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., ) gives
It was recognized a decade ago  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
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): . The linear and quadratic bias coefficient and 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 :
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., ).  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 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., ) 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 , but it has the disadvantage that the signal is de-localized: the number of possible configurations grows fast with the dimensionality of the -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.
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 [272, 647]. 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 , the effective linear bias parameter scales as . [272, 647, 392]. This is because the halo overdensity depends not only on the underlying matter density but also on the value of the auxiliary Gaussian potential .
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 which are comparable to the CMB ones [842, 971]. Similarly, planned galaxy surveys are expected to progressively improve upon existing limits [210, 209, 393]. For example, Euclid could reach an error on of (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 [799, 933]. For instance, orthogonal and folded models produce an effective bias that scales as 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 [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 . This is at the level of detectability by a survey like Euclid.
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 are potentially detectable by measuring the abundance of massive dark matter halos as traced by galaxies and galaxy clusters at . This approach has recently received renewed attention (e.g., [591, 407, 730, 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., . Note that while a positive skewness () 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.,  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.
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 [210, 209, 393]. 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 , this gives a marginalized error on the nonlinearity parameter of [210, 209]. Forecasts based on the most recent specifics for the Euclid surveys (see Table 21) are presented in  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 error of (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 . 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.
|Photometric survey||Spectroscopic survey|
|Surveyed area (deg)||15,000||15,000|
|Galaxy density (arcmin)||30||1.2|
|Number of redshift bins||12||12|
|Intrinsic ellipticity noise||-||0.247|
|Gaussian linear bias param.|
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 (30 with Planck priors) . 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 ( find 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 (4.5 with Planck priors) . This matches what is expected from both the Planck mission and the proposed BPol satellite.
Note that the forecast errors on are somewhat sensitive to the assumed fiducial values of the galaxy bias. In our study we have adopted the approximation . On the other hand, using semi-analytic models of galaxy formation,  found bias values which are nearly 10 – 15% lower at all redshifts. Adopting this slightly different bias, the constraint on 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
|Galaxy clustering (spectr. )||4.1 (4.0)||54 (11)||220 (35)|
|Galaxy clustering (photom. )||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)|
|Galaxy clustering (spectr. )||9.3 (7.2)||0.28 (0.21)|
|Galaxy clustering (photom. )||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, . 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 approximately scales as , , and 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 , the nonlinear matter power spectrum is computed using a halo model which has been tested against -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 ). 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.
The CMB bispectrum is very sensitive to the shape of non-Gaussianity; halo bias and mass function, the most promising approaches to constrain 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 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 , the halo-bias approach could give a lower statistical significance as the GR correction contribution has the opposite sign. If CMB detects 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., ). 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.
Living Rev. Relativity 16, (2013), 6
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