## [1107.5427] Large-scale clustering of galaxies in general relativity

 Authors: Donghui Jeong, Fabian Schmidt, Christopher M. Hirata Abstract: Several recent studies have shown how to properly calculate the observed clustering of galaxies in a relativistic context, and uncovered corrections to the Newtonian calculation that become significant on scales near the horizon. Here, we retrace these calculations and show that, on scales approaching the horizon, the observed galaxy power spectrum depends strongly on which gauge is assumed to relate the intrinsic fluctuations in galaxy density to matter perturbations through a linear bias relation. Starting from simple physical assumptions, we derive a gauge-invariant expression relating galaxy density perturbations to matter density perturbations on large scales, and show that it reduces to a linear bias relation in synchronous-comoving gauge, corroborating an assumption made in several recent papers. We evaluate the resulting observed galaxy power spectrum, and show that it leads to corrections similar to an effective non-Gaussian bias corresponding to a local (effective) fNL < 0.5. This number can serve as a guideline as to which surveys need to take into account relativistic effects. We also discuss the scale-dependent bias induced by primordial non-Gaussianity in the relativistic context, which again is simplest in synchronous-comoving gauge. [PDF]  [PS]  [BibTex]  [Bookmark]

Discussion related to specific recent arXiv papers
Dragan Huterer
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### [1107.5427] Large-scale clustering of galaxies in general re

This is a very impressive paper that appears to definitively solve the question of how GR effects affect the observed power spectrum on largest observable scales - a hot topic over the past few years. Well explained derivations and numerous examples are given to confirm the results - and those agree with (equally impressive) analyses by Baldauf et al 1106.5507, Challinor and Lewis 1105.5292, and Bonvin and Durrer 1105.5280.

The results are also in mild disagreement with the original analysis by Yoo et al 0907.0707. The issue seems to be that the linear bias relation is to be assumed in the synchronous gauge, but this then does not correspond to linear bias in the constant-redshift gauge, so appropriate care must be taken.

The bottom line is that the GR effects are small, but not completely negligible. Interestingly, they modify the large-scale P(k) in nearly the same way as nonzero primordial non-Gaussianity of the local type, with additional terms going as $k^{-2}$ and $k^{-4}$. Fortunately, the 'effective non-G' induced by GR is (f_{\rm NL})^{\rm eff}\lesssim 0.5, which is likely to stay well below the statistical error in LSS surveys for another decade. So basically the GR effects are likely to be deep within the noise until your favorite gigantic survey produces high-quality data.

One thing I missed has to do with effects of nonzero primordial non-Gaussianity at these huge scales. Does this paper implicitly confirm the results of Wands and Slosar 0902.1084 who find that non-Gaussianity of the local form does not get additional corrections near the Hubble scale? I am a bit confused, since the NG effect in this paper appears as-usual in the *synchronous* gauge (Eq 104 here), but what does that imply about it in the constant-redshift gauge, in which we presumably operate when we utilize spectroscopic LSS surveys? Or is this equation already in the constant-redshift gauge?

Finally, given that the results agree with Challinor and Lewis, I presume that CAMB-sources already produces P(k) consistent with the one described here?

Antony Lewis
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### Re: [1107.5427] Large-scale clustering of galaxies in genera

CAMB, CAMB sources, CMBFAST, CLASS etc all produce the synchronous-gauge P(k). But P(k) is not observable directly, what CAMB sources calculates is the directly-observable-in-principle C_l(z,z') of the source number counts. It's just like the CMB really - we don't measure P_\gamma(k) in the CMB, what we measure is C_l which includes a several different physical effects (which can't be described accurately solely in terms of local density sources at recombination, e.g. due to the ISW and doppler terms).

So we don't observe the number count P(k) in the constant redshift gauge, what we observe is a combination of the physical source number density and a load of geometric effects relating volumes of physical space to volumes in observed redshift-angle space. Bias relates the physical source density to the physical matter density perturbation, and is expected to be essentially the usual prescription in the synchronous gauge (i.e. all apart from the Yoo papers agree with Wands and Slosar, at least during matter domination and to the extent that the simple bias model works at all). To calculate what you observe you then need to also include all the geometric effects. Since what we observe is gauge invariant, there are lots of different equivalent ways to write it in terms of sums of different gauge-invariant terms, and they should all give the same answer (as this paper shows explicitly by doing it in a different gauge). Where bias enters this sum of terms is multiplying a gauge-invariant term that is equivalent to the synchronous-gauge density perturbation.

Fabian Schmidt
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### Re: [1107.5427] Large-scale clustering of galaxies in genera

Dragan Huterer wrote: One thing I missed has to do with effects of nonzero primordial non-Gaussianity at these huge scales. Does this paper implicitly confirm the results of Wands and Slosar 0902.1084 who find that non-Gaussianity of the local form does not get additional corrections near the Hubble scale? I am a bit confused, since the NG effect in this paper appears as-usual in the *synchronous* gauge (Eq 104 here), but what does that imply about it in the constant-redshift gauge, in which we presumably operate when we utilize spectroscopic LSS surveys? Or is this equation already in the constant-redshift gauge?
Hi Dragan,
Basically, the observed galaxy overdensity (\delta_g in constant-redshift gauge, if you want), is given by \delta_g^{obs} = \delta_g^{sc} + additional terms. This equation can be written in any gauge one wants, and the left hand-side will always be the same numerically. Synchronous-comoving gauge (sc) has the advantage that we know what \delta_g^{sc} looks like in the presence of PNG (on linear scales): it is simply (b + \Delta b(k)) \delta_m^{sc}, where \Delta b(k) is the well-known Dalal et al formula. This is because sigma_8 is the same everywhere on a constant-time slice only in synchronous gauge (in the Gaussian case).

However, some of the additional terms from various volume and redshift distortion effects also have a k^{-2}, and can thus be a confusion for the PNG signal if not accounted for. Luckily, we can account for them quite easily (for example, using CAMBsources) - we only need to know (or fit for) the linear bias and the source evolution.

Roy Maartens
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Affiliation: ICG Portsmouth & Physics, UWC

### [1107.5427] Large-scale clustering of galaxies in general re

Hi Dragan

The issue of non-gaussianity, and how it may be distinguished from bias on large scales, is covered in 1106.3999

Regards

Roy