PHYSICS OF NON-GAUSSIAN FIELDS AND THE COSMOLOGICAL GENUS STATISTIC

To help with the development of these ideas, several prevailing definitions of density thresholds are reconciled. The parameterization of density used in genus measurements is idiosyncratic, but well motivated; it is the simplest mapping independent of the underlying density distribution function. There are two (less preferable) alternatives: for density fields $\delta = \rho /\bar{\rho }- 1$ whose distribution functions f(δ) depart from Gaussian form—as the cosmological density field must over at least some scales—the parameter ν = δ/σ_ρ is a misleading mapping, imposing as it does an artificial symmetry about mean density. The often-mentioned lognormal model for the distribution function (Coles & Jones 1991) improves this insofar as it corresponds to the distribution function over a broader range of scales: using the parameter $\tilde{\delta }= \ln (1+\delta)$, the mapping is given by $\nu = \tilde{\delta }/ \sigma _{\tilde{\delta }}$. Except in highly evolved fields, this mapping is nearly equivalent to the most general parameterization:

$$\begin{equation} \nu \equiv \frac{f_G(\delta)-\bar{f_G}}{\sigma _{f_G}}\hbox{, where } f_G(\delta)\equiv \mathrm{Erf}^{-1}\left[\int _{-\infty }^\delta f(\delta ^{\prime })d\delta ^{\prime }\right]; \end{equation} \tag{ 1 }$$

and it is this variable that represents density in topological statistics. This choice of mapping insists that regions should be judged as comparably overdense and underdense when they occupy the same volume, rather than on the basis of their density.

To return a fair reconstruction of the density field from a density-dependent sampling (i.e., $\bar{n}\propto \rho \Rightarrow b\sim 1$) of N points, it is necessary to smooth the field with a filter exceeding a characteristic length scale of the point distribution. When this condition is not satisfied, the density field is not adequately reconstructed at that scale, and the genus measurement of the smoothed field will not be accurate. There is excellent motivation to understand and correct this effect, as doing so improves the utility of flux-limited samples for the purpose of the genus measurement, offering order-of-magnitude improvements in signal relative to a volume-limited sample. This subsection provides a parametric form for this correction and demonstrates its application to the case of a Gaussian random field with power-law power spectrum.

To examine this systematic, the same Gaussian random field realization is Poisson sampled n_r  =  100 times. The variance associated with the correction to the statistic is not explored in this work. Each sample is smoothed through a Gaussian window of scale parameter λ and the genus curve for the recovered field is measured. The quantity used here to parameterize the distortion present in the genus curve is the ratio of a characteristic length scale in the point distribution to the smoothing length:

$$\begin{equation} \Upsilon \equiv \frac{\lambda }{\bar{\ell }}; \end{equation} \tag{ 8 }$$

the quantity $\bar{\ell }$ can admit a variety of definitions, the choice of which is only important in so far as it be treated consistently. Among these are the radius of the average spherical volume per point; the side length of the average Euclidean volume per point; the empirical average separation between every pair of points; and the average nearest-neighbor separation of a Poisson distribution with the same sampling density; indeed, it seems that the definitions return length scales that are equivalent up to a constant factor. For the purpose of this work, the first of these

$$\begin{equation} \bar{\ell }\equiv \left(\frac{3V}{4\pi N}\right)^{1/3} \end{equation} \tag{ 9 }$$

is chosen.

It is hypothesized that the systematic is a function of ϒ, so that an additive correction factor to the raw genus curve can be defined:

$$\begin{equation} f(\nu ;\Upsilon) = \sum _{n=1}^{n_\mathrm{max}} a_n(\Upsilon)\psi _n(\nu); \end{equation} \tag{ 10 }$$

for a given input power spectrum, the task is to empirically derive the correction coefficients a_n(ϒ), which at best will require tabulation only to very small n_max. This proves to be the case for power spectra that have been tried: the correction function is accurately described once Hermite functions up to fourth order are included, with no significant gains beyond this. Furthermore, above values of ϒ ∼ 1, the correction is in the first Hermite mode at the 95% level, so that a formula can be given in terms of this quantity only. A figure showing the coefficient amplitude a₁ as a function of ϒ is given in Figure 1. Extension to arbitrary ϒ is achieved by interpolation or the use of the function:

$$\begin{equation} a_1(\Upsilon) = \beta _1\frac{1}{\Upsilon } + \beta _0;\quad \frac{a_1(\Upsilon)}{a_2(\Upsilon)} \simeq B_0\frac{1}{\Upsilon }; \end{equation} \tag{ 11 }$$

