Large-scale structure with gravitational waves. I. Galaxy clustering

Observed angular positions and redshifts of large-scale structure tracers such as galaxies are affected by gravitational waves through volume distortion and magnification effects. Thus, a gravitational wave background can in principle be probed through clustering statistics of large-scale structures. We calculate the observed angular clustering of galaxies in the presence of a gravitational wave background at linear order including all relativistic effects. For a scale-invariant spectrum of gravitational waves, the effects are most significant at the smallest multipoles ($2\le \ell \le 5$), but typically suppressed by six or more orders of magnitude with respect to scalar contributions for currently allowed amplitudes of the inflationary gravitational wave background. We also discuss the most relevant second-order terms, corresponding to the distortion of tracer correlation functions by gravitational waves. These provide a natural application of the approach recently developed in Schmidt and Jeong {arXiv:1204.3625 [Phys. Rev. D (to be published)]}.

Figure 1

Contributions to the observed galaxy angular power spectrum from inflationary gravitational waves, for a sharp source galaxy redshift of $\tilde{z}=2$, and using the tensor mode power spectrum defined in Sec. 2. The black solid line shows the total contribution, while the colored lines show contributions proportional to line-of-sight integrals of $h_{\parallel }$ (blue dotted), $h_{\parallel }^{\prime }$ (green short-dashed) and ${\nabla }_{\Omega }^2{h}_{\parallel }$ (magenta long-dashed). Here, we have assumed $b_e=2.5$, $Q=1.5$. The black square at $l=2$ indicates the result for $l=2$ if the observer term is neglected (see Sec. 4b).

Figure 2

Contributions to the kernel $F_l(k)$ for $l=2$ (thick) and $l=20$ (thin, only total contribution shown, scaled by ${10}^4$), for a sharp source redshift $\tilde{z}=2$ and the same parameters as in Fig. 1. Note that the separate contributions have nonzero weight for $k\to 0$, while the total $F_l^g$ (black solid) is only nonzero for $k\gtrsim {10}^{-4}h/\mathrm{Mpc}$, as required by causality.

Figure 3

Contributions of tensor modes to the angular galaxy power spectrum for a broad redshift distribution [Eq. (59)] expected for LSST, separated into different contributions as in Fig. 1 (again using $b_e=2.5$, $Q=1.5$). The thick lines show the exact calculation, while the thin lines show the Limber approximation using Eq. (61).

Figure 4

Total tensor mode contribution to the angular galaxy power spectrum as a function of source redshift, for a Gaussian redshift distribution centered on $\tilde{z}$ with an RMS width of $0.03(1+\tilde{z})$. Here, $b_e=2.5$, $Q=1.5$ fixed.

Figure 5

Relative impact on the total tensor mode contribution to galaxy clustering when changing $b_e$ from 2.5 to 4 and 1 (thick and thin green long-dashed, respectively), and when changing $\mathcal{Q}$ from 1.5 to 3 and 0 (thick and thin red dot-dashed, respectively). Here, a Gaussian redshift distribution centered on $\tilde{z}=2$ with RMS width $0.03(1+\tilde{z})$ was assumed.

Figure 6

Comparison between tensor mode contributions to the galaxy power spectrum (black solid) for $\tilde{z}=2$ (cf. Fig. 4), and scalar contributions for a linear bias $b=2$ (see Appendix  for details on the calculation of the scalar contributions).

Figure 7

Comparison between tensor mode and scalar contributions to the angular cross correlation between two widely separated redshift bins ($\tilde{z}=1$, ${\tilde{z}}^{\prime }=4$) and for $\mathcal{Q}=1$ so that most magnification contributions drop out ($\mathrm{\text{linear bias}}=2$; other parameters as in Fig. 4).