Probing Quadratically Coupled Ultralight Dark Matter with Pulsar Timing Arrays

Abstract

Ultralight dark matter may couple quadratically to Standard Model particles. Such quadratic interactions give rise to both coherent and stochastic signals in pulsar timing array (PTA) observations. In this work, we characterize these signals, including the effects of dark matter propagation in a finite-density medium, and assess the sensitivity of current and upcoming PTA observations to their detection. For coherent signals, we find that the sensitivity of current PTA observations competes with and sometimes exceeds that of other probes, such as equivalence principle tests and atomic clocks. For stochastic signals, we find that PTA sensitivities underperform equivalence principle constraints for both existing and upcoming PTA data sets.

Figures (6)

• Figure 1: Projected sensitivities to the dilaton couplings $d_i$ from NANOGrav 12.5-year (solid) and simulated data set of total 30-year observation baseline (dashed). Fully opaque lines denote unscreened sensitivities, whereas semi-transparent red lines indicate sensitivities screened by matter effect. The sensitivities at masses around $10^{-17}\,{\rm eV}$ arise from the stochastic fluctuations at $\omega \lesssim m_\phi \sigma^2$, while the constraints at lower masses arise from coherent fluctuations at $\omega = 2m_\phi$. Other constraints --- from MICROSCOPE equivalence principle tests Hees:2018fpgMICROSCOPE:2019jixMICROSCOPE:2022doy, BBN Stadnik:2015kiaSibiryakov:2020eirBouley:2022eer, and atomic clocks Hees:2016gopKennedy:2020bacBACON:2020ubhSherrill:2023zahFilzinger:2023zrs --- are shown.
• Figure 2: Projected sensitivities to the light QCD axion from the NANOGrav 12.5-year (solid) and simulated data set with a total 30-year observation baseline (dashed). Fully opaque lines denote unscreened sensitivities, whereas semi-transparent red lines indicate screened sensitivities. The limits at low mass arise from searches for coherent fluctuations at $\omega = 2m_\phi$, while the sensitivities at high mass arise from searches for stochastic fluctuations at $\omega \lesssim m_\phi \sigma^2$. Other limits, e.g. those from MICROSCOPE (App. \ref{['subsec:MICROSCOPE_Axion']} and Ref. Gue:2025nxq), atomic clock comparison tests Hees:2016gopKennedy:2020bacBACON:2020ubhSherrill:2023zahFilzinger:2023zrsOswald:2021vtc, oscillating nEDM Abel:2017rtm, SMBH spin-down Arvanitaki:2014wvaBaryakhtar:2020gaoHoof:2024quk, GW170817 Zhang:2021mks, BBN Blum:2014vsa, and white dwarf mass-radius relation Balkin:2022qer, are also shown.
• Figure 3: Form factors of the Doppler (blue), clock (orange), and pulsar spin (red) signals for repulsive quadratic interactions, as defined in Eqs. (\ref{['eq:form_factor_dop']})--(\ref{['eq:form_factor_psr']}). Dimensionless parameter $y$ quantifies the strength of the quadratic coupling. For $y \lesssim 1$, all three form factors approach unity, whereas for $y \gtrsim 1$, they exhibit power-law suppression due to the screening effect.
• Figure 4: Static QCD axion profile sourced by a dense object. We choose the $|\Delta m| \, R = 7$ and $m_\phi R = 10^{-2}$ for this plot, where $\Delta m$ is the finite density correction to the axion mass squared evaluated at $\theta_0 = 0$. Inside matter $r/R <1$, the profile approaches $\theta_0 = \pi$, while outside matter $r/R>1$, it behaves as $1/r$.
• Figure 5: An example of $V_{\rm nr}$ when the phase transition occurs. As discussed in the text, due to non-vanishing static profile, the in-medium mass squared takes a positive value, and hence, it acts as a repulsive potential rather than attractive one. Outside potential, we find $-1/r^2$ behavior, which can be neglected for the study of dark matter propagation in the parameter space of interest. For this result, we use the static profile obtained in the previous figure.