Cosmology with photometric weak lensing surveys ...Cosmologywith photometric weak lensing...

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BNL-113176-2016-JA Cosmology with photometric weak lensing surveys: Constraints with redshift tomography of convergence peaks and moments Andrea Petri, Morgan May, and Zoltán Haiman Submitted to Physical Review D November 2016 Physics Department Brookhaven National Laboratory U.S. Department of Energy USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25) Notice: This manuscript has been authored by employees of Brookhaven Science Associates, LLC under Contract No. DE-SC0012704 with the U.S. Department of Energy. The publisher by accepting the manuscript for publication acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes.

Transcript of Cosmology with photometric weak lensing surveys ...Cosmologywith photometric weak lensing...

Page 1: Cosmology with photometric weak lensing surveys ...Cosmologywith photometric weak lensing surveys:Constraints withredshift tomography of convergencepeaks andmoments Andrea Petri,1,2,*

BNL-113176-2016-JA

Cosmology with photometric weak lensing surveys:

Constraints with redshift tomography of convergence peaks and moments

Andrea Petri, Morgan May, and Zoltán Haiman

Submitted to Physical Review D

November 2016

Physics Department

Brookhaven National Laboratory

U.S. Department of Energy USDOE Office of Science (SC),

High Energy Physics (HEP) (SC-25)

Notice: This manuscript has been authored by employees of Brookhaven Science Associates, LLC under

Contract No. DE-SC0012704 with the U.S. Department of Energy. The publisher by accepting the

manuscript for publication acknowledges that the United States Government retains a non-exclusive, paid-up,

irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow

others to do so, for United States Government purposes.

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DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the

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agency thereof, nor any of their employees, nor any of their contractors,

subcontractors, or their employees, makes any warranty, express or implied, or

assumes any legal liability or responsibility for the accuracy, completeness, or any

third party’s use or the results of such use of any information, apparatus, product,

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The views and opinions of authors expressed herein do not necessarily state or

reflect those of the United States Government or any agency thereof.

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Cosmology with photometric weak lensing surveys: Constraints with redshift tomography of convergence peaks and moments

Andrea Petri,1,2,* Morgan May,2 and Zoltán Haiman31Department of Physics, Columbia University, New York, New York 10027, USA

2Physics Department, Brookhaven National Laboratory, Upton, New York 11973, USA3Department of Astronomy, Columbia University, New York, New York 10027, USA

(Received 3 May 2016; published 30 September 2016)

Weak gravitational lensing is becoming a mature technique for constraining cosmological parameters,and future surveys will be able to constrain the dark energy equation of state w. When analyzing galaxysurveys, redshift information has proven to be a valuable addition to angular shear correlations. We forecastparameter constraints on the triplet ðΩm; w; σ8Þ for a LSST-like photometric galaxy survey, usingtomography of the shear-shear power spectrum, convergence peak counts and higher convergencemoments. We find that redshift tomography with the power spectrum reduces the area of the 1σ confidenceinterval in ðΩm; wÞ space by a factor of 8 with respect to the case of the single highest redshift bin. We alsofind that adding non-Gaussian information from the peak counts and higher-order moments of theconvergence field and its spatial derivatives further reduces the constrained area in ðΩm; wÞ by factors of3 and 4, respectively. When we add cosmic microwave background parameter priors from Planck to ouranalysis, tomography improves power spectrum constraints by a factor of 3. Adding moments yields animprovement by an additional factor of 2, and adding both moments and peaks improves by almost a factorof 3 over power spectrum tomography alone. We evaluate the effect of uncorrected systematic photometricredshift errors on the parameter constraints. We find that different statistics lead to different bias directionsin parameter space, suggesting the possibility of eliminating this bias via self-calibration.

I. INTRODUCTION

Weak gravitational lensing is a promising technique toprobe the large scale structure of the Universe in which thetracers are intrinsically unbiased [1]. This technique hasthe potential of significantly improving the constraints onthe dark energy equation of state parameter w because it ismost sensitive to the matter density fluctuations at thenonlinear stage. Cosmology inferences from weak lensingobservations have been produced for past (CFHTLenS [2],COSMOS [3]) and current (DES [4]) surveys, and are beingplanned for future experiments as well (e.g. LSST [5],WFIRST [6], Euclid [7]). Because of the nonlinear natureof the density fluctuations probed by weak lensing,cosmological information might leak from quadratic sta-tistics (such as two-point functions and power spectra) intomore complicated non-Gaussian statistics, for which for-ward modeling requires numerical simulations of cosmicshear fields.Several different examples of these non-Gaussian sta-

tistics, and their cosmological information content, havebeen studied in the past as well (see [8–15] for a non-comprehensive list). The constraining power of weaklensing power spectra with the addition of redshift tomog-raphy information has been extensively investigated in theliterature (see e.g. [16–18]). In this work we concentrate on

the constraining power of a subset of non-Gaussianstatistics, combined with redshift tomography in a LSST-like survey. The authors of [19] investigated the cosmo-logical constraining power of shear peaks tomography.Previous work on redshift tomography with weak lensingMinkowski functionals is also present in the literature [8].Tomography relies on assigning accurate redshifts to

galaxies. We therefore also investigate the effects ofuncorrected photometric redshift systematics on parameterconstraints when using redshift tomography. This work isorganized as follows. In Sec. II we outline the shearsimulations we use in this work, followed by descriptionsof the convergence reconstruction procedure, forwardmodeling of galaxy shape and photometric redshift sys-tematics, and the parameter-inference techniques we usedto forecast constraints on cosmology. In Sec. III we presentour main results, which we discuss in Sec. IV. In Sec. Vwe present our conclusions as well as prospects forfuture work.

