Issue 
A&A
Volume 514, May 2010



Article Number  A79  
Number of page(s)  19  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/200912854  
Published online  26 May 2010 
A fitting formula for the nonGaussian contribution to the lensing power spectrum covariance
J. Pielorz  J. Rödiger  I. Tereno  P. Schneider
ArgelanderInstitut für Astronomie (AIfA), Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
Received 8 July 2009 / Accepted 10 December 2009
Abstract
Context. Weak gravitational lensing is one of the most
promising tools to investigate the equationofstate of dark energy.
In order to obtain reliable parameter estimations for current and
future experiments, a good theoretical understanding of dark
matter clustering is essential. Of particular interest is the
statistical precision to which weak lensing observables, such as
cosmic shear correlation functions, can be determined.
Aims. We construct a fitting formula for the nonGaussian part
of the covariance of the lensing power spectrum. The Gaussian
contribution to the covariance, which is proportional to the lensing
power spectrum squared, and optionally shape noise can be included
easily by adding their contributions.
Methods. Starting from a canonical estimator for the
dimensionless lensing power spectrum, we model first the covariance in
the halo model approach including all four halo terms for one fiducial
cosmology and then fit two polynomials to the expression found. On
large scales, we use a firstorder polynomial in the wavenumbers and
dimensionless power spectra that goes asymptotically towards
for
,
i.e., the result for the nonGaussian part of the covariance using
treelevel perturbation theory. On the other hand, for small scales we
employ a secondorder polynomial in the dimensionless power spectra for
the fit.
Results. We obtain a fitting formula for the nonGaussian
contribution of the convergence power spectrum covariance that is
accurate to
for the offdiagonal elements, and to
for the diagonal elements, in the range
and can be used for single source redshifts
in WMAP5like cosmologies.
Key words: cosmology: theory  largescale structure of Universe  gravitational lensing: weak  methods: numerical
1 Introduction
Weak gravitational lensing by the largescale structure, or cosmic shear, is an important tool to probe the mass distribution in the Universe and to estimate cosmological parameters. The constraints it provides are independent and complementary to those found by other cosmological probes such as cosmic microwave background (CMB) anisotropies, supernovae (SN) type Ia, baryon acoustic oscillations (BAO) or galaxy redshift surveys. The cosmic shear field quantifies the distortion of faint galaxy images that is induced by continuous light deflections caused by the largescale structure in the Universe (e.g., Schneider 2006; Bartelmann & Schneider 2001). Since this effect is too small to be measured for a single galaxy, large surveys with millions of galaxies are required to detect it in a statistical way. The cosmic shear signal has been successfully measured in various surveys, since the first detections of Kaiser et al. (2000); Bacon et al. (2000); Van Waerbeke et al. (2000); Wittman et al. (2000). Most recently, shear twopoint correlation functions were measured in the CanadaFranceHawaii Telescope Legacy Survey (CFHTLS) and were used to constrain the amplitude of dark matter clustering, , with uncertainty (Fu et al. 2008).
The next generation of galaxy surveys will greatly improve the precision with which weak lensing effects can be measured (Albrecht et al. 2006) enabling us to obtain, with accurate redshift information and tomographic measurements, precise constraints on the evolution of dark energy. However, the expected improvement in future data only leads to a significant improvement of the precision and accuracy of the cosmological interpretation if the systematic errors, the underlying physics, and the statistical precision of cosmic shear estimators are well understood. Systematics currently identified arise mainly from noncosmological sources of shear correlations, i.e., intrinsic alignments of galaxies (e.g., Schäfer 2009, for a review), and biases on the shear measurement (Massey et al. 2007; Semboloni et al. 2009). This paper addresses the issue of the statistical precision of cosmic shear estimators, determined by the covariance of the estimator. Since much of the scales probed by cosmic shear lie in the nonlinear regime, being affected by nonlinear clustering, the covariance depends on nonGaussian effects and has a nonGaussian, as well as a Gaussian, contribution. Indeed, even though the nonGaussianity of the shear field is weaker than that of the matter field due to the projection along the lineofsight, various studies indicate that the nonGaussian contribution to the covariance cannot be neglected when constraining cosmological parameters with weak lensing (Kilbinger & Schneider 2005; White & Hu 2000; Cooray & Hu 2001; Semboloni et al. 2007; Takada & Jain 2009; Scoccimarro et al. 1999).
Most cosmic shear results are based on the measurement of twopoint correlation functions of the shear field. Since, in general, the number of independent measurements is insufficient to infer the complete covariance directly from observations, one may derive it from raytracing maps of numerical Nbody simulations. This, however, requires a large number of realizations and, in addition, is very timeconsuming if an exploration of the covariance in the parameter space is needed. An alternative is to compute the covariance with an analytic approach. For shear twopoint correlation functions, Schneider et al. (2002) derived an expression for the Gaussian contribution to the covariance. Semboloni et al. (2007) fitted the ratio between that expression and a covariance computed with Nbody simulations, containing both Gaussian and nonGaussian contributions, providing thus a formula to compute the total covariance from the Gaussian term. In Fourier space, largescale modes are independent and, differently from real space, the Gaussian contribution to the covariance of the convergence power spectrum (i.e., of the Fourier transform of the twopoint shear correlation function) is diagonal and can be computed from the convergence power spectrum alone (Kaiser 1992; Joachimi et al. 2008), whereas the nonGaussian contribution can be computed from the trispectrum of the convergence (Scoccimarro et al. 1999). The trispectrum, on large scales, can be accurately derived in treelevel perturbation theory^{}, and, on small scales, is well represented by the onehalo term of a halo model approach. A nonGaussian part of the covariance consisting of a perturbation theory term and a onehalo term was used, e.g., in Takada & Jain (2009).
This paper aims at producing an accurate expression for the nonGaussian contribution of the covariance of the convergence power spectrum that is fast to compute, contributing thus to accurate estimates of cosmological parameters. Following Scoccimarro et al. (1999) and Cooray & Hu (2001), we start from a canonical estimator of the dimensionless convergence power spectrum and use it to derive an analytic expression for the corresponding covariance. The various spectra involved are evaluated using the halo model approach of dark matter clustering (Peacock & Smith 2000; Seljak 2000; Ma & Fry 2000; Scoccimarro et al. 2001; Cooray & Sheth 2002). The halo model approach assumes that all dark matter in the Universe is bound in spherical halos, and uses results from numerical Nbody simulations to characterize halo properties such as their profile, abundance and clustering behavior.
The evaluation of the covariance of the convergence power spectrum in the halo model approach is timeconsuming. In addition, it may be needed to repeat it for different cosmological models for the purpose of parameter estimation. To allow for a faster computation, we construct a fitting formula for the nonGaussian part of the convergence power spectrum covariance. On small scales, we fit the halo model result with a polynomial in the nonlinear dimensionless convergence power spectrum. On large scales, we fit the ratio between the halo model covariance and the perturbation theory covariance. We stress that it is a fit to the halo model covariance, not involving a covariance computed from Nbody simulations. The result is, however, calibrated by Nbody simulations, since they determine the halo model parameters.
The paper is organized as follows. We define in Sect. 2 the reference cosmology, considering the growth of matter perturbations. We introduce the convergence spectra, construct an estimator for the dimensionless convergence power spectrum, and derive an expression for its covariance in Sect. 3. In Sect. 4, we describe the halo model approach, and compute the covariance of the power spectrum. The covariance depends on the values of halo model parameters, which are also defined here. It also depends on the power spectrum, bispectrum and trispectrum of the correlations of halo centers. Expressions for these spectra, in the framework of perturbation theory, are given in the Appendix. Section 5 tests the accuracy of the halo model predictions, for both the power spectrum and its covariance, against two sets of raytracing simulations. Section 6 presents the fitting formula for the nonGaussian contribution to the covariance where its coefficients are given as function of source redshift. We conclude in Sect. 7.
2 Structure formation in a CDM cosmology
Throughout this work we assume a spatially flat cold dark matter model with a cosmological constant (
), as supported by the latest 5year data release of WMAP results (Komatsu et al. 2009). The expansion rate of the Universe,
,
in such models is described by the Friedmann equation
,
where
is the Hubble constant,
denotes the combined contributions from dark matter and baryons today in terms of the critical density
,
and
is the density parameter of the cosmological constant. The comoving distance to a source at a is then
where the scale factor is related to the redshift via the relation 1+z=1/a using the convention a(t_{0})=1 today.
In structure formation, the central quantity is the Fourier transform of the density contrast
,
which describes the relative deviation of the local matter density
to the comoving average density of the
Universe
at time t. We suppress the time dependence of
in the following. In this way, the mean density contrast is by
definition zero, and we can describe matter perturbations in the early
Universe as zeromean Gaussian random fields. In this case,
the statistical properties of the Fourier transformed density
field,
are completely characterized by the power spectrum
where is the ensemble average and denotes the Dirac delta distribution. Note that throughout this paper the tilde symbol is used to denote the Fourier transform of the corresponding quantity.
In linear perturbation theory, which is valid on large scales, the power spectrum at a scale factor a is characterized by
where the amplitude A is normalized in terms of , denotes the spectral index of the primordial power spectrum, and T(k) is the transfer function. Note that all Fourier modes of the matter density grow at the same rate, i.e., , where
is the growth factor which we normalize as D(a=1)=1. In the nonlinear regime, i.e., on small scales, different Fourier modes couple and the Gaussian assumption cannot be maintained. Thus we have to consider higherorder moments of the density field to describe its statistical properties. In perturbation theory, it is possible to find analytic expressions for these moments, which hold up to the quasilinear regime. In Appendix A, we derive the expressions for the bispectrum and trispectrum in treelevel perturbation theory, which are the Fourier transforms of the three and fourpointcorrelation functions, respectively.
3 Covariance of the convergence power spectrum
A central quantity in weak lensing applications is the twodimensional projection of the density contrast
on the sky, which is known as effective convergence
.
It is obtained by projecting the density contrast along the
backwarddirected lightcone of the observer according to
where denotes the redshiftdependent comoving distance, is the comoving distance to the horizon and the weight function G(w) takes into account the distribution of source galaxies along the lineofsight. We assume for simplicity that all background sources are situated at a single comoving distance , such that the weight function has the form
