Accessing the high- ℓ frontier under the Reduced Shear Approximation with k -cut Cosmic Shear
AAccessing the high- (cid:96) frontier under the Reduced Shear Approximationwith k -cut Cosmic Shear Anurag C. Deshpande, ∗ Peter L. Taylor, and Thomas D. Kitching Mullard Space Science Laboratory, University College London,Holmbury St. Mary, Dorking, Surrey, RH5 6NT, UK Jet Propulsion Laboratory, California Institute of Technology,4800 Oak Grove Drive, Pasadena, CA, 91109, USA (Dated: October 27, 2020)The precision of Stage IV cosmic shear surveys will enable us to probe smaller physical scales thanever before, however, model uncertainties from baryonic physics and non-linear structure formationwill become a significant concern. The k -cut method – applying a redshift-dependent (cid:96) -cut aftermaking the Bernardeau-Nishimichi-Taruya transform – can reduce sensitivity to baryonic physics;allowing Stage IV surveys to include information from increasingly higher (cid:96) -modes. Here we addressthe question of whether it can also mitigate the impact of making the reduced shear approximation;which is also important in the high- κ , small-scale regime. The standard procedure for relaxing thisapproximation requires the repeated evaluation of the convergence bispectrum, and consequentlycan be prohibitively computationally expensive when included in Monte Carlo analyses. We findthat the k -cut cosmic shear procedure suppresses the w w a CDM cosmological parameter biasesexpected from the reduced shear approximation for Stage IV experiments, when (cid:96) -modes up to5000 are probed. The maximum cut required for biases from the reduced shear approximation tobe below the threshold of significance is at k = 5 . h Mpc − . With this cut, the predicted 1 σ constraints increase, relative to the case where the correction is directly computed, by less than10% for all parameters. This represents a significant improvement in constraints compared to themore conservative case where only (cid:96) -modes up to 1500 are probed [1], and no k -cut is used. We alsorepeat this analysis for a hypothetical, comparable kinematic weak lensing survey. The key parts ofcode used for this analysis are made publicly available a . I. INTRODUCTION
Cosmic shear – the distortion of the observed ellip-ticities of distant galaxies resulting from weak gravita-tional lensing by the large-scale structure of the Universe(LSS) – is a powerful tool to better constrain our knowl-edge of dark energy [2]. Current weak lensing surveys[3–5] perform precision cosmology competitive with con-temporary Cosmic Microwave Background surveys. Up-coming Stage IV [2] cosmic shear surveys such as
Eu-clid [6], the Nancy Grace Roman Space Telescope [7],and the Rubin Observatory [8] will offer greater thanan order-of-magnitude leap in precision over the current-generation surveys [9]. Additionally, they will be ableto probe smaller scales than previously possible (see e.g.[1]).As a result of these improvements, we face new chal-lenges. One such issue is the small scale sensitivity prob-lem. This refers to the fact that the cosmic shear signalis sensitive to poorly understood physics at scales below k = 7 h Mpc − [10]. Nulling has previously been sug-gested as a potential solution [11]. An approach thathas shown promise in addressing this issue is to first ∗ [email protected] a https://github.com/desh1701/k-cut_reduced_shear https://roman.gsfc.nasa.gov/ apply the Bernardeau-Nishimichi-Taruya (BNT) nullingscheme [12], and then take a redshift-dependent angularscale cut. This technique is known as k -cut cosmic shear[13].Using k -cut shear to alleviate the small scale sensitivityproblem, we can push our analyses to include smaller andsmaller angular scales. For example, an appropriate k -cut would allow us to readily achieve the ‘optimistic’ casefor a