MEASURING PRIMORDIAL NON-GAUSSIANITY THROUGH WEAK-LENSING PEAK COUNTS

The inflationary paradigm is the leading theory of the early universe, which is of fundamental interest for cosmology and particle physics. Understanding the mechanism and energy scale of inflation remain major goals yet to be attained, despite the continuous and fervent efforts invested in this field.

A measurement of primordial gravitational waves would pin down the energy scale of inflation, but this still belongs to the not-so-near future. One possible way to discriminate between single- and multi-field inflation models is to test the Gaussianity of the primordial density fluctuations (Creminelli & Zaldarriaga 2004). The cosmic microwave background (CMB) has been so far the main and cleanest inflationary probe. Recent results from the Wilkinson Microwave Anisotropy Probe (WMAP; Komatsu et al. 2011) suggested tentative 1σ-level evidence of primordial non-Gaussianity of the local type, defined by the equation

$$\begin{equation} \Phi (\mbox{\bf x})=\phi (\mbox{\bf x})+f_{\rm NL}[\phi ^2(\mbox{\bf x})-\langle \phi ^2(\mbox{\bf x})\rangle ]. \end{equation} \tag{ 1 }$$

The parameter f_NL quantifies the local quadratic deviation of the Bardeen potential Φ from a Gaussian potential ϕ, and it is currently constrained to the value 32 ± 21(Komatsu et al. 2011).

It has long been suggested (Grinstein & Wise 1986; Lucchin & Matarrese 1988; Fry & Scherrer 1994; Matarrese et al. 2000) that low-redshift observables can also be used to measure primordial non-Gaussianity, despite the fact that the density field at such redshifts is strongly non-Gaussian due to the action of gravity. In the local non-Gaussianity models, there are mainly two effects on low-redshift observables as recently outlined in the comprehensive study of Dalal et al. (2008), and further explored in Desjacques et al. (2009), Grossi et al. (2009), Pillepich et al. (2010), Lam & Sheth (2009), and Smith et al. (2010). First,  f_NL induces a scale dependence in the bias of dark matter halos, which affects primarily the largest scales, i.e., k < 0.02 h Mpc^-1. Thus, one can in principle separate the f_NL scale dependence from the gravitational one, which occurs on smaller scales. Second, the abundance of massive halos is higher/lower for positive/negative values of f_NL (Matarrese et al. 2000; Lo Verde et al. 2008). As a final note, we emphasize the importance of high- and low-redshift constraints on non-Gaussianity as their comparison may provide insight into the scale dependence of f_NL (Byrnes et al. 2010; Shandera et al. 2010).

In this Letter, we shall numerically investigate the sensitivity of weak gravitational lensing (WL) peak counts to primordial non-Gaussianity of the local type. The potential of WL surveys to constrain f_NL has already been tackled (Amara & Refregier 2004; Pace et al. 2010; Fedeli & Moscardini 2010; Oguri & Takada 2011), though without considering shear peaks. Peak counts are a natural candidate for f_NL studies, since the largest of them are caused primarily by massive halos. For WL studies of peak statistics, we mention the works of Hamana et al. (2004), Hennawi & Spergel (2005), De Putter & White (2005), Marian et al. (2009), Kratochvil et al. (2010), Dietrich & Hartlap (2010), and Marian et al. (2010) and references therein. Should peak counts prove to be a sensitive f_NL probe, then one could easily use large future surveys like euclid (Refregier et al. 2010) or the Large Synoptic Survey Telescope (lsst; Abell et al. 2009) to obtain low-redshift constraints on primordial non-Gaussianity.

The observable that we use is the convergence field, i.e., the matter density projected along the line of sight and scaled by a geometrical factor. We study simulated WL convergence maps created from ray tracing through a suite of N-body simulations, generated with the publicly available code gadget (Springel 2005). A subset of these simulations was used and described in the work of Desjacques et al. (2009). Three values of f_NL are considered: 0, −100, and +100, while the other cosmological parameters are kept fixed. The cosmology matches the WMAP five-year data results (Komatsu et al. 2009). We have a total of 18 simulations, with six realizations per f_NL value. The initial conditions for each of the six sets of f_NL-model realizations are matched to reduce the cosmic variance on the comparison of the peak functions corresponding to each model. The box size is 1600 h⁻¹ Mpc, the number of particles is N = 1024³, and the softening length is l_soft = 40 h⁻¹ kpc.

