General Requirements on Matter Power Spectrum Predictions for Cosmology with Weak Lensing Tomography
PPreprint typeset in JHEP style - HYPER VERSION
General Requirements on Matter Power
Spectrum Predictions for Cosmology with
Weak Lensing Tomography
Andrew P. Hearin and Andrew R. Zentner
Pittsburgh Particle physics Astrophysics and Cosmology Center (PITT PACC)Department of Physics and Astronomy, University of Pittsburgh,Pittsburgh, PA 15260 USA
Zhaoming Ma
Brookhaven National Laboratory, Upton, NY 11973
Abstract:
Forthcoming projects such as DES, LSST, WFIRST, and Euclid aim tomeasure weak lensing shear correlations with unprecedented precision, constrainingthe dark energy equation of state at the percent level. Reliance on photometrically-determined redshifts constitutes a major source of uncertainty for these surveys.Additionally, interpreting the weak lensing signal requires a detailed understandingof the nonlinear physics of gravitational collapse. We present a new analysis of thestringent calibration requirements for weak lensing analyses of future imaging surveysthat addresses both photo-z uncertainty and errors in the calibration of the matterpower spectrum. We find that when photo-z uncertainty is taken into account therequirements on the level of precision in the prediction for the matter power spectrumare more stringent than previously thought. Including degree-scale galaxy cluster-ing statistics in a joint analysis with weak lensing not only strengthens the survey’sconstraining power by ∼ , but can also have a profound impact on the calibra-tion demands, decreasing the degradation in dark energy constraints with matterpower spectrum uncertainty by a factor of 2 − . Similarly, using galaxy clusteringinformation significantly relaxes the demands on photo-z calibration. We comparethese calibration requirements to the contemporary state-of-the-art in photometricredshift estimation and predictions of the power spectrum and suggest strategies toutilize forthcoming data optimally.
Keywords: dark energy theory, gravitational lensing, galaxy formation. a r X i v : . [ a s t r o - ph . C O ] A p r ontents
1. Introduction 12. Methods 3
3. Results 11 δ ln( P i ) Model 133.2.2 The Halo Model 163.3 Systematic Errors on Dark Energy 183.3.1 The δ ln( P i ) Model 183.3.2 Dependence on Multipole Range 223.3.3 Halo Model 26
4. Discussion 28
5. Conclusions 32
1. Introduction
Weak gravitational lensing of galaxies by large-scale structure is a potentially power-ful cosmological probe [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Forthcoming imaging surveyssuch as the Dark Energy Survey (DES), the survey to be conducted by the Large Syn-optic Survey Telescope (LSST), the survey of the European Space Agency’s Euclidsatellite [12], and the proposed Wide Field Infra-Red Survey Telescope (WFIRST)expect to exploit measurements of weak gravitational lensing of distant source galax-ies to constrain the properties of the dark energy [13, 14, 15, 16, 17, 18, 19, 20, 21,22, 23, 24, 25, 26, 27, 28, 29, 30]. Over the last several years, it has been recognizedthat the limited precision with which the matter power spectrum can be predicted– 1 –ay be one of several important, systematic errors that will need to be controlled inorder to realize this goal (e.g., Refs. [31, 32, 33, 34, 35, 36, 28, 37, 38, 39]).A large part of this uncertainty is due to the effects of baryons on lensing powerspectra, which were largely neglected in much of the early literature on cosmologicalweak lensing. Several groups, including our own, have begun numerical simulationprograms designed to address this issue with large-scale numerical simulations (e.g.,Ref. [39, 40, 41, 42, 43, 44]). With the notable exception of Ref. [33], there have notbeen detailed studies of the precision with which the matter power spectrum must bepredicted before it becomes a relatively small contributor to the error budget. In therun-up to large, computationally-intensive and human resource-intensive simulationcampaigns, we have studied the required precision with which the matter powerspectrum must be predicted in order to realize anticipated dark energy constraintsfrom cosmic weak lensing tomography. We present our results as a set of generalguidelines on the systematic errors on the matter power spectrum as a function ofscale and redshift.Another potentially dominant source of error for dark energy parameter estima-tors arises from the necessity of using approximate redshifts determined from photo-metric data rather than spectroscopic redshifts [45, 46, 47, 48, 49, 50, 51, 52, 53, 54].Photometric redshifts are significantly less precise than spectroscopic redshifts, andcan exhibit large biases. Interestingly, we find that the precision with which thepower spectrum must be predicted is very sensitive to the precision of photometricredshift determinations and vice versa, a result hinted at in the Appendix of Ref. [30].As part of our analysis, we model photometric redshift uncertainty and show how theprecision with which the power spectrum must be predicted varies with photometricredshift errors, and conversely.Briefly, we find that if prior information on the photometric redshift distribu-tion is weak, then dark energy constraints degrade 2 − δ ( k ) than if the photo-z distribution is characterized with high pre-cision. Thus we find that when photo-z uncertainty is taken into account the cal-ibration requirements on the theoretical prediction for the matter power spectrumare more stringent than previously thought. The complementarity of galaxy clus-tering statistics with weak lensing, well-studied in other contexts (see, for example,Refs. [25], [55], [56], [57], and [58]), has an ameliorating effect on power spectrummisestimations. Even when restricted to degree-scales, including galaxy correlationinformation can mitigate dark energy systematics induced by errors in the predic-tion for P δ ( k ) by up to 50%; alternatively, neglecting galaxy clustering statistics cancause the statistical constraints on dark energy parameters to degrade 2 − δ ( k ) or the photo-z distribution.This manuscript is organized as follows. In § δ ( k ) as well as our methods for estimatingstatistical and systematic errors. We present our results in § §
4. We conclude in §
2. Methods
A significant amount of the constraining power of weak lensing surveys will come fromscales on which the nonlinear effects of gravitational collapse cannot be neglected[28, 59, 60]. It is possible to excise data on relatively small scales, but such an excisionsignificantly degrades dark energy constraints from cosmological weak lensing [28, 59].Modeling nonlinear structure formation will be essential in order for forthcominggalaxy imaging surveys to realize their potential for constraining dark energy andmodified gravity [33, 61]. As this modeling is uncertain and can be an importantsource of error, we study the errors induced on dark energy parameter estimators byuncertainty in the underlying matter power spectrum and we quantify the relativeimportance of theoretical power spectrum errors as a function of wavenumber. Ourresults may serve as a guideline for computational programs aimed at predictingaccurate and precise matter power spectra for the purpose of comparing with weaklensing data.In the current and past literature, the three most commonly-used techniquesemployed for predicting the matter power spectrum in the mildly nonlinear regime arethe fitting formula of Peacock & Dodds [62], the Halo Model (see Ref. [63] for a review,as well as the many references therein), and the fitting formula of Smith et al. [64].As a rough look at the contemporary level of uncertainty in predictions for the matterpower spectrum P δ ( k ), and to set the stage for what is to come, we have plotted inFig. 1 the fractional difference in P δ ( k ) at several different redshifts predicted bythese three nonlinear evolution models. The differences in the predictions made bythese methods become significant on scales ( k (cid:38) . h Mpc − ), which coincides withthe scales at which the constraining power of weak lensing begins to peak. Whenthe baryonic physics of galaxy formation is taken into account (for example, as inRefs. [36, 39, 37]), among other possible effects the matter distribution within halosis known to change relative to N-body (dark matter-only) simulations. To illustratethe effect such a rearrangement may have on the matter power spectrum, in Fig. 1we additionally plot the fractional difference in P δ ( k ) that is induced when the HaloModel correctly predicts the nonlinear evolution but the mean concentration of darkmatter halos (quantified by the parameter c in Eq. 2.1) is misestimated by 20% . In this paper, we attempt to remain relatively agnostic about the types of errorsthat may be realized in predictions of the matter power spectrum at the percentand sub-percent levels, because such accurate predictions have not yet been made.We study two classes of models for uncertainty in the matter power spectrum. Ourfirst model is motivated by simulation results suggesting that one large source of– 3 – = 0 (cid:239) (cid:239) (cid:54) P ( k ) (cid:98) HMSM z = 0.6 (cid:239) (cid:239) (cid:54) P ( k ) (cid:98) HMPD z = 1.5 (cid:239) (cid:239) (cid:98)
SMPD z = 2.5 (cid:239) (cid:239) (cid:54) c Figure 1:
