Cosmological Information in Weak Lensing Peaks
Recent studies have shown that the number counts of convergence peaks in weak lensing (WL) maps, expected from large forthcoming surveys, can be a useful probe of cosmology. We follow up on this finding, and use a suite of WL convergence maps, obtained from ray-tracing N–body simulations, to study (i) the physical origin of WL peaks with different heights, and (ii) whether the peaks contain information beyond the convergence power spectrum . In agreement with earlier work, we find that high peaks (with amplitudes 3.5, where is the r.m.s. of the convergence ) are typically dominated by a single massive halo. In contrast, medium–height peaks () cannot be attributed to a single collapsed dark matter halo, and are instead created by the projection of multiple (typically, 4-8) halos along the line of sight, and by random galaxy shape noise. Nevertheless, these peaks dominate the sensitivity to the cosmological parameters , and . We find that the peak height distribution and its dependence on cosmology differ significantly from predictions in a Gaussian random field. We directly compute the marginalized errors on , and from the combination, including redshift tomography with source galaxies at and . We find that the combination has approximately twice the cosmological sensitivity compared to alone. These results demonstrate that contains non-Gaussian information complementary to the power spectrum.
pacs:PACS codes: 98.80.-k, 95.36.+x, 98.65.Cw, 95.80.+p
Weak gravitational lensing (WL) by large-scale cosmic structures has emerged as one of the most promising methods to constrain the parameters of both dark energy (DE) and dark matter (DM) (e.g. ref. (1); see also recent reviews in refs. (2); (3)). While linear and mildly nonlinear features in WL maps have been thoroughly explored, an important question that remains is: how much additional information lies in the nonlinear features of these maps? Motivated by this, we recently investigated a simple nonlinear statistic – counting peaks in WL maps directly as a function of their height and angular size ((4); hereafter Paper I). This statistic does not lend itself to straightforward mathematical analysis – it requires numerical simulations and has received relatively little attention (e.g. (5); (6)) until recent simulation work (7); (4).
In Paper I, we identified peaks in ray-tracing N-body simulations, defined as local maxima in two-dimensional convergence maps. We found that the number of peaks as a function of their height has a sensitivity to a combination of () competitive with other forthcoming cosmological probes. Dietrich and Hartlap (7) investigated peak counts as a function of and , and reached qualitatively similar conclusions.
One result identified in our study is that the cosmological
sensitivity arises primarily from medium–height peaks, with
amplitudes of , where is the r.m.s. of
the WL convergence . Dietrich and Hartlap include only higher
significance peaks in their analysis (with
Motivated by these findings, here we attempt to clarify the physical origin of the medium amplitude peaks, by identifying collapsed dark matter halos along sight–lines to individual peaks. The fact that the cosmological sensitivity is driven by relatively low–amplitude peaks raises a potential concern: these peaks may be dominated by galaxy shape noise, and/or may arise from random projections of large-scale overdensities in the mildly nonlinear regime. The counts of the medium peaks may then offer little information beyond conventional statistics, such as the power spectrum. Our second aim in this paper is therefore to investigate the origin of the cosmological information content of the WL peaks. To this end, we compare peak–height distributions in different cosmologies with those expected in corresponding Gaussian random fields (GRFs). Further improving on Paper I, we are able to provide marginalized constraints from the combination of peak counts and the power spectrum for the cosmological parameters , , and , not just parameter sensitivity, since we ran a much larger set of simulations.
This paper is organized as follows. In § II, we describe our calculational procedures, including the creation of the WL maps, the identification of collapsed halos, the prediction of peak counts in a GRF, and our statistical methodology to compare maps. In § III, we present our results, which include the matching of peaks and halos, matching peaks in different cosmologies, and the comparisons of the simulated peak counts to the Gaussian predictions. In § IV, we offer a detailed discussion of our main results, as well as of several possible caveats and extensions. Finally, in § V, we summarize our main conclusions and the implications of this work.
ii.1 Simulating Weak Lensing Maps
We generate a series of 80 cold dark matter N-body simulations for 7 different cosmological models with the code GADGET-2, which include DM only (no baryons). As our fiducial model, we adopt a CDM universe with the following parameters: cosmological constant = 0.74, matter density parameter = 0.26, Hubble constant , dark energy equation-of-state parameter , and a primordial matter power spectrum with a spectral index of and present-day normalization of . These values are consistent with the seven–year results by the WMAP satellite ((8)).
All simulations use DM particles, in a box with a
The interested reader is referred to Paper I for more details about
the simulations and the process of making the maps. We made one
important change, however, which must be high-lighted. While in
Paper I, the density and potential planes had a resolution of
(same as the resolution used for ray-tracing and the
final convergence map), here we adopted a higher resolution,
, for both the density and the potential planes. This
change has been proven to be important as lower resolution yields a
loss in power at large wave number, as demonstrated
by (9). In Fig. 1, we compare the 2D
angular power spectrum of the WL convergence field at
resolution with the theoretical power spectrum. The latter was
obtained by direct line–of–sight integration, using the Limber
approximation, and the fitting formulae for the nonlinear 3D matter
power spectrum from (10), calculated with the code
Once the maps have been created, we take the redshift-dependent r.m.s. of the noise in one component of the shear to be (13)
Note that this corresponds, in the weak lensing limit, to an r.m.s. ellipticity of ((14)).