II. METHODS

A. Cosmic shear simulations

We review the procedure used for generating simulatedshear catalogs. We consider a fiducial flat ΛCDMuniverse with parameters ðh;Ωm;ΩΛ;Ωb; w; σ8; nsÞ ¼ð0.72; 0.26; 0.74; 0.046;−1; 0.8; 0.96Þ [20,21]. We exam-ine different variations of the p ¼ ðΩm; w; σ8Þ triplet and*[email protected]

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run one N-body simulation for each choice of p, using thepublic code Gadget2 [22]. The simulations have acomoving box size of Lb ¼ 260 Mpc=h and contain5123 dark matter particles, which correspond to a massresolution of Mp ≈ 1010Msun per particle.The largestmode observed in ourN-body simulations cor-

responds to a wave number of kb ≈ 1=Lb ≈ 0.004hMpc−1.For the sake of recovering cosmological information fromweak lensing, this limitation does not create a concern, asseveral authors (see [17] for example) have shown thatmodesabove Lb contribute very little to parameter constraints.Moreover, the purpose of this work is to estimate theparameter constraints achievable in a weak lensing analysisincorporating tomography, not to produce simulations accu-rate enough for analyzing the data set that will be availablefrom LSSTand other surveys a decade hence. To analyze thedata sets that these surveys will produce, mode couplingsbetween large and small scales, which can cause effectssuch as super sample covariance [23–25], will need to beincluded. Baryonic effects will need to be included as well.Larger and more accurateN-body simulation techniques arecurrently under development in the community for thispurpose [26,27].The three-dimensional outputs of the N-body simula-

tions are sliced in sequences of two-dimensional lenses120 Mpc thick, which are lined up perpendicular to the lineof sight between the observer on Earth and a source atredshift zs. We make use of the multi-lens-plane algorithm[28,29] to trace the deflections of light rays originating atz ¼ 0 through the system of lenses out to redshift z. Toaccomplish this task, we make use of the LensTools [30,31]implementation of the multi-lens-plane algorithm. Anobserved galaxy position θ on the sky today correspondsto a real galaxy angular position βðθ; zsÞ, which can becalculated using the LensTools pipeline by solvingthe ordinary differential lens equations up to redshift zs.The Jacobian of βðθ; zsÞ is a 2 × 2 matrix that containsinformation about the cosmic shear field at θ integratedalong the line of sight:

∂βiðθ;zsÞ∂θj ¼

�1− κðθÞ− γ1ðθÞ −γ2ðθÞ

−γ2ðθÞ 1− κðθÞþ γ1ðθÞ

�: ð1Þ

The quantities that appear in Eq. (1) are the convergence κ,which is the source magnification due to lensing, and thecosmic shear γ, which is a measurement of the sourceellipticity due to lensing from large scale structure, assum-ing that the nonlensed shape is a circle.We simulate Ng ¼ 106 random galaxy positions

fθgg distributed uniformly in a field of view of sizeθ2FOV ¼ ð3.5 degÞ2, which correspond to a galaxy surfacedensity of ng ¼ 22 arcmin−2. The galaxies have a distri-bution in redshift which mimics the one expected in theLSST survey (which is shown in Fig. 2, along with theredshift binning we chose),

nðzÞ ¼ n0

�zz0

�2

exp

�−

zz0

�; ð2Þ

with z0 ¼ 0.3 and n0 a normalization constant fixed sothat nðzÞ integrates to the total number of galaxies Ng.The galaxies have a maximum redshift zmax ¼ 3. For eachgalaxy, we compute the cosmic shear at θg using Eq. (1),producing a shear catalog fγgg. Different random realiza-tions of a shear catalog fγggr can be obtained by rotatingand periodically shifting the large scale structure in theN-body snapshots according to the procedure explained in[32]. We produce Nr ¼ 16000 pseudoindependent realiza-tions of the shear catalog fγgg. These shear realizations alltogether cover 10 times the total survey area of LSST. Werepeat the above procedure for P ¼ 100 different combi-nations of the parameter triplet p, sampled according to aLatin hypercube scheme. The sampling procedure is thesame as described in [33,34]. The parameter space sam-pling we adopted for our simulations is shown in Fig. 1.For each of the parameter choices in Fig. 1, the N-body

initial conditions are generated using the same randomseed. In addition to these simulations, we produce simu-lated shear catalogs for a fiducial ΛCDM universe withp0 ¼ ð0.26;−1; 0.8Þ. In this case the randomization pro-cedure is based on five independent N-body simulations,and the same number Nr ¼ 16000 of pseudoindependentcatalog realizations is produced. This additional simulationset serves two purposes: it provides an independent data setfrom which to measure covariance matrices, and it providesa way to construct simulated observations that are inde-pendent of the simulations on which the cosmologicalforward model is trained. For the fiducial data set we choseto base the shear randomization procedure on five inde-pendent N-body simulations to ensure the independence ofthe Nr realizations for the purpose of estimating covariancematrices. Reference [32] recently showed that, even withonly one N-body simulation a few 104 independentrealizations can be produced.

