Improving three-dimensional mass mapping with weak gravitational lensing using galaxy clustering
Key Words.:Gravitational lensing:weak – (Cosmology:) large-scale structure – (Cosmology:) dark matter – Methods: data analysis
Context:The weak gravitational lensing distortion of distant galaxy images (defined as sources) probes the projected large-scale matter distribution in the Universe. The availability of redshift information in galaxy surveys also allows us to recover the radial matter distribution to a certain degree.
Aims:To improve quality in the mass mapping, we combine the lensing information with the spatial clustering of a population of galaxies that trace the matter density with a known galaxy bias (defined as tracers).
Methods:We construct a minimum-variance estimator for the 3D matter density that incorporates the angular distribution of galaxy tracers, which are coarsely binned in redshift. Merely all the second-order bias of the tracers has to be known, which can in principle be self-consistently constrained in the data by lensing techniques. This synergy introduces a new noise component because of the stochasticity in the matter-tracer density relation. We give a description of the stochasticity noise in the Gaussian regime, and we investigate the estimator characteristics analytically. We apply the estimator to a mock survey based on the Millennium Simulation.
Results:The estimator linearly mixes the individual lensing mass and tracer number density maps into a combined smoothed mass map. The weighting in the mix depends on the S/N of the individual maps and the correlation, , between the matter and galaxy density. The weight of the tracers can be reduced by hand. For moderate mixing, the S/N in the mass map improves by a factor for . Importantly, the systematic offset between a true and apparent mass peak distance (defined as -shift bias) in a lensing-only map is eliminated, even for weak correlations of .
Conclusions:If the second-order bias of tracer galaxies can be determined, the synergy technique potentially provides an option to improve redshift accuracy and completeness of the lensing 3D mass map. Herein, the aim is to visualise the spatial distribution of cluster-sized mass peaks. Our noise description of the estimator is accurate in the linear, Gaussian regime. However, its performance on sub-degree scales depends on the details in the galaxy bias mechanism and, hence, on the choice of the tracer population. Nonetheless, we expect that the mapping technique yields qualitatively reasonable results even for arcmin smoothing scales, as observed when this technique is applied to the mock survey with two different tracer populations.
The weak gravitational lensing effect is a well-established tool to infer properties of the projected large-scale matter distribution (e.g. Munshi et al., 2008; Schneider, 2006a, b). These therein exploited coherent shear distortions of distant galaxy images (defined as sources) result from the continuous deflection of light bundles by the intervening fluctuations in the large-scale gravitational field, which are most prominent and detectable around galaxy clusters. The lensing distortions probe the total matter content in the Universe, which makes them an excellent tool for studying the dark matter component, an essential ingredient of the standard cosmological model of cold dark matter with a cosmological constant (CDM, e.g., Dodelson, 2003).
The shear distortion pattern can be translated into a map of projected matter fluctuations. Early non-parametric mapping algorithms, which were refined later to obtain optimised methods for finite fields, achieved this only on the basis of a catalogue of source angular positions and ellipticities (e.g. Kaiser & Squires, 1993; Seitz & Schneider, 2001). With the advent of distance indicators of galaxies in wide field galaxy surveys, the purely geometric relation between shear magnitude and source (and lens) distance was incorporated into a new three-dimensional (3D) lensing algorithm to also recover information on the radial distribution of matter (Hu & Keeton, 2002; Bacon & Taylor, 2003; Simon et al., 2009; VanderPlas et al., 2011; Leonard et al., 2012). The best studied methodologies so far utilise linear inversion techniques, such as Wiener filtering or a radial matter-density eigenmode decomposition with a suppression of low signal-to-noise (S/N) modes. Owing to the relatively sparse and noisy sampling of the survey area with background sources, however, the resulting maps are usually very noisy, and significant detections are basically restricted to mass peaks of a galaxy cluster scale that has only moderate redshift accuracy. Moreover, the linear inversion utilises a radial smoothing with a broad smoothing kernel that (a) smears out localised peaks in a radial direction and (b) biases the peak distances (known as -shift bias; Simon et al. 2009), which potentially renders the resulting maps hard to interpret. To attain more realistic 3D maps, the radial elongation of peaks inside the map can be mended by regularising the inversion (Leonard et al., 2012), or by finding the maximum likelihood positions of one or a few individual mass peaks along the line-of-sight (l.o.s.) given the radial smoothing kernel and radial density profile in the map (Simon et al., 2012). However, this does not alleviate the principle problem of noisy maps and inaccurate peak distances. It merely provides more realistic estimators for the 3D mass map. Moreover, the noise properties of the maps are likely to be complex in regularised, non-linear methods.