For simplicity, in our analysis we assume that the source galaxies are located on a source plane at a fixed redshift, with . We apply a Gaussian smoothing to the maps. The noise in the convergence after the Gaussian smoothing then becomes
For reference, we note that at redshift , the above gives , very close to the r.m.s. of the convergence in the noise-free maps. The r.m.s. of the total convergence field, with noise included, is .
ii.2 Suite of Simulated Maps
In addition to the fiducial cosmology, we have run N-body simulations in six other models. Each of these differs from the fiducial model in a single parameter: we vary , , in both directions, with values = 0.750 and 0.850; = -0.8 and -1.2; and = 0.23 and 0.29. We assume the universe always stays spatially flat (i.e. ). The seven different cosmologies will hereafter be referred to as the fiducial, high-, low-, high-, low-, high-, and low- models, respectively, as summarized in Table 1. In each of these N-body runs, we create 1000 different WL maps with source galaxies at , and another 1000 maps with galaxies at . Each map covers a solid angle of degrees. All maps were created by mixing potential planes randomly among five different N-body runs, with independent realizations of the initial conditions, in the given cosmology. Finally, we created an additional 1000 control maps, using the planes from 45 additional independent N-body runs in the fiducial model. Having 9 times more strictly independent realizations allows us to compute the covariance matrix more accurately (needed for computing ; see below), and to check the robustness of our results to different realizations of the fiducial model.
|# of sims|
ii.3 Halo Finding
We use the publicly available AMIGA halo finder ((15); hereafter AHF) to identify collapsed halos in our N-body runs. AMIGA finds halos based on an iterative density refinement scheme. Its output consists of the 3D positions of the halos, and, importantly for us, the tagged set of particles belonging to each halo. The virial radius of a halo is such that when a sphere is placed at the halo’s location, with a radius , the overdensity is
where is the background baryon density, and where is adopted in this study. The mass of the halo is then simply given by
As a simple test of both our N-body simulations and our implementation of the halo finder, we reproduce the fitting formula (their equation B3.) for the halo mass function reported by Jenkins et al. (16). An example of this comparison is shown, in our fiducial model at , in Fig. 2. Overall, we find excellent agreement, with an accuracy of 25% or better up to a halo masses of . For larger masses, there is a large scatter.
One of the goals in this study is to identify halos contributing to each individual convergence peak. To do this, starting from each peak, we follow the light ray, and record the information (masses and location) of the halos found within a light cone centered on the peak, with a radius of 3 arcmin. This radius is chosen to be three times the smoothing scale. We have verified that doubling the radius of the light cone does not change our halo matching results below – in the sense that no additional halos are identified that contribute significantly to the total convergence of a peak (see detailed discussion below). To be consistent with the perpendicular projection of the particle density in each simulation snapshot, the light rays consist of a series of parallel line segments, which are perpendicular to the potential planes. The coordinates where the segments cross the potential planes are determined by the lensing deflection angle, computed and stored during the ray-tracing analysis. For the same reason, the light cones are composed of a series of parallel cylinders with a radius of 3 arcmin, centered on the corresponding light ray.
As explained above, we truncate the simulation snapshots, in order to generate more independent realizations of maps. As a result, occasionally, parts of halos that happen to be located near the plane of the truncation can be unphysically “cropped”. These cropped halos become important only when they are sufficiently massive to contribute to the convergence of a peak, and when they are located near (within a fraction of their virial radius) one of the two truncation planes (either in the front or the back). Given that halo virial radii are of order Mpc, and our truncated box size is 80Mpc, the probability that the latter condition is satisfied is . We therefore simply restrict our halo catalog only to halos that do not touch the edges. We have checked that neglecting the cropped halos does not significantly affect our results.
Our simulations resolve the density structure of individual halos with masses of . However, each map contains on the order of peaks, and more than a dozen halos can contribute to the total convergence of a single peak; therefore, computing the exact contribution of each halo to each peak is computationally impractical. Instead, we replace each halo by a spherically symmetric NFW (17) halo with the same virial mass . For a given impact parameter (defined as the angular distance between the halo center and the point of closest approach of the light ray corresponding to the peak), redshift , and mass , the contribution of the off-center halo to the convergence peak can then be calculated analytically.
More specifically, the density profile is assumed to follow
where is the radius from the halo center, and and are a characteristic radius and density. The profile is truncated at , inside which the mean overdensity with respect to the critical density of the universe at redshift is 200. We adopt the concentration parameter in this paper. The convergence due to the halo, given an extended redshift distribution of the background galaxies, is then given by
Here is the projected surface density of the halo (given explicitly in ref. (18); see their equations 26-27), is the comoving distance to redshift , is the surface number density of background galaxies per unit redshift, and is the mean total surface density. The latter is taken to be in this paper, with as the source redshift. Finally, we use a Gaussian window function to smooth the convergence induced by the halo,
where the center of the smoothing kernel () is set to the angular position of the light ray corresponding to a peak, and is the angle toward the halo center. The smoothing scale is chosen to be 1 arcmin, as in the simulated maps.
To check the accuracy of the NFW approximation for the convergence, we selected halos with masses in the range in one of the realizations of our fiducial model. For each halo, we record the value of the convergence in the map (with sources galaxies at ), in the pixel located in the direction toward the halo center. We then remove the halo from the simulation box, and repeat the ray-tracing procedure discussed above, to compute a new value at the same position, but without the halo. In Fig. 3, we show the actual difference , against the value expected based on the NFW halo model. As the figure shows, the NFW assumption works accurately, although it results in a slight underestimate of the convergence. The points in the figure yield an average fractional bias of and an r.m.s. scatter of .
ii.4 Gaussian Random Field Predictions
One of the key questions to be answered in this paper, is the extent to which the peak counts contain information beyond traditional measures, such as the power spectrum. For example, if the peaks were produced only by a combination of pure galaxy shape noise (which is Gaussian by assumption) and linear fluctuations in the matter density, then their statistics would be fully described by a Gaussian random field (GRF). The majority of the high peaks are known to be associated with collapsed, nonlinear objects, and their statistics will clearly be non-Gaussian. However, the extent to which lower-amplitude peaks are non-Gaussian is not clear ab-initio. As mentioned in the Introduction, these peaks contain most of the cosmological information, and therefore the departure in the statistics of these peaks from Gaussian predictions is important to understand and quantify.