where H(x) denotes the Heaviside step function. To take advantage of the Fourier properties, we analyze the statistical properties of the Fourier counterpart of . For the theoretical consideration of the convergence power spectrum covariance we need the second and the fourthorder moments, as will become apparent later. They are defined by
where the subscript ``c'' refers to the connected part of the corresponding moment and is a sum of Fourier wavevectors. The convergence power spectrum and trispectrum are calculated using the flatsky and Limber's approximation (Bernardeau et al. 2002; Kaiser 1998; Scoccimarro et al. 1999):
where and are the corresponding threedimensional matter power spectrum and trispectrum (Fourier transform of the fourpoint correlation function).
We are interested in estimating the dimensionless convergence power spectrum
and the corresponding covariance for wavevectors of different length . A natural choice for the estimator of the dimensionless convergence power spectrum is (Takada & Bridle 2007; Cooray & Hu 2001; Scoccimarro et al. 1999)
which is unbiased in the limit of infinitesimal small bin sizes, since . Here, denotes the solid angle of a survey with a fractional sky coverage of , and the integration is performed over the Fourier modes lying in the annulus defined by , where is the width of the ith bin. We denote the integration area formed by the annulus as .
The evaluation of the covariance of the estimator, Eq. (13), results in an expression of the form
where is given by Eq. (12), and denotes the Kronecker delta. The second term in square brackets is the binaveraged convergence trispectrum,
where is the nonlinear convergence trispectrum as defined in Eq. (11). To derive Eq. (14), we made use of the definitions of the convergence power spectrum and trispectrum in Eqs. (8) and (9), and of the discrete limit of the delta distribution . The derived expression consists of two terms: a Gaussian part , which scales as the convergence power spectrum squared and only contributes to the diagonal of the covariance matrix (see, e.g. Joachimi et al. 2008), and a nonGaussian part , which scales as the dimensionless binaveraged convergence trispectrum and introduces correlations between the wavevectors of different bins (see, e.g. Cooray & Hu 2001; Scoccimarro et al. 1999). Both terms are inversely proportional to the survey area A, but have a different behavior with respect to the bin width . While the Gaussian term decreases with increasing bin size, the nonGaussian term is independent of the binning, since the bin area cancels out after the integration.
We can analytically perform one of the integrations of the binaveraged trispectrum in Eq. (15). First, note that it only depends on the parallelogram configuration of the convergence trispectrum, i.e., setting
,
and
in Eq. (11).
Also, if we choose an appropriate coordinate system for the integration
over the wavevectors, the problem becomes symmetric under rotations
and we can parametrize the convergence trispectrum by the length of the
two sides of the parallelogram
and
and the angle between them,
.
Hence, we define
Making use of the symmetry properties of this problem, one angular integration becomes trivial and the integration in Eq. (15) simplifies to
If the binwidth is sufficiently small ( ), the integration area is , and we can make use of the mean value theorem. In this way, we approximate the integral in Eq. (17) by an angular average
Note that if is independent of the angle between and , an approximation of the covariance can be calculated without having to perform an integration at all. In particular, this is the case for the 1halo term of the threedimensional matter trispectrum, as we will see later in Eq. (42).
4 Halo model
We have seen that the covariance of the dimensionless convergence power spectrum estimator consists of two terms: a Gaussian part, which is proportional to the dimensionless convergence power spectrum squared and a nonGaussian part, which is the binaveraged dimensionless convergence trispectrum (see Eq. (14)). We will compute these terms using the halo model approach (Peacock & Smith 2000; Scoccimarro et al. 2001; Ma & Fry 2000; Seljak 2000; see also the comprehensive review by Cooray & Sheth 2002).
4.1 Overview
With the assumption that all dark matter is bound in sphericallysymmetric, virialized halos, the halo model provides a way to calculate the threedimensional polyspectra of dark matter in the nonlinear regime. In Sect. 4.5 below, we summarize the equations one obtains for the dark matter power spectrum and trispectrum.
In the halo model description, the density field at an arbitrary position
in space is given as a superposition of all N halo density profiles such that
where denotes the density profile of the ith halo with center of mass at and is the normalized profile. By parametrizing the halo profile in this way, we assume that the shape of the ith halo depends only on the halo mass m_{i} and the halo concentration parameter c_{i}, which we define below. The dark matter polyspectra of the density field follow then from taking the ensemble averages, , of products of the density at different points in space. Assuming that the number of halos is , where is the average number density of halos and V the considered volume, we compute the ensemble averages by integrating over the joint probability density function (PDF) for the N halos that form the field (Smith & Watts 2005), i.e.,
where denotes the PDF. If one considers that position and mass of a single halo are independent random variables, the PDF factorizes as
where n(m) is the halo mass function and p(cm) is the concentration probability distribution for halos given a mass m.
4.2 Ingredients
The halo model approach provides a scaledependent description of the statistical properties of the largescale structure. On small scales, the correlation of dark matter is governed by the mass profiles of the halos, whereas on large scales the clustering between different halos determines the nature of the correlation. As there are a multitude of models to describe the behavior on different scales, and an even larger number of parameters one has to set judiciously, there exists no such thing as a unique halo model. In order to have reproducible results, it is therefore necessary to specify ones choice of parameters. For this work, we will adopt the following parameters for the halo model:
 1.
 The average mass of a halo is defined as the mass within a sphere of virial radius
as
,
where
denotes the overdensity of the virialized halo with respect to the average comoving mass density
in the Universe. Typically, values for
are derived in the framework of the nonlinear spherical collapse model (e.g. Gunn & Gott 1972). Expressions valid for different cosmologies are summarized in Nakamura & Suto (1997). In our
implementation, we use the results which are valid for a flat CDMUniverse, i.e.,
where . We find for our fiducial WMAP5like cosmology .  2.
 Nbody simulations suggest that the density profile of
a halo follows a universal function. We choose to use the
NFW profile (Navarro et al. 1997), which is in good agreement with numerical results and has an analytical Fourier transform. It is given by
where is the amplitude of the density profile and characterizes the scale at which the slope of the density profile changes. For small scales ( ) the profile scales with , whereas for large scales it behaves as . The Fourier transform of the NFW profile is
where , we truncated the integration at in the second step, and introduced the concentration parameter in the third step. Additionally, we use for the sine and cosine integrals the definitions
 3.
 The abundance of halos of mass m at a redshift z is given by
where we introduced the dimensionless variable ,(27)
where D(z) denotes the redshiftdependent growth factor, is the smoothed variance of the density contrast, and denotes the value of a spherical overdensity that collapses at a redshift z as calculated from linear perturbation theory. In our work, we use the expression from Nakamura & Suto (1997), which is valid for a CDM Universe,
where . The quantity has only a weak dependence on redshift and we find for our fiducial model.The advantage of introducing is that part of the mass function can be expressed by the multiplicity function , which has a universal shape, i.e., is independent of cosmological parameters and redshift. In this work, we employ the Sheth and Tormen mass function (Sheth & Tormen 1999)
which is an improvement over the original PressSchechter formulation (Press & Schechter 1974). We use the parameter values p=0.3, q=0.707, and amplitude A(0.3)=0.322, which follows from mass conservation.  4.
 The concentration parameter
characterizes the form of the halo profile. From Nbody simulations one finds that the average, ,
depends on the halo mass (Bullock et al. 2001) like
where m_{*}=m_{*}(z=0) is the characteristic mass defined within the PressSchechter formalism as . In the following, we will use the values c_{*}=10 and as proposed by Takada & Jain (2003). This implies that more massive halos are less centrally concentrated than less massive ones. However, results from numerical Nbody simulations (Bullock et al. 2001; Jing 2000) indicate that there is a significant scatter in the concentration parameter for halos of the same mass. Furthermore, Jing (2000) proposes that such a concentration distribution can be described by a lognormal distribution
Typical values for the concentration dispersion range from to (Wechsler et al. 2002; Jing 2000). Note that the width of the distribution is independent of the halo mass. The variation of the halo concentration can be attributed to the different merger histories of the halos (Wechsler et al. 2002). We will analyze the impact of this effect on different spectra in Sect. 4.7. When we use only the mean concentration parameter, we have to replace the probability distribution of the concentration, needed for example in Eq. (21), by a Dirac delta distribution
 5.
 On large scales, the correlation of the dark matter density
field is governed by the spatial distribution of halos. Since the
clustering behavior of halos and matter density differ,
one introduces the bias factors b_{i}(m,z) such that
In this way, the halo density contrast, , is expressed as a Taylor expansion of the matter density contrast, . The bias parameters are in general derived based on the ShethTormen mass function introduced above. For the linear halo bias one obtains then
where p and q match the values used in the mass function. Expressions for higherorder bias factors can be found, e.g., in Scoccimarro et al. (2001). Since they only have a small impact on the quantities employed here, we take into account only the firstorder bias. In Fourier space we may then write
 6.
 To obtain the final correlation function, one has to perform integrations along the halo mass and optionally along the halo concentration, with limits formally extending from 0 to . In practice, we use the mass limits and . Masses smaller than give no significant contribution to the considered quantities, while, due to the exponential cutoff in mass, masses larger than are rare. For the concentration, we employ the integration limits and .
 7.
 Due to the cutoff in mass, the consistency relation (Scoccimarro et al. 2001)
does not hold. To cure this problem we consider a rescaled linear bias such that , where is the result of the integral in Eq. (36). In this way, one ensures that the halo term with the largest contribution to the correlation equals the perturbation theory expression on large scales (see Fig. 1).
4.3 Building blocks
Using the ingredients described in the previous section, it is
possible to define building blocks, which simplify significantly the
notation for expressing the polyspectra (Cooray & Hu 2001):
In the case i=0, we additionally define , for consistency.
Figure 1: Square configuration of the dimensionless convergence trispectrum against wavenumber for the range for . The upper panel displays the four individual halo terms, the sum of all four trispectrum halo terms (solid line) and the treelevel perturbation theory trispectrum , as indicated in the key. The lower panel shows the ratio between the indicated contributions and the complete trispectrum. The double dashed line illustrates the corresponding ratio of the approximation . Note that we consider the 4halo term only in the upper panel since it resembles the term on large scales. 