On the other hand, galaxy positions themselves are also tracers of the 3D matter density field and could therefore be employed to add extra information to the matter density maps that are obtained from 3D lensing. However, there are two complications here: (i) galaxies trace the matter density field only up to a systematic mismatch, which is generally dubbed galaxy bias, and (ii) a sampling by galaxy positions is affected by shot-noise (e.g. Dekel & Lahav, 1999; Martínez & Saar, 2002). The strategy of this paper is to refine the minimum-variance estimator in Simon et al. (2009) (STH09 hereafter) for the 3D matter density by adding the galaxy clustering information to the map making process. Since the minimum-variance estimators (Zaroubi et al., 1995) require second-order statistics of the input data to be specified, only the second-order bias parameters of the galaxy tracers have to be known (Gaussian bias or linear stochastic bias; Dekel & Lahav 1999). The galaxy bias as a function of scale and redshift could in principle be acquired in a self-consistent approach from the data by using lensing techniques (Schneider, 1998; van Waerbeke, 1998; Pen et al., 2003; Fan, 2003; Jullo et al., 2012; Simon, 2012), or with lesser certainty from simulations (Yoshikawa et al., 2001; Somerville et al., 2001; Weinberg et al., 2004). We therefore assume that it is basically known. The galaxy noise covariance within the minimum-variance estimator takes care of the galaxy sampling shot-noise. The outline of this paper is as follows. The Sections 2 and 3 present the details of the algorithm and a formalism to quantify its noise properties. We discuss the algorithm in the context of an idealised survey and then apply it to simulated data. In Section 4, we give details of the fiducial survey and the mock data. The results on the expected performance of the algorithm are presented in Section 5 and discussed in the final Section 6.
2 Independent reconstructions
We first consider the reconstruction of the matter density field and galaxy-number density field separately. The next section combines both into one 3D mass map.
2.1 Matter density on lens planes
We briefly summarise here the formalism already presented in STH09. We adopt the exact notation that is employed therein. For more details, we refer the reader to this paper.
We split the source catalogue into sub-samples where a redshift probability distribution (p.d.f.) is known. The complex ellipticities (Bartelmann & Schneider, 2001) of the sources belonging to the th sub-sample are binned on a 2D grid that covers the field-of-view of the survey area. This ellipticity grid is denoted by the vector , whose elements are the sorted pixel values of the grid. Every sub-sample uses the same grid geometry. The paper assumes that the weak lensing approximation is accurate enough for the lensing catalogue on the whole. That is, for the given source redshift and in the l.o.s. direction , the complex ellipticity, , is an unbiased estimator of the shear distortion, ,
where denotes the intrinsic (unlensed) complex ellipticity of a source image. Moreover, we assume a flat sky with a Cartesian coordinate frame.
We slice the light-cone volume, where the matter distribution is reconstructed, into slices. Within the slices we approximate the matter density contrast as constant along the line-of-sight. Every grid pixel defines a solid angle associated with a l.o.s. direction . Thus, the fluctuations of the matter density field inside a slice are fully described by the angular distribution of mean density contrasts on a plane (lens plane) and the width of the slice. The matter density contrast on a lens plane, , is binned with the same angular grid as the source ellipticities. We represent the grids, and , as vectors of equally ordered pixel values. We refer to a particular pixel by , where is the position of the pixel on the sky. Therefore, our algorithm represents the 3D-matter density contrast as an approximation by a discrete set of lens planes, which numerically limits the radial resolution, and a discrete set of pixels on the sky, limiting the angular resolution. The complete sets of ellipticity planes and lens planes are combined inside vectors of grids:
respectively. The brackets, which group together the vector arguments, should be understood as big vectors that are obtained by piling up all embraced vectors on top of each other.
In the weak lensing regime, the (pixelised) lensing convergence in the lowest-order Born approximation is the weighed projection of the density contrast on the lens planes:
where the coefficients express the response of the th convergence plane to the density contrast in the th lens plane. Namely,
The function denotes the p.d.f. of sources in comoving distance of the th source sub-sample, and sets the comoving radial boundaries of the th matter slice. We use for the Hubble radius and for the (comoving) angular diameter distance. The projection from a grid vector in -space to a grid vector in -space is hence denoted by the operator that is acting on .
The next step connects the convergence planes to the shear planes by a convolution of the lensing convergence on the grid
which introduces the operator to map to the corresponding shear plane (Hu & Keeton, 2002). In this sense, performs a linear transformation from - to -space.
Using this compact notation, we express the linear relation between the matter density (contrast) on the lens planes and the observed, binned ellipticity planes as:
Here, an additional vector denotes the binned intrinsic ellipticties of the sources of all source sub-samples. In the language of lensing, we consider this the noise term that dilutes the shear signal .
