Test of Weak Equivalence Principle with the Multi-band Timing of the Crab Pulsar

Abstract

Weak Equivalent Principle (WEP) can be tested through the parameterized post-Newtonian parameter γ, representing the space curvature produced by unit rest mass. The parameter γ 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 γ discrepancy set by such an extensively-observed and well-modeled source is as follows: γ radio − γ γ−ray < 3.28 × 10 −9 at the energy difference of E γ−ray /E radio ∼ 10 13 , γ radio − γ X−ray < 4.01 × 10 −9 at the energy difference of E X−ray /E radio ∼ 10 9 , γ radio − γ optical < 2.63 × 10 −9 at E optical /E radio ∼ 10 5 , and γ optical − γ γ−ray < 3.03 × 10 −10 at E γ−ray /E optical ∼ 10 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.

Shortcomings of Shapiro delay-based tests of the equivalence principle on cosmological scales

“…Here we show that the lattice sums in Eqs. (21) and (24) converge when summed on expanding cubes or spheres (centered at the origin). We first note that these series are not absolutely convergent, since the magnitudes of their terms do not fall off faster than β −3 as β → ∞.…”

Testing the Weak Equivalence Principle Using Optical and Near-infrared Crab Pulses

“…Due to our choice of wavelength bands, this upper limit lies in a region of parameter space which makes it complementary to similar photonic tests of WEP violation (Wei et al 2015;Tingay and Kaplan 2016;Zhang and Gong 2017). We summarize 3σ upper limits from a variety of recent astrophysical tests in Figure 3.…”

“…To translate our observed time delays into a measure of WEP violation, we follow recent works (Longo 1988;Wei et al 2015;Nusser 2016;Tingay and Kaplan 2016;Zhang and Gong 2017) in conservatively assuming that the observed delay ∆t is larger in magnitude than any delay due to a WEP violation. We note that this crucial assumption admits the possibility that a large WEP violation delay is being hidden by an almost equal intrinsic pulse delay in the opposite direction.…”

“…Finally, to calculate ∆γ, we employ a model of the gravitational potential U (r) experienced by the pulses as they travel across the Milky Way. We directly employ the model and parameters of Zhang and Gong (2017) who formulated it to derive similar limits for the Crab pulsar. The potential U (r) is modeled with two components, a Miyamoto-Nagai disc (Miyamoto and Nagai 1975) and a Navarro-Frenk-White (Navarro et al 1996) dark matter halo, and it is integrated from the Crab pulsar to Earth.…”

Cosmic transients, Einstein’s Equivalence Principle and dark matter halos

“…photons and neutrinos) or two (same) particles with different energies into account (therefore with different γ 1 and γ 2 ), it is possible to constrain EEP since the theories of gravity that are built in agreement with it predict γ 1 = γ 2 . The accuracy of the EEP has been tested through the difference of the γ values, ∆γ ≡ γ 1 − γ 2 , using several astronomical sources: supernova 1987A [11,12], gammaray bursts (GRBs) [10,[13][14][15][16], fast radio bursts (FRBs) [15,[17][18][19], the Crab pulsar [20][21][22], blazars [23,24] and gravitational wave (GW) sources [25][26][27][28]. Very recently, the association of GW and GRB observations was used to provide a new constraint on the ∆γ through the time delay between the gravitational and electromagnetic radiation produced by a binary neutron star merger [29].…”