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[1107.5427] Largescale 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 gaugeinvariant expression relating galaxy density
perturbations to matter density perturbations on large scales, and show that it
reduces to a linear bias relation in synchronouscomoving 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 nonGaussian 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 scaledependent bias induced
by primordial nonGaussianity in the relativistic context, which again is
simplest in synchronouscomoving gauge. 

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Dragan Huterer
Joined: 18 Jul 2005 Posts: 24 Affiliation: University of Michigan

Posted: August 01 2011 


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 constantredshift 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 largescale P(k) in nearly the same way as nonzero primordial nonGaussianity of the local type, with additional terms going as k^{ − 2} and k^{ − 4}. Fortunately, the 'effective nonG' induced by GR is (f_{NL})^{eff} 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 highquality data.
One thing I missed has to do with effects of nonzero primordial nonGaussianity at these huge scales. Does this paper implicitly confirm the results of Wands and Slosar 0902.1084 who find that nonGaussianity 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 asusual in the *synchronous* gauge (Eq 104 here), but what does that imply about it in the constantredshift gauge, in which we presumably operate when we utilize spectroscopic LSS surveys? Or is this equation already in the constantredshift gauge?
Finally, given that the results agree with Challinor and Lewis, I presume that CAMBsources already produces P(k) consistent with the one described here? 

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Antony Lewis
Joined: 23 Sep 2004 Posts: 1278 Affiliation: University of Sussex

Posted: August 04 2011 


CAMB, CAMB sources, CMBFAST, CLASS etc all produce the synchronousgauge P(k). But P(k) is not observable directly, what CAMB sources calculates is the directlyobservableinprinciple C_{l}(z,z') of the source number counts. It's just like the CMB really  we don't measure P_{γ}(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 redshiftangle 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 gaugeinvariant 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 gaugeinvariant term that is equivalent to the synchronousgauge density perturbation. 

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Fabian Schmidt
Joined: 02 Nov 2010 Posts: 1 Affiliation: Caltech

Posted: August 05 2011 


Dragan Huterer wrote: 
One thing I missed has to do with effects of nonzero primordial nonGaussianity at these huge scales. Does this paper implicitly confirm the results of Wands and Slosar 0902.1084 who find that nonGaussianity 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 asusual in the *synchronous* gauge (Eq 104 here), but what does that imply about it in the constantredshift gauge, in which we presumably operate when we utilize spectroscopic LSS surveys? Or is this equation already in the constantredshift gauge? 
Hi Dragan,
Basically, the observed galaxy overdensity (δ_{g} in constantredshift gauge, if you want), is given by δ_{g}^{obs} = δ_{g}^{sc} + additional terms. This equation can be written in any gauge one wants, and the left handside will always be the same numerically. Synchronouscomoving gauge (sc) has the advantage that we know what δ_{g}^{sc} looks like in the presence of PNG (on linear scales): it is simply (b + Δ b(k)) δ_{m}^{sc}, where Δ b(k) is the wellknown Dalal et al formula. This is because sigma_8 is the same everywhere on a constanttime 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. 

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Roy Maartens
Joined: 05 Oct 2010 Posts: 3 Affiliation: ICG Portsmouth & Physics, UWC

Posted: August 16 2011 


Hi Dragan
The issue of nongaussianity, and how it may be distinguished from bias on large scales, is covered in 1106.3999
Regards
Roy 

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