t)$ for the collapse of a momentum space Gaussian in three spatial dimensions (left), a collapsed object in three spatial dimensions (middle), and the merger of two collapsed objects in two spatial dimensions (right). In both the collapsing and merging case the parameter grows exponentially. In the stable case and in the nonlinear cases at early times the parameter grows quadratically. The growth of $Q$ for these systems corroborates the 1D results found in [46, 47], i.e. that quantum corrections grow exponentially during nonlinear growth and by a power-law for virialized systems and at very early times. For the collapsing and stable simulations $M_{\mathrm{tot}}={10}^{10}{M}_{\odot }$, $L=60\mathrm{kpc}$, and $\hslash /m=0.02{\mathrm{kpc}}^2/\mathrm{Myr}$, $t_c\sim 2$ (approximate collapse time), Gyr, $M=512^3$. For the merging simulation $M_{\mathrm{tot}}=1.2\times {10}^{10}{M}_{\odot }$, $L=120\mathrm{kpc}$, and $\hslash /m=0.02{\mathrm{kpc}}^2/\mathrm{Myr}$, $t_c\sim 0.6\mathrm{Gyr}$ (approximate merger time), $M=1024^2$.

Figure 11

Here we plot the distribution of complex angles and occupations of the ensemble of fields for a spatial overdensity in one spatial dimension. Two quantum simulations with $n_{\mathrm{tot}}\approx 6\times {10}^4$ and $n_{\mathrm{tot}}\approx 1\times {10}^6$ are plotted in black and cyan respectively. Each column represents a different time, $t$. The top row shows a histogram of the stream ensemble occupations numbers at $x=0$ and the bottom row a histogram of the stream ensemble complex angles. Shell crossing occurs at $t=1$. In these simulations $\tilde{\hslash }=2.5\times {10}^{-4}$ and $N_s=1024$.

Figure 12

We plot a normalized mass weighted amplitude variance [see Eq. (75)] in blue and a normalized mass weighted phase variance [see Eq. (76)] in orange for the gravitational collapse of an overdensity in a single spatial dimension. We set $N=512$, $M_{\mathrm{tot}}={10}^8{M}_{\odot }$, $L=60\mathrm{kpc}$, $\hslash /m=0.01$, $n_{\mathrm{tot}}={10}^6$.

Figure 13

The evolution of the gravitational collapse of initial overdensity for a quantum coherent state couple dot a tracer particle. The top row shows the phase space of the classical field evolution. Overlayed on top of this is the distribution representing the quantum state of the tracer particle (in black) and the particles classical phase space value (in red). Shell crossing occurs at $t=1$. In the bottom row, we plot the sample streams, used to approximate the Wigner distribution, values at $x=0$ (in black) with the classical field value (in red). We can see that as the quantum state spreads around the classical field value of the field it also spreads around the classical phase space position of the particle. In these simulations $\tilde{\hslash }=2.5\times {10}^{-4}$, $n_{\mathrm{tot}}\approx 6\times {10}^4$, $M_{\mathrm{tot}}=L=1$, and $N_s=512$.

Figure 14

Here we shows the results of using a test particle to estimate decoherence rates for the collapse of a overdensity in a single spatial dimension coupled to a test particle. The left plot show the uncertainty in the particles phase space position over time, and the right plot shows the $Q$ parameter, measuring the size of quantum corrections in the system. We see that the two grow similarly. Shell crossing occurs at $t=1$. In these simulations $\tilde{\hslash }=2.5\times {10}^{-4}$ and $N_s=512$, $n_{\mathrm{tot}}^s=6.7\times {10}^7$.

Figure 15

Here we plot $Q(t)$ for the gravitational collapse of an initial overdensity in a single and two spatial dimensions for two systems with the same dynamical times. The evolution is quite similar in both cases, reproducing the familiar results seen for this test problem [46]. For the 1D sim we set $N=512$, $M_{\mathrm{tot}}={10}^8{M}_{\odot }$, $L=60\mathrm{kpc}$, $\hslash /m=0.01$, $n_{\mathrm{tot}}={10}^6$. For the 2D sim we set $N=512^2$, $M_{\mathrm{tot}}={10}^8*L{M}_{\odot }$, $L=60\mathrm{kpc}$, $\hslash /m=0.01$, $n_{\mathrm{tot}}={10}^6$.