Fractional difference in P δ ( k ) as predicted by the three different models ofnonlinear evolution. The superscript indicates the fiducial model, the subscript the modelinducing the fractional change. For example, the curve labelled by δ HMSM represents thefractional change to the P δ ( k ) induced by using Smith et al. to compute nonlinear powerrather than the fiducial halo model; thus when this curve is positive the Smith et al. pre-diction for P δ ( k ) exceeds that of the halo model. The dashed curve labeled ∆ c shows thefractional change to P δ ( k ) induced by systematically misestimating the mean concentrationof dark matter halos by 20% . systematic error in nonlinear power spectrum predictions may arise from a systematicerror in predictions for the internal structures of dark matter halos. Significantrearrangement of dark matter may be a result of baryonic processes during galaxyformation, for example [34, 36, 37]. So, we suppose that halo abundances and haloclustering are well-known so that the halo model accurately predicts the gross shapeof P δ ( k ) , but that the distribution of matter within halos is uncertain. For ourpurposes, the average halo density profile may be taken as the Navarro-Frenk-White(NFW) density profile [65], ρ ( r ) ∝ (cid:18) c rR (cid:19) − (cid:18) c rR (cid:19) − . (2.1)The parameter c describes the concentration of the mass distribution towards thehalo center; the mass contained within the radius R defines the halo mass. The– 4 –verage halo concentration varies with halo mass and evolves in redshift; we modelthese dependences as a power laws, c ( m, z ) = c [ m/m ∗ , ] α (1 + z ) β , (2.2)where m ∗ , = 2 . × M (cid:12) /h is the typical mass of a halo that is just starting tocollapse at z = 0 . We allow the parameters α, β, c to vary about their fiducial valuesof c = 10 . , α = − . , and β = − , so that in this model uncertainty in P δ ( k )stems exclusively from uncertainty in the distribution of mass within gravitationally-collapsed, self-bound objects. Our fiducial values for α and β are chosen to matchresults from the Millennium Simulation [66], but our fiducial value of c is roughly afactor of two larger than the mean concentration of Millennium Simulation redshift-zero halos. This larger value is motivated by the results from Ref. [36] showing thatbaryonic physics typically produces a significant enhancement to the concentrationsof dark matter halos. We thus set our fiducial c according to Ref. [36] rather thanthe Millennium results.Our second class of model for uncertainty in the matter power spectrum P δ ( k, z ),is significantly more general. After choosing a technique for predicting a fiducial matter power spectrum on scales of interest, we parameterize uncertainty in P δ ( k ) asfollows. The range of scales between k = 0 . h Mpc − and k = 10 h Mpc − are binnedevenly in log( k ) , and uncertainty in the matter power spectrum on a scale k in the i th bin is parameterized via an additional parameter δ ln( P i ), such thatP δ ( k, z ) → P δ ( k, z ) (1 + δ ln( P i )) . The parameters δ ln( P i ) are then allowed to freely vary about their fiducial valuesof zero. We discuss our choice for the number of δ ln( P i ) parameters in § δ ( k ) uncertainty was first introduced in the context of weaklensing in [33]. As uncertainty in photometric redshifts is likely to be one of the chief contributionsto the error budget in future lensing measurements of dark energy, forecasts of darkenergy parameter constraints need to include photo-z uncertainty to realistically es-timate the constraining power of future surveys. While it is the primary goal of thismanuscript to study the influence of uncertainty in the matter power spectrum, ourprevious work has shown that photo-z calibration requirements can depend sensi-tively on the fiducial model for the nonlinear evolution of P δ ( k ) . In the Appendix ofRef. [30], we showed that as prior information on the photometric redshift distribu-tion decreases, constraints on w degrade ∼ δ ( k ) and the redshift distribution of the sources used to measure the weak lensingsignal. This suggests that uncertainty in photo-z’s and P δ ( k ) need to be treatedsimultaneously in order to accurately predict the calibration requirements for thematter power spectrum.We model the underlying redshift distribution as n ( z ) ∝ z exp − ( z/z ) , wherethe normalization is fixed so that (cid:82) ∞ n ( z )d z = N A , the mean surface density ofsources in the survey. For weak lensing studies of dark energy, the most ambitiousplanned experiments for the next ten years will be LSST and Euclid; these surveyscorrespondingly have the most stringent calibration requirements, and so we find ituseful to quantitatively phrase our results for future very-wide-area surveys such asthese. Thus unless stated otherwise we choose N A = 30 gal / arcmin , and z = 0 . , corresponding to a median redshift of unity. We emphasize here, though, that thequalitative trends in all of our results are unchanged by the particular details of thesurvey characteristics.We treat photometric redshift uncertainty in a relatively general manner, follow-ing the previous work of Ref. [45]. We assume that the source galaxies are binnedaccording to photometric redshift and that the true redshift distributions of the galax-ies within each of the photometric redshift bins, n i ( z ), are related to the overall, truegalaxy source redshift distribution, n ( z ), via n i ( z ) = n ( z ) (cid:90) z i high z i low d z ph P ( z ph | z ) , where z i low and z i high are the boundaries of the i th photometric redshift bin. Photo-zuncertainty is controlled by the function P ( z ph | z ) . We take this to be a Gaussian ateach redshift, P ( z ph | z ) = 1 √ πσ z exp (cid:20) − ( z − z ph − z bias ) σ z (cid:21) . (2.3)This may seem to be overly restrictive, but as the mean, z bias , and dispersion, σ z ,of this distribution can vary with redshift themselves, this parameterization allowsfor a wide variety of possible forms for the functions n i ( z ). We adopt a fiducialmodel for photometric redshift error in which σ z = 0 . z ), while z bias = 0 atall redshifts. We use these fiducial functions to set the values of σ z and z bias at31 control points, tabulated at intervals of ∆ z = 0 . z bias and σ z at each of these redshifts are free parameters in our forecasts,so that we model photo-z uncertainty with 2 ×
31 = 62 free parameters. This choiceof binning allows for maximal degradation in dark energy constraint in the absenceof prior information about the photometric redshift distribution of source galaxies– 6 –priors that would result from, for example, a photo-z calibration program). Thisindicates that our parameterization does not enforce correlations that yield better-than-expected dark energy constraints. The dark energy constraints in the absenceof prior information reduce to the same constraints that would be obtained with nobinning in photometric redshift.To model uncertainty in the fiducial photo-z distribution, we introduce priors onthe values of the dispersion and bias at the i th redshift control points, σ iz and z i bias , respectively. These priors are ∆ σ iz = σ iz (cid:115) ispec (2.4)∆ z i bias = σ iz (cid:113) N ispec , (2.5)where N ispec is the number of spectroscopic galaxies used in each of the 31 bins ofwidth δz = 0 . ispec equal to each other, effectivelyassuming that the calibrating spectra are sampled equally in redshift, whereas inpractice there will be looser constraints on sources at high redshift than at lowredshift. However, because the details of how a realistic calibration program willproceed remains uncertain at the present time, we use this simple model for priorinformation and postpone a refinement of this parameterization until the exact setof spectra that will be used to calibrate LSST and Euclid is better known.We emphasize here that N spec provides a convenient way to specify a one-parameterfamily of photo-z priors through Eqs. 2.4 & 2.5. The quantity N spec is likely not thetrue size of the spectroscopic calibration sample, but rather the equivalent size of asample that fairly represents the color space distribution of the sources used in thelensing analysis. We take the lensing power spectra of source galaxies binned by photometric redshiftas well as the galaxy power spectra and galaxy-lensing cross spectra as observablesthat may be extracted from large-scale photometric surveys. The Limber approxima-tion relates the power spectrum P s i s j ( k, z ) associated with the correlation functionof a pair of three-dimensional scalar fields, s i and s j , to its two-dimensional projectedpower spectrum, P x i x j ( (cid:96) ) : P x i x j ( (cid:96) ) = (cid:90) dz W i ( z )W j ( z )D ( z ) H ( z ) P s i s j ( k = (cid:96)/ D A ( z ) , z ) . (2.6)– 7 –q. 2.6 essentially describes how the two-dimensional scalar fields x i are observed asprojections of the three-dimensional scalar fields, s i . The angular diameter distancefunction is denoted by D A , and H ( z ) is the Hubble parameter.The weight function W i ( z ) specifies the projection of the 3D source fields ontothe 2D projected fields: x i ( ˆn ) = (cid:90) dz W i ( z )s i (D A ˆn , z ) . (2.7)For galaxy fluctuations, the weight function is simply the redshift distribution ofgalaxies in the i th tomographic bin, n i ( z ) , times the Hubble rate:W g i ( z ) = H ( z ) n i ( z ) . The weight function associated with fluctuations in lensing convergence is given by W κ i ( z ) = 32 H (1 + z )Ω m D A ( z ) (cid:90) ∞ z dz (cid:48) D A ( z, z (cid:48) )D A ( z (cid:48) ) n i ( z (cid:48) ) , where D A ( z, z (cid:48) ) is the angular diameter distance between z and z (cid:48) . In principle, neither the redshift distribution of the galaxies used for the galaxyclustering nor the tomographic binning scheme need be the same as that used forcosmic shear sources, but for simplicity we use the same underlying distribution andbinning for both so that the chief difference between the galaxy power spectrum P g i g j , the convergence power spectrum P κ i κ j , and the cross-spectrum P κ i g j , is theform of the weight functions. Above and throughout, lower-case Latin indices labelthe tomographic redshift bin of the sources. For a survey with its galaxies dividedinto N g redshift bins used to measure the galaxy clustering, and N s bins for thegalaxies used to measure cosmic shear, there will be N g ( N g + 1) / P g i g j , N s ( N s + 1) / P κ i κ j , and N s N g distinct cross-spectra P κ i g j . The matter power spectrum, P s i s j ( k, z ) = P δ ( k, z ) , sources the three-dimensionalpower in cosmic shear, whereas the source power for galaxy-galaxy correlations isthe 3-D galaxy power spectrum P s i s j ( k, z ) = P g i g j ( k, z ) . In all of our calculations werestrict galaxy correlation information to low multipoles (cid:96) ≤ k ≈ . h Mpc − , so it will suffice forour purposes to use a linear, deterministic bias to relate the mass overdensity, δ ( z ) , to the galaxy overdensity, δ g ( z ) = b ( z ) δ ( z ) . We allow for a very general redshift-dependent bias. To model uncertainty inthe galaxy bias function b ( z ) , we allow the bias to vary freely about its fiducial valueof unity in N b galaxy bias bins, evenly spaced in true redshift, so that uncertaintyin galaxy bias is encoded by N b parameters. We computed dark energy constraintsusing the Fisher analysis technique described in § N b ranging from 1 to 30 . We find that the dark energy constraints are insensitive to N b ranging from 1 − . – 8 –hroughout this manuscript, we present results pertaining to N b = 10 bins, so thatthe value of the galaxy bias function b ( z ) has independent, parametric freedom inredshift bins of width δz = 0 . . While finer binning is possible, particularly if thenumber of tomographic galaxy bins N g is increased, a further increase of parametricfreedom is unnecessary as galaxy bias is not a rapidly varying function of redshift(see, for example, Ref. [67]).As a further simplification, we set the fiducial value of the bias function to unityat all redshifts, b i ( z ) = b ( z ) = 1 at all z for all i. This choice of fiducial parametervalues is conservative, because the galaxies observed as part of high redshift sampleswill likely be biased [68], exhibiting relatively stronger correlations than matter, sowe underestimate signal-to-noise of galaxy clustering measurements in the fiducialcase.
