Lensing Effects on Gravitational Waves in a Clumpy Universe: Effects of Inhomogeneity on the Distance‐Redshift Relation

The distance-redshift relation determined by means of gravitational waves inthe clumpy universe is simulated numerically by taking into account the effectsof gravitational lensing. It is assumed that all of the matter in the universetakes the form of randomly distributed point masses, each of which has theidentical mass $M_L$. Calculations are carried out in two extreme cases:$\lambda\gg GM_L/c^2$ and $\lambda\ll GM_L/c^2$, where $\lambda$ denotes thewavelength of gravitational waves. In the first case, the distance-redshiftrelation for the fully homogeneous and isotropic universe is reproduced with asmall distance dispersion, whereas in the second case, the distance dispersionis larger. This result suggests that we might obtain information about thetypical mass of lens objects through the distance-redshift relation gleanedthrough observation of gravitational waves of various wavelengths. In thispaper, we show how to set limitations on the mass $M_L$ through the observationof gravitational waves in the clumpy universe model described above.Comment: 35 pages, 21 figures, ApJ accepted versio