B. Forward modeling of systematics

We next give an overview of the shear systematicsincluded in this work.The measured galaxy ellipticity ϵ is an estimate of the

cosmic shear γ due to large scale structure if the nonlensedgalaxy shape is a circle. Because the galaxies have intrinsicnoncircular shapes, the measured galaxy ellipticity ϵm is thesum of a cosmic shear term and the intrinsic ellipticity(galaxy shape noise) [1]

ϵm ¼ γ þ ϵn ð3Þ

where ϵn is a random Gaussian variable with zero mean andredshift dependent variance σnðzsÞ ¼ 0.15þ 0.035zs. Thisis equivalent to saying that the cosmic shear inferred from

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ellipticity observations γm can be written as the sum of thetrue cosmic shear plus a noise term γn with the samestatistical properties as ϵn. We add independent randomrealizations of the shape noise γn to each of the Nr shearcatalogs. Each shape noise realization is generated with adifferent random seed. The same random seeds are used togenerate shape noise catalogs across simulations withdifferent cosmological parameters fpig.In addition to shape noise contributions to the observed

galaxy ellipticity, we consider photometric redshift errorsas an additional contamination in the simulated catalogs. Inphotometric surveys such as LSST, the source redshift zs isestimated by measuring the source luminosity in a smallfinite set of optical frequency bands. Using this compressedluminosity information rather than the full spectrum

introduces biases in redshift estimation. Forward modelingof the cosmic shear using the procedure described inSec. II A, as well as the shape noise contributions, assumesa correct redshift distribution nðzÞ. An incorrect binning ofobserved galaxy redshifts according to the measuredphotometric distribution npðzpÞ can propagate the redshiftmeasurement errors all the way to cosmological parameterconstraints if the latter take advantage of redshift tomog-raphy. One of the goals of this work is to evaluate the size ofthis effect, assuming photometric redshift errors (photo-z)are left uncorrected.The study of photometric redshift errors is an active

subject of research, and includes investigation of tech-niques such as spectroscopic calibration, catastrophic errorsand cross-correlation techniques that we do not explore inthis work (see for example [35,36] for a more thoroughdiscussion). We model the effect of photo-z errors as aconstant bias term bphðzsÞ plus a random Gaussian com-ponent with variance σphðzsÞ,

zpðzsÞ ¼ zs þ bphðzsÞ þ σphðzsÞN ð0; 1Þ; ð4Þ

where N ð0; 1Þ is the standard normal distribution. We binthe Ng galaxies in our simulated catalogs into five redshiftbins zb, b ¼ 1…5. Several models have been proposed inthe literature for the photometric bias bphðzsÞ (see forexample [18]) and variance σphðzsÞ (see for example [36]).We chose the photo-z bias and variance functions in Eq. (4)to be the science requirements contained in the LSSTScience Book [36], namely bðzsÞ ¼ 0.003ð1þ zsÞ andσðzsÞ ¼ 0.02ð1þ zsÞ.We generate simulated observations by applying an

independent random realization of the photo-z correction(4) to each catalog realization in the fiducial cosmologyp0 and by rebinning the galaxies according to their

FIG. 1. Parameter space sampling chosen for our simulations. We show the sampling in the ðΩm; wÞ (left), ðΩm; σ8Þ (center) andðσ8; wÞ (right) projections of the parameter space. The fiducial cosmology has been marked in red.

FIG. 2. Galaxy distribution assumed throughout this work [seeEq. (2)], along with the choice of the redshift bins zb. Our galaxysample consists of Ng ¼ 106 galaxies. The figure shows thenumber of galaxies NgðzÞ at each redshift z.

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photometric redshifts zp. In the remainder of the paper weuse the following notation: we indicate a shear realization rin cosmology pwith shape noise added as γrðθg; zg;pÞ, andwe indicate a simulated observation in the fiducial cosmol-ogy as γobsðθg; zgÞ.

C. Convergence reconstruction

In this section we describe the procedure we use toconstruct convergence maps κ from the simulated shearcatalogs γ. We consider a two-dimensional square pixelgrid of area θ2FOV and with 512 pixels per side. Thiscorresponds to a linear pixel resolution of 0.5 arcmin. Weassign a shear value γðθp; zbÞ to each pixel θp according tothe following procedure:

γðθp; zbÞ ¼PNg

g¼1 γðθg; zgÞWðθg; θp; zg; zbÞPNg

g¼1Wðθg; θp; zg; zbÞ: ð5Þ

We chose a top-hat window function

Wðθg; θp; zg; zbÞ ¼�1 if θg ∈ θp; zg ∈ zb0 otherwise

: ð6Þ

The convergence κðθpÞ can be reconstructed from theE-mode of the shear field, which is evaluated from theFourier transform of the pixelized shear γðθp; zbÞ:

~κðl; zbÞ ¼�~γ1ðl; zbÞðl2

x − l2yÞ þ 2lxly ~γ

2ðl; zbÞl2x þ l2

y

�e−

l2θ2G

2 :

ð7Þ

We chose the Gaussian filter smoothing scale θG ¼0.5 arcmin to correspond to the linear pixel resolution.Inverting the Fourier transform yields the pixelized mapκðθp; zbÞ. We apply this procedure to both the shearrealizations γrðθg; zg;pÞ and the simulated observationsγobsðθg; zgÞ, yielding convergence realizations κrðθp; zb;pÞand simulated convergence observations κobsðθp; zbÞ.We measure a variety of summary statistics from the

pixelized convergence maps, which will then be used toforecast parameter constraints and biases. We considerthree kinds of summary statistics, namely the tomographicpower spectrum Pκκðl; zb; zb0 Þ, the tomographic peakcounts npkðν; zbÞ and a set of moments μðzbÞ. The tomo-graphic power spectrum is defined as

h~κðl; zbÞ~κðl0; zb0 Þi ¼ ð2πÞ2δDðlþ l0ÞPκκðl; zb; zb0 Þ: ð8Þ

Because the κ field is statistically isotropic, the expectationvalue hi, for each realization r, is taken over all modes lwith the same magnitude l ¼ jlj. Given the fact that oursimulation box is small, and we are ignoring nonlinearcouplings between large and small scale modes, we are

likely underestimating the κ power spectrum when per-forming ensemble averages based on a single N-body box.The authors of [37] estimated the effect of a varying boxsize on the 3D matter power spectrum, for boxes up to512 Mpc=h in size, and found the variations to be smallcompared to their sample variance, on spatial wavenumbers up to k ∼ 0.3hMpc−1.