We estimate the constraints from upcoming photometric surveys using the formalismof the Fisher information matrix. Useful references for this formalism include [69,70, 71, 72, 24]. The Fisher matrix is defined as F αβ = (cid:96) max (cid:88) (cid:96) min (2 (cid:96) + 1) f sky (cid:88) A , B ∂ O A ∂p α C − ∂ O B ∂p β + F P αβ . (2.8)The parameters of the model are p α and the O A are the observables described in § (cid:96) min = 2 for all observables. For the lensing spectra,we take P κκ observables we set (cid:96) max = 3000 so that the assumptions of weak lensingand Gaussian statistics remain relatively reliable [73, 74, 75, 76, 6]. For the galaxyclustering statistics, we eliminate small-scale information so that we do not needto model scale-dependent galaxy bias, which is potentially complicated in itself.Therefore, we set (cid:96) max = 300 for P gg and P κg , corresponding roughly to angularscales of ∼ not employ the use of Baryon Acoustic Oscillationfeatures in our galaxy power spectra. As we will see, the added benefit of a jointanalysis stems primarily from the increased ability to self-calibrate parameterizeduncertainty in P δ ( k ) and the distribution of photometric redshifts.In Eq. 2.8, C − is the inverse of the covariance matrix; our treatment of thecovariance matrix calculation, and its associated Fisher matrix, is very similar tothat in Ref. [77], to which we refer the reader for additional details. Briefly, thecovariance between a pair of power spectra P x i x j and P x m x n is given by Cov ( P x i x j , P x m x n ) = ˜ P x i x m ˜ P x j x n + ˜ P x i x n ˜ P x j x m , (2.9)– 9 –here in the case of either galaxy power or convergence power the observed spectra˜ P x i x j have a contribution from both signal and shot noise,˜ P x i x j ( (cid:96) ) = P x i x j ( (cid:96) ) + N x i x j , where N g i g j = δ ij N Ai is the shot noise term for galaxy spectra, with N Ai denoting thesurface density of sources, and N κ i κ j = δ ij γ N Ai is the shot noise for convergence.We calculate the observed cross-spectra ˜ P κ i g j without a contribution from shot noise,so that ˜ P κ i g j = P κ i g j , because galaxies are lensed by mass separated from them bycosmological distances, so the cross-correlation of the noise terms should be small. Weadopt a common convention of setting the intrinsic galaxy shape noise γ int = 0 . N A . The inverse of the Fisher matrix is an estimate of the parameter covariance nearthe maximum of the likelihood, i.e. at the fiducial values of the parameters. There-fore, the measurement error on parameter α marginalized over all other parametersis σ ( p α ) = (cid:112) [ F − ] αα . (2.10)Gaussian priors on the parameters are incorporated into the Fisher analysis via F P αβ in Eq. 2.8. If one is instead interested in unmarginalized errors, for example to testthe intrinsic sensitivity of the observables to a parameter p α , the quantity (cid:112) / [ F ] αα provides an estimate of the uncertainty on p α in the limit of zero covariance between p α and any of the other parameters in the analysis.The Fisher formalism can also be used to estimate the magnitude of a bias thatwould occur in parameter inference due to a systematic error in the observables.If ∆ O A denotes the difference between the fiducial observables and the observablesperturbed by the presence of the systematic error, then the systematic offset in theinferred value of the parameters caused by the error can be estimated as δp α = (cid:88) β [ F − ] αβ (cid:88) (cid:96) (2 (cid:96) + 1) f sky (cid:88) A , B ∆ O A C − ∂ O B ∂p β . (2.11)We assume a standard, flat ΛCDM cosmological model and vary seven cosmolog-ical parameters with fiducial values are as follows: Ω M h = 0 . , w = − , w a = 0 , Ω b h = 0 . , n s = 0 . , ln(∆ R ) = − . , and Ω Λ = 0 . . We utilize the followingmarginalized priors: ∆Ω M h = 0 . , ∆Ω b h = 0 . , ∆ n s = 0 . , ∆ln(∆ R ) = 0 . . These priors are comparable to contemporary uncertainty [78], so this choice shouldbe conservative. We have verified that strengthening these priors to levels of uncer-tainty that will be provided by Planck [79] does not induce a significant change toany of our results.We determined the number of independent parameters for the matter powerspectrum with an analysis of the off-diagonal elements of the inverse Fisher matrix.– 10 –he parameter covariance is Q αβ ≡ F − αβ (cid:113) F − αα F − ββ . (2.12)In general, − ≤ Q αβ ≤ , with Q αβ = ( − )1 corresponding to the case whereparameters p α and p β are perfectly (anti-)correlated and Q αβ = 0 correspondingto uncorrelated parameters. By increasing the number of matter power spectrumparameters until the Fisher matrix is no longer invertible in the absence of priorinformation on these parameters, we determined that a lensing-only analysis reachesa level of total information loss when ten δ ln( P i ) parameters are used. By studyingthe behavior of the off-diagonal Fisher Matrix entries as the number of matter powerspectrum parameters are increased, we find that this state of information loss occursafter values of Q αβ = ± . δ ln( P i ) , in agreementwith the method used in Ref. [33] to arrive at this conclusion (D. Huterer, privatecommunication). Because galaxy correlation observables provide additional informa-tion with which to self-calibrate matter power spectrum parameters, including P gg and P κ g allows for slightly finer binning in wavenumber, but for the sake of facili-tating a direct comparison between the different sets of observables we have limitedour analysis to ten parameters δ ln( P i ) , irrespective of whether we consider a jointanalysis or weak lensing alone.There is additional freedom in the choice of the number of tomographic bins oneuses to divide both the sources used to measure lensing as well as the sources usedto measure galaxy correlations. As has been noted in previous studies, for exampleRef. [45], dark energy information from lensing saturates at N s = 5 tomographic bins;this saturation point is determined by using the Fisher matrix (in the limit of perfectprior knowledge of all nuisance parameters, in our case the photo-z parameters σ iz and z i bias , and the δ ln( P i ) parameters) to compute the statistical constraints σ ( w , w a ) andincreasing the number of tomographic bins until the constraints cease to improve.We find that for the case of galaxy clustering this information saturation occurs at N g = 10 tomographic bins, although we note that this saturation point depends onthe maximum multipole used in the analysis. For example, Ref. [55] uses (cid:96) max = 2000for their galaxy clustering analysis and finds that additional information is availableby increasing the number of bins to N g = 40 . Our smaller N g saturation point is aconsequence of our conservative choice for (cid:96) max , and the lack of BAO informationimplied by this choice.