As the only input, the minimum-variance filter requires the signal covariance , which specifies the presumed two-point correlation between pixel values of on the lens plane(s) and the noise covariance , which quantifies the shear pixel noise variance and the correlation of noise between different pixels. Pixels that contain no sources have infinite noise. For the signal covariance, correlations between pixels that belong to different lens planes are set to zero. We note here that the signal covariance does not need to be the true signal covariance in the data, although the reconstruction may be sub-optimal as to map noise when it is not.
The signal covariance determines the degree of smoothing in the 3D map. The smoothing is uniquely defined by the linear transformation
and can be utilised for a comparison of the map to a theoretical matter distribution by (Simon et al., 2012). The radial smoothing is characterised by a radial point-spread function (p.s.f.) of the filter (STH09). After smoothing with the radial p.s.f., a peak in the true matter distribution does not necessarily peak at the same distance on average as in the smoothed map, which gives rise to the so-called redshift bias or -bias. Inside the filter, the constant tunes the level of smoothing by rescaling the noise covariance.
From a practical point of view, the Wiener filter consists of a series of linear operators that is applied step-by-step from the right to the left on the grids (Appendix B of STH09). Within this process, the signal covariance, , is a convolution or, equivalently, a multiplication in Fourier space of Fourier modes, , of the th lens plane with the angular signal power spectrum, , which is implicitly defined by
We approximate the power spectrum by using Limber’s equation in Fourier space:
where , is the Fourier transform of the pixel window function, is the 3D matter-density power spectrum at radial distance for wave-number , and is Dirac’s delta function (Kaiser, 1992). We denote the Fourier transforms of flat fields, , on the sky by , which is defined by
2.2 Galaxy numbers densities on lens planes
To improve the information in the 3D matter map and to possibly alleviate the -shift bias, we add the information gained from galaxy positions, which also probe the matter distribution (defined as tracers).
In this section, however, we first visit the problem of mapping the spatial galaxy number densities. For this purpose, we estimate the number density of galaxies projected onto the previously defined lens planes. Hence, we slice the full true 3D galaxy distribution into distance slices with distance limits . The galaxies are counted within each slice and angular grid pixel of the solid angle . Thereby, we receive the galaxy number density in the l.o.s. direction of the th slice, where is the number of counted galaxies. We compile the galaxy-number density values inside a grid vector , and we then arrange all grids inside a vector of grids:
This number density distribution of galaxies is what the following scheme seeks to recover from a galaxy sample with inaccurate distance information. Towards this goal, we split the observed galaxy sample utilising their redshift estimators, , into sub-samples with known radial p.d.f. ; denotes the redshift corresponding to . By projecting the th sample onto a 2D grid on the sky, one obtains the observed number density distribution
where flags mask pixels ( for mask), and
is the probability that a galaxy inside belongs to the slice . Owing to the redshift errors and masking, the observed distribution on the lens planes, , does not exactly match the true distribution . Therefore, denotes the expected fraction of galaxies on the th lens plane that is mapped onto the grid . Because of masking, the total number of galaxies is not necessarily conserved; that is . By a proper arrangement of the elements inside a matrix , the effect of on the entire 3D grid can be written as
We presume that galaxies sample an underlying smooth galaxy number density by a discrete Poisson process (e.g., Martínez & Saar, 2002). Therefore, the observable galaxy counts sample the underlying galaxy number density up to shot-noise, which is here formally expressed by the noise component .
By analogy with the matter density , we can find an minimum-variance filter to estimate the true distribution of galaxies on the lens planes; namely
As before, is the signal covariance, which is the angular clustering two-point correlation function of the galaxies on the lens planes, and denotes the shot-noise covariance. The degree of smoothing by the Wiener filter is tunable by using , which does not need to equal the parameter in Eq. 9. For the Poisson shot-noise covariance, we adopt a diagonal noise covariance, for , with for unmasked grid pixels , and infinite noise otherwise. The Wiener filter in the given form requires the inverse noise covariance, such that elements with infinite noise on the diagonal are zero. By , we denote the estimated mean number density of galaxies in pixel of the th sub-sample (see next section).
As for the matter density Wiener filter, a practical implementation of the Wiener filter in Eq. 19 consists of a series of linear operations applied to . The effect of is to multiply every angular mode of the th lens plane with the prior galaxy power spectrum , which we define relative to the matter power spectrum using the galaxy bias factor (e.g., Tegmark & Peebles, 1998):
where denotes the true mean number density of galaxies on the th lens plane. For this definition of the bias factor, the shot-noise contribution to the galaxy power spectrum is excluded as it is already accounted for in .
The angular bias factor is related to the 3D bias factor , where is the comoving 3D wave-number, by a projection that is approximated by Limber’s equation:
is given by Eq. 12. For this approximation, we assume that the number density of galaxies stays constant as function of inside the slice.