The peak count statistic npkðν; zbÞ is defined as thenumber of the local maxima of a certain height κmax ¼ νσ0,where σ0 is the standard deviation over all pixels. The set ofnine moments μðzbÞ is defined as follows (see [38–40]):

μ ¼ ðμ2; μ3; μ4Þμ2 ¼ ðhκ2i; hj∇κj2iÞμ3 ¼ ðhκ3i; hκj∇κj2i; hκ2∇2κiÞμ4 ¼ ðhκ4ic; hκ2j∇κj2ic; hκ3∇2κic; hj∇κj4icÞ: ð9Þ

In Eq. (9) the gradients ∇ are evaluated using finitedifferences between κ values at neighboring pixels andthe expectation values hi for each realization r are takenover the 5122 pixels in the map. The subscript c indicatesthat we consider only the connected parts of the quartic κmoments. In the definition of the peak counts and con-vergence moments we omitted the redshift index zb fornotational simplicity. In the next section, we describe thestatistical methods we use to infer cosmological parameterestimates p from simulated observations κobsðθp; zbÞ usingthe summary statistics Pκκðl; zb; zb0 Þ, npkðν; zbÞ and μðzbÞ.Concerns might arise on the accuracy with which oursimulations measure the summary statistics mentionedabove, given the small box size and the fact that we recyclea single N-body box for building our simulated sample.The authors of [32] studied the dependence of the powerspectrum and peak counts sample means as a functionof the number of independent N-body boxes and found thatthe variations are less than 10% in most cases, except forthe small scale power spectrum and the highest κ peaks, forwhich the variations are less than 20%.

D. Parameter inference

We adopt a Bayesian framework to forecast parameterconstraints. We indicate as d a summary statistic vector(which can be any of Pκκ, npk, μ or a combination of these).We label dðpÞ the sample mean of d over the Nr ¼ 16000

simulated realizations in cosmology p and we label dr thesummary statistic measured in realization r of the fiducialcosmology p0. Both dðpÞ and dr are measured while takinggalaxy shape noise into account. We further label dobs thesummary statistic measured in a simulated observation inwhich κ has been measured taking photo-z errors intoaccount. dobs is measured averaging a random sample ofNFOV ¼ 1600 realizations of the fiducial cosmology withphoto-z errors added. This number has been chosen to

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mimic the survey area of LSST ΩLSST ¼ NFOVθ2FOV.

Assuming no prior knowledge of the parameters p, wecan write the parameter likelihood L given the observationdobs using Bayes’ theorem:

−2 logLðpjdobsÞ¼ ðdobs−dðpÞÞTC−1ðdobs−dðpÞÞ: ð10Þ

The parameter likelihood (10) can be evaluated at everypoint p in parameter space by interpolating dðpÞ betweensimulation points fpig using a radial basis function (RBF)interpolation (see [31,33]).C is the d covariance matrix andis assumed to be p-independent. In practice we replace Cwith its estimated value C from Nr ¼ 16000 realizations ofthe summary statistics dr in the fiducial cosmology p0

without photo-z errors

dmean ¼1

Nr

XNr

r¼1

dr; ð11Þ

C ¼ 1

Nr − 1

XNr

r¼1

ðdr − dmeanÞðdr − dmeanÞT: ð12Þ

Cosmological parameter values p can be inferredfrom Eq. (10) by looking at the location at which thelikelihood LðpjdobsÞ is maximum. Parameter errors Δp canbe inferred from the likelihood confidence contours.Estimates of p, Δp can be obtained by approximatingthe model statistic dðpÞ dependency on parameters as linearin p, provided p is not too far from the fiducial model p0:

dðpÞ ¼ dðp0Þ þMðp − p0Þ þOðjp − p0j2Þ ð13Þ

where we defined ðMÞiα ¼ ð∂diðpÞ=∂pαÞp0as the first

derivative of the statistic diðpÞ with respect to cosmology.We evaluate M with finite differences on the smooth RBFinterpolation of the summary statistic dðpÞ. This linearapproximation allows for a fast estimate of p in termsof dobs:

p ¼ p0 þ ðMTΨMÞ−1MTΨðdobs − dðp0ÞÞ ð14Þ

Here Ψ≡ C−1 denotes the summary statistic’s precisionmatrix. With the linear approximation (13) the parameterlikelihood (10) is a multivariate Gaussian in p and its widthΣ can be estimated as

ðΣÞαβ ¼ −�∂2 logLðpÞ

∂pα∂pβ

�−1

p0

¼ ððMTΨMÞ−1Þαβ: ð15Þ

The square of the 1σ parameter errors Δp2 are the diagonalentries of Σ. The parameter covariance estimator (15) is thesame as one gets by adopting a Fisher matrix formalism forparameter forecasts (see [41]).

