The Study On Polarization Modes Of Gravitational Waves And Their Detection

The successful direct detection of gravitational waves is a key breakthrough in gravity.It not only opens a new window to test gravitational theory in strong field and nonlinear region,but also announces the new era of multi-messenger astronomy.To understand the nature of gravity,we need more precise measurements of the properties of gravitational waves.In General Relativity(GR),gravitational waves propagating with speed of light have plus and cross modes only.But in general metric theory of gravity,there might exist extra polarizations for gravitational waves and the speed might be different from the speed of light.Thus the characteristics of gravitational waves like propagating speed and polarization contents,can be used to test or exclude gravitational theories,and to verify whether GR is true.Different polarizations have distinguishable influence on the relative acceleration between nearby test particles,i.e.these effects are manifest in the geodesic deviation equation.In general metric theory of gravity,because Riemann tensor is gauge invariant in the linear approximation,we may use the geodesic deviation equation to define six basic polarizations: plus mode,cross mode,vector-x mode,vector-y mode,breathing mode and longitudinal mode.Using the geodesic deviation equation,we analyze the polarization contents in various theories of gravity,and propose the concept of single mixed state to resolve the controversy between the polarization states and the number of degrees of freedom.In BransDicke theory,the extra massless scalar field activates breathing mode;As for f(R)theory and Horndeski theory,the massive scalar field produces a mixed state of scalar polarization,which is a combination of breathing mode and longitudinal mode.In addition to the plus mode,cross mode and the mixed scalar mode,there are vector-x mode and vector-y mode in Einstein-aether theory which are produced by the dynamic vector field.These five polarizations have different velocities from the speed of light.There are six propagating degrees of freedom in Te Ve S theory.Along with two tensor modes and two vector modes,there are two mixed scalar modes,which are produced by two scalar fields respectively.Since the speed of the scalar modes in f(R)theory,Einstein-aether theory and some other theories is not equal to the speed of light,we argue that it is not suitable to use Newman-Penrose variables to analyze the polarization contents.Due to the distinguishable response of the detectors to different polarizations,we can use the observed data to differentiate polarizations and then test gravitational theories.In low frequency limit,the responses of Michelson-like interferometers to breathing mode and longitudinal mode are degenerated.However,it is viable to distinguish these two scalar modes for space-based detectors or pulsar timing array.Tianqin detector has distinct orbit and configuration from LISA detector,thus it has different sensitive sky locations for different polarizations.These sensitive regions will change with time,as the result of the detector's motion.Averaged response function reflects the sensitivity of the detector itself,which is useful for quick estimation of the detectability of various sources.We derive the semi-analytic formulas of the averaged response function of space-based detector for six polarizations.With these semi-analytic formulas,we can effectively reduce the time of calculation,comparing with numerical calculation.Besides,it is convenient to use the semi-analytic formulas to investigate the asymptotic behavior of the response function in low and high frequency limit.We also calculate the response function of pulsar timing array to different polarizations.Taking Einstein-aether theory as an example,we show that we can get the cross-correlation curve after observing a large amount of pulsars,and then set constraints on the polarizations and their speeds.With the increase number of observed gravitational wave events,it becomes possible to use these observed data to test gravitational theory.Residual test is one of important methods to verify GR directly.We obtain the residual by subtracting the GR-based waveform from the observed data.If the waveform model is a good representation of gravitational waves signal,then we expect that the residual is indistinguishable from the noise.Otherwise,the inconsistency between the residual and the noise might suggest the existence of new physics beyond GR.Having the amplitude in time domain and the Q-transformed energy of the residual and the noise model,we use chi-squared test to test if the residual is consistent with the noise.At the same time,we also calculate the Pearson correlation coefficients of data from two LIGO detectors to test the residuals.We find that all the signals in GWTC-1 are well described by the GR-based waveform.In other words,there is no significant evidence of the existence of extra polarizaitons.Furthermore,we can combine the data from the detector networks to construct the null stream,where all the tensor polarization contents vanish.Comparing the null stream with the noise-only data,we can check whether there is any extra polarization.