Test of Weak Equivalence Principle with the multi-band timing of the Crab Pulsar

Weak Equivalent Principle (WEP) can be tested through the parameterized post-Newtonian parameter $\gamma$, representing the space curvature produced by unit rest mass. The parameter $\gamma$ in turn has been constrained by comparing the arrival times of photons originating in distant transient events, such as gamma-ray bursts, fast radio bursts as well as giant pulses of pulsars. Those measurements normally correspond to an individual burst event with very limited energy bands and signal-to-noise ratio (S/N). In this letter, the discrepancy in the pulse arrival times of the Crab Pulsar between different energy bands is obtained by the phase difference between corresponding pulse profiles. This allows us to compare the pulse arrival times at the largest energy band differences, between radio and optical, radio and X-ray, radio and gamma-ray respectively. As the pulse profiles are generated by phase-folding thousands of individual pulses, the time discrepancies between two energy bands are actually measured from thousands of events at each energy band, which corresponds to much higher S/N. The upper limit of the $\gamma$ discrepancy set by such an extensively-observed and well-modeled source is as follows: $\gamma_{radio}-\gamma_{\gamma-ray}<3.28\times10^{-9}$ at the energy difference of $E_{\gamma-ray}/E_{radio}\sim10^{13}$, $\gamma_{radio}-\gamma_{X-ray}<4.01\times10^{-9}$ at the energy difference of $E_{X-ray}/E_{radio}\sim10^{9}$, $\gamma_{radio}-\gamma_{optical}<2.63\times10^{-9}$ at $E_{optical}/E_{radio}\sim10^{5}$, and $\gamma_{optical}-\gamma_{\gamma-ray}<3.03\times10^{-10}$ at $E_{\gamma-ray}/E_{optical}\sim10^8$. This actually measures the arrival times of freely-falling photons in the gravitational field of the Milky Way with the largest amount of events and with data of the highest S/N, which tests WEP at the energy band differences that has never been reached before.