Euclid -like survey; where e.g. the inclusion of angu-lar wave numbers of up to (cid:96) = 5000 [1] would be achiev-able. However, at these scales, two theoretical assump-tions cease to be valid; the reduced shear approximation,and the assumption that magnification bias can be ne-glected [14]. The latter of these is a selection effect, andcould potentially be addressed via a process like metacal-ibration [15, 16], in particular a ‘selection response’. Onthe other hand, relaxing the reduced shear approxima-tion requires the explicit calculation of the convergencebispectrum, which could be prohibitively computation-ally expensive for Stage IV experiments [14] and requiresa theoretical expression for the poorly understood matterbispectrum, including baryonic feedback. In this work,we demonstrate how the k -cut method preserves the re-duced shear approximation for a Stage IV survey even athigh- (cid:96) . Specifically, we examine the case of a Euclid -likeexperiment, as forecasting specifications for such a sur-vey are readily available [1]. This procedure bypasses theneed for the expensive computation of three-point terms,at the price of weakening cosmological parameter con-straints. We also repeat this analysis for a hypothetical a r X i v : . [ a s t r o - ph . C O ] O c t Tully-Fisher kinematic weak lensing survey [17, 18].This work is structured as follows: we begin by pre-senting the theoretical formalism, in Section II. We firstreview the standard, first-order cosmic shear power spec-trum calculation; including the contribution of non-cosmological signals from the intrinsic alignments ofgalaxies (IA) and shot noise. Then, we discuss the for-malism for relaxing the reduced shear approximation, aswell as giving an overview of the BNT transform and k -cut cosmic shear. The Fisher matrix formalism, used topredict the cosmological parameter constraints that willbe inferred from upcoming experiments, is then detailed.In Section III, we explain our modelling specifics and ourchoice of fiducial cosmology. Lastly, in Section IV, wepresent our results. We compare the cosmological pa-rameter biases resulting from making the reduced shearapproximation for two different matter bispectrum mod-els; showing that the correction calculation is robust tothe choice of model. Using the most up-to-date of thesemodels, we then demonstrate how a range of k -cuts affectthe predicted cosmological parameter constraints and thebiases from making the reduced shear approximation. II. THEORY
In this section, we first review the standard cosmicshear angular power spectrum calculation. Contributionsfrom IAs and shot noise are also described. Then, weexplain how the reduced shear approximation can be re-laxed. Next, we detail the BNT nulling scheme and k -cutcosmic shear procedure. Finally, the Fisher matrix for-malism is outlined. A. The First-order Cosmic Shear Power Spectrum
Weak lensing distorts the observed ellipticity of dis-tant galaxies. This change is dependent on the quantityknown as reduced shear, g : g α ( θ ) = γ α ( θ )1 − κ ( θ ) , (1)where θ is the source’s position on the sky, γ is the shear,a spin-2 quantity with index α , and κ is the convergence.Shear is the component of weak lensing which causes theanisotropic stretching that makes circular distributions oflight elliptical, and convergence is the isotropic increaseor decrease in the size of the image. In the weak lensingregime, | κ | (cid:28)
