3.4 Primordial Non-Gaussianity and Large-Scale Structure

As we have seen, even the simplest inflationary models predict deviations from Gaussian initial conditions. Confirming or ruling out the simplest inflationary model is an important goal and in this section we will show how Euclid can help achieving this. Moreover, Euclid data (alone or in combination with CMB experiments like Planck) can be used to explore the primordial bispectrum and thus explore the interaction of the fields during inflation.

3.4.1 Constraining primordial non-Gaussianity and gravity from 3-point statistics

Contrary to CMB research which mainly probes the high-redshift universe, current studies of the LSS focus on data at much lower redshifts and are more heavily influenced by cosmic evolution. Even if the initial conditions were Gaussian, nonlinear evolution due to gravitational instability generates a non-zero bispectrum for the matter distribution. The first non-vanishing term in perturbation theory (e.g., [218]) gives

B (⃗k1,⃗k2,⃗k3) = 2(P(k1)P (k2)J(⃗k1,⃗k2) + cyclicpermutations ) (3.4.1 )
where ⃗ ⃗ J(k1,k2) is the gravitational instability “kernel” which depends very weakly on cosmology and for an Einstein-de-Sitter universe assumes the form:
( ) ( )2 J(⃗k ,⃗k ) = 5+ ⃗k1-⋅⃗k2 k1-+ k2- + 2- ⃗k1-⋅⃗k2 . (3.4.2 ) 1 2 7 2k1k2 k2 k1 7 k1k2
This kernel represents the “signature” of gravity as we know it on the large-scale structure of the universe. Either a modification of the gravitational law or the introduction of a coupling between dark matter and another component (say dark energy) would alter the bispectrum shape from the standard form. The volume covered by Euclid will enable us to exploit this.

It was recognized a decade ago [924Jump To The Next Citation Point] that the contribution to the matter bispectrum generated by gravitational instability is large compared to the fossil signal due to primordial non-Gaussianity and that the primordial signal “redshifts away” compared to the gravitational signal. In fact, primordial non-Gaussianity of the local type would affect the late-time dark matter density bispectrum with a contribution of the form

BfNLlocal(⃗k ,⃗k ,⃗k ,z) = 2(f P(k )P (k )---ℱ-(⃗k1,⃗k2)--- + cyclicpermutations ). (3.4.3 ) 1 2 3 NL 1 2 D (z)∕D (z = 0)
where D (z ) is the linear growth function which in an Einstein–de Sitter universe goes like −1 (1 + z) and
2 ℱ = ---ℳ--(k3)---;ℳ (k) = 2-k-T-(k), (3.4.4 ) ℳ (k1 )ℳ (k2) 3 H20Ωm,0
T (k) denoting the matter transfer function, H0 the Hubble constant and Ωm,0 the matter density parameter. Clearly the contributions due to primordial non-Gaussianity and gravitational instability have different scale and redshift dependence and the two kernel shapes in configuration space are different, thus, making the two components, at least in principle and for high signal-to-noise, separable. This is particularly promising for high-redshift probes of the matter distribution like the 21-cm background which should simultaneously provide competing measures of fNL and a test of the gravitational law [731Jump To The Next Citation Point]. Regrettably, these studies require using a long-wavelength radio telescope above the atmosphere (e.g., on the Moon) and will certainly come well after Euclid.

Galaxy surveys do not observe the dark matter distribution directly. However, dark matter halos are believed to host galaxy formation, and different galaxy types at different redshifts are expected to populate halos in disparate ways [610, 975]. A simple (and approximate) way to account for galaxy biasing is to assume that the overdensity in galaxy counts can be written as a truncated power expansion in terms of the mass overdensity (smoothed on some scale): δg(x) = b1δDM (x) + b2(δ2DM − ⟨δ2DM⟩). The linear and quadratic bias coefficient b1 and b2 are assumed to be scale-independent (although this assumption must break down at some point) but they can vary with redshift and galaxy type. Obviously, a quadratic bias will introduce non-Gaussianity even on an initially Gaussian field. In summary, for local non-Gaussianity and scale-independent quadratic bias we have [924]:

[ ] ℱ (⃗k1,⃗k2) b2(z) B (⃗k1,⃗k2,⃗k3,z) = 2P (k1)P (k2 )b1(z)3 × fNL ---------+ J(⃗k1,⃗k2) + ------ + cyc. (3.4.5 ) D (z) 2b1(z) (3.4.6 )
Before the above expression can be compared against observations, it needs to be further complicated to account for redshift-space distortions and shot noise. Realistic surveys use galaxy redshifts as a proxy for distance, but gravitationally-induced peculiar velocities distort the redshift-space galaxy distribution. At the same time, the discrete nature of galaxies gives rise to corrections that should be added to the bispectrum computed in the continuous limit. We will not discuss these details here as including redshift space distortions and shot noise will not change the gist of the message.

From the observational point of view, it is important to note that photometric surveys are not well suited for extracting a primordial signal out of the galaxy bispectrum. Although in general they can cover larger volumes than spectroscopic surveys, the projection effects due to the photo-z smearing along the line-of-sight is expected to suppress significantly the sensitivity of the measured bispectrum to the shape of the primordial one (see e.g., [921]). [808] have shown that, if the evolution of the bias parameters is known a priori, spectroscopic surveys like Euclid would be able to give constraints on the fNL parameter that are competitive with CMB studies. While the gravitationally-induced non-Gaussian signal in the bispectrum has been detected to high statistical significance (see, e.g., [925, 533] and references therein), the identification of nonlinear biasing (i.e., b2 ⁄= 0) is still controversial, and there has been so far no detection of any extra (primordial) bispectrum contributions.

Of course, one could also consider higher-order correlations. One of the advantages of considering, e.g., the trispectrum is that, contrary to the bispectrum, it has very weak nonlinear growth [920], but it has the disadvantage that the signal is de-localized: the number of possible configurations grows fast with the dimensionality n of the n-point function!

Finally, it has been proposed to measure the level of primordial non-Gaussianity using Minkowski functionals applied either to the galaxy distribution or the weak lensing maps (see, e.g., [433, 679] and references therein). The potentiality of this approach compared to more traditional methods needs to be further explored in the near future.

3.4.2 Non-Gaussian halo bias

The discussion above neglects an important fact which went unnoticed until year 2008: the presence of small non-Gaussianity can have a large effect on the clustering of dark matter halos [272Jump To The Next Citation Point, 647Jump To The Next Citation Point]. The argument goes as follows. The clustering of the peaks in a Gaussian random field is completely specified by the field power spectrum. Thus, assuming that halos form out of linear density peaks, for Gaussian initial conditions the clustering of the dark matter halos is completely specified by the linear matter power spectrum. On the other hand, for a non-Gaussian field, the clustering of the peaks depends on all higher-order correlations, not just on the power spectrum. Therefore, for non-Gaussian initial conditions, the clustering of dark matter halos depends on the linear bispectrum (and higher-order moments).

One can also understand the effect in the peak-background-split framework: overdense patches of the (linear) universe collapse to form dark matter halos if their overdensity lies above a critical collapse threshold. Short-wavelength modes define the overdense patches while the long-wavelength modes determine the spatial distribution of the collapsing ones by modulating their height above and below the critical threshold. In the Gaussian case, long- and short-wavelength modes are uncorrelated, yielding the well known linear, scale-independent peak bias. In the non-Gaussian case, however, long and short wavelength modes are coupled, yielding a different spatial pattern of regions that cross the collapse threshold.

In particular, for primordial non-Gaussianity of the local type, the net effect is that the halo distribution on very large scales relates to the underlying dark matter in a strongly scale-dependent fashion. For k ≲ 0.02h Mpc −1, the effective linear bias parameter scales as k −2. [272, 647, 392Jump To The Next Citation Point]. This is because the halo overdensity depends not only on the underlying matter density but also on the value of the auxiliary Gaussian potential ϕ [392Jump To The Next Citation Point].