in the example case presented here, the maximum-likelihood estimates of the parameters are β₁ = −0.0548 and β₀ = 0.0166, and B₀ = 0.13. It is argued here that this formula is robust with respect to the choice of power spectrum, the absolute sampling density, and the resolution of the density field. Considering the form of the genus curve presented in Equation (2), it is well known that the power spectrum of the Gaussian random field enters only in the constant of proportionality. For small departures from Gaussianity, such as those studied by Matsubara (2003), the genus curve is modified by low-order Hermite polynomials, as is the case here, with the coefficients of these polynomials depending on the power spectrum via the skewness parameters. Crucially, because the Hermite mode coefficients a_n are calculated only with respect to the shape of the curve and not to its amplitude, the leading-order influence of the power spectrum is removed.

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The result is that the parameterized form of the correction presented in Equation (11) is applicable to a wide variety of power spectra, though the parameters themselves will need to be tuned in each case. It is foreseeable that an analytic form of these parameters could be derived in the same fashion as the second-order perturbation theory analysis of non-Gaussianity arising from gravitational evolution, though this is beyond the scope of the current exposition; an attempt to demonstrate both the generality of this form and the derive a connection between the parameters and the underlying power spectrum would be a worthy future advance.

It is clear, however, that this correction is not useful at arbitrarily small ϒ. Indeed, while the prevailing rule of thumb that ϒ ∼ 1 is sufficient to ensure a reasonable reconstruction seems to be borne out, there is nevertheless some correction required even then. While it is possible to go somewhat below this, the corrections quickly begin failing to converge; treatment when ϒ < 0.75 seems ill advised—better to deem the field insufficiently reconstructed. Above this limit, however, the correction formula holds to better than 1%. Further work will clarify the degree to which this correction holds for non-Gaussian (e.g., nonlinear) fields, exploring whether the small departures in Figure 1 are a breakdown of this scheme.

An extension of the sampling systematic correction is its application to flux-limited galaxy samples in which the survey selection function is well understood. When the selection function is known, it can be used to upweight those locations in the density field that have been relatively underobserved (for whatever reason), prior to smoothing. Nevertheless, these regions will still have lower sampling rates, which will induce a sampling systematic that varies with location. To take the specific example of redshift, at high redshifts the average pair separation $\bar{\ell }$ will often be relatively higher even with the selection upweighting taken into account. Because the genus is a locally additive quantity (after the reconstruction has taken place), the formula (11) can be used to correct the resulting measurement:

$$\begin{equation} f(\nu ;\Upsilon) = \int _\mathcal {V} a_1\left[\Upsilon (\mathbf {x}) \right]d\mathbf {x}. \end{equation} \tag{ 12 }$$

The improvement in signal to noise that results from using flux-limited samples depends on the specific selection properties of the survey, as the quality of the reconstruction of the density field is proportional to the number of galaxies used in its reconstruction; a volume-limited sample typically contains at most 10% of the total sample, which propagates directly into the resulting genus signal (see, e.g., James et al. 2009). However, the type of sample that should be constructed will ultimately depend on the scientific application: a flux-limited sample will, in general, contain greater variation of the galaxy mass function with redshift, which will impede the interpretation of measurements. The purpose of the study of the sampling systematic is to ensure that, when the selection and galaxy sample properties are well understood, Gaussian smoothing reconstruction can be used to study scales close to the mean inter-object separation.

The edge systematic has been treated in some depth by Melott & Dominik (1993), who pioneered a windowing correction that modifies the reconstructed field before genus measurement, rather than correcting the measurement after it is made. The motivation for the existing method is that power will be pushed through these edges during the reconstruction, so that boundary regions should be expected to be systematically underdense. This method works by creating a field of constant density value with the same geometry as the data, performing the same reconstruction procedure on it and ratio the smoothed data field by the smoothed constant field. This remains the best correction method for this effect, performing well for the current generation of surveys. These authors have noted that even when their correction method is applied, the measured genus curve should be scaled by a factor corresponding to the ratio of the total initial volume to the volume not pushed outside during the smoothing process, which is carried through to the computation of Hermite coefficients. As this affects all coefficients equally, their fractional contributions are unaltered.

