Constraining gravitational wave velocities using gravitational and electromagnetic wave observations of white dwarf binaries

Although the general theory of relativity (GR) predicts that gravitational waves (GWs) have exactly the same propagation velocity as electromagnetic (EM) waves, many theories of gravity beyond GR expect otherwise. Accurate measurement of the difference in their propagation speed, or a tight constraint on it, could be crucial to validate or put limits on theories beyond GR. The proposed future space-borne GW detectors are poised to detect a substantial number of Galactic white dwarf binaries (GWDBs), which emit the GW as semi-monochromatic signals. Concurrently, these GWDBs can also be identified as optical variable sources. Here we proposed that allocating a GWDB's optical light curve and contemporaneous GW signal can be used to trace the difference between the velocity of GW and EM waves. Simulating GW and EM wave data from 14 verification binaries (VBs), our method constrains propagation-originated phase differences, limiting the discrepancy between the speed of light ($c$) and GW ($c_{GW}$). Through the utilization of LISA's design sensitivity and the current precision in optical observation on GWDB, our study reveals that a four-year observation of the 14 recognized VBs results in a joint constraint that confines $\Delta c/c$ ($\Delta c = c_{\mathrm{GW}} - c$) to the range of $-2.1\times10^{-12}$ and $4.8\times10^{-12}$. Additionally, by incorporating this constraint on $c_{\mathrm{GW}}$, we are able to establish boundaries for the mass of the graviton, limiting it to $m_{\mathrm{g}}\le3\times10^{-23}\,e\mathrm{V}/c^{2}$, and for the parameter associated with local Lorentz violation, $\bar{s}_{00}$, constrained within the range of $-3.4\times10^{-11}\le\bar{s}_{00}\le1.5\times10^{-11}$.