3. Results
We begin by presenting our calculation of the scale-dependence of the sensitivity of– 11 –eak lensing (with and without galaxy correlations) to the matter power spectrum.Our second model for P δ ( k ) uncertainty is well-suited to this investigation: theconstraints on parameter δ ln( P i ) provide an estimate of the statistical significance ofthe weak lensing signal produced by correlations in the matter distribution on scales k ≈ k i . In the prevailing jargon, this calculation corresponds to self calibration of thematter power spectrum.The constraints from this computation are plotted as a function of scale in Fig. 2.The magenta curves at the bottom of the Fig. 2 pertain to unmarginalized constraintson the δ ln( P i ) . In other words, covariant uncertainty in cosmology, photometric red-shifts, and galaxy bias is not taken into account in the magenta curves. In plottingthe red and blue curves we illustrate our results when this covariance is accountedfor by marginalizing over all other parameters in the analysis. The red curves corre-spond to a calculation with N spec = 8000 , or ∆ σ z /σ z ≈ − . The blue curves pertainto a calculation with N spec = 2 × , which is sufficiently large that further increasesto N spec do not improve constraints on any of the parameters in our analysis, sopriors this tight effectively correspond to the case where the photo-z distribution isknown perfectly. The solid curves include galaxy correlation observables ( P gg and P κ g ) in addition to lensing observables and thus lie strictly below the dashed curves(lensing only). The step-like appearance of the curves reflects the coarse binning inwavenumber of our parameterization of P δ ( k ) uncertainty: a forthcoming very-wide-area, LSST- or Euclid-like survey is only able to constrain ∼
10 independent matterpower spectrum parameters (see § δ ( k ) , the results using either the halo model or the Peacock & Dodds [62] fittingformula are nearly identical, so conclusions drawn from Fig. 2 are quite robust todetailed changes in the fiducial model for nonlinear collapse.The minimum of the unmarginalized constraints in Fig. 2 at k ∼ h Mpc − occurs on the scale at which weak lensing is most intrinsically sensitive to matteroverdensities. This minimum occurs on a physical scale nearly an order of magnitudesmaller than the minimum of the marginalized constraints. This observation is anextension of the previous work of Ref. [33] and is itself an important result as itdemonstrates the need for precision in the prediction for P δ ( k ) over the full range ofnonlinear scales k (cid:46) h Mpc − . Because this shift in scale-dependence occurs evenfor the case of perfect prior knowledge on the photo-z parameters (N spec = 2 × ) , then it is not the effect of photometric redshift uncertainty that drives this shift inscale, but rather degeneracy with cosmological parameters. An analysis of the off-diagonal Fisher matrix elements shows that covariance of the δ ln( P i ) with dark energyparameters is chiefly responsible for this dramatic shift in the scale-dependence. Tobe specific, with p i = δ ln( P i ) and p j = w , w a , the corresponding Q ij (Eq. 2.12) havethe maximum magnitudes of any of the cosmological parameters in our parameterset and they attain their maxima at k ≈ . h Mpc − . – 12 – .1 1.0k (h/Mpc)0.000.050.100.150.200.25 (cid:109) ( (cid:98) l n P ( k ) ) Lensing OnlyJoint AnalysisN spec = 8000N spec = 2x10 Unmarginalized Constraints
Figure 2:
Statistical constraints on δ ln( P i ) , the parameters encoding uncertainty in thecalibration of the matter power spectrum, are plotted against the scale of the wavenumber.We plot unmarginalized constraints as magenta curves; the minimum of the magenta curvesat k ≈ h Mpc − illustrates that the intrinsic sensitivity of the weak lensing signal peaksat this scale. With the red and blue curves we plot marginalized constraints on the δ ln( P i )parameters for different levels of uncertainty in the distribution of photometric redshifts.The vertical axis values for the red and blue curves give the statistical precision with whicha future very-wide-area survey such as LSST or Euclid will be able to self-calibrate thetheoretical prediction for the matter power spectrum on a given scale. We proceed with results on the sensitivity of dark energy constraints to uncertaintyin predictions of the nonlinear evolution of P δ ( k ), incorporating possible additionaluncertainty from photometric redshift errors. Results for the δ ln( P i ) model appear in3.2.1 while those pertaining to the more restrictive ‘halo model” treatment of powerspectrum uncertainty are discussed in 3.2.2. δ ln( P i ) Model
In Figure 3 we depict contours of the degradation in the statistical constraints onthe dark energy equation of state associated with simultaneous uncertainty in P δ ( k )and photometric redshifts. The constraints on w and w a are shown in units of theperfectly calibrated limit, when power spectra and photometric reshifts are knownso well as to be inconsequential to the dark energy error budget; we denote theconstraints on w and w a in the limit of perfect calibration by σ perf ( w ) and σ perf ( w a ) , respectively. For example, we plot the ratio Ξ = σ ( w ) /σ perf ( w ) to illustrate the– 13 – N spec (cid:54) (cid:98) l n ( P i ) ! (cid:85) = (cid:109) (w ) / (cid:109) perf (w ) ! (cid:85) = . ! (cid:85) = . ! (cid:85) = . ! (cid:85) = . ! (cid:85) = . ! (cid:85) = . N spec (cid:54) (cid:98) l n ( P i ) (cid:85) = (cid:109) (w a ) / (cid:109) perf (w a ) (cid:85) = . (cid:85) = . (cid:85) = . (cid:85) = . (cid:85) = . (cid:85) = . Figure 3:
Contours of degradation in w and w a constraints appear in the top and bottompanels, respectively. The degradation is quantified by Ξ ≡ σ ( w , w a ) /σ perf ( w , w a ) , where σ perf is the level of statistical uncertainty of an LSST- or Euclid-like survey in the limitof perfect certainty on the photo-z distribution and the nonlinear evolution of P δ ( k ) . Theprecision in the calibration of the matter power spectrum P δ ( k ) appears on the verticalaxes, while the priors on the photo-z distribution, parameterized by N spec , appear on thehorizontal axes. Prior information about the functions z bias and σ z , which govern theuncertainty in the photo-z distribution P ( z ph | z ), is distributed uniformly in redshift ac-cording to ∆ z bias = √ σ z = σ z / (cid:112) N spec . Priors on the δ ln( P i ) parameters are distributeduniformly in among bandpowers in log( k ) . Dashed curves pertain to a survey using onlyweak lensing information, solid curves to a joint analysis that includes galaxy clustering.The gray, horizontal lines roughly bound the range of matter power spectrum uncertaintythat is attainable in advance of these surveys. – 14 – able 1:
Baseline Constraints
Observables σ ( w ) σ ( w a ) σ ( w piv )Weak Lensing Only 0 .
071 0 .
22 0 . .
058 0 .
19 0 . Notes. — Column (1) specifies whether or not observables employing galaxy clus-tering ( P gg and P κ g ) were used in the calculation. Columns (2), (3), and (4) givethe statistical constraints on w , w a , and w piv , respectively, that can be obtained byfuture very-wide-area surveys such as LSST or Euclid. Note that these constraintsaccount for statistical errors only and so represent the optimistic limit of achievabledark energy constraints for a survey with these characteristics.level of degradation of w constraints. For the sake of scaling our results to anabsolute statistical constraint, we summarize these baseline constraints in Table 1.We refer the reader to § δ ln( P i ), and the photometric redshift distributionsof sources. We quantify the precision of power spectrum prediction by a prior con-straint on the δ ln( P i ) parameters, ∆ δ ln( P i ). For example, a value of ∆ δ ln( P i ) = 0 . k i . For simplicity,we apply the same prior at all wave bands in order to produce Fig. 3. The assumedlevel of calibration of photometric redshifts is specified by the parameter N spec , asdiscussed in § δ ( k ) on scales relevant to lensing and large-scale galaxy clustering that maybe attainable by near-future numerical simulation campaigns. The precise valuesof the realized precisions will depend upon the resources dedicated to address thisissue as well as the ability of data to constrain baryonic processes that alter powerspectra and numerical simulations to treat these baryonic processes. For example,the Coyote Universe simulation campaign [42] has already achieved a 1% calibrationof the matter power spectrum to scales as small as k ≈ h Mpc − ; future resultsfrom, e.g., the Roadrunner Universe [80] will improve upon these results, althoughit is still not clear in detail how precisely P δ ( k ) will be calibrated to scales as smallas k ≈ h Mpc − , especially when uncertainty in baryonic physics is taken intoaccount.Fig. 3 contains contours for parameter degradation when using lensing data alone(dashed curves) as well as the corresponding constraint degradations when bothlensing and galaxy clustering observables are considered (solid curves). To be sure,– 15 –he additional information available to a joint analysis guarantees that σ perf ( w , w a )from the joint galaxy clustering and lensing analysis is less than the correspondingconstraint when considering lensing observables alone. In each case, we plot thedegradation Ξ = σ ( w , w a ) /σ perf ( w , w a ) relative to the idealized constraints for thattechnique. Accounting for this difference, Fig. 3 highlights the dramatic relaxationon the calibration requirements of both power spectrum predictions and photomet-ric redshifts provided by including galaxy clustering statistics. For the purpose ofstudying dark energy, this relaxation is the most significant advantage provided byutilizing galaxy correlations in the analysis.To further illustrate this point, consider particular examples that can be gleanedfrom Fig. 3. In the limit of perfect prior knowledge of P δ ( k ) and the photo-z distri-bution, the increase in constraining power that a joint analysis has over an analysisthat includes lensing observables only is merely ∼ − − spec ≈ ,
000 (arealistic proposition as we are assuming these calibrating galaxies to be distributedevenly in redshift) yields constraints on w a that are weakened by 75% due to pho-tometric redshift uncertainty, whereas if galaxy clustering information is neglectedN spec ≈ ,
000 will be required to protect against the same level of degradation indark energy constraints.