We consider a survey similar to euclid (Refregier et al. 2010) and to lsst (Abell et al. 2009) for the WL simulations with an rms σ_γ = 0.3 for the intrinsic image ellipticity, a source number density $\bar{n}_{{\rm gal}}=40\,{\rm arcmin}^{-2}$, and a redshift distribution of source galaxies given by ${\mathcal P}(z)=1.5\,z^2/z_0^3\,\exp [-(z/z_0)^{1.5}]$, where z₀ = 0.6. The median redshift of this distribution is z_med = 0.86.

From each N-body simulation we generate 16 independent fields of view. Each field has an area of $12 \times 12\,\rm deg^2$ and is tiled by 4096² pixels, yielding an angular resolution θ_pix = 10 arcsec and a total area of ${\approx} 14{,}000\, \rm deg^2$ for each f_NL model. The effective convergence κ in each pixel is calculated by tracing a light ray back through the simulation with a Multiple-Lens-Plane ray-tracing algorithm (Hilbert et al. 2007b, 2009). Gaussian shape noise with variance $\sigma _{\gamma }^2 /(\bar{n}_{{\rm gal}}\, \theta _{{\rm pix}}^{2})$ is then added to each pixel, which creates a realistic noise level and correlation in the filtered convergence field (Hilbert et al. 2007a).

For the peak finding we use an aperture filter (Schneider 1996), matching an Navarro–Frenk–White (NFW) profile (Navarro et al. 1997) convolved with a Gaussian function. Thus, we adopt the convergence model $\kappa _{\rm model}=\kappa _{\rm \scriptscriptstyle NFW} \circ f_{\rm Gauss}$, where f_Gauss is a Gaussian function of width f × l_soft. Here f = 1.5 for M < 7 × 10¹⁴ h⁻¹ M_☉, and f = 2 otherwise; the softening length l_soft was defined above. $\kappa _{\rm \scriptscriptstyle NFW}$ is the NFW convergence profile truncated at the virial radius, defined in Marian et al. (2010). This model agrees very well with the measured convergence profiles of the peaks in the maps. It is useful both when working with simulations, since it accounts for the lack of resolution below the softening scale, and when using real data, since shear data are difficult to obtain near the centers of clusters.

The amplitude of the smoothed field at a point x₀ is given by

$$\begin{equation} \hat{M}({\bf x_{0}})=\int d^{2}x \, W({\bf x_{0}}-{\bf x}) \kappa ({\bf x}), \end{equation} \tag{ 2 }$$

where W is our filter and κ is the convergence field. The filter W can be written as follows:

$$\begin{equation} W(x)={\mathcal C}_{\rm \scriptscriptstyle W}\, \frac{\kappa _{\rm model}(x) -\bar{\kappa }_{\rm model}(R)}{\sigma _{\gamma }^2\big/\bar{n}_{{\rm gal}}}, \end{equation} \tag{ 3 }$$

where ${\mathcal C}_{\rm \scriptscriptstyle W}$ is a normalization constant, R is the aperture radius, i.e., the radius over which the filter is compensated, and $\bar{\kappa }(x)=2/x^2\int _{0}^{x}dy y \kappa (y)$. We choose the normalization constant to be

$$\begin{equation} {\mathcal C}_{\scriptscriptstyle \rm W}= \frac{\sigma _{\gamma }^2}{\bar{n}_{{\rm gal}}} \,\frac{M_{\rm \scriptscriptstyle NFW}}{\int d^2x \,\kappa _{\rm model}^2(x)-\pi R^2\, {\bar{\kappa }_{\rm model}}^2(R)}\,. \end{equation} \tag{ 4 }$$