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4.4 Power spectrum
We can now compute the power spectrum from Eq. (20). The result consists of two terms, the 1halo and the 2halo terms,
(Seljak 2000). They are given by
where denotes the linear perturbation theory power spectrum, defined in Eq. (4), and we use the ingredients summarized earlieron. Note that, for convenience, we omit the redshiftdependence in the notation. Using the building blocks from Eq. (37), these terms can be written in the following compact form:
(40) 
The 1halo term, , denotes correlations in space between two points in the same halo, whereas the 2halo term, , takes into account correlations between two different halos. Hence, the 1halo term is dominant on small scales and the 2halo term is dominant on large scales. Note that the 2halo term converges to the linear power spectrum on large scales because of the consistency relation of the firstorder halo bias factor (see Eq. (36)) and of the limit for .
4.5 Trispectrum
We compute now the dark matter trispectrum in the halo model approach. As discussed in Sect. 3,
only parallelogram configurations of the trispectrum wavevectors
contribute to the covariance of the convergence power spectrum.
Restricting our
calculations to these configurations, we obtain four different halo
term contributions,
which are simpler than in the general case. In addition, we neglect terms involving higherorder halo bias factors since, on most scales, they provide only a small correction (Takada & Jain 2003; Ma & Fry 2000). We further note that the perturbative expansion of halo centers, used in the calculations, was shown to become inaccurate on nonlinear scales (Smith et al. 2007). The contributions to the trispectrum take the following forms, using the compact notation of the building blocks (see Cooray & Sheth 2002, for the expression of the halo model trispectrum including higherorder bias factors): The 1halo term, dominant on the smallest scales, is
The 2halo term has two contributions, , consisting of a term , which corresponds to correlations of three points within one halo and a fourth point in a second halo, and a term , which describes correlations involving two points in a first halo and the other two points in the second halo. They read,
=  
(43)  
=  (44) 
The 3halo term is given by
Finally, the 4halo term, dominant on large scales, is
and describes correlations of points distributed in four different halos. Note that, like for the power spectrum, the 2halo term is computed from the linear power spectrum. On the other hand, the 3 and 4halo terms depend on and , respectively, which are the lowestorder, nonvanishing, perturbation theory contributions to the bispectrum and trispectrum. Both spectra are derived in Appendix A.2.
4.6 Convergence spectra
The convergence power spectrum and trispectrum, needed to evaluate the covariance in Eq. (14), are computed by projecting and , according to Eqs. (10) and (11). Figure 1 shows the dimensionless convergence trispectrum, defined as for a square configuration where all wavenumbers have a length , where denotes the contribution from the corresponding halo term. The plot shows the individual contributions to the dimensionless (the projections of Eqs. (42)(46)), illustrating on which scales the individual terms are important, as well as the total contribution of all halo terms . We show, in addition, the dimensionless projected , which closely follows the 4halo term. We see that the commonly used approximation is accurate for large wavenumbers ( ) but has a deviation of about from the complete trispectrum for small wavenumbers ( ).
4.7 Stochastic halo concentration
The previous results were computed using the deterministic concentrationmass relation of Eq. (30). We now analyze the impact of scatter in the halo concentration parameter c on the covariance of the convergence power spectrum, using the stochastic concentration relation given by the the lognormal concentration distribution of Eq. (31).
Cooray & Hu (2001), analyzed the effect of a stochastic concentration on the threedimensional power spectrum and trispectrum and found that the behavior of the corresponding 1halo terms were increasingly sensitive to the width of the concentration distribution for smaller wavenumbers k. Furthermore, the effect of a stochastic concentration relation was more pronounced for the trispectrum than for the power spectrum, since the tail of the concentration distribution is weighted more strongly in higherorder statistics.
Performing the same analysis for the projected power spectrum and trispectrum, we find a similar trend as in the threedimensional case, but with a smaller sensitivity to the concentration width of the distribution on small scales. For , we find, in the case of the 1halo term of the power spectrum, a deviation from a deterministic concentration relation of about for wavenumbers larger than , whereas, for the 1halo term of the trispectrum, the deviation is of the order of in the same range. Thus, when considering the covariance of the convergence power spectrum, one should take into account the concentration dispersion in the 1halo term of the trispectrum but can safely neglect it for the power spectrum. Additionally, we find that a stochastic concentration has only a small impact on the 2halo terms of the power spectrum and trispectrum (Pielorz 2008).
From this analysis, we expect the effect of a concentration
distribution to be the strongest on the nonGaussian part of the
covariance, which depends on the trispectrum. To directly infer
the
impact of a concentration distribution on the covariance, we calculate
the 1halo contribution to the nonGaussian covariance, i.e., we
perform the bin averaging of Eq. (15),
for two different concentration dispersions , using our halo model implementation. Figure 2 shows the ratio
where denotes the deterministic concentration relation, for two values in two contour plots. In agreement with the previous results, the largest impact of the concentration dispersion occurs for wavenumbers larger than . For , the deviation of the covariance from the original deterministic concentration relation is small, of about . The effect becomes nonnegligible for , where we find a deviation of (for ) to (for ).
Figure 2: Contour plots of the ratio between the nonGaussian covariance 1halo term contribution (see Eq. (48)) computed with a concentration dispersion and with a deterministic concentration, against wavenumbers ranging from to . The left panel is for concentration dispersion , whereas in the right panel . 