1, so it is standard procedure to make thereduced shear approximation: g α ( θ ) ≈ γ α ( θ ) . (2)The convergence in tomographic redshift bin i is givenby: κ i ( θ ) = (cid:90) χ lim d χ δ [ S K ( χ ) θ , χ ] W i ( χ ) . (3) It is a projection of the density contrast of the Universe, δ , along the line-of-sight over comoving distance, χ , tothe survey’s limiting comoving distance, χ lim . The func-tion S K ( χ ) in equation (3) accounts for the curvature ofthe Universe, K , such that: S K ( χ ) = | K | − / sin( | K | − / χ ) K > χ K = 0 (Flat) | K | − / sinh( | K | − / χ ) K < (4) W i denotes the lensing kernel for tomographic bin i ,which is defined as follows: W i ( χ ) = 32 Ω m H c S K ( χ ) a ( χ ) (cid:90) χ lim χ d χ (cid:48) n i ( χ (cid:48) ) × S K ( χ (cid:48) − χ ) S K ( χ (cid:48) ) , (5)where Ω m is the dimensionless present-day matter den-sity parameter of the Universe, H is the Hubble con-stant, c is the speed of light in a vacuum, a ( χ ) is thescale factor of the Universe, and n i ( χ ) is the probabilitydistribution of galaxies within bin i .Under the flat-sky approximation [19], the spin-2 shearis related to the convergence via: (cid:101) γ αi ( (cid:96) ) = T α ( (cid:96) ) (cid:101) κ i ( (cid:96) ) , (6)where (cid:96) is the Fourier conjugate of θ , we make the ‘pref-actor unity’ approximation [19], and T α ( (cid:96) ) are trigono-metric weighting functions: T ( (cid:96) ) = cos(2 φ (cid:96) ) , (7) T ( (cid:96) ) = sin(2 φ (cid:96) ) , (8)in which the vector (cid:96) has angular component φ (cid:96) , andmagnitude (cid:96) .For an arbitrary shear field, two linear combinationsof the shear components can be constructed: a curl-free E -mode, and a divergence-free B -mode: (cid:101) E i ( (cid:96) ) = (cid:88) α T α (cid:101) γ αi ( (cid:96) ) , (9) (cid:101) B i ( (cid:96) ) = (cid:88) α (cid:88) β ε αβ T α ( (cid:96) ) (cid:101) γ βi ( (cid:96) ) , (10)where ε αβ is the two-dimensional Levi-Civita symbol,with ε = − ε = 1 and ε = ε = 0. The B -modeof equation (10) vanishes in the absence of higher-ordersystematic effects. This leaves the E -mode, for whichwe can define auto and cross-correlation power spectra, C γγ(cid:96) ; ij : (cid:68) (cid:101) E i ( (cid:96) ) (cid:101) E j ( (cid:96) (cid:48) ) (cid:69) = (2 π ) δ ( (cid:96) + (cid:96) (cid:48) ) C γγ(cid:96) ; ij , (11)with δ being the two-dimensional Dirac delta. Underthe assumption of the Limber approximation, where only (cid:96) -modes in the plane of the sky are taken to contributeto the lensing signal, the power spectra themselves are: C γγ(cid:96) ; ij = (cid:90) χ lim d χ W i ( χ ) W j ( χ ) S ( χ ) P δδ ( k, χ ) , (12)where P δδ ( k, χ ) is the matter power spectrum. Compre-hensive reviews of this standard calculation can be foundin [20, 21]. B. Intrinsic Alignments and Shot Noise
In reality, the angular power spectrum measured fromgalaxy surveys contains non-cosmological signals, in ad-dition to the cosmic shear contribution from equation(12). One such component is the result of the IA of galax-ies [22]. Galaxies that form in similar tidal environmentshave preferred, intrinsically correlated, alignments. Theobserved ellipticity of a galaxy, (cid:15) can be described tofirst-order as: (cid:15) = γ + γ I + (cid:15) s , (13)where γ is from cosmic shear, γ I is the IA contribution,and (cid:15) s is the galaxy’s source ellipticity in the absence ofany IA. A theoretical two-point statistic (e.g. the angu-lar power spectrum) calculated from equation (13) wouldthen consist of four kinds of terms: (cid:104) γγ (cid:105) , (cid:104) γ I γ (cid:105) , (cid:104) γ I γ I (cid:105) ,and a shot noise term from the uncorrelated part of theunlensed source ellipticities, (cid:15) s .Accordingly, the observed angular power spectra, C (cid:15)(cid:15)(cid:96) ; ij ,contain contributions from all these terms: C (cid:15)(cid:15)(cid:96) ; ij = C γγ(cid:96) ; ij + C I γ(cid:96) ; ij + C γ I (cid:96) ; ij + C II (cid:96) ; ij + N (cid:15)(cid:96) ; ij , (14)where C γγ(cid:96) ; ij are the cosmic shear spectra of equation (12), C I γ(cid:96) ; ij represent the correlation between the backgroundshear and the foreground IA, C γ I (cid:96) ; ij are the correlation ofthe foreground shear with background IA, C II (cid:96) ; ij are theauto-correlation spectra of the IAs, and N (cid:15)(cid:96) ; ij is the shotnoise. The C γ I (cid:96) ; ij spectra are zero except in the case ofwhen photometric redshifts cause scattering of observedredshifts between bins.The additional non-zero IA spectra can be described inan analogous manner to the shear power spectra, throughthe use of the non-linear alignment (NLA) model [23]: C I γ(cid:96) ; ij = (cid:90) χ lim d χS ( χ ) [ W i ( χ ) n j ( χ ) + n i ( χ ) W j ( χ )] × P δ I ( k, χ ) , (15) C II (cid:96) ; ij = (cid:90) χ lim d χS ( χ ) n i ( χ ) n j ( χ ) P II ( k, χ ) , (16)where P δ I ( k, χ ) and P II ( k, χ ) are the IA power spectra, and are defined as functions of the matter power spectra: P δ I ( k, χ ) = (cid:18) − A IA C IA Ω m D ( χ ) (cid:19) P δδ ( k, χ ) , (17) P II ( k, χ ) = (cid:18) − A IA C IA Ω m D ( χ ) (cid:19) P δδ ( k, χ ) . (18)Within these equations, A IA and C IA are free model pa-rameters to be determined by fitting to data or simula-tions, and D ( χ ) is the growth factor of density perturba-tions in the Universe, as a function of comoving distance.The shot noise, which is the last of the terms in equa-tion (14), is written as: N (cid:15)(cid:96) ; ij = σ (cid:15) ¯ n g /N bin δ K ij , (19)where σ (cid:15) is the variance of the observed ellipticities inthe galaxy sample, ¯ n g is the galaxy surface density ofthe survey, N bin is the number of tomographic bins used,and δ K ij is the Kronecker delta. The shot noise term iszero for cross-correlation spectra because the ellipticitiesof galaxies at different comoving distances should be un-correlated. Equation (19) assumes that the tomographicbins used in the survey are equi-populated. C. Relaxing the Reduced Shear Approximation
The procedure for relaxing the reduced shear approx-imation involves Taylor expanding equation (1) around κ = 0, and retaining terms up to and including second-order: [14, 24, 25]: g α ( θ ) = γ α ( θ ) + ( γ α κ )( θ ) + O ( κ ) . (20)This expression for g α is then substituted for γ α in equa-tion (9). Recomputing the power spectrum, we recoverequation (12) plus a second-order correction term: δC RS (cid:96) ; ij = (cid:90) ∞ d (cid:96) (cid:48) (2 π ) cos(2 φ (cid:96) (cid:48) − φ (cid:96) ) × B κκκij ( (cid:96) , (cid:96) (cid:48) , − (cid:96) − (cid:96) (cid:48) ) , (21)in which B κκκij are the two-redshift convergence bispectra.Under the assumption of an isotropic universe, we arealways free to set φ (cid:96) = 0.The convergence bispectra can then be safely expressedsubject to the Limber approximation [26] as projectionsof the matter bispectra, B δδδ : B κκκij ( (cid:96) , (cid:96) , (cid:96) ) = B κκκiij ( (cid:96) , (cid:96) , (cid:96) ) + B κκκijj ( (cid:96) , (cid:96) , (cid:96) )= (cid:90) χ lim d χS ( χ ) W i ( χ ) W j ( χ ) × [ W i ( χ ) + W j ( χ )] × B δδδ ( k , k , k , χ ) . (22)The analytic form of the matter bispectrum is not fullyknown. Instead, expressions are typically obtained byfitting to N-body simulations [27–29]. In this work, weexamine two such approaches.The first approach starts from second-order perturba-tion theory [30], and then fits the resulting expression tosimulations. We denote this approach by SC, after thefirst work to propose this methodology [27]. Here, thematter bispectrum can be written as: B δδδ ( k , k , k , χ ) = 2 F eff2 ( k , k ) P δδ ( k , χ ) P δδ ( k , χ )+ cyc. perms. , (23)with: F eff2 ( k , k ) = 57 a ( n s , k ) a ( n s , k )+ 12 k · k k k (cid:18) k k + k k (cid:19) b ( n s , k ) b ( n s , k )+ 27 (cid:18) k · k k k (cid:19) c ( n s , k ) c ( n s , k ) , (24)where a, b , and c are fitting functions given in [27].A more contemporary approach adopts the Halofit formalism [31] for the matter power spectrum, to alsodescribe the matter bispectrum [29]. We denote this ap-proach by BH, as this technique is known as