View Image

Figure 47: For illustration purposes this is the effect of a local fNL of ±50 on the z = 0 power spectrum of halos with mass above 1013M ⊙.

The presence of this effect is extremely important for observational studies as it allows to detect primordial non-Gaussianity from 2-point statistics of the galaxy distribution like the power spectrum. Combining current LSS data gives constraints on fNL which are comparable to the CMB ones [842, 971]. Similarly, planned galaxy surveys are expected to progressively improve upon existing limits [210Jump To The Next Citation Point, 209Jump To The Next Citation Point, 393Jump To The Next Citation Point]. For example, Euclid could reach an error on fNL of ∼ 5 (see below for further details) which is comparable with the BPol forecast errors.

The scale dependence of the halo bias changes considering different shapes of primordial non-Gaussianity [799Jump To The Next Citation Point, 933Jump To The Next Citation Point]. For instance, orthogonal and folded models produce an effective bias that scales as k− 1 while the scale dependence becomes extremely weak for equilateral models. Therefore, measurements of the galaxy power spectrum on the largest possible scales have the possibility to constrain the shape and the amplitude of primordial non-Gaussianity and thus shed new light on the dynamics of inflation.

On scales comparable with the Hubble radius, matter and halo clustering are affected by general-relativity effects: the Poisson equation gets a quadratic correction that acts effectively as a non-zero local fNL [94, 731]. This contribution is peculiar to the inflationary initial conditions because it requires perturbations on super-horizon scales and it is mimicked in the halo bias by a local fNL = − 1.6 [922]. This is at the level of detectability by a survey like Euclid.

3.4.3 Number counts of nonlinear structures

Even a small deviation from Gaussianity in the initial conditions can have a strong impact on those statistics which probe the tails of the linear density distribution. This is the case for the abundance of the most extreme nonlinear objects existing at a given cosmic epoch, massive dark matter halos and voids, as they correspond to the highest and lowest density peaks (the rarest events) in the underlying linear density field.

Thus small values of fNL are potentially detectable by measuring the abundance of massive dark matter halos as traced by galaxies and galaxy clusters at z ≳ 1 [649]. This approach has recently received renewed attention (e.g., [591, 407, 730Jump To The Next Citation Point, 609, 275, 918, 732] and references therein) and might represent a promising tool for Euclid science. In Euclid, galaxy clusters at high redshift can be identified either by lensing studies or by building group catalogs based on the spectroscopic and photometric galaxy data. The main challenge here is to determine the corresponding halo mass with sufficient accuracy to allow comparison with the theoretical models.

While galaxy clusters form at the highest overdensities of the primordial density field and probe the high-density tail of the PDF, voids form in the low-density regions and thus probe the low-density tail of the PDF. Most of the volume of the evolved universe is underdense, so it seems interesting to pay attention to the distribution of underdense regions. For the derivation of the non-Gaussian void probability function one proceeds in parallel to the treatment for halos with the only subtlety that the critical threshold is not negative and that its numerical value depends on the precise definition of a void (and may depend on the observables used to find voids), e.g., [490]. Note that while a positive skewness (fNL > 0) boosts the number of halos at the high mass end (and slightly suppress the number of low-mass halos), it is a negative skewness that will increase the voids size distribution at the largest voids end (and slightly decrease it for small void sizes). In addition voids may probe slightly larger scales than halos, making the two approaches highly complementary.

Even though a number of observational techniques to detect voids in galaxy surveys have been proposed (see, e.g., [247] and references therein), the challenge here is to match the theoretical predictions to a particular void-identification criterion based on a specific galaxy sample. We envision that mock galaxy catalogs based on numerical simulations will be employed to calibrate these studies for Euclid.

3.4.4 Forecasts for Euclid

A number of authors have used the Fisher-matrix formalism to explore the potentiality of Euclid in determining the level and the shape of primordial non-Gaussianity [210Jump To The Next Citation Point, 209Jump To The Next Citation Point, 393Jump To The Next Citation Point]. In what follows, unless specifically mentioned, we will focus on the local type of non-Gaussianity which has been more widely studied so far.