In Figure 4 we display the degradation in the statistical constraints on w (left panels)and w a (right panels) due to uncertainty in the halo concentration parameters c , α , and β [Eq. (2.2)]. The precision level of the calibration of c appears as thehorizontal axis in the top panels, α in the middle panels, and β the bottom panels.Along each row of panels in Fig. 4, only the parameter labeled on the horizontal axeshas uncertain prior knowledge. In other words, the remaining two concentrationparameters are treated as perfectly known for simplicity. As in Fig. 3, each curve isnormalized to the constraint that would be realized in the limit of perfect knowledgeof the halo structure parameters, σ cperf . Note however, that this baseline constrainthas been recomputed for each assumed value of photometric redshift uncertainty,as specified by the color-coding of each curve, so that the degradation representsonly that amount of additional degradation due to uncertainty about halo structure.Thus for each level of photometric calibration precision appearing in the legend,N spec = 5000 , , , and 10 , the vertical axis value gives the degradation inthe w ( w a ) constraints strictly due to uncertainty in the parameter labeled on thehorizontal axis. – 16 – .01 0.10 1.00 (cid:54) c / c (cid:109) ( w ) / (cid:109) c pe r f ( w ) (cid:54) c / c (cid:109) ( w a ) / (cid:109) c pe r f ( w a ) (cid:54) (cid:95) / (cid:95) (cid:109) ( w ) / (cid:109) c pe r f ( w ) N spec = 10 N spec = 10 (cid:54) (cid:95) / (cid:95) (cid:109) ( w a ) / (cid:109) c pe r f ( w a ) N spec = 5000N spec = 10 (cid:54) (cid:96) / (cid:96) (cid:109) ( w ) / (cid:109) c pe r f ( w ) Weak Lensing OnlyJoint Analysis (cid:54) (cid:96) / (cid:96) (cid:109) ( w a ) / (cid:109) c pe r f ( w a ) Figure 4:
Plot of the degradation in dark energy constraints as a function of priors on haloconcentration parameters, defined by c ( m, z ) = c [ m/m ∗ , ] α (1 + z ) β . Degradation of theconstraints on w are shown in the left panels, while degradation of w a constraints are shownin the right panels. The top panels show degradation as a function of the fractional prioruncertainty in the normalization of the mass concentration relation c . The middle row ofpanels shows degradation as a function of the power law index describing the dependenceof concentration on mass, α , and the bottom panels show the degradation as a function ofthe power-law index describing the dependence of concentration on redshift, β . Differentlevels of priors on photo-z parameters are color-coded as labeled in the middle panels. In all cases, the degradation of dark energy parameters is relatively modest. Inparticular, the degradation induced by halo structure uncertainty alone is (cid:46) ∼ In this section we explore the related problem of systematic errors on the dark energyequation of state parameters induced by uncertainty in predictions of the matterpower spectrum. To this point, we have already addressed degradation in darkenergy parameters induced by treating uncertainty in the matter power spectrum asa statistical uncertainty. In such a calculation, the underlying assumption is that amodel for the power spectrum is accurate but the parameters of the model are knownwith imperfect precision. Our aim in this section is to elucidate dark energy equationof state errors in the related circumstance of a systematic error on power spectrumpredictions. The underlying framework is that a model for the power spectrum existsand is assumed to be correct (or at least, that it contains the true power spectrumwithin its parameter set), but the model misestimates the power spectrum over somerange of wavenumbers. This situation may be the most relevant to forthcoming dataanalyses if, as a specific example, the dominant errors in power spectrum predictionsstem from systematic errors in the numerical treatment of baryons. The result ofparameter inferences that use such systematically-offset theoretical power spectra willbe systematically-offset dark energy equation of state estimators. In this section, wequantify the systematic errors in dark energy equation of state parameters inducedby systematic offsets in the matter power spectrum. Not surprisingly, the results ofthe previous section will be a useful guide for anticipating and understanding theresults of this section. As before, we begin with our more general δ ln( P i ) model andlater address systematic errors in the halo-based model. δ ln( P i ) Model
We begin our treatment of systematic errors with our δ ln( P i ) model. This modeltreats the predicted power spectrum as a sequence of bandpowers distributed evenlyin log( k ). The generality of this model enables us to address a specific, importantconcern, namely: How does the severity of the induced systematic error in the darkenergy equation of state depend on the comoving scale k at which the prediction forP δ ( k ) is erroneous? In so doing, we can prescribe how well the theoretical matter– 18 –ower spectrum should be calibrated as a function of scale, setting a specific goal forlarge-scale simulation efforts.In Fig. 5 we have plotted the systematic error induced on w (top panel) and w a (bottom panel) by a 5% error in P δ ( k ) at the comoving scale k labeled along thehorizontal axis. The induced systematic errors have been calculated according toEq. 2.11, where the systematic shift to the observables is induced by introducing a5% systematic error in P δ ( k ) over a range of wavenumbers with a width that is 10%of k err , the scale at which the error is centered, viz.∆P δ ( k )P δ ( k ) = (cid:26) .
05 : 0 . k err ≤ k ≤ . k err σ cperf , the statistical uncertainty of the parameter assuming perfect prior knowl-edge of each of the δ ln( P i ) but with photo-z uncertainty at the level given by thecolor-coding of the curve and the observables included in the analysis.Each curve approaches zero at large and small scales. Very large scale lensingcorrelations contribute little constraining power on dark energy, and so it should beexpected that scales larger than k (cid:46) − h Mpc − should contribute comparablylittle to the error budget; errors in P δ ( k ) on scales smaller than k (cid:38) h Mpc − arecomparably tolerable because we make no use of correlation information from multi-poles greater than (cid:96) max = 3000 . From the wavenumber-range containing the peaks incurves in Fig. 5 we can see that the most significant dark energy biases induced bysystematic errors in P δ ( k ) come from errors on scales 0 . h Mpc − (cid:46) k (cid:46) h Mpc − . This finding is consistent with the results presented in Fig. 2, in which we can seethat this is the same range of scales over which the tightest statistical constraintscan be obtained by self-calibrating the parameters δ ln( P i ) . Both of these figures thusillustrate that the weak lensing information about dark energy that will be availableto future very-wide-area surveys such as LSST or Euclid comes primarily from grav-itational lensing events produced by perturbations on scales 0 . h Mpc − (cid:46) k (cid:46) h Mpc − , in good agreement with previous results [33].Although almost all of the curves are limited to systematic biases δ ( w , w a ) (cid:46) σ ( w , w a ) , we remind the reader that the y-axis value gives the magnitude of thebias when the systematic error in P δ ( k ) is isolated to just a single bin of wavenumbersof width ∆ k/k = 0 . , and when the magnitude of the power spectrum error is 5% . However, we only chose these particular values for the sake of making a definiteillustration, and so for the results in Fig. 5 to be useful to the calibration programit will be necessary to scale the systematics we predict according to the particulardetails of the matter power spectrum error whose consequences are being estimated.We give several examples of this below to illustrate the utility of our calculations.– 19 – .1 1.0k (h/Mpc) (cid:239) (cid:239) (cid:98) w / (cid:109) c pe r f ( w ) Joint AnalysisLensing Only (cid:239) (cid:239) (cid:98) w a / (cid:109) c pe r f ( w a ) N spec = 10 N spec = 10 Figure 5:
Plot of the systematic error in the inferred value of w (top panel) and w a (bottom panel) induced by a systematic misestimation of the matter power spectrum P δ ( k )over a small range of wavenumbers. The value of the induced systematic is scaled in unitsof the statistical uncertainty of an LSST- or Euclid-like survey, and is plotted against thescale at which P δ ( k ) has been incorrectly predicted by 5% over a range of wavenumbersof width ∆ k/k = 0 . . See the text for a detailed description of how these results can berescaled for P δ ( k ) errors spanning a range of wavenumbers of a different width. Resultspertaining to an analysis that only uses weak lensing information appear as dashed curves,a joint analysis that includes galaxy clustering as solid curves. We show results for twodifferent levels of photo-z calibration, with the red curves corresponding to N spec = 10 , and the blue curves to N spec = 10 . – 20 –uppose there is a 3% error in P δ ( k ) made over a range of ∆ k/k = 0 . , centeredat k ≈ h Mpc − . For definiteness, consider a lensing-only analysis with photo-zuncertainty modeled by N spec = 10 . The y-axis value of the corresponding (dashed,red) curve at k ∼ h Mpc − is δw = 0 . σ ( w ) . This value needs to be rescaled bya value of 3 / δ ( k ) and the 5% error plotted in Fig. 5; additionally, scaling by afactor of 4 is necessary to account for the fact that the range of scales over whichthe error is operative spans 4 of our bins , giving an estimate of δw ≈ . σ ( w ) . If a 3% P δ ( k ) error, again spanning 4 of our ∆ k/k = 0 . k ≈ . h Mpc − , the estimation method of the first example naively implies thatthere is zero systematic error associated with such a power spectrum misestimationbecause the error is centered at a wavenumber at which δw changes sign, and sothe contributions to the net dark energy systematic to the left and right of k ≈ . h Mpc − appear to cancel. Such cancellations are not necessarily spurious, and infact a very general formalism for choosing a set of nuisance parameters specificallydesigned to take advantage of this phenomenon has recently been proposed [81].However, because it may not be known if the sign of the matter power spectrumerror inducing the biases also changes sign over the range of wavenumbers on whichthe error is made, it may not be possible to exploit this sign change to minimize thenet effect of the error. In such a case, we advocate assuming the worst-case scenario,that biases produced by errors in multiple bins in wavenumber conspire to contributeadditively; by construction this will yield a conservative estimate for the systematicinduced by the error in P δ ( k ) . Thus for this example, the absolute value of the y-axisvalues should be used to estimate the net dark energy systematic. The magnitude ofthe dashed, red curve for this particular systematic peaks at δw = 0 . σ ( w ) at theendpoints of its operative range, 0 . h Mpc − (cid:46) k (cid:46) h Mpc − ; the curve approachesthese maxima from its value of zero at k = 0 . h Mpc − and so we approximate thisas an 0 . σ ( w ) error spanning 4 of our bins. Thus for a 3% systematic error inP δ ( k ) spanning the range 0 . h Mpc − (cid:46) k (cid:46) h Mpc − , our final estimate is givenby δw = [0 . × × (3 / σ ( w ) = 0 . σ ( w ) . For power spectrum errors made over very broad ranges of wavenumber, thesimple linear scaling of ∆ k/k is no longer appropriate and one must rely on the fullmachinery of our calculation to integrate the absolute value of δw /σ ( w ) over thescales over which the error is operative. This is also useful to conservatively estimatean ultimate target goal for the matter power spectrum calibration effort. When per- Simple, linear scaling is a very good approximation when correcting for the magnitude of theP δ ( k ) error, but this prescription is only approximately correct when rescaling according to thewidth of the range of scales over which the error is made. We find that the derivatives of lensingpower spectra with respect to δ ln( P i ) parameters are stable to roughly factor-of-five changes innumerical step-size, and so simple, linear scaling will be appropriate so long as the width of thewavenumber range is less than ∆ k/k (cid:46) . . – 21 –orming this integration on scales 1 h Mpc − (cid:46) k (cid:46) h Mpc − , the worst-case estimateis a systematic error of δ ( w ) ≈ − σ ( w ); if one instead integrates a 5% P δ ( k ) errorover the entire range of wavenumbers 0 . h Mpc − (cid:46) k (cid:46) h Mpc − , a worst-case es-timate of the coherently contributing biases ranges from 9 − σ. To ensure that darkenergy systematics are kept at or below the level of statistical constraints, 0 .