As a simple test, we directly compare our simulated peak counts with those expected in a GRF. Fortunately, the peak counts in a two–dimensional GRF, and their distribution in height, are predictable analytically (19). In fact, they depend only on the first and second derivatives of the correlation function on small scales (or, equivalently, the first two moments of the power spectrum). For completeness, we reproduce the relevant equations here. The differential number of maxima per unit solid angle, , with height in the range to , where is measured in units of the standard deviation of the random field, is given by
and where is the continuous 2d power spectrum of the convergence field, and is its two-point correlation function. Integrating Eq. (10) over gives the total number of peaks regardless of their height,
To verify the accuracy of these analytic formulae, we produced 200 numerical maps of GRFs, by generating 200 independent random realizations of the theoretical 2d weak lensing power spectrum (in our fiducial model, assuming a source redshift ). We first generate a 2d complex random field in space, with the real and imaginary parts of Fourier modes distributed independently, following Gaussians with a standard deviation of . Here is the power spectrum (20). We then perform a discrete Fourier transform to produce maps in real space. The GRF maps have a size of , to mimic the actual WL maps. Noise is then added according to Eqs. (1) and (2), with =15, and a 1 arcmin smoothing is applied. To avoid edge effects, we discard pixels located within 40 pixels ( the smoothing scale) of the map edges.
To reproduce the random GRF realizations as closely as possible, we do not calculate , , from the input power spectrum; instead, we measure these directly from the maps, using the finite difference derivatives (ignoring the constant coefficients in Eq. (14))
where is the 2d convergence field, and , and are the averages of the corresponding first and second derivatives. For reference, this leads to a prediction of 1680 peaks, compared to the actual number 1685 found in the noiseless maps; the prediction based on the analytic calculation of the ’s is slightly worse, 1649. We have checked that the situation is similar in the true WL maps: measuring numerically gives a slightly more accurate prediction for the total number of peaks than calculating analytically from the power spectrum through Eq. (14).
In Fig. 4, we show predictions from the analytic formulae (Eqs. (10)-(14)), and the mean peak counts in our 200 mock GRF maps. This tests our reproduction of the formulae, as well as the accuracy of our numerical measurements of , , . The agreement is excellent, with residuals of only percent over most of the range shown.
The main advantage of creating mock numerical realizations of the GRF maps is that we can use these to measure the (co)variance in the Gaussian peak counts. This covariance matrix is necessary to compute the values between pairs of cosmologies in the Gaussian case (see discussion below).
ii.5 Statistical Comparisons
The basic statistical task in this paper is to assign a significance of the difference between a pair of maps, given the stochastic fluctuations in the maps over many realizations. This is required in order to quantify how well two cosmological models can be distinguished with the peak counts.
The simplest statistical test consists of computing between a pair of cosmologies and , using the mean number of peaks in each bin, averaged over all realizations,
where is the difference between the average number of peaks in bin in cosmology and in cosmology . Note that here can label bins of different peak–heights, but can also include different source galaxy redshifts or smoothing scales. Here denotes the covariance matrix of the number of peaks in cosmology ,
where is the number of peaks in bin in the realization (i.e. convergence map) of the cosmology , and is the total number of realizations.
This could be interpreted directly as a likelihood or confidence level only if (i) the peak count distribution in each bin were Gaussian, and (ii) the mean peak counts depended linearly on the cosmological parameters. As long as the change in parameters is small, a Taylor expansion to the first order is a good approximation, and therefore the second condition is unlikely to be strongly violated in our case. We will verify below that condition (i) is satisfied to a good accuracy, as well. To be specific, we examine directly the distribution of the quantity defined as
where is the difference between the number of peaks in bin in realization of a test cosmology , and the average number of peaks in the same bin in the fiducial cosmology . We will show that closely follows a true chi-squared distribution.
Unless stated otherwise, in this paper, we use five bins to calculate . The bin boundaries are chosen by visual inspection, using two rough criteria: (i) the difference in the peak height distributions in the two cosmologies should not change sign within any of the bins, and (ii) the numbers of peaks in each bin should be as comparable as possible. The influence of the number of bins and the bin boundaries on our results will be discussed in § IV below.
ii.6 Fisher Matrix and Marginalized Error
With the assumptions that (i) the observables, i.e. the mean number of peaks in each bin , depend linearly on the cosmological parameters ; and (ii) that for a fixed , the probability distribution of follows a Gaussian, the marginalized error on each parameter can be calculated using the Fisher matrix (e.g. ref. (21)).
The Fisher matrix for point in the parameter space is given by the matrix trace
where the Greek indices refer to model parameters, a comma preceding an index denotes a partial derivative with respect to the corresponding parameter, is the covariance matrix of ’s, and is the expectation value of . In this paper, we will consider only the second term in Eq. (22) above. The constraints through this term arise from the dependence of the mean number of peaks on the cosmological parameters. In principle, additional constraints could be available from the first term, which represents the dependence of the (co)variances on cosmology (see refs. (22) and (23) for related points in the context of cluster counts). Our results suggest that this dependence is relatively weak; in practice, however, we do not have a sufficient number of independent realizations to accurately evaluate this dependence. From the Fisher matrix, the marginalized error on the parameter is calculated simply as . (In contrast, the square root of Eq. (19) divided by the difference between model parameters corresponds to the parameter sensitivity with all other parameters fixed.)
iii.1 The Physical Origin of Peaks and of their Cosmology Dependence
In general, peaks in the convergence field can arise for different reasons. They could be caused by (i) one or more collapsed halos along the LOS; (ii) large-scale, mildly overdense filaments, seen in projection; (iii) non–linear “protoclusters” that are on their way to collapse, but have not yet virialized and settled into an equilibrium structure (24); (25); and (iv) pure galaxy shape noise. In reality, peaks can be produced by a combination of the above effects.
