Galactic Shapiro Delay to the Crab Pulsar and limit on Einsteins Equivalence Principle Violation

We calculate the total galactic Shapiro delay to the Crab pulsar by including the contributions from the dark matter as well as baryonic matter along the line of sight. The total delay due to dark matter potential is about 3.4 days. For baryonic matter, we included the contributions from both the bulge and the disk, which are approximately 0.12 and 0.32 days respectively. The total delay from all the matter distribution is therefore 3.84 days. We also calculate the limit on violations of Einsteins equivalence principle by using observations of nano-shot giant pulses from the Crab pulsar with time-delay $<0.4$~ns as well as using time differences between radio and optical photons observed from this pulsar. Using the former, we obtain a limit on violation of Einsteins equivalence principle in terms of the PPN parameter $Delta gamma < 2.41times 10^{-15}$. From the time-difference between simultaneous optical and radio observations, we get $Delta gamma < 1.54times 10^{-9}$. We also point out differences in our calculation of Shapiro delay and that from two recent papers (arXiv:1612.00717 and arXiv:1608.07657), which used the same observations to obtain a corresponding limit on $Delta gamma$.