BiHalofit .In this paradigm, the matter bispectrum constitutes one-halo (1h), and three-halo (3h) terms: B δδδ ( k , k , k , χ ) = B h ( k , k , k , χ )+ B h ( k , k , k , χ ) . (25)These terms are then determined by fitting to N-bodysimulations. A full description of these can be found inAppendix B of [29]. D. k -cut Cosmic Shear Given that the shear angular power spectrum is a pro-jection of the matter power spectrum, to remove sensi-tivity to physical scales below a certain k-mode we mustremove angular scales above the corresponding (cid:96) -mode.One may imagine that, in the regime of the Limber ap-proximation, this could simply involve removing infor-mation where (cid:96) > kχ . However, in reality lensing kernelsare broad; meaning that lenses across a range of distancesand scales contribute power to the same (cid:96) -mode. Con-sequently, this simple method of removing scales is noteffective on its own [10].A solution comes in the form of the BNT nullingscheme [12]. In this formalism, the observed tomographicangular power spectrum can be re-weighted in such a waythat each redshift bin retains only the information aboutlenses within a small redshift range. This procedure canbe illustrated by first considering three discrete sourceplanes. Then, the BNT weighted convergence, assumingflatness, can be written as: κ BNT = 3Ω m H c (cid:90) χ β d r δ ( χ ) a ( χ ) w ( χ ) , (26) where χ β is the comoving distance to source plane i , and: w ( χ ) = (cid:88) β,χ β >χ p β χ β − χχ β , (27)where p β are the weights for planes β = { , , } with χ < χ < χ , for the three bin case. In the BNT scheme,weights are then chosen such that w ( χ < χ ) = 0. Thiscoupled with the fact that lenses with χ > χ will notcontribute to the re-weighted convergence, means that κ BNT will only be sensitive to lenses with comoving dis-tances in the range χ ≤ χ < χ . This argument can begeneralized [32] for an arbitrary number of continuoussource bins; leading to the construction of a weightingmatrix, M , that can be applied to the observed tomo-graphic angular power spectra: C BNT (cid:96) = M C (cid:96) M T , (28)where C (cid:96) is a matrix of the C (cid:96) ; ij for all tomographicbin combinations, at the given (cid:96) -mode, and C BNT (cid:96) is itsBNT-nulled counterpart.For a given k -cut, we remove information where (cid:96) >k cut χ mean i from the tomographic BNT-nulled angularpower spectrum of bin i . Here, we use the mean comov-ing distance of the redshift bin rather than the minimumdistance to the bin in order to avoid removing the firstbin entirely. This has negligible impact on reduction insensitivity to small scales [13]. E. Fisher Matrices and Bias Formalism
The cosmological parameter constraints for a given sur-vey can be predicted by using the Fisher matrix formal-ism [33]. The Fisher matrix is given by the expectationof the Hessian of the likelihood. By safely assuming aGaussian likelihood [34, 35], we can rewrite the Fishermatrix in terms of only the mean of the data vector, andthe covariance of the data. For the cosmic shear case,we note that the mean of the shear field is zero. Underthe Gaussian covariance assumption, the specific Fishermatrix for a cosmic shear survey is then (see e.g. [1] fora detailed derivation): F τζ = f sky (cid:96) max (cid:88) (cid:96) = (cid:96) min ∆ (cid:96) (cid:18) (cid:96) + 12 (cid:19) × tr (cid:20) ∂ C (cid:96) ∂θ τ C (cid:96) − ∂ C (cid:96) ∂θ ζ C (cid:96) − (cid:21) , (29)where f sky is the fraction of sky included in the survey,∆ (cid:96) is the bandwidth of (cid:96) -modes sampled, the sum is overthese blocks in (cid:96) , and τ and ζ refer to parameters ofinterest, θ τ and θ ζ . The predicted uncertainty for a pa-rameter, τ , is then calculated with: σ τ = (cid:113) F ττ − . (30)This formalism can be adapted to show how biased thepredicted cosmological parameter values will be when asystematic effect in the data is neglected [36]: b ( θ τ ) = (cid:88) ζ F − τζ f sky (cid:88) (cid:96) ∆ (cid:96) (cid:18) (cid:96) + 12 (cid:19) × tr (cid:20) δ C (cid:96) C (cid:96) − ∂ C (cid:96) ∂θ ζ C (cid:96) − (cid:21) , (31)where δ C (cid:96) is a matrix with every tomographic bin combi-nation of the systematic effect, δC (cid:96) ; ij , evaluated at mode (cid:96) . In this work, these systematic effect terms are givenby the reduced shear correction of equation (21). III. METHODOLOGY