Relatively high–amplitude () lensing peaks have been studied thoroughly in the past (e.g. (26); (27); (28)), and it is known that a large fraction of these peaks is attributable to a single collapsed massive halo. Our motivation to revisit this topic is that the cosmological information is contained primarily in the lower–amplitude peaks, whose origin has not yet been clarified. As described in section II, we start with each individual peak, and identify all halos along the sightline. Likewise, for each halo, we identify peaks that are located within a 3 arcmin distance from the 2d sky position of the halo.
We would also like to know why the number of peaks changes with cosmology. To help clarify this, we examine realizations of pairs of models with different values of with quasi identical initial conditions. In these pairs of models, we use the same random seeds to generate the amplitudes and phases for the Fourier modes of the density and velocity field at redshift , and then scale these back to the starting redshift of the simulation, with the linear growth factor. Since the power spectra differ only by an overall normalization, and since the growth factor is independent of , the initial conditions, as well as the final WL maps, maintain very similar patterns. Therefore we can attempt to match individual peaks in the two cosmologies, to see what happens to a given peak, when, say, is increased.
Results in this section are based on 50 realizations of the noisy maps in the fiducial and in the low- models, with source galaxy redshift , and 1 arcmin smoothing. The maps have an angular size , pixels. To avoid edge effects, 30 pixels( 3 times the smoothing scale) from all four edges of each map are discarded, leaving an area of . The fiducial model is used for studying the origins of the peaks, and a comparison between the fiducial and the low– model is used for studying the cosmology-sensitivity.
We first perform an analysis of peak-halo matching, closely following Hamana et al. (27), for both high and medium peaks. High peaks are defined as those with , where is the maximum value of the peak, in units of the standard deviation of the noise field. We restrict our halo catalog to halos that are expected (based on the NFW approximation; Eqs. (5)-(6) with ) to produce a peak with a height of . Similarly, for medium peaks with , we restrict our halo catalog to those halos with . We carry out the peak-halo matching by searching for a matched pair candidate within a radius of 1.8 arcmin from the peak position or from the halo center. This maximum angular separation is chosen, as in (27), so that it is larger than the smoothing radius of 1 arcmin, while it is still smaller than the angular virial radius of a massive halo at , where the lensing kernel has the most weight. If there is more than one candidate pair within this radius, we follow (27) and adopt the closest one as the primary candidate.
|class||matching result||number of matches|
|(high peaks)||(medium peaks)|
|halo peak||526 (0.93)||2653 (4.7)|
|halo with no paired peak||230 (0.41)||19609 (35)|
|peak with no paired halo||2264 (4.0)||24709 (44)|
|halo peak||2 (0.0035)||90 (0.16)|
|halo peak||12 (0.021)||194 (0.34)|
The 50 noisy maps contain a total of 2,802 high peaks and 27,556 medium peaks. For comparison, the halo catalogs contain 758 massive halos with and 22,352 medium-sized halos with . Following Hamana et al. (27), we sort the results of the matches into the following five categories: () both a halo and a peak are each other’s primary pair candidate; () a halo without a paired peak () a peak without a paired halo; () a halo has a matched peak, but is not the primary matched halo of that peak; and () a peak has a matched halo, but it is not the primary matched peak of that halo. Table 2 shows the number of matches, separately for high and medium peaks, falling into each category.
Bearing in mind differences in redshift, noise, and peak height thresholds, our results are in reasonable agreement with (27). Overall, we have found, on average, 5.0 deg high peaks and 1.3 deg massive halos. The majority () of the massive halos produce a one-to-one matching peak, although these account only for of the total population of high peaks. More specifically, (27) finds 23 deg high peaks and 8.1 deg massive halos. Their matches per in categories , , and are 5.9, 2.1, 15 accordingly. In general, we have found fewer peaks and halos than (27), which can be attributed to our lower and values. Our threshold of high peaks is also not identical to theirs. However, we have checked that the number of halos above a threshold is in good agreement with their equation 15 (27), when our cosmological parameters are used. We have found a slightly lower completeness of the halos (less fraction of halos in category and more fraction in category ), and also a lower purity (more fraction of peaks in category ). We have chosen a higher source redshift, ((27) used ), this makes projection effects more important, and may explain why our completeness is lower. Our noise is slightly larger, and our simulations and final maps have higher resolution, compared to (27) – these effects tend to increase the number of peaks relative to number of halos, and to reduce the purity of identifying halos.
The most important result in Table 2 is that, in difference from massive halos, only a small fraction () of the 22,352 medium-sized halos produce a medium peak with a one-to-one pair, accounting for less than of all medium peaks. We conclude that the close agreement in the number of medium peaks and halos (27,556 vs. 22,352) is a coincidence - and noise and projection effects are much more important for medium peaks than for high peaks.
We next extend the above analysis, by identifying all halos, down to low masses, by using a larger, 3 arcmin cone around the LOS toward each peak, and by computing the expected contribution of each halo to the total convergence at the position of the peak (based on the NFW approximation). We then rank the halos according to their values (starting from highest and going down to the lowest). We add the noise at the peak location to this ranked list, and ask the following question: starting from the highest value, how far down this ranked list do we need to sum the contributions, before they account for of the total peak height?
In the upper two panels of Fig. 5, we plot the distribution of this quantity; a “0” indicates that noise is the single largest contributor, and already accounts by itself for most of the peak, “1” indicates that at least 1 halo had to be included, etc. These panels clearly show that the large majority of high peaks are dominated by a single halo, which accounts for at least half of the peak amplitude. Most of these halos fall below the expected threshold . The high peaks thus have a much better one-to-one match with halos than Table 2 implies, once lower–mass halos are included. On the other hand, the large majority of medium peaks are dominated by noise.
