Constraints on ultralight scalar dark matter from pulsar timing

We perform a Bayesian analysis of pulsar-timing residuals from the NANOGrav pulsar-timing array to search for a specific form of stochastic narrow-band signal produced by oscillating gravitational potential (Gravitational Potential Background) in the galactic halo. Such oscillations arise in models of warm dark matter composed of an ultralight massive scalar field ($m\sim 10^{-23}\mathrm{eV}$), recently considered by Khmelnitsky and Rubakov [J. Cosmol. Astropart. Phys. 2 (2014) 019]. In the monochromatic approximation, the stringent upper limit (95% C.L.) on the variable gravitational potential amplitude is found to be ${\mathrm{\Psi }}_c<1.14\times 10^{-15}$, corresponding to the characteristic strain $h_c=2\sqrt{3}{\mathrm{\Psi }}_c<4\times 10^{-15}$ at $f=1.75\times 10^{-8}\mathrm{Hz}$. In the narrow-band approximation, the upper limit of this background energy density is ${\mathrm{\Omega }}_{\mathrm{GPB}}<1.27\times 10^{-9}$ at $f=6.2\times 10^{-9}\mathrm{Hz}$. These limits are an order of magnitude higher than the expected signal amplitude. The applied analysis of the pulsar-timing residuals can be used to search for any narrow-band stochastic signals with different correlation properties. As a by-product, parameters of the red noise present in four NANOGrav pulsars ($\mathrm{J}1713+0747$, $\mathrm{J}2145-0750$, $\mathrm{B}1855+09$, and $\mathrm{J}1744-1134$) have been evaluated.

Figure 1

The typical form of the cumulative distribution of ${\mathrm{\Omega }}_{\mathrm{GPB}}$ in one frequency bin numerically reconstructed by the MCMC method. The blue color (light gray) shows the range containing 95% of all Makov chain points. The quantile 0.95 of the ${\mathrm{\Omega }}_{\mathrm{GPB}}$ cumulative distribution function (CDF) gives the 95% confidence level ${\mathrm{\Omega }}_{\mathrm{GPB}}<{\mathrm{\Omega }}_{0.95}$.

Figure 2

An upper limit on the amplitude of the variable gravitational potential ${\mathrm{\Psi }}_c$ due to the massive scalar field oscillations as a function of the central frequency $f$. Shown is the case of narrow-band approximation (the black line), the monochromatic signal approximation with the Earth term only (the thin gray dashed line), and the monochromatic approximation using both the Earth and pulsar terms (the gray dashed-dot line); the lines are shown for the 95% confidence level. Data from eight pulsars from the NANOGrav Project with white-noise rms residuals were used. The dashed line shows the model amplitude (24).

Figure 3

Left panel: An upper limit on the ${\mathrm{\Omega }}_{\mathrm{GPB}}$ of the ultralight scalar field as a function of frequency $f$; the solid curve corresponds to the 95% confidence level. The dashed line shows the model value (25). Right panel: The same limit in terms of the local dark matter density ${\rho }_{\text{DM}}$. The dashed line shows the local galactic dark matter density $0.3\mathrm{GeV}{\mathrm{cm}}^{-3}$. Data from 12 pulsars from the NANOGrav Project have been used.

Figure 4

Estimation of the red-noise parameters ARN and γRN for pulsars J1713+0747, J2145-0750, B1855+09, and J1744-1134 using the Markov chain Monte Carlo method. Different confidence levels are shown in color: black, 68% confidence level; gray, 95% confidence level; and light gray, 99.7% confidence level. The plots were obtained using the R statistical package [43].