Vector and tensor contributions to the luminosity distance

We compute the vector and tensor contributions to the luminosity distance fluctuations in first order perturbation theory, and we expand them in spherical harmonics. This work presents the formalism with a first application to a stochastic background of primordial gravitational waves.

Figure 1

We show the tensor power spectrum rescaled by ${\ell }^4$ for the fluctuations in the luminosity distance for different values of the source redshift ($z=0.5$, dotted pink line; $z=1$, dot-dashed blue line; $z=2$, dashed green line; $z=3$, long-dashed orange line; and $z=4$, solid red line). In the figure we have set $r=1$.

Figure 2

We show the tensor power spectrum rescaled by ${\ell }^4$ for the fluctuations in the luminosity distance as a function of the source redshift for different values $\ell$ ($\ell =60$, dotted blue line; $\ell =40$, dot-dashed green line; $\ell =20$, dashed orange line; and $\ell =10$, solid red line). Also here $r=1$.

Figure 3

We show the contributions from the different terms of Eq. (27) to the tensor power spectrum rescaled by ${\ell }^4$ for the fluctuations in the luminosity distance as a function of the source redshift for $\ell =40$. (First term: dotted red line; second term: dot-dashed brown line; third term: dashed blue line; and correlator of the first and second terms: solid green line). The third term is always subdominant. We plot the correlator between the first and the second terms (solid line) with the opposite sign since it is negative. The second and the first terms have opposite signs, and they cross at $z_c\simeq 1.65$.