When the dimension Nb of the summary statistics spaceis large, numerical issues can arise in the estimation of theparameter error bars if the covariance matrix C is measuredfrom simulations. When Nr independent realizations areused to estimate C, its inverse Ψ is biased by a constantfactor (see [42–44]) which can be taken into account. Whenthe bias correction is applied, we can calculate theexpectation value of the covariance estimator (15) (seeagain [44]):

hΣi ¼�Nr − Nb þ Np − 1

Nr − Nb − 2

�Σ ð16Þ

where Σ is the asymptotic covariance one obtains with aninfinite number of realizations andNp ¼ 3 is the number ofparameters we are estimating. The scatter of the parameterestimates (14) on the other hand scales as [44]

hCovðpÞi ¼�

Nr − 2

Nr − Nb þ Np − 2

�Σ: ð17Þ

Although Eqs. (16) and (17) agree in the limit Nr → ∞,they can be different when a finite number of realizations isused. The degradation factor in the parameter covarianceestimate in (16) is of order 1þ ð1þ NpÞ=Nr, while thescatter of the estimates p is of order 1þ ðNb − NpÞ=Nr.These numbers can be very different if Nb is large. Thismeans that the parameter error bar estimate (16) is tooconservative if Nb=Nr is of order unity. This could be thecase with the inclusion of tomography information. If webin the single-redshift summary statistic with Nst intervals,and consider Nz redshift bins, this can lead to a summarystatistic vector of size Nb ¼ OðNstN2

zÞ for the powerspectrum and Nb ¼ OðNstNzÞ for the remaining statistics.This can become quickly comparable with Nr ¼ 16000once more redshift bins or a finer binning of the summarystatistic are considered. In order to avoid these errordegradation issues, we apply dimensionality reductiontechniques to the summary statistics we are considering.Even if these techniques might not play a vital role inthis work, as the maximum Nb=Nr ratio we use is of order1%, they will definitely be relevant in future experimentswhen using finely binned summary statistics or whencombining different cosmological probes. We explain thedimensionality reduction techniques we adopted in the nextparagraph.

E. Dimensionality reduction

We apply a principal component analysis (see [41] forexample) to reduce the dimensionality of our summarystatistics while preserving the cosmological informationcontent. The model statistic dðpÞ can be regarded asa P × Nb matrix dpi. Consider the whitened model matrix

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Δpi ¼ PdpiPPq¼1 dqi

− 1: ð18Þ

We perform a singular value decomposition (SVD) of Δ:

Δ ¼ LΛR ð19Þ

where L is P ×Q, Λ ¼ diagðΛ1;…;ΛQÞ,R is Q × Nb andQ ¼ minðP;NbÞ. Rni is the ith component of the nth basisvector in statistics space. The singular value Λn is thevariance of the whitened summary statistic along the nthbasis vector. We assume that only summary statisticprojections on the first Nc basis vectors contain relevantcosmological information, whereNc < Nb is a number thathas to be determined from the simulations. Let RðNcÞ be amatrix made of the first Nc rows of R (we assume that thesingular values Λi are sorted from highest to lowest). Wedefine the Principal Component Analysis (PCA) projectionof a summary statistic d on Nc principal components as

dnðNcÞ ¼XNb

i¼1

ðRðNcÞÞni�P

diPPp¼1 dpi

− 1

�: ð20Þ

Through the above procedure, we hope to capture thecosmological information contained in d by projecting it onthe Nc < Nb principal components that vary the most withcosmology parameters.

F. Priors from CMB experiments

In this paragraph we describe how we included priorknowledge of cosmological parameters from previouscosmic microwave background (CMB) observations, suchas Planck [21]. CMB experiments provide tight constraintson ðΩm; σ8Þ, but they are not sensitive to dark energyparameters such as w. Nevertheless, prior knowledge ofΩmand σ8 could in principle help in breaking degeneraciesbetween these parameters and w in weak lensing observa-tions. The CMB parameter prior probability function can bewritten as

−2 logLCMBðpÞ ¼ ðp − p0ÞTFCMBðp − p0Þ ð21Þ

where we assumed that the best fit parameters are thesame p0’s that appear in Eq. (13). Parameter constraintsfrom the Planck CMB experiment are made availableto the public via the parameter Monte Carlo Markovchains (MCMC) published on the Planck LegacyArchive [45]. We can use these MCMC data to estimatethe parameter covariance matrix ΣCMB on the parametermultiplet ðΩm;Ωbh2; θMC; τ; w; ns; σ8Þ, marginalized overthe Planck nuisance parameters. We then compute theparameter prior Fisher matrix FCMB ¼ Σ−1

CMB. Fixing thevalues of all parameters but ðΩm; w; σ8Þ and applyingthe prior to the weak lensing parameter likelihood (10)

is equivalent to taking the ðΩm; w; σ8Þ slice of FCMB, which

we call FðΩm;w;σ8ÞCMB , and computing the parameters’ con-

straints subject to the CMB prior as

ΣlensþCMB ¼ ðMTΨMþ FðΩm;w;σ8ÞCMB Þ−1: ð22Þ

In the next section we describe the main results ofthis work.