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5 Comparison with Nbody simulations
In Sect. 4, we computed the power spectrum and trispectrum of the dark matter fluctuations. Projecting them according to Eqs. (10) and (11), and inserting the result in Eqs. (12), (14), and (15), we obtained the covariance of the convergence power spectrum estimator.
To test the accuracy of the halo model predictions for the statistics of the dark matter density field, we compare the dimensionless convergence power spectrum and the corresponding covariance, calculated in the halo model approach, with results from two different raytracing simulations.
Table 1: Cosmological parameters used to set up the initial power spectrum.
Table 2: Parameters used for generating the two Nbody simulations.
5.1 Virgo and Gems simulation
For our comparison, we chose one simulation from Jenkins et al. (1998) and ten simulations from Hartlap et al. (2009), which we denote in the following as Virgo and Gems simulation, respectively. The Virgo simulation was carried out in 1997 by the VirgoConsortium for a CDM cosmology (see Table 1) with particles in a periodic box of comoving side length (see Table 2). It uses the PP/PMcode HYDRA, which places subgrids of higher resolution in strongly clustered regions. Structures on scales larger than can be considered as well resolved. The Gems simulations were set up in cubic volumes of comoving side length with 256^{3} particles (see Table 2). Note that the simulations employ either the BBKS (Bardeen et al. 1986) or the EH (Eisenstein & Hu 1998) transfer function. The cosmology chosen reflects the WMAP5 results (Komatsu et al. 2009) and thus has a lower value for than the Virgo simulation (see Table 1). It uses the GADGET2 code to simulate the evolution of dark matter particles (Springel 2005) and has a softening length of .
5.2 Raytracing
The output of numerical simulations are threedimensional distributions of particles in cubic boxes over a range of redshift values. In order to compare the results with the predicted convergence power spectrum from the halo model, we make use of the multiplelensplane raytracing algorithm (see e.g., Jain et al. 2000; Hilbert et al. 2009) to construct effective convergence maps. The basic idea is to introduce a series of lens planes perpendicular to the central lineofsight of the observer's backward light cone. In this way, the matter distribution within the light cone is sliced and can be projected onto the corresponding lens plane. By computing the deflection of light rays and its derivatives at each lens plane, one simulates the photon trajectory from the observer to the source by keeping track of the distortions of ray bundles. In this way, the continuous deflection of light rays is approximated by a finite number of deflections at the lens planes. As a result, one obtains the Jacobian matrix for the lens mapping from source to observer and can construct convergence maps.
For both simulations, a similar number of around 200 effective convergence maps were produced, with source galaxies situated at a single redshift of either or . Table 2 summarizes the parameters used for producing the resulting convergence maps with the multiplelensplaneraytracingalgorithm. These are: the side length, , of the cubic simulation box, the number of particles, , used for the simulation, their mass, , and the number of available convergence maps, , with area .
The maps produced with the Gems simulation have an area of 16 , while the ones from Virgo are much smaller, with 0.25 .
Figure 3: Dimensionless convergence power spectrum against wavenumber for galaxy sources at redshift ( upper panels) and ( lower panels). Points with error bars show measurements from the two sets of numerical simulations: Gems (left plots) and Virgo (right plots). Both simulations use a similar particle setup but differ in the area of the maps (see Tables 1 and 2). They are compared with the corresponding results obtained with fitting formulae from Smith et al. (longdashed line), PeacockDodds (shortdashed line), and the halo model predictions for a deterministic halo concentration (solid line). See text for a discussion. 

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5.3 Convergence power spectrum
To test the accuracy of the halo model approach in describing the nonlinear evolution of dark matter, we compare the dimensionless projected power spectrum, computed in the halo model approach, to the ones estimated from the numerical Nbody simulations.
The dimensionless convergence power spectra of the simulations are
measured from the realspace twodimensional convergence maps of
length
and gridsize
.
For this, we first apply a Fast Fourier Transform^{} to obtain
on each gridpoint. Then, we estimate the power spectrum at a wavenumber
for the kth convergence map by averaging over all Fourier modes in the band
}, i.e.,
where denotes the number of modes in each band of width . Finally, the ensemble average of the dimensionless power spectrum is obtained by averaging over the results of the various convergence maps, i.e., . The error bars for the power spectrum estimate are computed from the dispersion over the convergence maps used, and are due to sample variance. Since modes with a length similar to the side length of the convergence map are only poorly represented, the sampling variance is largest for small . The effect is stronger for the Virgo simulation than for the Gems simulation, since the Gems convergence maps have a much larger area .
Figure 3 shows a good agreement between the halo model and the Nbody simulation convergence power spectra. We also include, in Fig. 3, the convergence power spectra computed using the fitting formulae of Peacock & Dodds (1996) and Smith et al. (2003). Both, the halo model and the fitting formulae, show a better agreement with simulations for the lower source redshift case. On small scales, the halo model and the Smith et al. fitting formula agree well with the simulations, whereas the PeacockDodds fitting formula has too little power, in particular in the case of the Gems simulation. On intermediate scales ( ), the halo model is less accurate than the Smith et al. fitting formula, suggesting that the halo model suffers from the halo exclusion problem on these scales, as described, e.g., in Tinker et al. (2005). This means that, while in simulations halos are never separated by distances smaller than the sum of their virial radii, this is not accounted for in the framework of our halo model and is probably the cause for the deviation. The good agreement of the Smith et al. formula is not surprising, since it is based on simulations of similar resolution than the ones we consider here. In contrast, a similar comparison using the convergence power spectrum estimated with the Millennium Run (Hilbert et al. 2009) clearly favors the halo model prediction over of the two fitting formulae, with both fitting formulae strongly underestimating the power on intermediate and small scales.
The good overall accuracy of the halo model results was expected since its ingredients, such as the mass function and the halo profile, were obtained from Nbody simulations. We note that, in this analysis, we used a deterministic concentration parameter, since the use of a stochastic one has only a small effect on the small scales of the convergence power spectrum.
5.4 Covariance of the convergence power spectrum
The similarity between simulation and halo model power spectra was to some extent expected. The ability to make an accurate description of higherorder correlations provides a stronger test of the halo model. Due to its important role in calculating the error of the power spectrum and for parameter estimates, we focus in this section on the accuracy of the covariance of the dimensionless convergence power spectrum.
We need an appropriate estimator for the power spectrum covariance of the simulations. As each simulation provides
different maps, we have
realizations
of the power spectrum. From these we estimate the covariance by
applying the unbiased sample covariance estimator which has
for our purpose the form:
where is the projected power spectrum estimate of the kth effective convergence map at a wavenumber (see Eq. (49)). We evaluate for both, Virgo and Gems simulations, for the case .
The halo model results, , are calculated as described in Sect. 4 and include all terms of the nonGaussian covariance. In their computation, we use our fiducial halo model with the ingredients summarized in Sect. 4.2, and use the cosmological parameters values corresponding to each simulation, given in Table 1, for the case . We consider both, a deterministic concentrationmass relation and a stochastic one, with dispersion for the 1halo term of the trispectrum.
Figure 4: Relative difference (see Eq. (51)) between the halo model prediction for the covariance of the convergence power spectrum and the results from the Virgo simulation ( ) against wavenumbers . The binning scheme is linear with a constant bin width of ranging from to . The left panel displays the full covariance, including all four halo terms for the trispectrum, and uses a deterministic concentrationmass relation denoted by . The right panel illustrates the same covariance but with a stochastic concentration distribution of width for the 1halo term of the trispectrum. 

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Figure 5: Relative difference (see Eq. (51)) between the halo model prediction for the covariance of the convergence power spectrum and the results from the Gems simulation ( ) against wavenumbers . The binning scheme is linear with a constant bin width of ranging from to . The left panel displays the full covariance, including all four halo terms for the trispectrum, and uses a deterministic concentrationmass relation denoted by . The right panel illustrates the same covariance but with a stochastic concentration distribution of width for the 1halo term of the trispectrum. 

