New perspective on galaxy clustering as a cosmological probe: General relativistic effects

We present a general relativistic description of galaxy clustering in a Friedmann-Lemaître-Robertson-Walker universe. The observed redshift and position of galaxies are affected by the matter fluctuations and the gravity waves between the source galaxies and the observer, and the volume element constructed by using the observables differs from the physical volume occupied by the observed galaxies. Therefore, the observed galaxy fluctuation field contains additional contributions arising from the distortion in observable quantities and these include tensor contributions as well as numerous scalar contributions. We generalize the linear bias approximation to relate the observed galaxy fluctuation field to the underlying matter distribution in a gauge-invariant way. Our full formalism is essential for the consistency of theoretical predictions. As our first application, we compute the angular auto correlation of large-scale structure and its cross correlation with CMB temperature anisotropies. We comment on the possibility of detecting primordial gravity waves using galaxy clustering and discuss further applications of our formalism.

Figure 1

Power spectra of perturbation variables computed at $z=0$ in the conformal Newtonian and the synchronous gauges. Vertical lines show the comoving line-of-sight distance ($k=1/r(z)$; light gray) at each redshift indicated in the legend and the horizon scale ($k=H_0$; dark gray) today. Near the horizon scale, even power spectra of matter fluctuations in two gauges differ dramatically, showing that gauge effects are substantial and it is nontrivial to relate perturbation variables to observable quantities. Two distinct choices of gauge conditions cannot be used simultaneously around the horizon scale (e.g., Newtonian gauge equations with synchronous gauge transfer function outputs from CMBFast or CAMB).

Figure 2

Systematic errors in theoretical predictions of the auto correlation (left) of the QSO sample and its cross correlation (right) with CMB temperature anisotropies. Attached bottom panels show the mean $\Delta {\chi }^2$ of the measurements, when only the cosmic variance is considered. As in the standard practice, the theoretical predictions of the angular correlations are computed by using ${\delta }_{\mathrm{std}}=b{\delta }_m+(5p-2)\kappa$ with ${\delta }_m$ in the synchronous gauge and $\kappa$ in the conformal Newtonian gauge, and the angular correlations computed with our full expression for ${\delta }_{\mathrm{obs}}$ in Eq. (36) are compared to the predictions with ${\delta }_{\mathrm{std}}$. Projection along the line-of-sight suppresses the large-scale modes, where the matter fluctuations in two gauges in Fig. 1 differ substantially. Note that at $l=2$ the correct theoretical prediction is larger by a factor of 1.8 than that one would incorrectly predict in the standard method, and it is $1.2\mathrm{\text{−}}\sigma$ away from the estimated cosmic variance shown as shaded regions. Since the signal-to-noise ratio of the cross correlation measurements is largest at low angular multipoles (measurements uncertainties are large at $l>10$), the systematic errors in the standard method could bias the inferred cosmology.