In order to examine whether k -cut cosmic shear canbe used to minimise the impact of the reduced shear ap-proximation on Stage IV surveys, we adopt forecastingspecifications for a Euclid -like survey [1]. The k -cut tech-nique enables the inclusion of information from smallerangular scales, making the ‘optimistic’ scenario for sucha survey, where (cid:96) -modes up to 5000 are studied, moreachievable. Accordingly, we compute the reduced shearcorrection, and carry out the corresponding k -cut anal-ysis, up to this maximum (cid:96) . This is compared to the‘pessimistic’ case for such an experiment where only (cid:96) -modes up to 1500 are included, and no k -cut is necessary[1].The fraction of sky that will be covered by a Euclid -likesurvey is f sky = 0 .
36. The intrinsic variance of unlensedgalaxy ellipticities is modelled with two components, eachof value 0.21, so that the root-mean-square intrinsic el-lipticity is σ (cid:15) = √ × . ≈ .
3. The surface densityof galaxies will be ¯ n g = 30 arcmin − . We examine thecase where the data consists of ten equi-populated red-shift bins with limits: { } .The galaxy distributions within these tomographicbins, assuming they are determined with photometricredshift estimates, are modelled according to: N i ( z ) = (cid:82) z + i z − i d z p n ( z ) p ph ( z p | z ) (cid:82) z max z min d z (cid:82) z + i z − i d z p n ( z ) p ph ( z p | z ) , (32)where z p is measured photometric redshift, z − i and z + i are edges of the i -th redshift bin, z min and z max definethe range of redshifts covered by the survey, and n ( z ) isthe true distribution of galaxies with redshift, z , whichis defined using the expression [6]: n ( z ) ∝ (cid:18) zz (cid:19) exp (cid:20) − (cid:18) zz (cid:19) / (cid:21) , (33)wherein z = z m / √
2, with z m = 0 . p ph ( z p | z ) exists to TABLE I. Parameter values in this investigation in order todescribe the probability distribution function of the photo-metric redshift distribution of sources, in equation (34).Parameter Value c b z b σ b c o z o σ o f out account for the probability that a galaxy at redshift z ismeasured to have a redshift z p , and is given by: p ph ( z p | z ) = 1 − f out √ πσ b (1 + z ) exp (cid:40) − (cid:20) z − c b z p − z b σ b (1 + z ) (cid:21) (cid:41) + f out √ πσ o (1 + z ) × exp (cid:40) − (cid:20) z − c o z p − z o σ o (1 + z ) (cid:21) (cid:41) . (34)In this equation, the first term on the right-hand sidedescribes the multiplicative and additive bias in redshiftdetermination for the fraction of sources with a well mea-sured redshift, while the second term accounts for the ef-fect of a fraction of catastrophic outliers, f out . The valuesassigned to the parameters in this equation are stated inTable I. Then, the galaxy distribution as a function ofcomoving distance is n i ( χ ) = N i ( z )d z/ d χ .Kinematic lensing has been proposed as a method toreduce shape noise in weak lensing by an order of magni-tude [17]. It is predicated on spectroscopic measurementsof disk galaxy rotation and use of the Tully-Fisher rela-tion in order to control for the intrinsic orientations ofgalaxy disks. Here, we study the effect of k -cut cosmicshear on the hypothetical TF-Stage III survey describedin [17]. This survey includes (cid:96) -modes up to 5000, has f sky = 0 .