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We compare the halo model with the simulations covariance matrices considering their relative deviation
We note that although the number of available convergence maps for our simulations () is too small to obtain a percentage level accuracy for the covariance estimate (Takahashi et al. 2009), we assume that the covariance from the simulations is the reference one, and put it in the denominator of Eq. (51). The comparisons are displayed in Figs. 4 and 5 for the Virgo and Gems simulation, respectively. In the case of the Virgo simulation with , the best agreement is a relative deviation of found on intermediate scales between and . For small scales the halo model underestimates the simulation by or more. On large scales, i.e., for wavenumbers , the halo model overestimates the simulation by around . However, since the simulation covariances suffer from a large sampling variance due to the small size of the convergence maps, the comparison is not meaningful on these scales. The agreement improves for , in this case there is a larger range of scales where the deviation is small. For the Gems simulation the agreement is much better. There is now an interval of scales ( ) where the deviation is in the range . Throughout all the scales probed, the deviation on the diagonal of the covariances is around . However, for offdiagonal components, at small scales, the agreement is still poor. Finally, the improvement of considering is less significant than in the Virgo case.
Although the halo model predictions strongly deviate from the simulation estimates of the covariance on small scales, this analysis does not necessarily imply a poor accuracy of those predictions. Indeed, the simulation covariances estimated in this analysis have a strong scatter. In particular, in the case of the Gems simulation, we found from bootstrap subsamples of 50 convergence maps that the resulting covariances can deviate up to from the average covariance of the complete sample (Pielorz 2008). This is in agreement with the results by Takahashi et al. (2009) who need to use 5000 simulations to obtain an estimate of the matter power spectrum covariance at a subpercent level accuracy.
There are however very recent indications that Eq. (14) indeed underestimates the covariance of the convergence power spectrum on small scales (Sato et al. 2009). In particular, sample variance in the number of halos in a finite field is not accounted for by Eq. (14). Indeed, the mass function yields a mean number density, but there are fluctuations on the number of halos due to the largescale mass fluctuations. Sample variance in the number of clusters in a volumelimited survey (Hu & Kravtsov 2003) has been considered in cluster abundance studies (e.g., Vikhlinin et al. 2009). In the halo model framework, the sample variance in the number of halos was derived in Takada & Bridle (2007) and its contribution to the covariance of the convergence power spectrum was found, in Sato et al. (2009), to boost the nonGaussian errors of a 25 survey by one order of magnitude on scales . The increase is reduced for larger survey areas.
6 Fitting formula for the covariance of the convergence power spectrum
Future weak lensing surveys will provide much more precise measurements of the convergence power spectrum. In order to obtain robust constraints on cosmological parameters, accurate estimates of both the power spectrum and its covariance are needed.
6.1 Methodology
The number of measured power spectra is, in general, insufficient to infer the complete covariance directly from observations. One has thus to derive it either from raytracing maps of numerical Nbody simulations or with an analytic approach. A drawback of the first method is that it requires a large number of realizations and, in addition, is very timeconsuming if an exploration of the covariance in the parameter space is needed.
In the previous sections, we derived the covariance, and in particular its nonGaussian part, with an analytic approach. This computation is, however, timeconsuming and it might be useful to obtain an accurate covariance in a faster way. A first approach would be to rely on stronger approximations. For example, a commonly used approximation consists on evaluating the nonGaussian covariance from , instead of using the full trispectrum. We saw in Sect. 4 that this approximation recovers the full trispectrum for scales but deviates by on scales for square configurations (compare also with Fig. 6).
An alternative approach that we consider in the following, is to find a fitting formula for the halo model covariance that can be subsequently used without the need for implementing the halo model. We will provide a fit only for the nonGaussian part of the halo model covariance, since the Gaussian part only depends on the nonlinear convergence power spectrum and can thus be accurately computed without relying on the halo model. The inclusion of nonGaussian errors increases the total covariance and one might think of fitting the ratio between the nonGaussian and Gaussian terms. However, this is not a good quantity to fit, since the Gaussian contribution is diagonal and binningdependent. In contrast, a similar ratio was fitted in the real space, where the Gaussian term is nondiagonal and binningindependent, using a nonGaussian contribution measured from Nbody simulations (Semboloni et al. 2007). In Fourier space, there is some relevant analogous information contained in the treelevel perturbation theory trispectrum. Indeed, pursuing the analogy with the realspace fit, is a nondiagonal and binningindependent quantity, with a lower amplitude than the full trispectrum, approaching it at large scales. We thus compute , the covariance predicted in treelevel perturbation theory on large scales (see Eq. (A.27) and Appendix A for a detailed derivation) and compare it with our covariance computed with the full halo model trispectrum, , defined in Eq. (14). The ratio between the two covariances is shown in Fig. 6. In agreement with Fig. 1, the covariance predicted by treelevel perturbation theory contributes only to the complete nonGaussian covariance along the diagonal. On very large scales , the ratio lies between 1.1 and 2. On smaller scales, decreases fast and is no longer useful for fitting purposes.
Figure 6: Ratio against wavenumbers . The covariance in treelevel perturbation theory resembles the nonGaussian covariance on very large scales (i.e., ) to approximately except along the diagonal and the closest offdiagonals. Here a good approximation of the nonGaussian covariance requires at least one additional halo term. For wavenumbers , an accurate description of the nonGaussian covariance requires at least the 1halo term. 

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This discussion motivates us to use two different fitting formulae to
model the nonGaussian covariance over the whole range of scales. On
large scales (
), we model the ratio
as a polynomial in the wavenumbers,
,
and the dimensionless nonlinear convergence power spectrum
.
More precisely, we assume
where and which ensures the symmetry property . In this way, for large scales as predicted by the halo model (see Fig. 6). On smaller scales, we use the fact that the degree of nonGaussianity is expected to depend on the amplitude of the dimensionless power spectrum; if the latter is small, only small nonGaussianities may be expected. In Eq. (52), the nonlinear clustering on these scales is thus encoded in the firstorder polynomial in wavenumbers and power spectra.
On small scales, i.e. for
,
we model directly the nonGaussian covariance amplitude using a secondorder polynomial in the power spectrum, such that
The scaledependence of the nonlinear regime of the covariance of the threedimensional power spectrum is also to some extent captured in powers of the nonlinear power spectrum^{}. This assumption is weaker when dealing with projected spectra; due to the redshift projection of the Limber equation, such a relation is not so tight for the corresponding twodimensional spectra. We verify that nevertheless considering only terms up to secondorder in the power spectrum in Eq. (53) already produces an accurate fitting formula in this range of scales, allowing for a better fit with a smaller number of parameters than by using an arbitrary function of wavenumbers.
To ensure a smooth transition from small to large wavenumbers, we
consider a linear combination of the two matrices defined in Eqs. (52) and (53) with weightings of
thirdorder in the wavenumbers, such that the full nonGaussian covariance becomes
with the transition scale fixed at . We use thus a model with nine free parameters, which is a compromise between expressive power and simplicity, and perform the fit using the range . More precisely, we find the coefficients for the final fit formula by performing two least square fits to Eqs. (52) and (53), respectively. To obtain the coefficients for small (large) scales we computed the corresponding covariance as described above within the halo model approach with 61 bins in the range ( ) and employed the condition ( ) for the fit.
The covariances used in the fitting procedure, both treelevel perturbation theory and halo model covariance, depend on perturbation theory polyspectra. These were evaluated from the expressions derived in Appendix A using a linear matter power spectrum computed with the EisensteinHu transfer function (Eisenstein & Hu 1998) and assuming the WMAP5like fiducial model shown in Table 3. The nonlinear convergence power spectrum, used in the polynomial expressions, was evaluated from the same linear power spectrum. In addition, the halo model nonGaussian covariance was evaluated using the input parameters as described in Sect. 4.2, including a stochastic concentrationmass relation with for the 1halo term of the trispectrum.
Table 3: Fiducial model used for the fitting procedure.
Table 4: Bestfit values for the parameters (p_{0},p_{1},p_{2}) of the redshiftfit for the set of coefficients .
Figure 7: Coefficients a_{i} of the fitting formula obtained for various source redshifts. Left (right) panel shows the coefficients of the large (small) scales fitting formula. For convenience, scaled versions of the coefficients are shown, as indicated in the key. Note for example that a_{5}, a_{7} and a_{8} change sign. The lines show the redshift fit, Eq. (55), with the parameters given in Table 4. They provide a good fit for (indicated by the vertical line). 