Though the detailed specification of this systematic effect is unclear, it is known that a minimum smoothing length exceeding the scale of individual pixels must be applied to any pixelated density field. When this scale of smoothing is not met, the genus measurement is distorted symmetrically about median density in the manner shown in Figure 2. In work by Hamilton et al. (1986), the effect of finite (octahedral) pixel size was developed analytically, though without explicit invocation of a smoothing scale (cf. Equation (21) of that work). This computation shows that the expected expression (Equation (2) in the present work) is recovered in the limit of vanishing pixel size r and that residual terms $\mathcal {O}(r^2)$ are well described by quantities formulated from the higher spectral moments of the power spectrum. Melott et al. (1989) have developed this idea in the context of the two-dimensional genus statistic, but with application to cubical pixels of the kind most commonly employed in genus calculations. This impressive computation, to sixth order in the pixel size, shows clearly how polynomials in ν, such as the Hermite polynomials, carried to progressively higher orders with coefficients expressed in terms of spectral moments, can account for the behavior observed in the large pixel-size regime of genus computations.

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We present here a treatment that complements the formalisms that have been developed previously, both by explicitly mixing consideration of the smoothing scale and by presenting a form of that correction that carries to all orders. For this systematic, the bias induced in the genus curve is not a function of a single Hermite mode, but of the full Hermite spectrum of the measurement. This complicated form (see the right-hand inset in figure) suggests that treatment for this systematic can be achieved by the inclusion of corrections to Hermite coefficients at all orders. The proposed formula for this correction is

$$\begin{equation} a_n = \left\lbrace \begin{array}{@{}rl}0, & n=1,3,5,\ldots \\ (-1)^{n/2}e^{-\gamma n} & n = 2,4,6,\ldots \end{array} \right. ;\qquad \gamma \approx \gamma _1n+\gamma _0, \end{equation} \tag{ 13 }$$

where the maximum-likelihood parameter values are γ₁ = 3.75 and γ₀ = 0.325 for the field presented in this example. The left-hand inset in Figure 2 demonstrates the variation of this systematic for several smoothing lengths (λ = 0, 0.5, and 1.0 pixels), tending toward the delta function δ(n − 2) in the limit λ → ∞. Even when λ exceeds the pixel size, some effect remains. In practice, this systematic is not often an issue because the smoothing scale will naturally exceed the pixellation length by a sufficient factor in order to ensure that the sampling systematic is addressed, but further work can develop a treatment for cases where this is not achievable.

A promising direction in which to extend this result is to consider expressing the coefficients of the correction in terms of the spectral moments of P(k), as has been done in earlier work. While not attempted here, some guidance can be found by comparing the terms of the expansions presented in those previous results to the forms of Hermite polynomials, matching coefficients as polynomials of spectral moments.

4.1.1. Perturbation Theoretic Treatment

An influential formalism for the study of the topology at weakly nonlinear scales has been provided by Matsubara (1994) and developed energetically by Matsubara (2003). In this perturbative scheme, carried out in powers of the variance of the density field, the modes corresponding to n = 1, 3, and 5 appear with greater relative strength at scales below some scale, λ ∼ 10 Mpc h⁻¹ at the present epoch. This scheme is naturally re-expressed in terms of the current work, as a prediction for the spectrum of Hermite coefficients, yielding

$$\begin{equation} g_\mathrm{NL}(\nu ;\lambda) \propto e^{-\nu ^2/4}\sum _{n=1}^5 a_n(\lambda)\psi _n(\nu) +\mathcal {O}(\sigma ^2) \end{equation} \tag{ 14 }$$

with

$$\begin{equation} \left[\begin{array}{@{}c@{}}a_1 \\ a_2 \\ a_3 \\ a_4 \\ a_5\end{array}\right] = \left[\begin{array}{@{}c@{}}S^{(2)}\sigma (\lambda)\\ \sqrt{2}\\ \sqrt{6}S^{(1)}\sigma (\lambda) \\ 0 \\ \sqrt{(10/3)}S^{(0)}\sigma (\lambda)\end{array}\right], \end{equation} \tag{ 15 }$$

where S^{(0, 1, 2)} are the skewness parameters of Matsubara (2003) and σ is the rms of the evolved density field at spatial scale λ. By expressing the non-Gaussian evolution in terms of the relative contribution of each mode, the result is independent of the power spectrum of the density field, which enters only as the overall normalization of the genus curve and so scales all modes equally; all other coefficients are zero, to this order in the perturbation expansion.

These two approaches to non-Gaussian effects on cosmic structure topology lead naturally to the suggestion that this study be carried out in terms of the Hermite spectrum of the genus curve. The ansatz for this program is that the distinct processes inducing non-Gaussianity in the cosmological density field—survey selection effects, bias, redshift-space distortions, nonlinear gravitational evolution, and primordial physics—have differing spectral imprints that will be delineated. In the remainder of this communication, results applying this decomposition to the study of nonlinear evolution and selection effects are presented.