12, with an intrinsic ellipticity of σ (cid:15) = 0 . n g = 1 . − .We consider the survey to have ten equi-populated red-shift bins with limits: { } . A kinematic sur-vey will not have IA contributions. The galaxy distribu-tion for such a survey is modelled by: N i ( z ) ∝ z α e − (cid:16) zz (cid:17) β , (35)with α = 29 . z = 1 . × − , and β = 0 . w w a CDM fiducial cosmol-ogy. Allowing for a time-varying dark energy equation-of-state, the model consists of the following parameters:the present-day total matter density parameter Ω m , thepresent-day baryonic matter density parameter Ω b , theHubble parameter h = H / − Mpc − , the spec-tral index n s , the RMS value of density fluctuations on TABLE II. Fiducial values of w w a CDM cosmological param-eters adopted in this work. Values were selected to match [1].Cosmological Parameter Fiducial ValueΩ m b h n s σ (cid:80) m ν (eV) 0.06 w − w a h − Mpc scales σ , the present-day value of the darkenergy equation of state w , the high-redshift value ofthe dark energy equation of state w a , and massive neu-trinos with a sum of masses (cid:80) m ν (cid:54) = 0. We choose thesame fiducial parameter values as presented in [1]. Weexplicitly state these in Table II. The BNT matrices arecalculated using the code of [32]. Additionally, to cal-culate the matter power spectrum, we use the publiclyavailable CAMB cosmology package [37], with Halofit [31] and corrections from [38] used to compute the non-linear contributions. Comoving distances are computedwith
Astropy [39, 40]. To obtain the matter bispec-trum of the BH approach, we employ the publicly avail-able C code supplied with [29]. The IA power spectraare modelled with the parameter values: A IA = 1 . C IA = 0 . m , Ω b , h, n s , σ , w , w a , and A IA ,for consistency with [1]. IV. RESULTS AND DISCUSSION
In this section, we demonstrate the effect k -cut cos-mic shear has on addressing the biases resulting fromthe reduced shear approximation, for a Euclid -like ex-periment and a hypothetical kinematic survey. We be-gin by comparing the cosmological parameter biases, forthe standard calculation with no k -cut, found when thereduced shear approximation is relaxed with either theSC or BH bispectrum models. Next, the change in pa-rameter constraints and biases for the BNT transformedpower spectra with a range of k -cuts are shown; first fora Euclid -like survey, and then a kinematic lensing survey.
A. Comparing Matter Bispectrum Models
The ratio of the reduced shear correction of equation(21) calculated using the BH bispectrum, relative to the https://github.com/pltaylor16/x-cut https://camb.info/ http://cosmo.phys.hirosaki-u.ac.jp/takahasi/codes_e.htm ‘ . . . . . δ C R S ; B H ‘ / δ C R S ; S C ‘ Bin 0.001 – 0.418Bin 0.678 – 0.789Bin 1.019 – 1.155Bin 1.576 – 2.50
FIG. 1. Ratio of reduced shear corrections calculated with twodifferent matter bispectrum models. The first of these usesthe approach of [27] and is labelled by SC, whereas the secondis the
BiHalofit model [29] and is denoted by BH. The cor-rection terms for four different auto-correlation spectra acrossthe survey’s anticipated redshift range are presented, and arerepresentative of all the spectra. The most extreme disagree-ment between the models occurs at (cid:96) = 89, where they dis-agree by 27%. We note that the reduced shear correctionis negligible at these scales, and only becomes significant atscales above (cid:96) ∼ correction calculated using the SC bispectrum is shownin Figure 1. Here, the correction terms for the auto-correlation of four bins, with redshift-limits: 0.001 –0.418, 0.678 – 0.789, 1.019 – 1.155, and 1.576 – 2.50, areshown in order to illustrate the difference between thetwo models. The consequent difference in the predictedparameter biases from using the two models is stated inTable III.From Figure 1, we see that the two approaches producecorrection terms that differ at most by 27%. At low- (cid:96) andat all but the highest redshifts, the BH model produces TABLE III. Cosmological parameter biases predicted if thereduced shear correction is neglected for two different matterbispectrum models. The SC model uses the fitting formulaeof [27], while BH is the
