Detecting dark-matter waves with a network of precision-measurement tools

Virialized ultralight fields (VULFs) are viable cold dark-matter candidates and include scalar and pseudoscalar bosonic fields, such as axions and dilatons. Direct searches for VULFs rely on low-energy precision-measurement tools. While previous proposals have focused on detecting coherent oscillations of the VULF signals at the VULF Compton frequencies for individual devices, here I consider a network of such devices. Virialized ultralight fields are essentially dark-matter waves and as such they carry both temporal and spatial phase information. Thereby, the discovery reach can be improved by using networks of precision-measurement tools. To formalize this idea, I derive a spatiotemporal two-point correlation function for the ultralight dark-matter fields in the framework of the standard halo model. Due to VULFs being Gaussian random fields, the derived two-point correlation function fully determines $N$-point correlation functions. For a network of $N_D$ devices within the coherence length of the field, the sensitivity compared to a single device can be improved by a factor of $\sqrt{N_D}$. Further, I derive a VULF dark-matter signal profile for an individual device. The resulting line shape is strongly asymmetric due to the parabolic dispersion relation for massive nonrelativistic bosons. I discuss the aliasing effect that extends the discovery reach to VULF frequencies higher than the experimental sampling rate. I present sensitivity estimates and develop a stochastic field signal-to-noise ratio statistic. Finally, I consider an application of the formalism developed to atomic clocks and their networks.

Figure 1

(a) Dark-matter wave observatory based on a global network of existing low-energy precision-measurement laboratories (red dots) around the globe. (b) Satellite mission for probing VULF DM correlation function; both the distance between the satellites and the angle between galactic velocity ${\mathbf{v}}_g$ and separation $\mathbf{d}$ vectors can be varied. (c) Terrestrial experiment with fixed nodes utilizes the daily variation of the angle between the galactic velocity and two-node separation vector.

Figure 2

VULF dark-matter line profile for colocated linear portal-sensitive devices or individual apparatus for $\eta =v_g/v_{\mathrm{vir}}=1$. Dashed vertical line marks the position of the Doppler-shifted Compton frequency ${\omega }_{\varphi }^{\prime }={\omega }_{\varphi }+m_{\varphi }{v}_g^2/2\hslash$. The profile value is strictly zero for $\omega <{\omega }_{\varphi }^{\prime }-{\eta }^2/2{\tau }_c$ due to the dispersion relation for massive particles. The maximum value of $F\left(\omega \right)\approx 0.18{\tau }_c$ is attained at $\omega \approx {\omega }_{\varphi }^{\prime }+0.22/{\tau }_c$ and the full width at half maximum is $\mathrm{\Delta }{\omega }_{\varphi }\approx 2.5/{\tau }_c$.

Figure 3

Spatial dependence of the VULF two-point correlation function for $\eta =v_g/v_{\mathrm{vir}}=1$. The red solid curve illustrates the dependence for the case when the device separation vector $\mathbf{d}$ is aligned with the galactic velocity ${\mathbf{v}}_g$ and the dashed blue curve when $\mathbf{d}\perp {\mathbf{v}}_g$. The galactic orientation angle is defined as ${\theta }_g={cos}^{-1}({\mathbf{k}}_g·\mathbf{d})$.

Figure 4

Effects of DFT aliasing on the observable power spectral density of dark-matter field. Reduced PSD is defined as $\langle |{\tilde{\varphi }}_p{|}^2\rangle /N{\mathrm{\Phi }}_0^2$ and is shown as a function of DFT frequency detuning from 4 Hz. The aliased Doppler-shifted Compton frequency for all curves is 4 Hz. The Nyquist frequency in the simulation is 5 Hz. See the text for other simulation parameters. The top panel is for low-frequency Doppler-shifted Compton frequencies and the bottom panel shows PSDs for high-frequency fields. The data sampling rate is the same for all shown curves.

Figure 5

Projected constraints on the electromagnetic gauge modulus $d_e$ for high Compton frequencies. The excluded parameter space from the equivalence principle (EP) tests [10] is shown as a shaded region. The solid orange line is the projected limit from Eq. (25) for an optical clock compared to the state-of-the-art cavity [40] (see the text for details). The dashed green line is drawn neglecting the bandwidth-limiting effect of the filter function (24).