4.1.2. Examination of Horizon Run Simulation

In this section, the formalism of Section 3 is applied to genus measurements in the Horizon Run of Kim et al. (2009), a dark-matter-only N-body simulation of some 70 billion particles across a volume of L³ = (6.592 Gpc h⁻¹)³, from which dark matter halos are drawn following the methodology of Kim & Park (2006). The simulation volume studied here is a periodic cube at redshift zero, with halos of all masses down to resolved limit used, though, following convention, each object is assigned equal weight, rather than a weight proportional to its mass, during the reconstruction process.

The topology of structure in this simulation is examined across scales from 5 to 65 Mpc h⁻¹. At all scales, the spatial resolution of the density field and number of resolution elements in the measurement are held fixed. To achieve this, subvolumes extending from the origin to L_max = 100λ Mpc h⁻¹ are extracted from the full simulation and embedded within an array of size N³ = 400³. Periodicity is assumed during the density field reconstruction.

Figure 3 shows the genus curves measured from the Horizon Run subhalo catalog on scales ranging from 5 to 65 Mpc h⁻¹. It is surprising to see an asymmetry toward isolated high-density structures even at the largest scales of reconstruction, which is well into the linear regime; this result confirms the analysis carried out by Kim et al. (2009), which is in contrast to earlier studies. To explore this measurement further, the Hermite decomposition of the genus curves is presented in Figure 4, in which the contribution from each mode is normalized to give its fractional contribution to the total: $\tilde{a}_n = |a_n| / \sum _m |a_m|$. The decomposition expresses concisely the degree of nonlinear evolution in the density field as reconstructed from these objects. It is to be expected that part of the non-Gaussian signal in the decomposition arises from the use of biased tracers of the density field, though this is unlikely to manifest itself in a scale-dependent fashion at the scales studied here in the absence of any primordial non-Gaussianity, so that the relative evolution of the odd-numbered Hermite modes (Figure 4) is taken to be a description of the modification to topology induced by nonlinear gravitational evolution. At the shortest scales, the sampling density falls below that required for direct reconstruction of the field, so that the systematic correction derived in the previous section is applied to the measurement. Even with this correction applied, it is clear that the mode corresponding to a pure Gaussian random field is underrepresented relative to the expectation from second-order perturbation theory, particularly evident at the largest scales.

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The peculiar motion of galaxies affects the genus measurement only through its impact on the reconstructed density field. No longer a systematic to be overcome, this coupling between the density and velocity fields yields information about the processes of gravity and structure formation. The linear treatment of this coupling, the Kaiser limit, has been shown to generate no distortion to the shape of the genus measurement for a Gaussian random field; like the survey boundary correction, this affects the amplitude of the genus curve, so that the fractional contribution from Hermite modes is unchanged. Matsubara (1996) has speculated that the short scale distortion effects, which act orthogonally to those in the Kaiser limit, will also alter just the amplitude of the curve, a claim that has been examined in some simulations (Park et al. 2005b). Given the interest directed toward redshift-space phenomena in the field at present, it seems plausible that in the intermediate limit between these two regimes could host effects that do alter the shape of the genus curve, though it is not clear that such scales can be examined in the current generation of redshift surveys.

Broad classes of inflationary models generate significant non-Gaussianity in the early-time cosmological density field. The customary formalism for this effect is through the (Bardeen) curvature perturbation in the matter era:

$$\Phi = \phi + f_\mathrm{NL}(\phi ^2 - \langle \phi \rangle ^2),$$

where ϕ is a purely Gaussian random variable and f_NL is taken to characterize the magnitude of non-Gaussianity in the field (Komatsu & Spergel 2001). The bispectrum of the primordial curvature perturbation, identically zero in a Gaussian random field, will peak in regions of Fourier space for which the closed triplet of wavenumbers $\lbrace \mathbf {k}_1, \mathbf {k}_2, \mathbf {k}_3\rbrace$ exhibits one of a handful of ratios, giving rise to a classification of primordial non-Gaussianity in terms of triangle configurations. Of concern here are the "squeezed" (cf. "local") (k₁ ≪ k₂ ∼ k₃), equilateral (k₁ ∼ k₂ ∼ k₃), and enfolded (k₁ ∼ k₂ ∼ k₃/2) configurations, corresponding respectively to inflation scenarios involving multiple fields, horizon crossing, and approximate shift symmetry in very general single-field models (Wagner et al. 2010).

