Cosmology in the non-linear regime : the small scale miracle
Fabien Lacasa ⋆1, 2
1: Institut d’Astrophysique Spatiale (IAS), Bâtiment 121, F-91405 Orsay, Université Paris-Sud 11 and CNRS, UMR 8617, France
2: Département de Physique Théorique and Center for Astroparticle Physics, Université de Genève, 24 quai Ernest Ansermet, CH-
1211 Geneva, Switzerland
February 14, 2022
Interest rises to exploit the full shape information of the galaxy power spectrum, as well as pushing analyses to smaller non-linear scales. Here I use the halo model to quantify the information content in the tomographic angular power spectrum of galaxies Cgal(iz), l for future high resolution surveys : Euclid and SKA2.
I study how this information varies as a function of the scale cut applied, either with angular cut lmax or physical cut kmax. For this, I use analytical covariances with the most complete census of non-Gaussian terms, which proves critical. I find that the Fisher information on most cosmological and astrophysical parameters follows a striking behaviour. Beyond the perturbative regime we first get decreasing returns : the information keeps rising but the slope slows down until reaching a saturation. The location of this plateau is a bit beyond the reach of current modeling methods : k ∼ 2 Mpc−1 and slightly depends on the parameter and redshift bin considered. I explain the origin of this plateau, which is due to non-linear effects both on the power spectrum, and more importantly on non-Gaussian covariance terms. Then, pushing further we see the information rising again in the highly non-linear regime, with a steep slope. This is the small scale miracle, for which I give interpretation and discuss the properties. Hints are shown that this information should be disentanglable from the astrophysical content, and could improve Dark Energy constraints. Finally, more hints are shown that high order statistics may yield significant improvements over the power spectrum in this regime, with the improvements increasing with kmax. Data and notebooks reproducing all plots and results will be made available at
https://github.com/fabienlacasa/SmallScaleMiracle
Future surveys of the large scale structure such as Euclid (Lau- reijs et al. 2011), LSST (Abell et al. 2009) and SKA (Maartens et al. 2015) will allow high resolution mapping of the distribu- tion of galaxies in the Universe. Exploiting the most out of these data sets would require to (i) use the full shape of the statisti- cal measurements, in contrast for instance with targeted BAO extraction, and (ii) push the analyses to the smallest accessible scales.
Full shape information of the galaxy power spectrum can in- deed extract information from faint features (e.g. Tansella 2018) but also from the general slope (to constrain nS ), and from the amplitude (to constrain σ8 and the growth of structure) if used in conjunction with weak lensing or higher order correlation (Hoff- mann et al. 2015). This has been shown to encode more con- straining power than usual BAO and RSD analyses (Loureiro et al. 2019; Tröster et al. 2019).
Pushing to small scales is challenging for future surveys be- cause they are entering the non-linear regime of structure forma- tion where the physics of the dark matter halos become relevant. There is however a wealth of evidence that the halo properties do encode cosmological information to constrain Dark Energy and Gravity (Balmès et al. 2014; Lopes et al. 2018, 2019; Contigiani et al. 2019; Ryu & Lee 2019). This motivates rising interest to use non-linear scales for cosmological constraints (Lange et al. 2019, e.g.).
See:
https://arxiv.org/pdf/1912.06906.pdf
There has been a revival of interest in the question as to whether the cosmological expansion also proceeds at smaller scales. There is a tendency to reject such an extrapolation by confusing it with the intrinsically un-
observable ”expansion” (let us refer to this as ”pseudo-expansion”) described above. By contrast, the metric of Friedman–Robertson–Walker (FRW) in general relativity is intrinsically dynamic with the increase (decrease) of proper distances correlated with red–shift (blue–shift). It does so on any scale provided the light t ravel time is much longer than the wave period. Thus, the cosmological metric alone does not dictate a scale for expansion and in principle, it could be present at the smallest practical scale as real– a s opposed to pseudo–expansion, and observable in principle.
However, it is reasonable to pose the question as to whether there is a cut–off at which
systems below this scale do not partake of the expansion. It would appear that one would
be hard put to justify a particular scale for the onset of expansion. Thus, in this debate,
we are in agreement with Anderson (1995) that it is most reasonable to assume that the
expansion does indeed proceed at all scales. However, there is a certain ironical quality
attached to the debate in the sense that even if the expansion does actually occur at all
scales, we will show that the effects of the cosmological expansion o n smaller spatial and
temporal scales would be undetectable in general in the for eseeable future and hence
one could just as comfortably hold the view that the expansion occurs strictly on the
cosmological scale.
The question of whether the expansion of the universe affects local systems like
clusters of galaxies or planetary systems was first raised many years ago and has received
continued scrutiny (McVittie 1933; J¨arnefelt 1940, 1942; Pachner 1963; Dicke & Peebles
1964; Callan et al. 1965; Irvine 1 965; Noerdlinger & Petrosian 1971) with the most
recent consideration by Anderson (1995) who extends the question to the stellar scale
and even below this. The recurrent attention paid to this issue indicates that to this
point a definitive answer is still lacking. However, it is our sense that the prevalent
perception is that the physics of systems which are small compared to t he radius of
curvature of the cosmological background is essentially unaffected by the expansion of
the universe.
See:
https://www.researchgate.net/public...f_the_Cosmological_Expansion_on_Local_Systems
See:
https://www.researchgate.net/public...al_expansion_on_local_dynamics_and_kinematics
See:
https://www.intechopen.com/chapters/77754
See:
https://arxiv.org/pdf/1309.3503.pdf
See:
https://css2024.math.cas.cz/
Observations of the local environment have shown that it is characterised by an array of galaxies arranged in structures which are separating according to the Hubble flow. At each point in space Einstein’s Equation ensures that it is only the local density of matter which will control the rate of change of volume of a small element of space around that point, the local matter density changes the local scale factor. Remote massive objects will distort the space element as they approach but will return it to its original state after they have passed, not affecting the local scale factor. The matter density probability distribution at each point can be estimated by considering the density distribution over space, it is peaked strongly at zero. The vast majority of points have zero density which means that the local scale factor will be expanding progressively faster than anywhere where the matter density is positive. It is this mechanism where dense regions expand more slowly than empty regions which is essential to trigger the formation of structure. However the main consequence in the context of the local cosmology is that emptiness is becoming ever more frequent. Using a maximum probability algorithm that the best estimate of the cosmic scale factor must be the most frequent local scale factor sets the cosmic scale factor as that for emptiness.
Frequently, the quantitative effect of the large-scale cosmological expansion on local systems is studied in the light of Newtonian approach, and the General Relativity Theory is neglected. We have obtained the equation of motion for a particle in the field of a mass M embedded in a FLRW universe with scalar factor 𝑎(𝑡) in the light of Post-Newtonian approximation. For distances less than 𝑅!" (all the planets in the Solar System), the relativistic perturbation is greater than the cosmological perturbation, hence, it should be considered in the influence of the cosmological expansion on local systems. The product 𝐻!𝑐 = 6.99𝑥10!!"𝑚𝑠!! (Pioneer anomaly) is obtained and it is due to the combination of the cosmological expansion and the Post-Newtonian approximation. It may lead to solving the mystery.
The new critical radius (𝑅!) at which the acceleration due to the cosmological expansion has the same magnitude as the two-body attraction is given by the general solution of equation of motion, and it is valid for local systems.
Hence, the Post-Newtonian approximation should be considered in the influence of the cosmological expansion on local systems.
Hartmann352
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