III. RESULTS

In this section we present the main results of this work.Figure 3 shows the robustness of the dimensionalityreduction technique we adopted for the three summarystatistics considered, namely the convergence powerspectrum Pκκðl; zb; zb0 Þ; peak counts npkðν; zbÞ; andmoments μðzbÞ. To measure the power spectrum we chose15 uniformly spaced multipole bands in the rangeðlmin;lmaxÞ ¼ ð200; 2000Þ. There are only 15 independentðzb; zb0 Þ combinations (5 diagonal þ10 off-diagonal),which lead to a total of Nb ¼ 15ðmultipolesÞ ×15ðredshiftÞ ¼ 225 power spectrum measurement bands,including cross redshift information. We bin the conver-gence peak counts in 30 uniformly spaced ν bins inthe range ðνmin; νmaxÞ ¼ ð−2; 7Þ, for a total of Nb ¼30ðpeak heightsÞ × 5ðredshiftÞ ¼ 150measurement bands.The total size of the moments summary statistic vectoris Nb ¼ 9ðmomentsÞ × 5ðredshiftÞ ¼ 45.The forecast error bars on w are calculated according to

Eq. (15), where the covariance matrix C and its inverse Ψhave been estimated from Nr ¼ 16000 realizations of eachsummary statistic in the fiducial cosmology.Figure 4 shows a comparison between the w constraints

obtained using single-redshift bins, with and without PCAdimensionality reduction, and compares these single-redshift constraints with the ones obtained using redshifttomography. When we calculate parameter inferences usingthe convergence power spectrum Pκκ, we can cross-checkthe results obtained with our simulations with the onesobtained with the analytical code NICAEA [46]. This codeallows us to predict the convergence power spectrum as afunction of cosmological parameters p, for an arbitrarygalaxy redshift distribution nðzÞ. Parameter inferences canbe obtained from the NICAEA predictions for Pκκðl; zi; zjÞ(where fzig are the centers of the redshift bins) usingEq. (15). To proceed in the calculations we approximate thePκκ covariance matrix with the one we would obtain in thelimit in which the κðθÞ field is Gaussian:

hδPlðz1; z2ÞδPlðz3; z4Þi

¼ Plðz1; z3ÞPlðz2; z4Þ þ Plðz1; z4ÞPlðz2; z3Þfskyδlbinð2lþ 1Þ ð23Þ

where Plðz1; z2Þ is shorthand for Pκκðl; z1; z2Þ;δPlðz1; z2Þ ¼ Plðz1; z2Þ − Plðz1; z2Þ is the scatter in the

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P estimator; δlbin is the width of the linearly spacedmultipole bands; and fsky ¼ θ2FOV=4π is the sky coveragefraction of one field of view. In this approximation the crossvariance terms between different multipoles are assumed tobe zero.Figure 5 shows the 1σ confidence contours on the

ðΩm; wÞ doublet calculated from Eq. (15) after the PCAdimensionality reduction performed according to Eq. (20),for a variety of choices of statistic andNc. We also show theimprovements on the confidence contours when combiningdifferent summary statistics after the corresponding dimen-sionality reductions have been performed. The constraintsin the ðΩm; w; σ8Þ parameter space for a variety of summarystatistics are summarized in Tables I (weak lensing only)and II (with priors from Planck added).Figure 6 shows the effect of ignoring photo-z errors on

parameter constraints. To evaluate this effect we constructdifferent simulated observations, with and without photo-zerrors, and compare the results of the parameter fitaccording to Eq. (14). Using our simulation suite, weconstruct 20 simulated observations: the summary statisticin each observation is calculated by taking the mean of a

random sample of NFOV ¼ 1600 realizations of the sum-mary statistic in the fiducial cosmology (randomly chosenamong the ensemble ofNr ¼ 16000 that are available in theensemble). The estimated covariancematrix C is scaled by afactorNFOV to take into account the construction process ofthe simulated observations. This procedure allows us toforecast the results a LSST-like survey would obtain. Westress that, because of the small size of our simulation boxthe covariance estimate that we obtain is likely not accurateenough to produce constraints from LSST data. Full treat-ment of observables’ covariance matrices, along with largerN-body simulations and super sample covariance effects,will be investigated in future work.

IV. DISCUSSION

In this section we discuss our findings. Figure 3 showsthat our dimensionality reduction technique is robust. Inparticular, for all the summary statistics we consider, aplateau in the w error is reached for a sufficiently highnumber of principal components Nc. We also see that forsingle-redshift statistics, this plateau is reached for ∼5

FIG. 3. 1σ errors on w, marginalized over ðΩm; σ8Þ, as a function of the number of the principal components Nc, using the powerspectrum (top left), peak counts (top right) and moments (bottom). The thin colored lines refer to single-redshift summary statistics,while the thick black line shows the case in which the joint redshift information is included.

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components for the power spectrum and the moments, and∼10 components for the peak counts. Moreover, Fig. 4shows that, at least for the four highest redshift binsfzb∶b ≥ 2g, most of the cosmological information con-tained in the full (pre-PCA) summary statistic vector can becaptured with a limited number of principal componentsNc < Nb. The minimum number of components necessaryto capture most of the available information increases whenincluding redshift tomography, and can reach ∼30 for thepower spectrum and moments and ∼40 for the peak counts.Figure 4 also clearly shows that, when considering a

single-redshift bin and a single summary statistic, most ofthe information on w is contained in the highest redshiftgalaxies. PCA does not seem to capture all the informationin the lowest redshift bin, even when enough componentsare included to reach the plateau in Fig. 3. This can beattributed to the fact that PCA is not scale invariant [41],because there is freedom in choosing the whitening scale inEq. (18). Our choice of the whitening scale seems to affect

significantly the first redshift bin, with the effect beingmitigated for the highest redshift bins. The top left panel ofFig. 4 also shows reasonable agreement between the resultswe obtain with our simulations and the ones we calculatewith the analytical code NICAEA.

