Recognizing Axionic Dark Matter by Compton and de Broglie Scale Modulation of Pulsar Timing

Light axionic dark matter, motivated by string theory, is increasingly favored for the “no weakly interacting massive particle era”. Galaxy formation is suppressed below a Jeans scale of $\simeq {10}^8{M}_{\odot }$ by setting the axion mass to $m_B\sim {10}^{-22}\mathrm{eV}$, and the large dark cores of dwarf galaxies are explained as solitons on the de Broglie scale. This is persuasive, but detection of the inherent scalar field oscillation at the Compton frequency ${\omega }_B=(2.5\text{months}{)}^{-1}(m_B/{10}^{-22}\mathrm{eV})$ would be definitive. By evolving the coupled Schrödinger-Poisson equation for a Bose-Einstein condensate, we predict the dark matter is fully modulated by de Broglie interference, with a dense soliton core of size $\simeq 150\mathrm{pc}$, at the Galactic center. The oscillating field pressure induces general relativistic time dilation in proportion to the local dark matter density and pulsars within this dense core have detectably large timing residuals of $\simeq 400\mathrm{nsec}/(m_B/{10}^{-22}\mathrm{eV})$. This is encouraging as many new pulsars should be discovered near the Galactic center with planned radio surveys. More generally, over the whole Galaxy, differences in dark matter density between pairs of pulsars imprints a pairwise Galactocentric signature that can be distinguished from an isotropic gravitational wave background.

Figure 1

The left panel shows predicted soliton profiles for a range of $m_B$ (dashed curves), including the profile of a massive simulated halo of $10^{11}{M}_{\odot }$ (solid red curve), corresponding to the simulation on the right (using the yt package [12]), where the granular de Broglie scale structure is visible on all scales, including the dense central soliton. Also indicated is a NFW profile (blue dashed curve) that fits well the azimuthally averaged galaxy profile (solid red curve).

Figure 2

Predicted timing signal between pulsar pairs, for a light scalar field of $m_B\sim 0.8\times {10}^{-22}\mathrm{eV}$. The top panels show the relative timing signal between local pulsars at 8 kpc, and pulsars close to the Galactic center at a radius of 50 and 500 pc. In the bottom left and right panels both members of the pair are located at the same Galactocentric radius of 0.5 and 8 kpc, with relative phases chosen indicated in the inset box, bottom right, illustrating that the signal can cancel in such cases. The shaded regions indicate how the density modulation on the de Broglie scale can enhance or diminish the pairwise relative timing signal.

Figure 3

Characteristic strain measured between pairs of pulsars, with Galactocentric radii chosen as in Fig. 2 and compared with the expected sensitivities from the current and forthcoming PTA experiments (adapted from [9, 21]), with the corresponding oscillation frequency shown above. We highlight the relatively high signal strength we calculate for pulsars within the Galactic soliton region as a function of axion mass in a series of curves, demonstrating that this signal is already detectable for central Galactic pulsars. For comparison, the diagonal dotted line is the prediction obtained by [9] for local pulsars, assuming a smooth density distribution, with the blue shaded region representing the wider range we predict that includes our de Broglie scale DM density modulation. The local upper limit obtained by [22] is also shown (black square with arrow).