The most promising avenue is exploiting the scale-dependent bias on very large scales in studies of galaxy clustering at the two-point level. Early Fisher forecasts for the Euclid redshift survey found that, for a fiducial model with fNL = 0, this gives a marginalized 1σ error on the nonlinearity parameter of ΔfNL ≃ 2 [210, 209]. Forecasts based on the most recent specifics for the Euclid surveys (see Table 21) are presented in [393Jump To The Next Citation Point] and summarized in Table 22 below. Updated values of the galaxy number counts and of the efficiency in measuring spectroscopic redshifts correspond to a marginalized 1σ error of f ≃ 4– 5 NL (depending a little on the detailed assumptions of the Fisher matrix calculation), with a slightly better result obtained using the Euclid spectroscopic sample rather than the photometric one (complemented with multi-band ground-based photometry), at least for a fiducial value of fNL = 0 [393Jump To The Next Citation Point]. The forecast errors further improve by nearly a few per cent using Planck priors on the cosmological parameters determined with the power spectrum of CMB temperature anisotropies.


Table 21: Specifications of the surveys used in the Euclid forecasts given in Table 22. The redshift distributions of the different galaxy samples are as in Section 1.8.2 (see also [393Jump To The Next Citation Point]).
  Photometric survey Spectroscopic survey
Surveyed area (deg2) 15,000 15,000
Galaxy density (arcmin−2) 30 1.2
Median redshift 0.8 1.0
Number of redshift bins 12 12
Redshift uncertainty σz∕(1+ z) 0.05 0.001
Intrinsic ellipticity noise γ - 0.247
Gaussian linear bias param. √----- 1 + z √----- 1 + z

The amplitude and shape of the matter power spectrum in the mildly nonlinear regime depend (at a level of a few per cent) on the level of primordial non-Gaussianity [877, 730, 392]. Measuring this signal with the Euclid weak-lensing survey gives ΔfNL ≃ 70 (30 with Planck priors) [393Jump To The Next Citation Point]. On the other hand, counting nonlinear structures in terms of peaks in the weak-lensing maps (convergence or shear) should give limits in the same ballpark ([627] find ΔfNL = 13 assuming perfect knowledge of all the cosmological parameters).

Finally, by combining lensing and angular power spectra (and accounting for all possible cross-correlations) one should achieve ΔfNL ≃ 5 (4.5 with Planck priors) [393Jump To The Next Citation Point]. This matches what is expected from both the Planck mission and the proposed BPol satellite.

Note that the forecast errors on fNL are somewhat sensitive to the assumed fiducial values of the galaxy bias. In our study we have adopted the approximation √ ------ b(z) = 1 + z [753]. On the other hand, using semi-analytic models of galaxy formation, [698] found bias values which are nearly 10 – 15% lower at all redshifts. Adopting this slightly different bias, the constraint on fNL already degrades by 50% with respect to our fiducial case.

Euclid data can also be used to constrain the scale dependence of the nonlinearity parameter (see Table 23). To this purpose, we consider a local model of primordial non-Gaussianity where

( )nf f = f(piv)⋅ -k-- NL , (3.4.7 ) NL NL kpiv
with fiducial values −1 kpiv = 0.02h Mpc, (piv) fNL = 30, and nfNL = 0. In this case, the combination of lensing and clustering data gives Δ αs,m = 0.18 (0.14 with Planck priors) and Δf (NpLiv)≃ 9 (7 with Planck priors) [393Jump To The Next Citation Point]. These constraints are similar to what is expected from future studies of the CMB bispectrum with Planck [809].