5% accu-racy in the prediction for P δ ( k ) over the entire range of 0 . h Mpc − (cid:46) k (cid:46) h Mpc − will be required of the simulations calibrating the matter power spectrum. Note thatthese requirements are somewhat more restrictive than estimations from previouswork [33]. The results presented above in § (cid:96) max , the maximum mul-tipole used in the cosmic shear analysis. Naturally, as (cid:96) max increases the matterpower spectrum must be modeled with greater precision and to smaller scales. Fig-ure 6 represents a simple illustration of this point. Each curve in Fig. 6 pertainsto a cosmic shear-only experiment in the limit of perfect knowledge of the photo-zdistribution, but with different choices for (cid:96) max color-coded according to the legend.The axes are the same as those in the top panel of Fig. 5, and so this plot shows how w biases induced by P δ ( k ) errors change with the choice for the maximum multipoleused in the lensing analysis.At the end of § δ ( k ) systematicsare kept at or below the level of the statistical constraints. Briefly, one assumes theworst case scenario, that the sign of the P δ ( k ) errors conspire to contribute coherentlyto the dark energy bias. In this case, to estimate the most severe dark energy biasthat could be induced by an error in P δ ( k ) made over a range of wavenumbers,one simply adds the absolute value of the relevant curve over the relevant range ofwavenumbers. To obtain a more optimistic estimation, one could suppose that theerrors in each bin are perfectly uncorrelated, in which case they may be treated asindependent Gaussian random variables so that their net contribution to the errorbudget is computed by adding the individual contributions in quadrature. In eitherthe pessimistic or optimistic case, one obtains a precision requirement by finding themagnitude of the P δ ( k ) error that would result in the sum described above equalto unity, since this would imply that the net systematic bias on the dark energyparameter is equal to the statistical constraint on that parameter (recall that thecurves in Figures 5 and 6 plot the systematics in units of the statistical uncertaintyof the survey). Ideally, of course, the goal of the P δ ( k ) calibration program is toachieve sufficient precision such that the systematics are well below the level of The exact number depends on the level of photo-z calibration as well as the choice of observables. – 22 – .1 1.0k (h/Mpc) (cid:239) (cid:239) (cid:98) w / (cid:109) c pe r f ( w ) l max = 3000 l max = 1000 l max = 300 Figure 6:
Plot of the systematic error in the inferred value of w (top panel) induced bya systematic misestimation of the matter power spectrum P δ ( k ) . This plot is identical toFigure 5 except the curves illustrate weak lensing-only results in the limit of perfect corecalibration (N spec → ∞ ) but with different choices for the maximum multipole. Naturally,the sensitivity of the experiment to small-scale P δ ( k ) errors decreases as (cid:96) max decreases. statistical constraints, and so setting this sum to unity simply provides a guidelinefor the calibration.In Fig. 7 we have performed the calculation of the power spectrum precisionrequirement as a function of (cid:96) max . All curves pertain to weak lensing-only experi-ments , with the level of photo-z precision coded according to the legend. The topthree curves correspond to the optimistic precision estimate (errors in each bin areadded in quadrature), the bottom three curves to the pessimistic estimate (the abso-lute value of the error in each bin are added). The requirements plotted in Fig. 7 areset according to the magnitude of the systematics in w ; the w a -based requirementsare very similar.The optimistic calculation assumes that the P δ ( k ) errors at different wavenumberare completely independent; the pessimistic calculation assumes that the errors areperfectly correlated. Because the level of correlation between matter power spectrumerrors at different wavenumbers will not be known, we stress that for a given (cid:96) max the only way to guarantee that P δ ( k ) errors do not contribute significantly to thedark energy error budget is to attain the level of precision illustrated by the bottomcurves. We have not illustrated the significance of including galaxy clustering statistics because it is arelatively minor effect, as evidenced by Fig. 5. – 23 – equired Precision for P (cid:98) (k)
300 750500 1000 1500 2000 2500 3000 l max (cid:98) r eq P (cid:98) ( k ) / P (cid:98) ( k ) N spec = 3 x 10 N spec = 7 x 10 N spec = 1 x 10 Figure 7:
Plot of the required precision with which the matter power spectrum mustbe predicted as a function of (cid:96) max . The top curves represent an optimistic estimate, thebottom curves a pessimistic estimate. All curves represent results for a cosmic shear-onlyanalysis; calculations for different levels of photo-z precision are color-coded according tothe legend. To guarantee that matter power spectrum errors do not contribute significantlyto the dark energy error budget, the precision in the prediction for P δ ( k ) must reach thelevel illustrated by the bottom curves. One uses the results in Figure 8 to estimate thewavenumber to which the level of precision plotted here must be attained. See text fordetails concerning the calculation of these estimates. In Fig. 7, the crossing of the P δ ( k ) precision requirement curves pertaining todifferent levels of N spec may seem somewhat counterintuitive; one might expect thatimproving photo-z uncertainty can only lead to more stringent demands on the ac-curacy of the prediction for the matter power spectrum. This intuitive expectationis supported by a theorem proved Appendix A of Ref. [53], in which the authorsdemonstrate that the net ∆ χ induced by a systematic error is always reduced bythe addition of (unbiased) prior information. However, as shown in Appendix B ofthe same paper when marginalizing over multiple parameters, ∆ χ per degree of free-dom may increase. In our case, for large maximum multipoles ( (cid:96) max (cid:38)
00 750500 1000 1500 2000 2500 3000 l max k m a x r eq ( h / M p c ) Figure 8:
Plot of k maxreq , the wavenumber to which the matter power spectrum must bepredicted, as a function of (cid:96) max . The quantity k maxreq is insensitive to the level of photo-zprecision, and so the same k maxreq pertains to all values of N spec . One uses the results ofFigure 7 to estimate the level of precision in the prediction for P δ ( k ) that needs to beattained for all k < k maxreq . a mild increase in the w bias, a fact which we have traced to a mild difference inthe degeneracy between w and Ω Λ for different values of N spec . From Fig. 6 it is evident that for smaller choices of (cid:96) max dark energy biases areless sensitive to matter power spectrum errors on small scales. For each (cid:96) max thereis a maximum wavenumber, k maxreq , such that systematic errors in P δ ( k ) for k > k maxreq do not produce significant biases in dark energy parameters. We estimate k maxreq byfinding the bin in wavenumber at which the induced w bias becomes less than 10%of the maximum magnitude that δw attains in any bin. Because of the steepness ofthe scaling of δ ( w ) /σ ( w ) with k on scales smaller than the wavenumber at whichthe systematics attain their maximum magnitude, we find that our k maxreq estimationsare insensitive to the choice for this percentage. We present our results for k maxreq as afunction of (cid:96) max in Fig. 8.The results in this section provide a set of concrete benchmarks for the campaignof numerical simulations designed to calibrate the prediction for the matter powerspectrum, as well as a guideline for choosing the maximum multipole that should– 25 –e included in any cosmic shear analysis. For a given (cid:96) max , one uses the resultspresented in Fig. 7 to estimate the precision with which P δ ( k ) must be predicted onall scales k < k maxreq , where the k maxreq estimate appears in Fig. 8. Of course choosingsmaller values of (cid:96) max naturally decreases the constraining power of the survey, andso the results we present here can be used to inform the optimal choice for (cid:96) max that balances the need for statistical precision against the threat of matter powerspectrum systematics. In Fig. 9 we illustrate our results for the propagation of systematic errors in haloconcentration parameters through to w (top panel) and w a (bottom panel). Eachcurve corresponds to a calculation in which a single concentration parameter, either c (red), α (green), or β (blue), is systematically offset upwards of its fiducial valueby 10% , while assuming that all of the concentration parameters are known withperfect accuracy and precision. The value of the induced systematic error on w ( w a )has been normalized by the statistical constraints on the parameter at the level ofphoto-z calibration specified by the horizontal axis value for N spec . We propagatesystematic errors via Eq. 2.11, as in § spec increases. The physical interpretation of this trend appliesto nearly all of the results presented in this manuscript, and so we discuss it in de-tail in §
4. Briefly, as photo-z priors are relaxed the cosmological interpretation ofthe weak lensing signal must rely more heavily on precise knowledge of the matterdistribution. In the context of the halo model this implies that errors in halo con-centrations have more drastic consequences for dark energy parameter inference atlower values of N spec . Dark energy systematics are also less severe when galaxy clustering information isincluded; for each halo parameter, and at every level of photo-z calibration, the solidcurves are smaller in magnitude than the dashed. This trend is to be expected as itsanalogues have manifested in previous sections: the additional information availablein galaxy clustering statistics mitigates the consequences for cosmology of errors inthe matter power spectrum prediction (see Ref. [25] for the analogous benefit ofmitigating photo-z systematics by including galaxy correlations). Notice that thereis a proportionally greater mitigation of the systematics at lower values of N spec ; theinterpretation of this observation is somewhat subtle. When photo-z priors are weak,galaxy clustering information plays a more important role in the self-calibration ofthe photo-z parameters , σ iz and z i bias . As discussed in the preceding paragraph,systematic errors induced by incorrect predictions of halo concentrations are more This is an important observation in itself and has been noted elsewhere in the literature (forexample, Ref. [25]). Most of the so-called “complementarity” of weak lensing and galaxy clusteringstems from these signals calibrating each other’s nuisance parameters. – 26 – N spec (cid:98) w / (cid:109) c pe r f ( w ) (cid:54) c = 10% (cid:54) (cid:95) = 10% (cid:54) (cid:96) = 10%lensing alonejoint analysis N spec (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:98) w a / (cid:109) c pe r f ( w a ) Figure 9:
In the top panel we plot the systematic error on w against N spec , the quantityencoding the precision with which the photo-z distribution has been calibrated. The samequantity is plotted for w a in the bottom panel. Dark energy systematics δ ( w , w a ) havebeen scaled by σ cperf ( w , w a ) , the statistical constraints on the parameter at the level ofN spec given by the horizontal axis value of N spec . The systematic errors are induced by 10%misestimations of our three halo concentration parameters c , α, and β. Results for eachparameter are color-coded according to the legend in the top panel. The concentrationparameters are defined by c ( m, z ) = c [ m/m ∗ , ] α (1 + z ) β . Dashed curves correspond to re-sults for an analysis using weak lensing information only, solid curves an analysis employinggalaxy clustering information together with weak lensing in a joint analysis. severe when information about the photo-z distribution is limited. Therefore, becauseincluding galaxy clustering information at low N spec has a greater impact on thestatistical constraints than when N spec is very large, there is a concomitantly greatermitigation of the induced systematics associated with halo concentration errors at– 27 –ower values of N spec . In the limit of very large N spec , when uncertainty in the photo-z distribution canbe neglected, a 10% error in the mean halo concentration c induces a systematic erroron dark energy parameters that is comparable to or worse than the statistical con-straints. This estimation is consistent with previous results (Refs. [28, 82]). For haloconcentration errors made at other levels, we can estimate the induced systematicerror on the inferred value of dark energy parameters via simple linear scaling . Forexample, as can be seen in Fig. 9, in the limit of very large N spec a 10% misestimationof the parameter c induces a systematic error δw = 1 . σ ( w ) for a weak lensing-onlyanalysis; thus a 25% error on c induces a δw = [1 . × (0 . / . σ ( w ) = 3 σ ( w )systematic. For the sake of concreteness we conclude this section with the followingrough guideline that is based on Fig. 9 evaluated at N spec ≈ for a joint analysis:in order to guarantee that dark energy systematics induced by halo concentrationerrors are kept at or below the statistical constraints, the parameter c must be cal-ibrated to an accuracy of 5% or better, the parameter α to an accuracy of 12% orbetter, and β to better than 25% .
4. Discussion
We have studied the significance of matter power spectrum uncertainty for weak lens-ing measurements of dark energy. Our results can serve as an updated guideline forthe calibration requirements on theoretical predictions of P δ ( k ) and photometric red-shift distributions. The photo-z requirements revise those in Ref. [45], who modelednonlinear evolution using the Peacock & Dodds fitting formula, and were thus overlypessimistic about photo-z calibration. We also revise the requirements for precisionin the prediction of P δ ( k ) outlined in Ref. [33], who assumed perfect knowledge ofthe distribution of photometric redshifts and were thus overly optimistic.Both of our models for uncertainty in the theoretical prediction for the matterpower spectrum have been studied previously. Our second model, in which we allowthe value of P δ ( k ) to vary freely about its fiducial values in ten bins of bandpower(see § δ ( k ) uncertainty, 2) includinggalaxy clustering statistics in the set of observables, and 3) by taking into accountuncertainty in the distribution of photometric redshifts. Our motivation to treatuncertainty in P δ ( k ) and the photo-z distribution simultaneously comes from resultspresented in the Appendix of Ref. [30], where the authors showed that the photo-zcalibration requirements vary significantly depending on the assumed fiducial model The stability of the derivatives of our observables with respect to halo concentration parametersover a broad range of numerical step sizes ensures the accuracy of this simple linear scaling. – 28 –f the P δ ( k ) in the nonlinear regime. This result suggests a nontrivial interplaybetween the photo-z and matter power spectrum calibration demands.The cause of this interplay has a simple physical interpretation. In weak lensing,there is a degeneracy between the redshift of a galaxy whose image is distorted andthe typical size of the overdensity responsible for most of its lensing: at fixed angularscale, correlations in the image distortions of sources at high redshift are produced(on average) by overdensities that are larger in comoving size than those producingcorrelations at low-redshift. The more precisely the photo-z distribution is known,the narrower the range of possible wavenumbers contributing to the lensing signal.As priors on photo-z parameters are relaxed, the redshifts of the sources are knownwith decreasing precision, and so more information about the power spectrum isrequired in order to compensate. Thus at lower values of N spec , dark energy parameterinference at a fixed level of statistical uncertainty requires more precise knowledgeof the matter power spectrum.Consequences of this basic physical picture appear throughout this manuscript.For example, in Figures 5 and 9, appearing in § § spec . Another example appears in Figure4 of § w and w a degrademore rapidly with uncertainty in halo concentrations when N spec is small relative tolarger values of N spec . Similarly, the statistical constraints on dark energy discussedin § δ ln( P i ) at lower values of N spec . The nontrivial relationship between uncertainty in P δ ( k ) and in the distribution ofphotometric redshifts clearly illustrates that a detailed and accurate study of thecalibration requirements on future imaging surveys requires the simultaneous accountof these contributions to the dark energy error budget that we present here.The significance of uncertainty in halo concentrations for the dark energy pro-gram has also been studied previously [28, 82]. Again, we have generalized theircalculations and our conclusions are in good agreement with their results, wherecommensurable. In particular, Ref. [28] studied the prospects for future weak lens-ing surveys to self-calibrate halo concentration parameters while simultaneously con-straining dark energy parameters. Even with only very modest prior information onphoto-z parameters, we agree with the conclusion in Ref. [28] that the prospect forfuture imaging surveys to self-calibrate uncertainty in halo concentrations is verypromising, especially when galaxy correlation statistics are employed in a joint anal-ysis: the statistical degradation on w from self-calibrating the mean, redshift-zerohalo concentration c is less than 6% for N spec = 5000 (corresponding roughly to∆ σ z /σ z ≈ − ). The degradation in the constraints is even milder when the photo-z distribution is more precisely characterized. However, this result is provisional inthat it relies on halo concentration being the most significant mode in which thepower spectrum is uncertain. – 29 –ur halo model-based treatment of P δ ( k ) uncertainty is well-motivated by Ref. [36],who carried out a suite of numerical cosmological simulations including hydrodynam-ics with a variety of energy feedback mechanisms. One of the salient conclusions ofRef. [36] is that the effects of baryonic physics on the matter power spectrum canbe well-modeled as an enhancement to the mean concentration of dark matter halos.Thus we chose our first model of uncertainty in P δ ( k ) with the intention to study therequirements of the dark energy program for precision in our ability to predict theeffects of baryons on the large-scale distribution of matter. However, recent resultsfrom the OverWhelmingly Large Simulation (OWLS) project [83] suggest that anenergy feedback mechanism modeling the effects of AGN is necessary to reproducethe characteristics of groups of galaxies [84]. In a recent study based on these results[39], the authors found that a multicomponent halo model with a gas profile that isindependent from the dark matter profile can accurately model the power spectrain the OWLS project. We note, however, that not even N-body simulations haveachieved the desired precision ( < . . h Mpc − (cid:46) k (cid:46) h Mpc − required of the P δ ( k ) calibration [42]. Nonetheless, these results are intrigu-ing and suggest that a more complicated model than the one we consider here maybe necessary to fully encapsulate the baryonic modifications to the matter powerspectrum. We leave the development and exploration of such a model as a task forfuture work.When the assumption that the halo model accurately characterizes all the grossfeatures of P δ ( k ) is relaxed, self-calibrating the matter power spectrum is very likelyto be infeasible. To see this, we turn back to our second, more conservative modelof matter power spectrum uncertainty. As shown in Fig. 2, even in the limit ofperfectly precise prior knowledge on the photo-z distribution, future surveys areunable to self-calibrate the value of P δ ( k ) to better than 7% on any scale, renderingP δ ( k ) as a dominant component in the error budget and increasing errors on w and w a by a factor of 3 or more. As discussed in § δ ( k ) are kept at or below the level ofstatistical uncertainty of an LSST- or Euclid-like survey, the theoretical predictionfor the matter power spectrum will need to be accurate to at least 0 .