Since noise does not contain any cosmological information, in the bottom two panels of Fig. 5, we repeat the same exercise, except that the noise contribution is excluded, and we show the number of halos required to account for the total halo contribution. These panels show that while the high peaks are typically dominated by a single halo, the contributions from a second (or higher-rank) halos is often ( of cases) as important. In the case of medium peaks, however, it is very rare ( of cases) for a single halo to dominate the cumulative halo contribution. Instead, there is a broad distribution, but typically (in half the cases) halos are required to account for half of this total halo contribution. As a sanity check, we have computed the analogous distribution for random directions on our maps (i.e., not toward peaks). For these random directions, as shown in Figure 6, the distribution has an even broader shape, centered at 8, and generally shifted toward larger numbers of halos. This reassures us that the medium peaks still do preferentially pick out directions toward conjunctions of halos. We also examined the masses and redshifts of these dominant halos. We have found that the masses range between , and found no correlations in redshift (i.e., the contributing halos are not part of a single structure). Finally, we find that simply adding up the expected contribution of all halos along the LOS to a peak always overproduces the value of the peak (not surprising, since this neglects the deficit from underdense regions).
We next wish to clarify why there are fewer medium-height peaks when is increased (and vice versa). One can intuitively guess that increasing simply increases the ”scatter” due to large scale structures. In the linear regime, changing simply changes the local (3d) density contrast, by the same factor everywhere. Pretending that this holds in the nonlinear regime, it is easy to see that the set of peaks would be invariant under changing – however, positive peaks would be enhanced, and negative peaks (i.e., maxima residing inside large-scale voids) would become yet more negative. This would broaden the peak-height distribution, and reduce the number of peaks near . Of course, this picture is over-simplified: as is increased, the change in the density field is not a simple re-scaling; furthermore, peaks can be destroyed and new peaks can be created. Indeed, we already know that the total number of peaks changes with (Paper I).
To understand the dominant effect, we attempt to follow and match individual peaks in a pair of models with different . We have found that a direct matching of peaks is not possible, because the locations of the peaks tend to shift (by several to more than 20 pixels) and therefore the correspondence between peaks in two different maps remains ambiguous (except for the very highest and most conspicuous peaks). Instead, we proceed by using halos as intermediate proxies for the peaks. Starting from a peak in the first cosmology, we identify the halo that contributes most to this peak. For this analysis, we consider only those peaks for which the matched primary halo contributes at least 10% of the total halo contribution (otherwise it is unfair to use the halo as a proxy for the peak). We next search the entire halo catalog in the second cosmology, and identify the “same” halo, by finding the one that shares most common particles with the halo in the first cosmology. Finally, we search through the peaks in the second cosmology within a cone of 3 arcmin around the halo, requiring that their height is within the range of the original peak in the first cosmology, and that the “same” halo contributes at least 10% of the total halo contribution. If no such peak is found, then the match is declared unsuccessful. If more than one such peak is found, we select the one to which the matched halo contributes the most.
Fig. 7 shows the results of the above matching procedure between the fiducial and the low- models, for both high peaks (upper two panels) and medium peaks (lower two panels). Focusing on the high peaks first, there are 1,987 high peaks in the lower model, and 2,802 high peaks in the fiducial model. In the upper-left panel, we start with the peaks in the low- model, and show their matches in the fiducial model. In the upper-right panel, we reverse the direction, and start with the fiducial model. We find an “unambiguous” match (in the sense defined in the preceding paragraph) for 87.1% and 80.8% of the peaks, respectively. Proceeding to the two lower panels, we show the results for the medium peaks. There are 29,097 medium-height peaks in the low- model; only 56.2% of these have a matching peak in the fiducial model. Likewise, starting from the 27,556 medium-height peaks in the fiducial model (bottom panel), we find that 55.7% of these have a match in the low- cosmology.
In conclusion, most high peaks are matched to a peak in the other cosmology, although the peak values in the two cosmologies differ by an amount comparable to . In contrast, about half of the medium-height peaks do not have a clear match in the other cosmology. This is despite the fact that we use the same realization of the noise map in both cosmologies (i.e., we avoid creating an entirely different set of peaks by a different noise-realization), and despite our rather lenient definition of a “match”. We speculatively interpret this result as follows. Since the medium peaks are typically created by the sum of pure noise and many halos in projection, their existence and their amplitude are both sensitive to small changes in the spatial distribution and masses of these halos. In indirect support of the above conclusion, we have identified the following trend: on average, the proxy-halo contributes 49.6% for high peaks that have a match, vs. 33.8% for those that do not. Similarly, for medium peaks, the proxy halo contributes 25.1% vs. 18.9% for match vs. unmatched peaks. This shows that whenever the dominant halo accounts for a smaller fraction of the peak , it is more ”fragile” and is less likely to have a match in the other cosmology. Finally, we have found that when is increased, then the matched halo in the higher- cosmology has typically grown more massive, with an increase by 15% and 12% on average for matched and unmatched high peaks, 16% and 14% for matched and unmatched medium peaks. This trend, however, does not hold in the values of the matched peaks; as Fig. 7 shows, the peak heights have a significant scatter, but a relatively low bias, between the cosmologies. This could be explained by the fact that underdense voids become even more underdense when is increased, which tends to “cancel” the increase in caused by the fattening of halos. To be specific, we find a fractional bias and an r.m.s. scatter . when high peaks in lower model are matched to high peaks in fiducial model. In the reverse direction, the bias and scatter are -0.088 and 0.17. These results agree with the upper two panels in Fig. 7, showing a positive bias when is increasing, and a negative bias when is decreasing. The bias for the medium peaks is less clear than for the high peaks. In fact, the bias is positive in both matching directions, with the number for increasing (0.068) more positive than the number for decreasing (0.032). This is because the medium peaks are dominated by noise, rather than halos, and the peaks which increase in height because of the positive noise have a larger chance to survive as a peak than those that are hurt by the noise. The scatter for medium peaks is , about twice the scatter for high peaks.
|class||low- fiducial||fiducial low-|
|exit to low||3303||3420|
|stay in bin||7683||7515|
|exit to high||5373||4408|
Finally, we examine the “movement” of the peaks in height as is varied, in order to test our hypothesis, stated above, that an increase in tends to evacuate peaks from near the (or near the maximum of the peak-height distribution). For example, we divide the 29,097 medium-height peaks in the low- model into several cases. Approximately half (12,738) of these peaks are unmatched: they “disappear” when is increased (equivalently, these are peaks that “appear” when one starts from the fiducial model, and decreases ). The remaining 16,359 matched peaks are further divided into middle, lower and high cases, based on whether they remain in the original medium- bin (7,683), exit this bin toward higher (5,373) or to lower (3,303), in the other cosmology. These results, as well as the corresponding results in the reverse matching direction, are summarized in Table 3.