There are two possibilities for improving the constraints:the use of redshift tomography and the combination ofdifferent statistics. Table I shows that the area and volumeof the ðΩm; wÞ ellipse and ðΩm; w; σ8Þ ellipsoid shrink by afactor of 8 when redshift tomography is added to the powerspectrum, while the improvement is more modest for theremaining statistics (negligible for the peaks, and a factor of2 for the moments). Combining the power spectrum and thepeak counts in the highest redshift bin leads to a factor-of-10 improvement in the ðΩm; wÞ and ðΩm; w; σ8Þ 68% con-fidence intervals, with tomography further shrinking thecontours by an additional factor of 2. Combining the powerspectrum and the moments in the highest redshift binprovides 20 times tighter constraints on ðΩm; wÞ and

FIG. 4. Comparison between single-redshift w constraints (marginalized over Ωm and σ8) obtained without PCA (black bars) and withPCA (colored bars) as a function of the redshift bin zb. We show constraints obtained from the power spectrum (top left), peak counts(top right) and moments (bottom). Note the different scales on the vertical axis in the three different panels. The black dashed lines in thetop left panel refer to parameter constraints on w obtained with the public code NICAEA [46], assuming a Gaussian covariance model forthe power spectrum, as specified by Eq. (23).

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ðΩm; w; σ8Þ, with tomography yielding an additional factor-of-2 improvement. Table I also shows that power spectrumtomography can help break the degeneracy between Ωmand σ8. The same is not true for peaks and momentstomography, although combining these statistics with thepower spectrum yields better constraints by a factor of 2and 3 respectively on Ωm and σ8.Table II shows that parameter priors from Planck yield a

factor-of-6 improvement on the ðΩm; wÞ and ðΩm; w; σ8Þ

68% confidence intervals, even when a single-redshift binis considered. When the Planck priors in Eq. (22) areincluded, the improvements in constraints when addingredshift tomography or combining different statistics aremore moderate. Tomography improves power spectrumconstraints by a factor of 3. Adding moments improves byan additional factor of 2, and adding both moments andpeaks improves by almost a factor of 3 over powerspectrum tomography alone.

FIG. 5. 1σ tomographic constraints on the ðΩm; wÞ parameter space, marginalized over σ8, obtained using Eq. (15). The covariancematrix C and its inverse Ψ have been computed from 16000 summary statistics realizations, and have been scaled by a factorNFOV ¼ 1600 to mimic the constraining power of a LSST-sized survey. The thick line ellipses in the right panel refer to ðΩm; wÞobtained from the weak lensing statistics considered in this work, but subject to Planck priors as described in Eq. (22). The thin solidlines in the left and right panels are the same.

TABLE I. Constraints on the ðΩm; w; σ8Þ parameter triplet using different summary statistics and redshift information. Each columnðΔΩm;Δw;Δσ8Þ contains the 1σ constraints on a particular parameter, marginalized over the other two. The last two columns containrespectively the area of the ðΩm; wÞ 68% confidence level ellipse (marginalized over σ8) and the volume of the 68% confidence levelellipsoid in ðΩm; w; σ8Þ space, both calculated as the square root of the determinant of the relevant Σ minors.

Statistic ΔΩm Δw Δσ8 106areaðΩm; wÞ 109volume ðΩm; w; σ8ÞPower spectrum (z5) 0.0222 0.0286 0.0298 632 654Power spectrum (tomo.) 0.0038 0.0213 0.0060 76 74Peaks (z5) 0.0049 0.0316 0.0050 98 99Peaks (tomo.) 0.0042 0.0271 0.0043 93 122Moments (z5) 0.0027 0.0276 0.0026 48 39Moments (tomo.) 0.0020 0.0214 0.0020 28 21Power spectrumþ peaks (z5) 0.0040 0.0209 0.0044 58 53Power spectrumþ peaks (tomo.) 0.0021 0.0153 0.0026 27 26Power spectrumþmoments (z5) 0.0023 0.0190 0.0025 32 26Power spectrumþmoments (tomo.) 0.0016 0.0150 0.0019 18 14Power spectrumþ peaksþmoments (z5) 0.0020 0.0127 0.0024 21 17Power spectrumþ peaksþmoments (tomo.) 0.0015 0.0121 0.0018 14 11

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Figure 5 shows that peaks and moments contain cos-mological information that is not contained in the powerspectrum, because a similar improvement cannot beobtained by simply increasing the number of PCA com-ponents in the power spectrum dimensionality reductionprocedure. This is consistent with previous work (see forexample [33]).Figure 6 quantifies the effect of uncorrected photo-z

errors on the ðΩm; wÞ constraints. Because of the stochasticnature of the observations, parameter values p estimatedfrom Eq. (14) are affected by statistical fluctuations. InFig. 6 we show 20 random draws from the probability

distribution of δp ¼ pphoto-z − pno-photo-z. We can concludethat the p estimator is biased if hδpi ≠ 0. Figure 6 clearlyshows that photo-z errors cause parameter biases at morethan 1σ significance level when using the power spectrumand the moments, while no bias is observed for the peakcount statistic within its 68% confidence region. Peakhistogram shapes are more robust to this kind of systematiceffect since the peak locations are determined by theinformation coming from neighboring galaxies, while thephoto-z errors have no spatial correlation. Photo-z errorsare more likely to alter point estimates of the κ distributionand larger scale correlations which affect the powerspectrum.We also observe that photo-z errors bias the constraints in

slightly different directions, leaving open the possibility ofidentifying and correcting this bias through self-calibrationtechniques.