5% or betteron all scales k (cid:46) h Mpc − . These results reinforce the necessity of an aggressivecampaign of numerical cosmological simulations if surveys such as LSST or Euclidare to achieve their potential as dark energy experiments.To contextualize these findings with the current state-of-the-art in numericalsimulations, we first compare this requirement to the results from the Coyote Universeproject [41], a suite of nearly 1,000 N-body (gravity only) simulations spanning 38fiducial w CDM cosmologies. To date, this is the most ambitious campaign of N-bodysimulations yet performed with the aim to robustly calibrate P δ ( k ) over the full rangeof scales relevant to weak lensing. The power spectrum emulator based on theirresults has recently been completed [85], and in Ref. [42] the authors demonstrate– 30 –hat results from their simulations can be used to model P δ ( k ) with sub-percentaccuracy on all scales k (cid:46) h Mpc − . In addition to the need to expand the rangeof scales over which this level of precision has been attained, the Coyote Universedoes not account for the nonlinear effects of neutrino mass [86], which have beenrecently established [87] to introduce percent-level changes to the standard Smith etal. method of prediction. Moreover, Coyote’s 1% precision only applies to w CDMcosmologies; extending this level of precision to dynamical dark energy models is anactive area of current research on in this field, for example, Refs. [44, 88, 89, 90].These complications aside, the Coyote Universe project is, by itself, insufficient tocompletely calibrate the matter power spectrum because N-body simulations neglectthe effect that baryonic gas has on P δ ( k ) . Suites of hydrodynamical simulationssuch as the OWLS project [91] discussed above will be essential contributions to thecalibration program. Continued improvement both in N-body and hydrodynamicalsimulations will clearly be necessary in order to meet the calibration requirementswe present here.
In updating the photo-z precision requirements for future imaging surveys we havequantified the uncertainty in the photo-z distribution in terms of N spec . We reiter-ate here an important difference between the meaning of N spec in our forecasts andelsewhere in the literature. In this work, the quantity N spec defines a one-parameterfamily of priors on the photo-z distribution via Eqs. 2.4 & 2.5. In practice the actualnumber of galaxies in the calibration sample will likely need to be larger than N spec , for example because it will be challenging to obtain a calibration sample that fairlyrepresents the color space distribution of the galaxies in the imaging survey. More-over, even if such a representative sample is obtained in a particular patch of sky,sample variance due to the relatively narrow sky coverage of current and near-futurecalibration samples has a significant impact on the accuracy of the calibration [92].We sought to provide general guidelines for a broad range of future imaging surveys,and so we have not attempted to model how these important, survey-specific issuesaffect the calibration requirements. Instead, in our formulation the photo-z preci-sion requirements are formally specified in terms of the necessary amount of priorknowledge on the photo-z distribution, which in turn is encoded by the parameterN spec . Our modeling of galaxy clustering has several simplifying assumptions that arerelevant to the calibration of future imaging surveys. In modeling galaxy bias asa function of redshift only, we have implicitly assumed perfect knowledge of howgalaxy bias depends on wavenumber. Uncertainty in the scale-dependence of thegalaxy bias is degenerate with uncertainty in P δ ( k ) , and so the improvement in theconstraining power of a survey provided by including galaxy clustering will degradewhen accounting for uncertainty in the scale-dependence of galaxy bias. However, we– 31 –ave restricted the range of angular scales on which galaxy correlations are exploitedso that we probe only very large scales ( ∼ h − Mpc), and so we expect correctionsaccounting for this scale-dependence are small.The benefit of galaxy correlations to the photo-z calibration is diluted by lensingmagnification bias [53, 93], which induces a spurious correlation between sources thatare well separated in redshift space. This effect thus threatens the ability of galaxycross-correlations to detect and calibrate outliers in the photo-z distribution and willneed to be accounted for in order to fully realize the potential of galaxy clustering.We intend to generalize our results to include these effects in a future paper.
5. Conclusions
We have studied the matter power spectrum calibration requirements for future very-wide-area weak lensing surveys such as LSST or Euclid. While our findings applyto all planned imaging surveys designed to use weak lensing to study dark energy,we have phrased our conclusions in terms of these particular surveys because theircalibration demands are the most stringent. Our results generalize previous find-ings by simultaneously accounting for photometric redshift uncertainty, as well as bystudying the significance of galaxy clustering information. We explored two differ-ent models for uncertainty in the nonlinear physics of gravitational collapse, whichwe describe in detail in § δ ( k ), but that the internal structure of halosis uncertain. In our second, more agnostic model, we allow the value of P δ ( k ) tovary freely about its fiducial value in ten logarithmically-spaced bins of bandpowerspanning the range 0 . h Mpc − ≤ k ≤ h Mpc − . We conclude this manuscript byproviding a brief summary of our primary results.1. Future imaging surveys will be unable to self-calibrate the value of P δ ( k ) tobetter than 7% on any scale. This renders infeasible the possibility of com-pletely self-calibrating the theoretical prediction for the matter power spec-trum because systematic errors at such levels would induce unacceptably largebiases in the inferred value of the dark energy equation of state. Moreover, themarginalized constraints on P δ ( k ) are the tightest at scales k ≈ . h Mpc − , nearly an order of magnitude larger in size than where the unmarginalizedconstraints computed in Ref. [33] attain their minimum, emphasizing the ne-cessity of a precise calibration of P δ ( k ) over the full range of wavenumbers 0 . h Mpc − (cid:46) k (cid:46) h Mpc − .
2. To ensure that systematics are kept at levels comparable to or below the sta-tistical constraints on w and w a , P δ ( k ) must be accurately predicted to aprecision of 0 .
5% or better on all scales k (cid:46) h Mpc − in advance of futureweak lensing observations that will be made by LSST or Euclid.– 32 –. The required precision for the P δ ( k ) prediction as well as the scale to which thisprecision must be attained depend sensitively on (cid:96) max , the maximum multipoleused in the cosmic shear analysis. Figures 7 and 8 together provide a concreteguideline that can be used to directly inform the optimal choice for (cid:96) max thatbalances the need for statistical precision against the threat of matter powerspectrum systematics.4. In keeping with the results in the Appendix of Ref. [30], we find that thephoto-z calibration requirements are less stringent by a factor of ∼ δ ( k ) is modeled with the Smith et al. fitting formularelative to Peacock & Dodds, significantly relaxing the demands for photo-zprecision that appear in Ref. [45].5. Dark energy constraints degrade ∼
40% more slowly with photo-z uncertaintywhen including galaxy correlations in a joint analysis with weak lensing, evenwhen the clustering information is restricted to degree-scales and with coarsetomographic redshift binning so that Baryonic Acoustic Oscillation features arenot resolved.6. Including galaxy clustering statistics (again, even when Baryon Acoustic Os-cillation information is neglected) also significantly relaxes the calibration re-quirements and mitigates the severity of systematic errors induced by erroneouspredictions for P δ ( k ) , especially when prior information on the photo-z distri-bution is weak. Dark energy systematics can be reduced by up to 50% byincluding galaxy clustering information; the statistical constraints on w and w a can degrade 2 − δ ( k ) uncertainty when galaxycorrelations are neglected.7. The redshift-zero, mean halo concentration, c , must be accurately predictedwith a precision of 5% or better to keep systematics in dark energy parame-ters below the level of statistical constraints. If internal halo structure is thedominant mode of P δ ( k ) uncertainty, then the prospect for self-calibrating c are quite promising, as this would only degrade the dark energy constraints by5 − .
8. The matter power spectrum calibration requirements are more stringent whenthe distribution of photometric redshifts is known with less precision. Thiseffect is due to a degeneracy between source redshift and lens size, and is thechief motivation for a simultaneous account of these sources of uncertainty.We find that the constraints on w and w a degrade 2 − δ ( k ) uncertainty for spectroscopic calibration samples with the statisticalequivalent of N spec ≈ spec ≈ . – 33 –. The requirements for the precision with which P δ ( k ) need be predicted are, ingeneral, less stringent for DES than for LSST by a factor of a few. To ensurethat matter power spectrum systematics do not contribute significantly to thedark energy error budget for DES, we find that if correlations up to a maximummultipole of (cid:96) max = 3000 are used in the lensing analysis then P δ ( k ) will need tobe calibrated to an accuracy of 2% or better on scales k (cid:46) h Mpc − . The DESrequirements scale with (cid:96) max in a similar fashion to the scaling of the LSSTrequirements summarized in Fig. 8. The difference between the requirementsis driven by the relative depth of these two surveys: the constraining poweron dark energy provided by cosmic shear measurements derives chiefly fromsmall-scale correlations ( k (cid:38) h Mpc − ) where shot noise is most significant.The shallower depth of DES ( z med = 0 . , N A = 15 gal / arcmin ) results inthese modes being less informative, and so DES suffers less from uncertaintyin small-scale information. Acknowledgments
We thank Wayne Hu, Martin White, Dragan Huterer, Jeff Newman, Chris Purcell,Douglas Rudd, Shahab Joudaki, and Bob Sakamano for useful discussions. ARZ andAPH are supported by the Pittsburgh Particle physics, Astrophysics, and CosmologyCenter (PITTPACC) at the University of Pittsburgh, and by the US National ScienceFoundation through grant AST 0806367. ZM is supported through Department ofEnergy grant DOE-DE-AC02-98CH10886.
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