If x₀ is the location of a peak created by an NFW cluster of mass $M_{\rm \scriptscriptstyle NFW}$ and redshift z, and the convergence field is smoothed with a filter tuned to precisely such a cluster, i.e., κ_model in Equation (3) corresponds to the same $M_{\rm \scriptscriptstyle NFW}$ and z, then this filter returns a maximum signal-to-noise ratio (S/N) at x₀. At this location, the amplitude of the smoothed map is

$$\begin{equation} \hat{M}({\bf x_{0}}| M_{\rm \scriptscriptstyle NFW}, z)=M_{\rm \scriptscriptstyle NFW}. \end{equation} \tag{ 5 }$$

The peak is assigned the mass $M_{\rm \scriptscriptstyle NFW}$. If the peak does not correspond to a real halo, and it is the result of line-of-sight projections, then it can still be assigned an "effective" mass. In practice, we smooth the convergence field with filters of various masses, which yield different amplitudes (larger and smaller than the filter mass) at the location of peaks. We interpolate these amplitudes to determine the filter mass that would satisfy Equation (5).

We choose R to be the virial radius of the cluster to which the filter is tuned: $R=(3M_{\rm \scriptscriptstyle NFW}/800\,\pi \,\bar{\rho })^{1/3}$, where $\bar{\rho }$ is the mean density of the universe. Thus, a filter at a given redshift can be specified either through the mass $M_{\rm \scriptscriptstyle NFW}$ or through its size R. We adopt the mass convention of Sheth–Tormen (Sheth & Tormen 1999), with an overdensity defined as 200 × the mean density (not the critical density), and the concentration parameter of Gao et al. (2008). We evaluate the S/N of the mass estimator in a simplified scenario where we ignore projection effects and consider only the intrinsic ellipticity noise. In this case, the variance of the estimator is given by ${\rm Var}(\hat{M})=M_{\rm \scriptscriptstyle NFW}\,{\mathcal C}_{\rm \scriptscriptstyle W}$, and the S/N is obtained by combining this with the above equations. Note that while the mass estimator does not depend on the shape noise due to the normalization constant $\mathcal C$, the S/N scales with it as aperture filters usually do: ${\rm S/N} \sim (\bar{n}_{{\rm gal}}/\sigma _{\gamma }^2)^{1/2}$.

Finally, in the case where no shape noise is included, we use the same filter as in our previous work (Marian et al. 2010). This filter can also be obtained by formally taking $\sigma _{\gamma }^2/\bar{n}_{{\rm gal}}=1$ in Equations (3) and (4) above.

The analysis of the convergence maps is carried out in two situations: with and without shape noise. We adopt a very conservative approach in which we consider known only the redshift distribution of the source galaxies and the shape noise level, without any other information on the sources or on the detected peaks. We also do not resort to tomographic techniques. While this is an overly pessimistic scenario for a next-generation lensing survey like euclid or lsst, our goal here is to provide a proof of concept of the possibility of using WL peak counts to constrain primordial non-Gaussianity, rather than the final quantitative answer to this question.

We perform a hierarchical smoothing of the maps with filters of various sizes, from the largest down to the smallest, as described in Marian et al. (2009, 2010). This approach removes the problem of "peaks-in-peaks" and it also naturally eliminates the dependence of the measured peak function on a particular filter scale. Since the median redshift of the source distribution is 0.86, we adopt a fixed redshift of 0.3 for the matched filter described above. For this redshift, the scale of the filter varies from corresponding masses of 2 × 10¹⁵ h⁻¹ M_☉ to 3 × 10¹³ h⁻¹ M_☉ in the absence of shape noise, and 10¹⁴ h⁻¹ M_☉ in the presence of it. The latter lower-limit choice of the filter scale is due to the fact that shape noise contaminates seriously the smaller peaks; imposing a minimum S/N threshold alleviates but does not remove the contamination. Peaks are detected in the smoothed maps as local maxima and are assigned a mass as described above.

Figure 1 presents the main result of this work. The upper panels illustrate the peak functions measured in the f_NL = ±100 cosmologies, in the absence (left panel) and presence of shape noise (right panel). For each f_NL model, the points are the average of the peak counts measured in all 96 fields and the error bars represent errors on the mean. In the case of shape noise, we select only peaks with an S/N>3. We note that the two average peak functions are clearly distinct at the high-mass end, i.e., for M>3 × 10¹⁴ h⁻¹ M_☉. This is better seen in the lower panels which show the difference of the peak abundance measured in the f_NL = ±100 cosmologies, relative to the f_NL = 0 peak abundance. We have minimized the impact of the matched initial conditions of the simulations on the error bars: the fractional difference is computed as an average of the f_NL ± 100 peak abundance in each field ratioed to the average Gaussian peak abundance of 96 fields (which represents our best model for the true Gaussian peak abundance), as opposed to the average of the ratio of the non-Gaussian and Gaussian peak functions measured in the same field. The latter would have removed the cosmic variance of the fractional difference, because of the matched initial conditions of the simulations from which we built the convergence maps.

