Estimation of spectrum and parameters of relic gravitational waves using space-borne interferometers

We present a study of spectrum estimation of relic gravitational waves (RGWs) as a Gaussian stochastic background from output signals of future space-borne interferometers, like LISA and ASTROD. As the target of detection. the analytical spectrum of RGWs generated during inflation is described by three parameters: the tensor-scalar ratio, the spectral index and the running index. The Michelson interferometer is shown to have a better sensitivity than Sagnac and symmetrized Sagnac. For RGW detection, we analyze the auto-correlated signals for a single interferometer, and the cross-correlated, integrated as well as un-integrated signals for a pair of interferometers, and give the signal-to-noise ratio (SNR) for RGW, and obtain lower limits of the RGW parameters that can be detected. By suppressing noise level, a pair has a sensitivity 2 orders better than a single for one year observation. SNR of LISA will be 4-5 orders higher than that of Advanced LIGO for the default RGW. To estimate the spectrum. we adopt the maximum likelihood (ML) estimation, calculate the mean and covariance of signals, obtain the Gaussian probability density function (PDF) and the likelihood function, and derive expressions for the Fisher matrix and the equation of the ML estimate for the spectrum. The Newton-Raphson method is used to solve the equation by iteration. When the noise is dominantly large. a single USA is not effective for estimating the RGW spectrum as the actual noise in signals is not known accurately. For cross-correlating a pair, the spectrum cannot be estimated from the integrated output signals either, and only one parameter can be estimated with the other two being either fixed or marginalized. We use the ensemble averaging method to estimate the RGW spectrum from the un-integrated output signals. We also adopt a correlation of un-integrated signals to estimate the spectrum and three parameters of RGW in a Bayesian approach. For all three methods, we provide simulations to illustrate their feasibility