By examining the table, we conclude there is indeed a preferentially larger scatter in the direction out of the “medium” bin, when going from the lower to the fiducial model, compared to the reverse direction: . This table further reveals that there are two distinct reasons for the decrease in medium-height peaks. Approximately 2/3rd of the total decrease (of 29,097-27,556 = 1,541 1500 peaks), or 900 peaks, can be attributed to the above mentioned ”scatter” due to the increased density contrast – i.e. more peaks moving out of the bin than into the bin. The remaining 1/3rd of the decrease is due to losing peaks, i.e. 500 more peaks are destroyed than created, as is increased. Based on the preceding discussion, we speculate that this latter affect is caused by the projections of multiple halos, which can create and destroy the relatively low amplitude peaks.
In summary, the results in this section suggest that medium-height peaks are almost always dominated by pure galaxy shape noise, but they receive a significant contribution from collapsed halos, with typically 4-8 halos in projection along the LOS. The halos drive the cosmological sensitivity of these peaks in two ways: by (i) changing the amplitudes of the noise peaks, and by (ii) destroying and creating new peaks. Between these last two effects, in the case of , we found that the first is twice as important as the second.
iii.2 Comparison to Gaussian Predictions
Our next task is to examine whether (i) the statistics of the peaks, and (ii) their cosmology-sensitivity differs significantly from predictions in a Gaussian random field. The degree of any departure from a GRF is especially important to quantify for the medium peaks, since the results of the last section suggest that these are heavily dominated by pure Gaussian noise.
Our main results are shown in Fig. 8, which directly compare the peak counts in our simulated maps with those in a GRF. The GRF predictions are computed from the theoretical formula as discussed above, but using the (moments of) the power spectrum , , that were measured from the corresponding simulated maps. In each panel, we also compare the high-, fiducial, and low- models. In the bottom of each panel, we also show (i) the fractional difference between the GRF and the fiducial model and (ii) how the change in the peak counts between pairs of cosmologies is different in the GRF and our simulated maps. The source galaxies are assumed to be at , and all results shown in the figure include 1 arcmin smoothing. We plot the mean number of peaks in convergence bins of width , averaged over 1000 realizations. In the top two panels, we exclude noise from the maps; in the bottom two panels, noise is included. Finally, in the right two panels, we have scaled the convergence field by its r.m.s. value (these histograms use a bin width of ). This removes information that arises from alone. If the sole effect of changing was to change the heights of individual peaks by a constant factor, then this would result in a re-scaling of the peak-height probability distribution; the re-scaling by clarifies the relative importance of this effect.
As these figures show, in the noiseless case, the peak height distributions are very different from the Gaussian predictions and are reminiscent of the skewed one-point function of , which has a sharp drop at low demagnification, and a long tail to high magnification (e.g. ref. (6) and references therein). This correspondence of the high-tails is not entirely surprising; indeed, a pixel with a very high value is likely to mark a peak. When noise is added, the distributions near their peaks look much more similar to the Gaussian predictions. However, there is still a large non-Gaussian deficit of the lowest peaks (with the most negative ) and a clear excess of the highest () peaks. Importantly, however, there also remains a clear difference in the peak-height distributions even for the medium-height () peaks. Finally, as illustrated in the bottom insert in each panel, the cosmology-sensitivity of our peak histograms is also different from that in a GRF. These last points are encouraging, and suggest that the medium peaks do contain non-Gaussian information.
iv.1 Sensitivity to Cosmological Parameters
|noisy us||F and High-||-0.0028||0.0217||0.0407||0.0695|
|noisy sc||F and High-||0.2650||0.6682||1.3550||3.3013|
|noisy us||F and High-||-0.0050||0.0200||0.0383||0.0627|
|noisy sc||F and High-||0.4618||0.9950||1.5750||3.0556|
|noisy us||F and High-||-0.0019||0.0190||0.0347||0.0551|
|noisy sc||F and High-||0.2565||1.1450||2.4939||3.0368|
|Sim||F and High-||5.16||0.46||5.89||4.29|
|GRF||F and High-||10.65||0.23||5.87||3.16|
|Sim||F and Low-||5.01||0.34||5.09||3.67|
|GRF||F and Low-||9.93||0.16||4.98||2.58|
|Sim||F and High-||3.61||0.033||4.02||2.46|
|GRF||F and High-||7.68||0.014||3.77||2.01|
|Sim||F and Low-||4.39||0.053||4.44||2.56|
|GRF||F and Low-||8.79||0.043||4.08||2.11|
|Sim||F and High-||0.98||0.47||0.65||0.27|
|GRF||F and High-||0.93||0.017||0.46||0.14|
|Sim||F and Low-||0.44||0.27||0.36||0.16|
|GRF||F and Low-||0.54||0.004||0.26||0.08|
The number counts have been found (Paper I) to depend sensitively on a combination of (,). Here we vary and separately, in order to clarify the sensitivity to each of these parameters; we also consider variations in . We use , defined in Eq. (19) above, to measure the significance of the difference in the peak counts , caused by the changes in these parameters. We used fiducial and other cosmological maps to calculate the change in , but we used the control maps to compute the covariance matrix. Having 9 times more strictly independent realizations (45 control maps vs 5 realizations in the fiducial model) allows us to compute the covariance matrix more accurately. To isolate the sensitivity from beyond a change in the r.m.s. , we also compute the ’s between the scaled peak height distributions . In these analyses, we use five convergence bins whose locations are chosen by visual inspection, as explained above. Examples of bin boundaries we used are listed in Table 4.