Table 22: Forecast 1σ errors for the nonlinearity parameter fNL based on two-point statistics (power spectra) of the Euclid redshift and weak-lensing surveys. Results are obtained using the Fisher-matrix formalism and marginalizing over eight cosmological parameters (Ω Λ, Ωm, Ωb, h, ns, σ8, w0, wa) plus a large number of nuisance parameters to account for galaxy biasing, nonlinear redshift-space distortions and shot noise (see [393Jump To The Next Citation Point] for details). Results within parentheses include the forecast priors for the cosmological parameters from the power spectrum of CMB temperature anisotropies measured with the Planck satellite (note that no prior is assumed on fNL). The label “Galaxy clustering” refers to the anisotropic power spectrum P (k∥,k⊥) for spectroscopic data and to the angular power spectrum C ℓ for photometric data. The combined analysis of clustering and lensing data is based on angular power spectra and includes all possible cross-correlations between different redshift bins and probes. nonlinear power spectra are computed using the halo model. This introduces possible inaccuracies in the forecasts for weak lensing data in the equilateral and orthogonal shapes (see main text for details).
Bispectrum shape local orthogonal equilateral
Fiducial fNL 0 0 0
Galaxy clustering (spectr. z) 4.1 (4.0) 54 (11) 220 (35)
Galaxy clustering (photom. z) 5.8 (5.5) 38 (9.6) 140 (37)
Weak lensing 73 (27) 9.6 (3.5) 34 (13)
Combined 4.7 (4.5) 4.0 (2.2) 16 (7.5)


Table 23: Forecast 1σ errors for a scale-dependent local model of primordial non-Gaussianity [393Jump To The Next Citation Point]. Details of the forecasts are as in the previous Table 22.
Δf (pNiLv) ΔnfNL
Galaxy clustering (spectr. z) 9.3 (7.2) 0.28 (0.21)
Galaxy clustering (photom. z) 25 (18) 0.38 (0.26)
Weak lensing 134 (82) 0.66 (0.59)
Combined 8.9 (7.4) 0.18 (0.14)

In the end, we briefly comment on how well Euclid data could constrain the amplitude of alternative forms of primordial non-Gaussianity than the local one. In particular, we consider the equilateral and orthogonal shapes introduced in Section 3.3.2. Table 22 summarizes the resulting constraints on the amplitude of the primordial bispectrum, fNL. The forecast errors from galaxy clustering grow larger and larger when one moves from the local to the orthogonal and finally to the equilateral model. This reflects the fact that the scale-dependent part of the galaxy bias for k → 0 approximately scales as k −2, k −1, and k0 for the local, orthogonal, and equilateral shapes, respectively [799, 933, 802, 305, 306]. On the other hand, the lensing constraints (that, in this case, come from the very nonlinear scales) appear to get much stronger for the non-local shapes. A note of caution is in order here. In [393Jump To The Next Citation Point], the nonlinear matter power spectrum is computed using a halo model which has been tested against N-body simulations only for non-Gaussianity of the local type.17 In consequence, the weak-lensing forecasts might be less reliable than in the local case (see the detailed discussion in [393]). This does not apply for the forecasts based on galaxy clustering which are always robust as they are based on the scale dependence of the galaxy bias on very large scales.

3.4.5 Complementarity

The CMB bispectrum is very sensitive to the shape of non-Gaussianity; halo bias and mass function, the most promising approaches to constrain fNL with a survey like Euclid, are much less sensitive. However, it is the complementarity between CMB and LSS that matters. One could envision different scenarios. If non-Gaussianity is local with negative fNL and CMB obtains a detection, then the halo bias approach should also give a high-significance detection (GR correction and primordial contributions add up), while if it is local but with positive fNL, the halo-bias approach could give a lower statistical significance as the GR correction contribution has the opposite sign. If CMB detects f NL at the level of 10 and a form that is close to local, but halo bias does not detect it, then the CMB bispectrum is given by secondary effects (e.g., [620]). If CMB detects non-Gaussianity that is not of the local type, then halo bias can help discriminate between equilateral and enfolded shapes: if halo bias sees a signal, it indicates the enfolded type, and if halo bias does not see a signal, it indicates the equilateral type. Thus even a non-detection of the halo-bias effect, in combination with CMB constraints, can have an important discriminative power.


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