2.3 True mean galaxy numbers
The true galaxy number densities in Eq. 20 have to be derived from the data itself. For an estimator of , we go back to Eq. 17, which relates the observed number of galaxies, , to the true number on the lens planes, . For an ensemble average of this relation, we expect
wherein all elements equal the same number owing to the statistical homogeneity of the galaxy-number density fields, hence
Summing over all pixels with in total of the th tracer sample yields
averages over the area of the grid. Inverting the former equation, gives
For an unbiased estimator of on the right hand side, we insert the observed galaxy number densities, which is . The value of , which is utilised for the noise covariance in the foregoing section, is computed from Eq. 23 and the estimated .
In the simple case of negligible redshift errors, we find , where denotes the Kronecker symbol. In this case, we consequently find
for the number of unmasked pixels. Moreover, we find for a number of galaxies within the th sub-sample and a survey area . Thus, the galaxy number density is scaled up by to account for the mask.
However, the estimator in Eq. 26 has one caveat, since is basically a convolution of with the redshift error of galaxies. A deconvolution through possibly results in oscillating and negative values for . We therefore regularise Eq. 26 by a constrained solution of that maximises the likelihood:
under the condition that for all . We determine this solution numerically. The additional covariance can be used to give different weights to the observed values, such as by weighing the number of galaxies in each galaxy sample in order to account for the galaxy shot-noise. For equal weights, we simply set .
3 Combined reconstruction
In this section, we combine the information on the 3D matter density in the lensing data and the galaxy distribution.
3.1 Minimum-variance estimator
Up to now, we have considered the galaxy number density and matter density fields separately. However, contains information about and vice versa, as galaxies trace the matter distribution to a certain degree. On a statistical level, this relation is reflected by a non-vanishing cross-correlation,
for pairs of pixels on the same lens plane, which has not entered our formalism thus far. Slices are thought to be wide enough, such that correlations between pixels belonging to different lens plane are negligible.
We combine the - and -grids inside one new vector,
where the combined noise vector is
In this compact notation, the action of a matrix
on a product vector is defined as
In this sense, the projection matrix is
Following the usual assumptions of a minimum-variance filter, the optimal filter for estimating from in this combined problem is
which uses the short-hand notations,
The galaxy shot-noise and the intrinsic ellipticities of the sources, which are comprised in , are assumed to be uncorrelated. By choosing different tuning parameters , the impact of the Wiener smoothing can be adjusted independently for the matter and galaxy map.
The novelty of the combined reconstruction is that tracer number and matter density maps exchange information, if the cross-correlation matrix is non-vanishing. In a practical implementation of the filter 37, we apply step by step linear operations to the grids stored inside as before. As with the previous operators and , the application of amounts to a multiplication of angular grid modes with the cross-correlation power spectrum, , determined by
(See the next section for details on the implementation.) We define with respect to the matter power spectrum by employing the galaxy-matter cross-correlation factor (Tegmark & Peebles, 1998). The angular function is approximately related to the 3D correlation factor according to
To understand the mode of operation of the minimum-variance filter in Eq. 37, it is instructive to recast it into the mathematically equivalent form:
where . Step-1 involves no Wiener smoothing to construct the maps; no matrix is involved in this step. As this is usually too noisy, we apply an additional smoothing to these maps by virtue of the Wiener filter in Step-2. This filter linearly combines and averages pixel values in the maps based upon the expected S/N in the unbiased maps. It is Step-2, the analogue of the matrix in Sect. 2.1, that introduces biases into the maps, especially through a radial smoothing. Moreover, only Step-2 formally mixes pixels from the mass map and the tracer number density map by means of the off-diagonal matrix . Therefore, Step-1 makes independent mass and tracer maps that are only later combined in Step-2, according to our prior knowledge of their correlation. Setting results in a unity matrix for Step-2 or no smoothing.
Analogous to a lensing-only reconstruction, the Wiener filter thus applies a radial and transverse smoothing to the map to increase the signal-to-noise ratio. The smoothing makes the maps biased estimators of the matter and galaxy-number density fields. The smoothing is, however, uniquely defined by
which and can be applied to theoretical maps of the matter and galaxy number density for a quantitative comparison to the data.
3.2 Fourier space representation
For shear and galaxy number noise homogeneous over infinite grids with no gaps, the estimator in Eq. 37 takes a simple form in Fourier space. Under these idealistic conditions, the angular modes of all lens planes combine to
which are only linear functions of the - and -modes of the same ; there is no mixing between modes of different . Therefore, a reconstruction is then done most easily in Fourier space by
where are the observable input grids. The tuned covariance matrix of the (homogeneous) noise is
where is the mean source number density of the th source sample (out of in total ); is their intrinsic shape noise variance, and is the Poisson shot-noise power (white noise). Possible noise contributions owing to intrinsic alignments of sources are ignored here, hence has no off-diagonal elements. Furthermore, one has
where (Kaiser & Squires, 1993). For , we set . Here, does not depend on . The signal covariance is