Our main results are shown in Table 5, and can be enumerated as follows.
Raw cosmology sensitivity. The simulated noisy values in the unscaled maps are significant (), and suggest that the cosmological sensitivity of the peak counts is competitive with other methods (after scaling to the full size of an all-sky survey, such as LSST; this extrapolation is discussed further below). The sensitivity for is about an order of magnitude weaker () than for the other parameters. However, this is the case for other observables, such as the power spectrum, as well. As shown below (see Table 14 and related discussion) the peak counts and the power spectrum individually have similar sensitivities to all three parameters; they can furthermore be combined to improve the marginalized errors by a factor of two on all three parameters.
Can we “scale out” the cosmological information? By comparing the scaled and unscaled cases in the noisy maps, we see that scaling the maps by the variance reduces the values only by a modest amount. In these maps, only a small fraction of the parameter-sensitivity arises through changes in . Interestingly, the situation is different in the raw, noiseless maps. Nearly all of the sensitivity in these maps are attributable : the values diminish significantly after the scaling. This result is somewhat counter-intuitive, and implies that there is a “non-linear” interaction between noise and physical structures. More precisely, the result can be re-stated as follows: before adding noise, the cosmology-induced changes are very similar to a uniform ’stretching’ of the peak height distribution along the -axis. However, once the noise is added, the cosmology-induced changes are no longer described by such stretching. In hindsight, this is not entirely surprising: given that noise has almost no effect on the highest peaks, and has increasingly larger impact on the lower peaks, it is to be expected that the addition of noise spoils the “linear” stretching.
The impact of noise. We find, furthermore, that the addition of the noise increases the values for and for , while for varying , noise hurts. This is similar to a result we found in Paper I, namely that noise increases the change in the number of peaks. While in the unscaled maps, the increase in is modest, in the scaled maps, the increase is very significant (the interpretation of this is already explained in the preceding paragraph). This result – i.e. that the addition of pure noise helps increase the – is also somewhat counter-intuitive, and will be discussed in detail in § IV.5 below.
Which peaks drive the sensitivity? To answer this question, we calculate values separately for peaks with low, medium and high amplitudes. In the scaled maps, the low range is chosen to be below -0.8. The medium range is wide, centered on the mode of the peak height distribution. The high range is defined to be above the height at which the peak-height distributions start to differ significantly in any given pair of cosmology. This is inferred visually from logarithmic scaled peak height distributions, such as those shown in the right two panels in Fig. 8,and typically falls at , or . On the unscaled maps, we use the same boundaries as above, converted to values using the fiducial model. Both in the low and high ranges, we adopt two (adjacent) bins, and in the medium range, we use either two or three bins (depending on where the peak-height distributions in a given pair of cosmologies cross). Table 6 summarizes the bin boundaries.
|noisy low||F and High-||-0.2697||-0.031||-0.0250|
|noisy medium||F and High-||0.0150||0.0214||0.0390||0.0460|
|noisy high||F and High-||0.1070||0.1240||1.4000|
|noisy low||F and High-||-0.2697||-0.031||-0.0250|
|noisy medium||F and High-||0.0156||0.0313||0.0469|
|noisy high||F and High-||0.1000||0.1150||1.4000|
|low||F and High-||0.62||0.035||0.32||0.025|
|medium||F and High-||2.55||0.024||3.30||0.87|
|high||F and High-||2.85||0.069||2.13||0.36|
|frac.||F and High-||1.17||0.28||0.98||0.29|
|low||F and Low-||0.34||0.05||0.21||0.02|
|medium||F and Low-||2.92||0.04||3.00||0.62|
|high||F and Low-||1.73||0.09||1.27||0.32|
|frac.||F and Low-||1.00||0.52||0.88||0.26|
|low||F and High-||0.53||0.009||0.23||0.01|
|medium||F and High-||1.53||0.004||2.65||0.70|
|high||F and High-||1.30||0.01||0.94||0.08|
|frac.||F and High-||0.93||0.68||0.95||0.33|
|low||F and Low-||0.36||0.01||0.25||0.025|
|medium||F and Low-||2.16||0.007||3.00||0.70|
|high||F and Low-||1.04||0.003||0.79||0.093|
|frac.||F and Low-||0.81||0.36||0.91||0.32|
We show the resulting values from each type of peak in Table 7, for models varying and . As the difference caused by is small, we do not discuss it here. We also list the ratio , where the numerator refers to the values calculated here, and the denominator to the total computed above from the entire range. Even though the low/medium/high ranges we use are disjoint, this ratio can exceed unity (if in the original , the bins were non-ideally placed). The table shows that in the noisy maps, by far the largest contribution comes from peaks in the medium range. These are followed in importance by the high and the low peaks. We also see that in the noisy unscaled case, the low, medium and high ranges together account for essentially all () of the total unscaled . In the scaled case, they add up to a smaller fraction () of the total.
How robust are the results? One may ask whether the values (e.g. listed in Table 5) are robust under changes of the random realizations of the underlying maps. This is a potential concern especially when the values are low. We used our control maps to re-compute both the covariance matrix, and the change in , and to see how the values change. We found values of are very stable, and change by . For , the change is , and for the smallest , (occurring in scaled noiseless maps), the change is . Our finite number of realizations is therefore only adequate to give an order-of-magnitude estimate of the distinction between these pairs of maps. We note that increasing the number of bins also changes the values (increasing them by ; see section IV.7), but this change is systematic, and does not influence our conclusions.
iv.2 Distinction from a Gaussian Random Field
We next turn to the question of whether the cosmology sensitivity offers information beyond a pure GRF. We know that the high peaks are non-Gaussian, whereas the medium peaks, which drive the sensitivity, appear to follow the GRF predictions more closely (though still visibly deviate from them, even in the noisy maps).
