Discovering the Dispersion of Gravitational Waves using Multi-Band Observation including Deci-Hertz: A Unique Probe to Cosmic Acceleration

The dispersion in the speed of gravitational waves is a novel way to test the general theory of relativity and understand whether the origin of cosmic acceleration is due to any alternative theory of gravity. Several alternative theories of gravity predict dispersion in the gravitational wave signal in a frequency-dependent deviation from the speed of light at lower frequencies than accessible from current ground-based detectors. We show how a multi-band observation of gravitational wave signal combining deci-Hertz gravitational wave signal from LGWA (Lunar Gravitational Wave Antenna) with ground-based detectors such as Cosmic Explorer or Einstein Telescope, and also including LISA (Laser Interferometer Space Antenna), we can probe the energy scale associated with effective theory of modified gravity scenarios by combining only $\mathcal{O}(10)$ high signal to noise ratio (SNR) with a precision of approximately $8.6\%$. This precision will further improve with the inclusion of more events as $\sqrt{N}$. In the future, this measurement will shed light on an unexplored domain of fundamental physics and will bring deeper insights into the phenomenon of cosmic acceleration. The operation of the gravitational wave detector in the deci-Hertz frequency band is key to exploring this frontier of fundamental physics.