V. CONCLUSIONS

In this work we have studied cosmological parameterconstraint forecasts for a LSST-like galaxy survey using theconvergence power spectrum and a range of non-Gaussianstatistics. We make use of redshift tomography to improvethe constraints relative to their single-redshift counterparts.We also investigate the effects of uncorrected photo-zsystematic effects on the inferred cosmology. Our mainfindings can be summarized as follows:

(i) Principal component analysis is a robust techniqueto keep the dimensionality of the parameter spaceunder control and to avoid the numerical pitfallsexplained in [43,44,47] and more recently in [32]. Inparticular, we find that only a few components(5–10) are necessary to characterize the cosmologi-cal information content in single-redshift statistics,while more components (30–40) are necessary when

TABLE II. Constraints on the ðΩm; w; σ8Þ parameter triplet using different summary statistics and redshift information, includingFisher priors from Planck according to Eq. (22). Each column ðΔΩm;Δw;Δσ8Þ contains the 1σ constraints on a particular parameter,marginalized over the other two. The last two columns contain respectively the area of the ðΩm; wÞ 68% confidence level ellipse(marginalized over σ8) and the volume of the 68% confidence level ellipsoid in ðΩm; w; σ8Þ space, both calculated as the square root ofthe determinant of the relevant Σ minors.

Statistic ΔΩm Δw Δσ8 106areaðΩm; wÞ 109volume ðΩm; w; σ8ÞPower spectrum (z5) 0.0076 0.0274 0.0084 94 96Power spectrum (tomo.) 0.0028 0.0129 0.0035 32 31Peaks (z5) 0.0048 0.0237 0.0050 57 57Peaks (tomo.) 0.0041 0.0204 0.0042 55 70Moments (z5) 0.0024 0.0172 0.0025 27 21Moments (tomo.) 0.0019 0.0149 0.0020 18 14Power spectrumþ peaks (z5) 0.0040 0.0184 0.0043 40 36Power spectrumþ peaks (tomo.) 0.0021 0.0127 0.0025 20 19Power spectrumþmoments (z5) 0.0022 0.0145 0.0025 22 17Power spectrumþmoments (tomo.) 0.0015 0.0120 0.0018 14 11Power spectrumþ peaksþmoments (z5) 0.0019 0.0110 0.0023 16 13Power spectrumþ peaksþmoments (tomo.) 0.0015 0.0104 0.0018 12 9

FIG. 6. Distribution of parameter estimates in the ðΩm; wÞparameter space using the power spectrum (red), peak counts(green) and moments (blue) to fit 20 LSST-like simulatedobservations. The parameter deviations ðδΩm; δwÞ are obtainedcomparing parameter estimates from Eq. (14) with and withoutphoto-z errors. The colored squares and the ellipses correspondrespectively to the mean and 1σ level of the ðδΩm; δwÞ distri-bution, assuming the latter is Gaussian.

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tomography is included. Nevertheless we find thatthe number of required components Nc is signifi-cantly smaller than the full summary statistic spacedimensionality before performing PCA.

(ii) When considering a single-redshift bin, most of thecosmological information on w is contained in highredshift galaxies. Constraints can be improved withredshift tomography or combining different non-Gaussian statistics with the power spectrum. Theimprovement originating from the combination ofdifferent statistics is attributed to the complementaryinformation that non-Gaussian statistics carry, as asimilar improvement cannot be obtained from asingle statistic.

(iii) Redshift tomography on the power spectrum shrinksthe ðΩm; wÞ 68% confidence ellipse by a factor of 8;combining the peak counts with the power spectrumin the highest redshift bin leads to factor-of-10 betterconstraints, while adding the moments instead re-duces the size of the ðΩm; wÞ ellipse by a factor of20. When redshift tomography is added on top ofthese statistics combinations, an additional factor-of-2 improvement is observed. Constraint improve-ments adding redshift tomography and combinationsof different statistics are less dramatic when priorsfrom CMB experiments are included in the analysis.

(iv) Uncorrected photo-z systematics can bias parameterconstraints obtained from the power spectrum andthe moments, but in slightly different parameterdirections, leaving open the possibility of somewhateliminating this bias via self-calibration.

This work explores the advantage of deep galaxy surveyssuch as LSST, which have access to shape and redshiftinformation of high z galaxies and provide valuablecosmological information on the dark energy equation of

state. We also stress the fact that redshift tomography can insome cases provide more stringent constraints on param-eters but, for this technique to be viable, accurate knowl-edge of galaxy redshifts is necessary. Future work needs toaddress the requirements for photometric measurements’accuracy when using non-Gaussian statistics, as well as theself-calibration techniques that can be used when differentsummary statistics are available in addition to the powerspectrum.

ACKNOWLEDGMENTS

We thank Hu Zhan, Salman Habib, Jeffrey Newman,Colin Hill and Licia Verde for useful discussions. We alsothank Martin Kilbinger and Lloyd Knox for comments onan earlier version of this manuscript. Most of the calcu-lations were performed at National Energy ResearchScientific Computing Center (NERSC). We thank theLSST Dark Energy Science Collaboration (DESC) forthe allocation of time, and for many useful discussions.Part of the simulations in this work were also performed atthe National Science Foundation Extreme Science andEngineering Discovery Environment facility, supportedby Grant No. ACI-1053575, and at the New YorkCenter for Computational Sciences, a cooperative effortbetween Brookhaven National Laboratory and StonyBrook University, supported in part by the State ofNew York. This work was supported in part by the U.S.Department of Energy under Contract No. DE-SC00112704, and by the National Science FoundationGrant No. AST-1210877 (to Z. H.) and by the ResearchOpportunities and Approaches to Data Science (ROADS)program at the Institute for Data Sciences and Engineeringat Columbia University (to Z. H.).

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