We begin by directly quantifying the difference between the GRF and the simulated peak-height distributions (shown in Fig. 8). In Table 8, we show the values between the maps and the corresponding Gaussian predictions. In order to isolate the non-Gaussian effects in the mean peak counts, the covariance matrix is evaluated in the fiducial model (i.e. the covariance matrix in the GRF is not used). The large numbers in this table reveal that overall, the peaks are highly non-Gaussian, even in the noisy case. Similar to Table 7, we study the significance of the non-Gaussianity separately for low, medium and high peaks. We chose the ranges and the bin boundaries by the same procedure as described for Table 7. Our results are summarized in Table 9, and quantify the expectation that both the medium and high peaks differ significantly from the GRF predictions, even in the noisy maps (although the noiseless maps are more non-Gaussian). In the noisy maps, the significance of the non-Gaussianity for the low peaks is relatively low, but this is likely a result of the relatively small number of these low peaks. The table also shows that the three disjoint regions together account only for about 20-30% of the total . This implies that the peak height distribution departs from the GRF prediction everywhere (and no values are unimportant for the total ).
The above demonstrates that one can (easily) tell the difference of each map from a GRF. We next ask whether the cosmology-induced changes also differ from those in the Gaussian case. To answer this, we first calculate the between a pair of cosmologies, using the expectation values of the GRF peak counts in both cosmologies, computed from Eq. (10), with derived from the simulated maps in the corresponding models. These results are listed below the cosmological values in Table 5. In order to isolate non-Gaussian effects in the mean peak counts from those in the covariance matrix, in these GRF calculations, we again use the covariance matrix from the simulated fiducial model. Therefore, these GRF values do not represent the absolute distinguishability of the two GRF maps; they are meant only to be compared to the ’s from the corresponding cosmological simulations.
As the comparisons of two adjacent rows in the table shows, the noisy ’s in the simulations are generally close, overall, to the corresponding values predicted in the GRF. This, of course, does not necessarily mean that the information is the same as in a GRF - indeed, we found above that the mean counts deviate from the GRF predictions even in the noisy case. However, the cosmology-induced differences in the peak counts are, apparently, similar in magnitude to that in a GRF. When we repeat the GRF calculations with the covariance matrix adopted from the mock GRF maps, we find that, typically, the values increase by about a factor of 4. We find that in the GRF case, the standard deviation in the peak counts in each bin is close to Poisson shot noise . In our maps, the fluctuations are typically larger, by up to a factor of 4, which explains the corresponding reduction in the . Interestingly, the variance in the 3D space density of clusters has been found to exceed Poisson noise by a similar factor (23), providing physical intuition for this result.
Finally, comparing the simulated and GRF–predicted values in the noiseless case, we see that non-Gaussianity reduces the significance between cosmologies before scaling, but increases the significance after scaling. This result makes sense: the peak counts in the Gaussian case follow an almost strict linear scaling with , hence much of the difference disappears after such scaling (although the linear scaling is not, in fact, exact, see Eqs. (10)-(11)).
iv.3 Information Beyond the Power Spectrum
In the last section, we found that the peak height distribution is very different from the expectation in a GRF with the same power spectrum, especially for high peaks. While this is encouraging, we next study directly how much information is beyond the power spectrum. This is important, since in general, a random field can be non–Gaussian, but could still be fully characterized by its power spectrum. For example, one can imagine that the 3–point (and higher order) correlation functions are pre-specified functions of the power spectrum.
We here measure the 2d convergence power spectrum directly from the map produced in each realization, and treat it as another observable, in addition to the peak height distribution. The factor is included to make the observables in each bin close in magnitude (which helps make matrix inversion more stable). To be more specific, we first computed the power spectrum in 200 equal–sized finer bins, with width . The power was evaluated by taking the Fourier transform of the convergence field and averaging the power in each finer bin in the radial direction. To calculate the ’s, we considered only the range , and divided this range into 5 equal-sized bins (linear in ), and assigned each of the 200 finer bins into one of these 5 bins (for computing marginalized errors, we use 15, rather than 5 bins; see below). Using these five equally-spaced bins, we evaluate the mean within each bin , as well as the elements of the new covariance matrix. is then computed between pairs of cosmological models, analogous to Eq. (19) for the peak counts.
In order to find the information beyond the power spectrum, we first calculate the using the power spectrum alone. We then combine the five peak counts and the five power spectra into a vector of 10 observables, and compute the cross–terms, , to obtain the elements in the off-diagonal blocks of the covariance matrix. This allows us to calculate from the combination, taking into account their correlations.
|Peak Counts ()||Fiducial||5.16||5.89|
|Power Spectrum ()||and||17.06||8.12|
|Peak Counts ()||Fiducial||5.01||5.09|
|Power Spectrum ()||and||13.03||5.76|
|Peak Counts ()||Fiducial||3.61||4.02|
|Power Spectrum ()||and||17.69||6.15|
|Peak Counts ()||Fiducial||4.39||4.44|
|Power Spectrum ()||and||16.49||5.61|
|Peak Counts ()||Fiducial||0.98||0.65|
|Power Spectrum ()||and||0.92||0.29|
|Peak Counts ()||Fiducial||0.44||0.36|
|Power Spectrum ()||and||0.64||0.19|
The results are shown in Table 10. Comparing the individual ’s first, we see that in the noisy maps, the sensitivity of peak counts is roughly comparable to the power spectrum, although about weaker for and , and about twice stronger for . In the noiseless maps, the power spectrum is more sensitive, especially for and . This shows, interestingly, that the power spectrum sensitivity is much more degraded by noise than the peak counts. This is not surprising, given that the constraints from the power spectrum are dominated by linear fluctuations on relatively large scales (29); (30), with noise adding linearly to the large-scale structure signal. In other words, unlike for peak counts, adding noise does not change the signal (i.e., the difference between two cosmologies), as long as the galaxy noise is independent of cosmology, whereas the noise increases the variances .
Inspecting next the combined