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6.2 Redshift dependence
All quantities,
,
,
and
,
were evaluated for several values of source redshifts (assuming a single source redshift plane), in the range
.
For each redshift, we perform the fit and find a set of bestfit values for the coefficients
.
Next, we fit the bestfit values of each coefficient as a function of
redshift. Some of the bestfit values are increasing functions of the
source redshift, while others decrease with redshift, and others still
are nonmonotonic. They all are, however, monotonic in the redshift
range of interest for current and future weak lensing surveys,
.
In this range, the bestfit values
are accurately fitted with
Table 4 shows the resulting values of the 27 parameters, which define our fitting formula for the halo model covariance of the convergence power spectrum. To obtain these results we use the 3parameter fit as defined in Eq. (55) which is valid in the range . The behavior of the coefficients with redshift is shown in Fig. 7. We note that a few of them change sign, with most remaining always positive. In addition, their amplitudes may be quite distinct, since they are applied to quantities of quite different amplitudes, such as wavenumber, power spectrum or power spectrum squared. The coefficients of the fitting formula for large scales, Eq. (52), all decrease with redshift in absolute values. This compensates the increase of the power spectrum with redshift, allowing for a decrease of the ratio with redshift, as expected.
Figure 8: Accuracy of the fitting formula for . Upper left panel: relative deviation , defined in Eq. (56). Other panels: diagonal cuts through the fitted nonGaussian covariance, showing the covariance on the diagonal ( upperright panel), for ( lowerleft) and for ( lowerright). All three panels show the fitted and the halo model nonGaussian covariances (as well as the Gaussian for the upperright panel) as function of the lowest scale in the upper part, and the deviation in percent in the lower part, where the dashed lines mark the level. For the Gaussian covariance we employed a logarithmic binning with 61 bins in a range from to . 

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6.3 Accuracy of the fitting formula
To test the performance of the fitting, we calculate the relative
deviation between the nonGaussian covariance computed with the fitting
formula and the halo model one. The deviation,
is computed for every wavenumber pair and shown in Fig. 8, for the case of . The upper left panel of Fig. 8 shows the absolute value of the deviation for each element of the covariance matrix, while the other panels show the deviation along cuts through diagonals of the covariance.
The fit works quite well on the offdiagonal elements that are close to the diagonal, showing an average overestimation of . When moving along any offdiagonal, from larger to smaller scales, (lower panels of Fig. 8), we move from the first fit, Eq. (52), where the deviation is mostly positive, to the second one, Eq. (53), where the deviation is mostly negative. The transition occurs at the local maximum, which indicates that none of the fitting formulae should be extrapolated to the other region. The fits break down on the smallest scales, with the deviation increasing very rapidly when the largest scale reaches . Smaller scales would clearly require higherorder terms in Eq. (53).
As we move away from the diagonal the fit gets increasingly worse, in particular the deviations are larger than in the region shown in black in the upper left panel of Fig. 8, which correlates very small with very large scales. The reason for this is that this region was effectively not fitted, since it is not contained in any of the two blocks fitted by Eq. (54). Restricting to the range where both scales are between , roughly of the elements show deviations between and , with the average of the absolute deviations being . This range contains also of outliers where the deviations are larger than . The outliers occur in the lowamplitude correlations between the largest and smallest scales.
The fit is worse on the diagonal than on the first offdiagonals (where ). On the diagonal, the fit always underestimates the covariance. Scales in the range show a deviation between and , with an average of . The accuracy degrades at larger scales, which is not a problem since for the nonGaussian contribution to the total covariance is negligible, as seen in Fig. 8 (upper right panel). Adding the Gaussian contribution to the fitted nonGaussian one, the underestimation in the diagonal elements is always better than , with an average of .
In summary, the fitting formula for the cosmic variance, including Gaussian and nonGaussian contributions, has an average accuracy of in the offdiagonal and in the diagonal. It is valid when both scales are in the range , corresponding to in real space. This is roughly the range used in the latest results from CFHTLSWide (Fu et al. 2008). This range includes the scales where nonGaussianity is relevant, i.e., where the cosmic shear error budget is both dominated by cosmic variance and has important contributions from nonlinear clustering (see Sect. 6.4).
6.4 Impact of the accuracy of the fitting formula on parameter estimations
We study the impact of the fitting formula accuracy on the estimation
of cosmological parameters, using a Fisher matrix approach. For this,
we need to take into account not only the Gaussian and nonGaussian
contributions to the covariance, but also the noise in the observed
power spectrum. In practical applications, the convergence field
in Eq. (13)
is obtained
from the observed ellipticities of the source galaxies. The intrinsic
ellipticity field (i.e., in the absence of a gravitational
lensing effect) is assumed to have zero mean and rms of
per
component. This shape noise contaminates the observed power spectrum.
Assuming that the intrinsic ellipticities of different galaxies do not
correlate, the shape
noise contribution to the covariance of the power spectrum is diagonal
and given by the new terms arising in Eq. (14) when replacing the power spectrum, in that expression, by the observed one defined as (Kaiser 1992),
where n is the number density of source galaxies.
We consider the following three surveys: a mediumdeep weak lensing survey covering an area of with , like the current CFHTLSWide, a wider survey of similar depth covering with , like the planned Dark Energy Survey (DES)^{}, and a wide and deep survey with and , like the proposed E UCLID^{}. For all three we assume and compute the covariance using the halo model and the developed fit formula in the range . Additionally, we add the covariance of a Gaussian contribution using a logarithmic binning with 61 bins in a range from to . The wider surveys will measure correlations on scales larger than , which we do not consider here for comparison purposes. Note also that for very large scales the flatsky approximation breaks down.
Figure 9: Left panel: relative contributions to the diagonal of the convergence power spectrum covariance. The binningdependent Gaussian to nonGaussian ratio (solid line; see caption of Fig. 8 for the employed binning scheme) and the shape noise to cosmic variance ratio for two surveys with and (dotted), are shown as function of multipole . The ticks at the intersection of the line with the ratio curves enclose the approximate range where the nonGaussian contribution is important for the E UCLIDlike survey. Right panel: Fisher ellipses in the plane ( contours) calculated with the halo model covariance (solid) and the fitted covariance (dashed) for the DESlike (large ellipses) and E UCLIDlike (small ellipses) surveys. Compared to the halo covariance case, the fitted covariance error ellipses are decreased (enlarged) by () for the DES (E UCLID) case. 

