General relativity has predicted the existence of gravitational waves (GW), which are waves of the distortions of space-time with two degrees of polarization and the propagation speed of light. Alternative theories predict more polarizations, up to a maximum of six, and possible deviation of propagation speed from the light speed. The present paper reviews recent proposals to test the gravity theories in the radiation regime by observing GWs using pulsar timing arrays.
Deep Dive into Testing Gravity Theories In The Radiative Regime Using Pulsar Timing Arrays.
General relativity has predicted the existence of gravitational waves (GW), which are waves of the distortions of space-time with two degrees of polarization and the propagation speed of light. Alternative theories predict more polarizations, up to a maximum of six, and possible deviation of propagation speed from the light speed. The present paper reviews recent proposals to test the gravity theories in the radiation regime by observing GWs using pulsar timing arrays.
Two important characteristics of GWs are important to differentiate the validity of gravity theories in the radiative regime; the polarization and dispersion of GW in vacuum. In alternative metric theories, GW can have up to six possible polarization states, four more then are allowed by GR, Furthermore, the propagation speed of GW can deviate from the predication of GR that GW propagates at light speed in vacuum, i.e. the effective graviton mass is zero.
Pulsar timing array is a unique technique to detect nano-Hertz GWs by timing multiple millisecond pulsars, which are very stable celestial clocks [1]. It turns out that a stochastic GW background leaves an angular-dependent correlation in pulsar timing residuals for widely spaced pulsars [2]: the correlation C(θ ) between timing residual of pulsar pairs is a function of angular separation θ between the pulsars. One can analyse the timing residual and measure such a correlation to detect GWs [1]. Lee et al. (2008Lee et al. ( , 2010) ) have found that the exact form of C(θ ) is very different from the one of GR, if the GW has extra polarization state or graviton mass is not zero. Thus by measuring the correlation function, we can directly test gravity theories in the radiative regime.
A GW introduces extra signal in pulsar timing data. Let the unit vector of the GW propagation direction be êz , GW frequency be f , the direction from the observer to the photon source (pulsar) be ni . The GW induced frequency-shift of a pulsar timing signal is [3,4,5]
where the D is the displacement vector from the observer to the pulsar, h i j (t, 0) and h i j (t -|D|/c, D) are the metric perturbations by GW at the Earth and at the pulsar when the received pulse was emitted, ω is the angular frequency for the pulsar pulse, f cut = m g c 2 /h is the cut-off frequency of GW due to the graviton mass m g , and the k g is the GW wave vector given by [4]
The induced pulsar timing residuals R(t) are given by the temporal integration of above the frequency shift at Earth given above, thus
ω dτ. The spatial metric perturbation h i j (t, r) induced by a stochastic GW background is a superposition of monochromatic GWs with random phases and amplitudes. It is [6]
where Ω is the solid angle, index i, j run from 1 to 3, h P is the amplitude of the GW propagating in the direction of êz per unit solid angle per frequency of polarization state P, and the polarization tensor ε P ab of GWs are given in details in Lee et al. (2008). The superscript P takes value of ‘+, ×’ for the two Einsteinian modes of GW polarization, ‘b’ and ’l’ for the breathing and the longitudinal mode respectively, and ‘sn, se’ for the shear modes.
Such stochastic GW background leaves a correlation between timing residuals of pulsars pairs [2,3]. Such correlation, C(θ ), depends on the angular distance θ between two pulsars as well as on the polarization of GW and graviton mass. Lee et al. (2008) have calculated the pulsar timing correlation function for all the polarization modes of GW. For the Einsteinian modes and for the breathing mode, the cross-correlation function C P (θ ) is independent of earth-pulsar distances and independent of the GW characteristic strain spectrum. In contrast, for the modes that are not purely transverse, the shear and longitudinal modes, the cross correlation functions depend on the specifics of the strain spectra and on the pulsar distribution in distance.
Fig. 1 shows the correlation function according to different classes of GW polarization. Clearly by comparison of these ’theoretical’ correlation curves with observations we can test the polarization state of GWs.
Lee et al. (2010) have calculated the pulsar timing correlation function for a GW background with none-zero-mass graviton. They noted that the pulsar timing crosscorrelation function for a massive GW background depends on the graviton mass, specific power spectra of the GW background, and on the observation schedule. The 5-year and 10-year correlation functions are reproduced in Fig. 2, where the graviton
The label of each curve indicates the corresponding graviton mass in unit of electron-volts (eV). The left panel are the correlation functions for a 5-year bi-weekly observations. The right panel shows correlation functions for 10-year bi-weekly cases. We take α = -2/3 for these results. These correlation are normalized such that the C(0) = 0.5 for two different pulsars.
with the same mass introduces more deviation to the 10-year correlation function than it does to the 5-year one. Intuitively speaking, the necessary conditions for a positive detection of a graviton mass should be: 1. The GW is strong enough such that the GW can be detected; 2. the physical effects of alternative theories should be strong enough to see the deviation from GR. These intuition is confirmed by simulations in [3,4], which show that the high detection rate is achieved only if one has enough pulsar and if the graviton mass is large enough
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