Simulations of Shapiro, Gravitational, and Doppler time delays in pulsar networks for ultralight dark matter

TL;DR

This study investigates how de Broglie-scale granules in ultralight dark matter perturb pulsar timing through Shapiro time delays, gravitational redshift, and Doppler shifts. It combines simulations of mock pulsar arrays in a fluctuating ULDM background with analytic quasi-particle estimates to derive RMS time delays and their spectral shapes, showing distinct $P(f)$ forms for each effect. The results indicate that Shapiro delays and redshift produce characteristic temporal power spectra that, if detected, would provide smoking-gun evidence for ULDM, though current pulsar timing arrays lack sufficient sensitivity for typical parameter choices. The work highlights that longer observing times dramatically improve constraints and mass reach, offering a potential path to probe ULDM in the mass range $m ackground obreak ext{ around } obreak 10^{-17}$ eV, while acknowledging caveats related to substructure and multi-field extensions.

Abstract

The study of ultralight dark matter helps to constrain the lower bound of the mass in minimally coupled dark matter models. The granular structure of ultralight dark matter density fields produces metric perturbations which have been identified as a potentially interesting probe of this model. For dark matter masses $m \gtrsim 10^{-17} \, \mathrm{eV}$, these perturbations would fluctuate on timescales comparable to observational timescales. In this paper, we estimate the expected time delay these fluctuations would generate in simulated pulsar signals. We simulate arrays of mock pulsars in a fluctuating granular density field. We calculate the expected Shapiro time delay, gravitational redshift, and Doppler shift and compare analytical estimates with the results of simulations. Finally, we provide a comparison with existing pulsar observation sensitivities.

Figures (12)

• Figure 1: A density slice through a typical dark matter halo. Here we plot the log density. We can see the granule structure resulting from interference patterns in the classical field. The normalization of the field is arbitrary.
• Figure 2: The projected spatial ultralight dark matter density is shown on the left. The superimposed line in red on the left plot is a projected hypothetical path between a pulsar and Earth. The value of the potential interpolated at each position is potted on the right. Here the field mass $m = 10^{-22} \, \mathrm{eV}$.
• Figure 3: The Shapiro time delay (left), gravitational redshift (center), and Doppler shift (right) for a simulated pulsar at $1 \, \mathrm{kpc}$ from Earth in a ULDM density field at the local dark matter density. The mass of the dark matter field in this simulation was $m = 10^{-22} \, \mathrm{eV}$.
• Figure 4: The average Shapiro time delay measured for pulsars at a distance of $1 \, \mathrm{kpc}$ (left) and the average gravitational potential fluctuations (center), $\braket{\Phi}_\mathrm{rms}$, the average Doppler shift per root time (right), for a set of plane-wave box simulations, described in section \ref{['sec:ICs']}, as a function of the field mass $m_{22}$. The data points for the Shapiro delays correspond to the average over all the pulsars at a single moment in time. The large scatter around the prediction is due to the random potential fluctuations around the observer at that time in each simulation. The quasi-particle approximation for each effect equation is shown in red. For Shapiro delays the prediction is equation \ref{['eqn:dt_sh']}, for the potential rms the prediction is given by equation \ref{['eqn:phi_rms_approx']}, and for the Doppler shifts the prediction is given by equation \ref{['eqn:doppler_z_over_T']}. The quasi-particle approximation provides a reasonably accurate estimation in each situation.
• Figure 5: The average Shapiro time delay (left) and gravitational redshift (right) measured for pulsars in a plane-wave box simulation described in section \ref{['sec:ICs']} as a function of the pulsar-Earth separation, $x$. The quasi-particle approximation, equation \ref{['eqn:phi_rms_approx']}, is shown in red. Time delay values are shown as blue dots for specific pulsar-Earth pairs, with the average as a function of $x$ plotted in blue. The quasi-particle approximation provides a reasonably accurate estimation of the Shapiro time delays in simulations. Note that the amplitude of the Shapiro delay grows with the distance approximately as a random walk. However, the redshift values grow quickly over a few de Broglie wavelengths and then quickly plateau near the predicted value. The field mass for this simulation was $7.5 \, m_{22}$. Pulsar and Earth position pairs were chosen randomly in a box of size $L = 2.2 \, \mathrm{kpc}$.