A study of the influence of primordial non-Gaussianity on the topology of large-scale structure has been undertaken on the scales above 100 Mpc h⁻¹ by Hikage et al. (2006), who present formulae for the genus statistic (among others) from perturbation theory. In comparing these analytic results to N-body simulations, they note that the impact of primordial non-Gaussianity should be slight in proportion to the degree of gravitationally induced non-Gaussianity in the field. Nevertheless, the discovery by Dalal et al. (2008) of a scale-dependent galaxy bias induced by local-type primordial non-Gaussianity provides sufficient motivation to attempt an exploration of this effect through the formalism presented in this work.

Here, the impact of primordial non-Gaussianity on the topology of the late-time density field has been evaluated empirically with the simulation data set of Wagner et al. (2010). These data are the output of a sequence of N-body simulations with generic non-Gaussian conditions, where the departure from Gaussianity is characterized completely by the bispectrum of the initial conditions. Power spectra, halo mass functions, and bias properties of these simulation data have been presented by Wagner et al. (2010) and Wagner & Verde (2012). In this suite, halo catalogs derived from an evolved density with initial non-Gaussianity f_NL = {0, ±100, ±250, ±500} have been produced, in each of the local, equilateral, and enfolded configurations. Snapshots have been provided at z ∼ {0, 0.33, 0.67, 1, 1.5, 2}; in this work, we consider halo catalogs from the redshift one density field with a view to optimizing the number density of halos and (reciprocally) the degree of nonlinear evolution in the underlying density field. Concurrent work by Wales et al. (2012) incorporates an examination of the redshift dependence of these measurements.

The halo catalog output of these simulations is transformed into a counts-in-cells density field in the same manner as the data in the previous section. To evaluate the departures from non-Gaussianity present in the genus measurement of these data, the magnitude of the base (n = 2) mode of the decomposition, relative to that of the Gaussian case, is studied across the range of scales from 15 to 65 Mpc h⁻¹, in steps of 5 Mpc. The scale at which ϒ = 1 is around 12 Mpc, so all scales examined are free from the impact of the sampling systematic. Genus curves are measured for each (f_NL, configuration) pairing, in each of the eight available simulations, before the Hermite decomposition is performed on each measurement. The variance between the measurements of the Hermite coefficients for the eight simulations is taken to be an estimator of the variance of the coefficient statistic.

Figure 5 shows the relationship between f_NL and the relative-to-Gaussian strength of the n = 2 Hermite mode. This is an interesting result that has several possible interpretations. Perhaps the most satisfactory is that, as the amplitude of the genus curve is a functional of the (biased) power spectrum, even though the underlying dark matter power spectrum is fixed for a given redshift, configuration, and realization, the scale dependence of the quantity a₂(f_NL)/a₂(gauss) is a measure of scale-dependent bias. A prediction of this hypothesis is that all other such ratios a_n(f_NL)/a_n(gauss) should show the same effect; however, these data do not allow such a test to be carried out at sufficient significance; e.g., Figure 6 shows the same measurement as Figure 5 with the lower significance modes of the genus curve. In this hypothesis, it is significant that the local configuration shows this scale dependence. Figure 7 shows how the other configurations of the bispectrum do not result in the same behavior. In some models it is predicted that the equilateral configuration should show a 1/k scale-dependent bias, which would be expected to manifest here. Even though this is weaker than the 1/k² dependence of the scale-dependent bias in the local configuration, it is somewhat surprising that only the local configuration displays this behavior—however, this seems to be compatible with the results of Wagner et al. (2010).

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An alternative explanation that is tractable by the means presented in this work is that, if the bispectrum manifests in the genus curve in a manner that is not simply related to the amplitude, then the results in Figure 8 constrain the mechanism of this action. It might be hoped that, given the exact prediction for the genus curve of a Gaussian random field, many distinct types of non-Gaussianity could be examined, even phenomenologically. This is a topic that requires a wider set of simulation data to examine and that will be explored in the future.

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To return briefly to the hypothesis of scale-dependent bias manifesting in the amplitude of the genus curve: recent work by several authors has given rise to methods of constraining dark energy and modified gravity by means of the genus curve amplitude (Wang et al. 2012; Zunckel et al. 2011), based on the development of a standard ruler from large-scale structure by Park & Kim (2010). This class of tests probes the cosmological parameters via the linear regime density field as well as the comoving volume element, both of which determine the genus amplitude. Cosmological density fields that possess some primordial non-Gaussianity will also preserve their initial (non-Gaussian) topology into the linear regime. The impact of primordial non-Gaussianity, as measured and interpreted in the present work, can be used as part of the standard ruler technique, by exposing an additional scale-dependent term (that is, beyond the usual scale dependence of the genus curve amplitude) that does not manifest within the comoving volume element. This improves the possibility of distinguishing these phenomena using the genus curve.