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The left panel of Fig. 9 compares the different terms contributing to the diagonal of the covariance, by showing the Gaussian to nonGaussian ratio and the shape noise to cosmic variance ratio, where by cosmic variance we denote the sum of the Gaussian and nonGaussian contributions. The ratio between the Gaussian and nonGaussian terms is independent of the survey and, for our particular choice of binning, nonGaussianity starts to affect the diagonal around , where its amplitude is of the Gaussian amplitude, and dominates from onwards. Shape noise, including both pure shape noise and the coupling with cosmic variance, also becomes important on small scales, but in a surveydependent way. It is as large as the cosmic variance on for surveys with 10 (40) galaxies per arcmin squared. The vertical lines in Fig. 9 (left panel) show, for the E UCLIDlike survey, the range where the nonGaussian contribution to the diagonal is nonnegligible, i.e., where it accounts for more than of the cosmic variance while having an amplitude of at least of the shape noise. This range is roughly , or approximately . This is a rough estimate of the minimum range where the fitting formula is required to have a good accuracy.
In addition, the accuracy of the offdiagonal terms is crucial, since
the nonGaussianity is the sole contribution there. To evaluate
the required range of validity of the fitting formula,
in a way that includes the offdiagonal elements and is
independent of bin width, we define the signaltonoise ratio (Takada & Jain 2009),
For each survey, we compute the signaltonoise ratio (SNR) using all scales between , the largest scale where the fitting is valid, to successive values of . The SNR increases with , as more scales are included in Eq. (58). The increasing rate, however, decreases with increasing , tending to zero when additional scales do not carry additional cosmological information. The value of for which this saturation occurs increases with decreasing shape noise, and gives a good indication of the range where the fitting formula is required to have a good accuracy in order to get accurate estimates of cosmological parameters. We find that, when increasing from to , the SNR increases by a factor of 1.03 for the CFHTLSlike survey and by a factor of 1.09 for the E UCLIDlike survey. Hence, even for the latter there is no much gain in reaching .
We consider now the Fisher information matrix, which to first order,
neglecting the cosmology dependence of the covariance matrix, is
given by
where the derivatives are taken w.r.t. a set of cosmological parameters . The Fisher matrix defines the error ellipsoid in parameter space, with yielding the marginalized error on , and giving the error on assuming the other parameters are perfectly known.
We compute Eq. (59) for both the fit and halo model covariances, for each of the three surveys. We perform the derivatives at the fiducial model of Table 3, varying only 2 cosmological parameters . For each survey, we compare the areas of the two error ellipses (which defines the inverse of the figureofmerit) thus obtained. For all three cases, there is a good agreement between the two ellipses.
For the two cases with large shape noise, CFHTLS and DES, we find that the fitting formula underestimates the Fisher ellipse, as compared to the halo model covariance. This is expected because with an enhanced diagonal the correlations between bins are weaker, and the result is dominated by the accuracy of the diagonal, where the fitting formula underestimates the covariance, as we saw earlier on. The deviation is however weak, the areas of the ellipses obtained using the fitting formula are smaller than the halo model result, and the deviation is uniformly distributed on the parameter space (see Fig. 9, right panel, large ellipses). This implies that the deviation on the marginalized constraints is much smaller, and we find that the fitting formula underestimates the errors on both and by only , for both surveys. This corresponds to a deviation of of the parameters values, for CFHTLS (DES). In contrast, for the E UCLID case, where the covariance has larger correlations, the fitted covariance produced an ellipse slightly larger than the halo model one, by about (see Fig. 9, right panel, small ellipses).
6.5 Covariance of realspace estimators
For practical purposes it is sometimes more convenient to study
realspace correlations rather than correlations in Fourier space. We
therefore define an estimator of a general secondorder cosmic shear
measure which is related to the convergence power spectrum
estimator by
where W(x) is an arbitrary weight function. A wellknown example of this equation are the shear twopoint correlation functions and with weight functions and , respectively (see discussion in Joachimi et al. 2008). Using the definition Eq. (60), we find for the relation between the covariance of the realspace estimators to the covariance of the Fourierspace estimators:
=  
(61) 
which is related to the covariance of the dimensionless power spectrum used in the fitting formula by . Inserting the result of the dimensionless power spectrum covariance given in Eq. (14) yields
(62) 
We find that the Gaussian part of the realspace covariance is independent of the binning scheme and is nondiagonal in contrast to the covariance in Fourier space.
7 Conclusions
We present a fitting formula for the halo model prediction of the nonGaussian contribution to the covariance of the dimensionless power spectrum of the weak lensing convergence. The formula was constructed assuming a CDM cosmology with WMAP5like cosmological parameters. In particular, it was obtained for , and other parameter values as shown in Table 3. It is valid for a scale range of , corresponding to in real space and can be used for surveys with galaxy source redshifts . In this range, it reproduces the results of a full implementation of the halo model approach, with a scaleaveraged accuracy of in the offdiagonal, and in the diagonal elements. The formula also allows us to recover the halo model error ellipses within . The range of validity of the formula and its level of accuracy render it applicable to low shape noise scenarios from next generation weak lensing surveys.
To use the formula, shown in Eq. (54), one needs three quantities:
 The nonGaussian contribution to the covariance of the convergence power spectrum in treelevel perturbation theory , shown in Eq. (A.27). This involves the computation of the convergence trispectrum in treelevel perturbation theory, Eq. (A.26), which requires the calculation of the linear power spectrum and of the F_{2} and F_{3} coupling functions (Eqs. (A.10) and (A.12)).
 The nonlinear convergence power spectrum.
 The 9 coefficients of the fit, which are obtained by inserting the 27 values given in Table 4 in Eq. (55), for the required redshift.
The work presented in this paper is based on the assumption that the halo model is a powerful approach to probe nonlinear clustering. We tried to test this assumption against results from Nbody simulations, but our comparisons were inconclusive. Indeed, such analysis requires raytracing simulations with both large number of convergence maps and large convergence map area. Only then it would be possible to minimize the effect of sampling variance in the simulations, which hindered our attempted tests. Such simulations would also allow us to consider error bars for the estimate of the simulation covariance. There are however indications that the halo model approach underestimates the nonGaussianity of the covariance and there are attempts to include additional contributions (Sato et al. 2009; Takada & Jain 2009).
The reliability of the halo model for higherorder polyspectra also needs to be studied in more detail. Some work in this direction are the analyses of the impact of the triaxiality of the halo profiles (Smith et al. 2006; Smith & Watts 2005), and halo exclusion effects (Tinker et al. 2005). Moreover, the issue of halo substructure (Dolney et al. 2004) and of the effect of a stochastic concentration parameter have to be understood properly. This paper also addresses this last issue. We analyze the impact of a stochastic concentration parameter on the covariance of the convergence power spectrum. We found that the effect can safely be neglected for the Gaussian contribution, with the convergence power spectrum varying only slightly, and at small scales, for concentration scatters of . For the nonGaussian contribution the effect is more pronounced due to the higher sensitivity of the trispectrum to a stochastic concentration relation. In the case of the 1halo term of the trispectrum, we find it useful to take into account a concentration dispersion of . The deviation to a deterministic concentrationmass relation is larger than for wavenumbers .
Although the fitting formula we obtained provides a more thorough estimate for the nonGaussian contribution to the power spectrum covariance than the earlier approximation of Semboloni et al. (2007) (global accuracy of along the diagonal, which becomes less for the offdiagonals), there is still room for improvements. One drawback of our approach is that it requires the computation of the convergence trispectrum in treelevel perturbation theory. Furthermore, it is only applicable to a small range of WMAP5like cosmologies and is only valid in the interval . A possible way to avoid these problems and extend the accuracy of the fitting formula might be to construct it entirely from its threedimensional counterpart, the threedimensional covariance of the matter power spectrum. This has the advantage that perturbations of different length scales are not additionally mixed due to projection effects, which might allow us to cover a wider range of cosmologies. The desired projected covariance could then be obtained by performing an additional integration along the redshiftspace. We will address this issue in a future paper.
AcknowledgementsThe authors thank Christoph Lampert for invaluable discussions, and Jan Hartlap for providing his raytracing simulations of the Gems and Virgo simulations. Together with Martin Kilbinger both provided useful comments to the manuscript. J.P. would like to thank the TRR33. J.R. is supported by the Deutsche Forschungsgemeinschaft under the project SCHN 342/71 within the Priority Programme SPP 1177 ``Galaxy Evolution''. I.T. is supported by the Marie Curie Training and Research Network ``DUEL''.
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Appendix A: Cosmological perturbation theory
On large scales, different Fourier modes evolve independently from
each other and thus conserve the Gaussian behavior of the density
perturbation field
.
It is therefore
convenient to work in Fourier space and Fourier transform the fields as
well as the nonlinear fluid equations (consisting of continuity, Euler
and Poisson equation) that describe their evolution in an
expanding Universe. Contrary to linear perturbation theory, there is a
coupling between different Fourier modes mediated by the coupling
function
on smaller scales. In this case, the fluid equations for the density contrast and the irrotational peculiar velocity field
in Fourier space are given by (e.g., Bernardeau et al. 2002)
where we introduced the two fundamental mode coupling functions
For an Einsteinde Sitter (EdS) cosmology it is possible to find a perturbative ansatz that separates the scale and time dependencies, whereas for a general CDM model it is impossible to find a separable solution to Eqs. (A.1) and (A.2). However, Scoccimarro et al. (1998) showed that it is possible to find a separable solution in any order if one makes an approximation that is valid at percentage level. One indeed finds then the same recursion relation as in the EdS case. The ansatz is then
Therefore, we find that the whole information on cosmological parameters is encoded in the growth function due to its dependence on the Hubble parameter (see Eq. (5)).
A.1 Coupling functions
The nth order density contrast and the divergence of the peculiar velocity in Eq. (A.4) is given by
and the nth order coupling functions F_{n} and G_{n} are obtained by the following recursion relations (Jain & Bertschinger 1994)
(A.7)  
(A.8) 
where and . The initial conditions for these recursion relations are and . To get the functions and that are symmetric in its arguments, one must perform the following symmetrizing procedure
where the sum is taken over all possible permutations of the set . These equations enable us to calculate the density contrast in the nth order of perturbation theory by using the iterative equations for the coupling functions.
The calculation of the secondorder coupling functions is straightforward. The result is
The thirdorder coupling function is given by
(A.11) 
where . Employing Eq. (A.9), we find the symmetric function
From now on the symmetry superscript `` (s)'' will be omitted because we will only deal with symmetric coupling functions. For the calculation of the trispectrum in the halo model approach as described in Sect. 4.5, one needs perturbation theory. More precisely, we need the subsequent components
=  
(A.13) 
where we have defined the difference vector and the sum of the vectors .
We already mentioned in the previous section that it is possible to
find a solution for an arbitrary cosmology if one makes a small
approximation. In the literature one can find closed solutions for the
second and thirdorder coupling functions. The secondorder coupling
function changes to
(A.14) 
where for (Bernardeau et al. 2002). For our fiducial choice of , we get . Thus, within a few percent correction to the first and last term, the coupling functions are independent of cosmological parameters.
A.2 Correlation functions
A.2.1 Bispectrum
The dark matter bispectrum is defined as
Since the connected bispectrum vanishes for Gaussian random fields, it is the first intrinsically nonlinear moment. Inserting the perturbative expansion (A.4) for each term results generally in an infinitely large sequence of correlators. The lowest nonvanishing order is the socalled treelevel contribution to the bispectrum. We find for the correlator in tree level
(A.16) 
Replacing the secondorder density contrast with Eq. (A.5) results in
=  
=  (A.17) 
where we applied Wick's theorem to express the fourpoint correlator of Gaussian fields in terms of products of power spectra, and performed the two integrations over the Dirac delta distributions. The results for the other two terms of the treelevel bispectrum are simply obtained by permutations of the arguments. Finally, the treelevel bispectrum is given by
The factor 2 follows from using the symmetrized version of the secondorder coupling function.
A.2.2 Trispectrum
The dark matter trispectrum is defined as the connected fourpoint function in Fourier space:
where . We find that there are two different nonvanishing contributions to the tree level:
=  
(A.20) 
In total we find 6 terms of the first type and 4 terms for the second type, where the rest is obtained by permutations. All other contributions either vanish or are built of higherorder terms. Note that for the second type of terms we need the results from perturbation theory up to the third order. The calculation of each term is a tedious but straightforward calculation. We obtain for the first term of the expansion
(A.21) 
where the sixpoint correlator resolves into 15 terms consisting of power spectra products. Performing the integrations over the arising delta functions yields in the end 8 different terms. Similarly, we find for the second type of terms
(A.22) 
The other terms are easily obtained by permutations, however, we present here the complete result to avoid confusion with a shorthand notation that is introduced afterwards. The trispectrum of cold dark matter is in first nonvanishing order given by (Fry 1984):
where
and
where , and .
For the covariance matrix one only needs the parallelogram configuration. This imposes the condition
and
on the wavevectors. In this case Eq. (A.23) simplifies to
where , , and . Consequently, we define the nonGaussian contribution to the covariance in treelevel perturbation theory as
where A denotes the survey area, the integration area and G(w) the lensing weight function (see Eq. (7)). More details on the notation can be found in Sect. 3.
Footnotes
 ... theory^{}
 We refer by treelevel perturbation theory to the lowest, nonvanishing order of the considered quantity in perturbation theory.
 ... Transform^{}
 For the FFT we use an algorithm from the GNU scientific library (see http://www.gnu.org/software/gsl/ for more details).
 ... spectrum^{}
 Using the hierarchical ansatz, which is approximately valid in the highly nonlinear regime, the threedimensional dark matter trispectrum is proportional to the power spectrum cubed.
 ... (DES)^{}
 http://www.darkenergysurvey.org/
 ... UCLID^{}
 http://www.dunemission.net/
All Tables
Table 1: Cosmological parameters used to set up the initial power spectrum.
Table 2: Parameters used for generating the two Nbody simulations.
Table 3: Fiducial model used for the fitting procedure.
Table 4: Bestfit values for the parameters (p_{0},p_{1},p_{2}) of the redshiftfit for the set of coefficients .
All Figures
Figure 1: Square configuration of the dimensionless convergence trispectrum against wavenumber for the range for . The upper panel displays the four individual halo terms, the sum of all four trispectrum halo terms (solid line) and the treelevel perturbation theory trispectrum , as indicated in the key. The lower panel shows the ratio between the indicated contributions and the complete trispectrum. The double dashed line illustrates the corresponding ratio of the approximation . Note that we consider the 4halo term only in the upper panel since it resembles the term on large scales. 

