On the propagation of gravitational waves in a $\Lambda$ CDM universe

Abstract

We study here how the presence of non-zero matter density and a cosmological constant could affect the observation of gravitational waves in pulsar timing arrays. Conventionally, the effect of matter and cosmological constant is included by considering the redshift in frequency due to the expansion. However, there is an additional interesting effect due to the change of coordinates from the ones describing the geometry of the region where waves are produced to the ones used to measure the pulsar timing residuals. This change of coordinates is unavoidable as the strong gravitational field in a black hole merger distorts clocks and rules. Harmonic waves produced in such a merger become anharmonic when detected by a cosmological observer. The effect is tiny but appears to be nevertheless observable for the type of gravitational waves to which pulsar timing arrays are sensitive and for the favoured values of the cosmological parameters.

A nontrivial footprint of standard cosmology in the future observations of low-frequency gravitational waves

“…These results agree with the previous works [34][35][36], and they show that, regardless the equation of state, the first term in the expansion is always at order √ ρ 0 . Using this fact, we can expand the First Friedmann equation at first…”

“…Thus, the main objective of this research is to find the coordinate transformation between {t, r} and {T, R} and thus, taking advantage of the principle of covariance and relativity, we will able to describe the propagation of gravitational waves as seen by a cosmological observer. For a compendium with all the results found before the writing of this paper, including the coordinate transformations of a de Sitter Universe, besides other implications, see [36].…”

“…Note that there is still gauge freedom, but we can completely fix it using the Transverse-Traceless gauge, in which hµν = h µν . Therefore, the metric perturbation can be decomposed [36] as…”

“…If the source is located at a large (but non cosmological) distance, the perturbation will be described approximately by a combination of harmonic functions [35,36]…”

“…Furthermore, in those papers was proposed that the magnitude of residual time in a PTA experiment could change due to the action of the cosmological constant. A wide explanation and a review of this phenomenon can be found in the work of Alfaro et al [36], where the action of non-relativistic matter was included into the phenomenon, showing that Dark Matter increases the effect of Λ on the propagation of gravitational waves. In that sense, we expect that each cosmological component of the Universe affects the propagation of GWs similar to the case of Λ, reason of why we will follow the idea of using the Hubble constant as the main control parameter.…”