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Just like in the case of the three-dimensional halo mass function (see, for example, Desjacques et al. 2009; Smith et al. 2010), the peak functions for the f_NL models show a deviation from the Gaussian case. Unlike the three-dimensional studies which have presented halo mass functions measured at a single redshift, the peak functions that we show here combine peaks in the redshift range of the source distribution, and therefore are not as regular and symmetric as their three-dimensional counterparts. The asymmetry is most likely due to modifications of the S/N of peaks by line-of-sight projections and also by shape noise contamination. The trend is similar however, with high-mass peaks displaying the largest deviation. For both f_NL models, this is about 10%–15% for the largest mass bins, i.e., M>4 × 10¹⁴ h⁻¹ M_☉.

To quantify the significance of the deviation, we perform a χ²-test. For the fiducial model f_NL = 0, we estimate the covariance of the counts in a field. We use the covariance of the mean to obtain the χ². For both f_NL ± 100 we find a probability <0.1% to fit the f_NL ± 100 peak functions with an f_NL = 0 peak function. This is also true if we consider only the diagonal elements (the variance of the mass bins) instead of the full covariance matrix, and also if we vary the mass bins. We also use the measured counts to estimate the Fisher error that a 14,000 deg² WL survey would yield on f_NL. Assuming all other cosmological parameters known, the forecasted error is Δf_NL ≈ 13 for the fiducial value f_NL = 0. In the above we considered that the peak abundance scales linearly with f_NL, similar to the three-dimensional halo abundance. The values of the χ² and the Fisher error are largely maintained also if we minimize the impact of the matched initial conditions of the simulations, by using the first three simulations to compute the f_NL = +100 peak function and the last three for the f_NL = −100 function.

Though these results are already very encouraging, it is possible to improve measurements of primordial non-Gaussianity from WL surveys even if one considers only peak counts. The most important is probably the use of tomography, as numerical and analytical studies of the three-dimensional mass functions have shown that deviations of the halo abundance from the Gaussian case significantly increase with redshift. Tomography would allow peaks to be separated not only in terms of their S/N–mass but also of their redshifts, thus acquiring more sensitivity to f_NL. On the other hand, one has to beware a possible degeneracy with the amplitude of the matter power spectrum, σ₈. This can be solved by either using as a prior very good knowledge of σ₈ from other probes, such as the CMB (The Planck Collaboration 2006), WL, and large-scale structures or by combining several observables sensitive to both f_NL and σ₈, as exemplified in Oguri & Takada (2011). Our immediate goals are to further study how WL can be used to constrain primordial non-Gaussianity, to build improved f_NL estimators from WL observables, and to forecast f_NL constraints based on these observables.

For now we convey a simple, yet powerful statement: future WL surveys could detect primordial non-Gaussianity of the local type from at least one statistic-peak count. In particular, surveys like euclid and lsst should be able to provide complementary information to the f_NL information obtained from the ongoing CMB mission Planck (The Planck Collaboration 2006).

We kindly thank V. Springel for making public gadget-2 and for providing his B-FoF halo finder. L.M., S.H., and P.S. are supported by the Deutsche Forschungsgemeinschaft (DFG) through the grant MA 4967/1-1, through the Priority Programme 1177 "Galaxy Evolution" (SCHN 342/6 and WH 6/3), and through the Transregio TR33 "The Dark Universe." R.E.S. and V.D. were partly supported by the Swiss National Foundation under contract 200021-116696/1, the WCU grant R32-2008- 000-10130-0, and the UZürich under contract FK UZH 57184001. R.E.S. also acknowledges support from a Marie Curie Reintegration Grant and the Alexander von Humboldt Foundation.