Open with DEXTER  
In the text 
Figure 2: Contour plots of the ratio between the nonGaussian covariance 1halo term contribution (see Eq. (48)) computed with a concentration dispersion and with a deterministic concentration, against wavenumbers ranging from to . The left panel is for concentration dispersion , whereas in the right panel . 

Open with DEXTER  
In the text 
Figure 3: Dimensionless convergence power spectrum against wavenumber for galaxy sources at redshift ( upper panels) and ( lower panels). Points with error bars show measurements from the two sets of numerical simulations: Gems (left plots) and Virgo (right plots). Both simulations use a similar particle setup but differ in the area of the maps (see Tables 1 and 2). They are compared with the corresponding results obtained with fitting formulae from Smith et al. (longdashed line), PeacockDodds (shortdashed line), and the halo model predictions for a deterministic halo concentration (solid line). See text for a discussion. 

Open with DEXTER  
In the text 
Figure 4: Relative difference (see Eq. (51)) between the halo model prediction for the covariance of the convergence power spectrum and the results from the Virgo simulation ( ) against wavenumbers . The binning scheme is linear with a constant bin width of ranging from to . The left panel displays the full covariance, including all four halo terms for the trispectrum, and uses a deterministic concentrationmass relation denoted by . The right panel illustrates the same covariance but with a stochastic concentration distribution of width for the 1halo term of the trispectrum. 

Open with DEXTER  
In the text 
Figure 5: Relative difference (see Eq. (51)) between the halo model prediction for the covariance of the convergence power spectrum and the results from the Gems simulation ( ) against wavenumbers . The binning scheme is linear with a constant bin width of ranging from to . The left panel displays the full covariance, including all four halo terms for the trispectrum, and uses a deterministic concentrationmass relation denoted by . The right panel illustrates the same covariance but with a stochastic concentration distribution of width for the 1halo term of the trispectrum. 

Open with DEXTER  
In the text 
Figure 6: Ratio against wavenumbers . The covariance in treelevel perturbation theory resembles the nonGaussian covariance on very large scales (i.e., ) to approximately except along the diagonal and the closest offdiagonals. Here a good approximation of the nonGaussian covariance requires at least one additional halo term. For wavenumbers , an accurate description of the nonGaussian covariance requires at least the 1halo term. 

Open with DEXTER  
In the text 
Figure 7: Coefficients a_{i} of the fitting formula obtained for various source redshifts. Left (right) panel shows the coefficients of the large (small) scales fitting formula. For convenience, scaled versions of the coefficients are shown, as indicated in the key. Note for example that a_{5}, a_{7} and a_{8} change sign. The lines show the redshift fit, Eq. (55), with the parameters given in Table 4. They provide a good fit for (indicated by the vertical line). 

Open with DEXTER  
In the text 
Figure 8: Accuracy of the fitting formula for . Upper left panel: relative deviation , defined in Eq. (56). Other panels: diagonal cuts through the fitted nonGaussian covariance, showing the covariance on the diagonal ( upperright panel), for ( lowerleft) and for ( lowerright). All three panels show the fitted and the halo model nonGaussian covariances (as well as the Gaussian for the upperright panel) as function of the lowest scale in the upper part, and the deviation in percent in the lower part, where the dashed lines mark the level. For the Gaussian covariance we employed a logarithmic binning with 61 bins in a range from to . 

Open with DEXTER  
In the text 
Figure 9: Left panel: relative contributions to the diagonal of the convergence power spectrum covariance. The binningdependent Gaussian to nonGaussian ratio (solid line; see caption of Fig. 8 for the employed binning scheme) and the shape noise to cosmic variance ratio for two surveys with and (dotted), are shown as function of multipole . The ticks at the intersection of the line with the ratio curves enclose the approximate range where the nonGaussian contribution is important for the E UCLIDlike survey. Right panel: Fisher ellipses in the plane ( contours) calculated with the halo model covariance (solid) and the fitted covariance (dashed) for the DESlike (large ellipses) and E UCLIDlike (small ellipses) surveys. Compared to the halo covariance case, the fitted covariance error ellipses are decreased (enlarged) by () for the DES (E UCLID) case. 

Open with DEXTER  
In the text 
Copyright ESO 2010
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