Bihalofit model [29]. The differ-ence between the two approaches is also stated, and is notsignificant. Here σ denotes the 1 σ uncertainty.Cosmological SC Model BH Model Absolute Difference inParameter Bias/ σ Bias/ σ Biases/ σ Ω m -0.32 -0.28 0.04Ω b -0.011 -0.0056 0.0044 h n s σ w -0.40 -0.33 0.07 w a FIG. 2. Change in the 1 σ cosmological parameter constraints predicted for a Euclid -like survey, when a range of k -cuts areapplied. These results are for the ‘optimistic’ case for such a survey, where (cid:96) -modes up to 5000 are included. Unsurprisingly,the constraints weaken as lower k -cuts are taken; corresponding to more information being removed. The black dashed line at k = 5 . h Mpc − marks the maximum k -cut required for biases from the reduced shear correction to not be significant.FIG. 3. Change in cosmological parameter biases with changing k -cuts, when the reduced shear correction is neglected, for a Euclid -like survey. The values are reported as a fraction of the 1 σ uncertainty of the respective parameter. A parameter isconsidered to be significantly biased if the bias is greater than 0.25 σ . Beyond this point, the biased and unbiased confidenceregions overlap less than 90%. These results are for the ‘optimistic’ case for a Euclid -like survey, where (cid:96) -modes up to 5000are included. The black dashed line at k = 5 . h Mpc − marks the maximum k -cut required for biases from the reduced shearcorrection to not be significant. The brown dotted line denotes the threshold for a bias to be significant. Generally, a lower k -cut corresponds to smaller biases, as sensitivity is reduced to regions where the reduced shear correction is largest. a correction smaller than the SC one. The BH correc-tion then increases until the two models produce com-parable results at (cid:96) ∼ (cid:96) -mode, theBH model once again produces a smaller correction valuethan the SC approach. For the highest redshift bins, thesame trend persists. However, in this case the correctionsstart off being comparable, before the BH term becomesgreater than the SC correction. After peaking at scalesof (cid:96) ∼ (cid:96) -modes, where the reduced shear correction is typically negligible [14]. Additionally, these differences are likelyto be dwarfed by baryonic model uncertainties.Despite these differences, Table III shows that the re-sulting cosmological parameter biases from the two mod-els are not significantly different. Accordingly, althoughthe BH and SC models can differ significantly at calcu-lating the matter bispectrum for certain scales and con-figurations [29], the reduced shear correction calculationcan be considered robust to the choice of matter bispec-trum model. For all results that follow, we use the BHmatter bispectrum model. TABLE IV. Predicted parameter uncertainties, and biases from neglecting the reduced shear approximation, for a
Euclid -likesurvey under three different scenarios. The ‘optimistic’ scenario is when (cid:96) -modes up to 5000 are included, and no k -cut ismade, while the ‘maximum k -cut’ columns denote the situation where (cid:96) -modes up to 5000 are included, but a k -cut is takenat k = 5 . h Mpc − , as this is the maximum k -cut to achieve non-significant biases. Finally, the ‘pessimistic’ case is whenonly (cid:96) -modes up to 1500 are included, and no k -cut is taken. The ‘maximum k -cut’ option is able to suppress the biases to thepoint of not being significant, while still achieving more precise constraints than the ‘pessimistic’ option. Here σ denotes the1 σ uncertainty.Cosmological Optimistic ( (cid:96) max = 5000) Maximum k -cut Pessimistic ( (cid:96) max = 1500) Optimistic Maximum k -cut PessimisticParameter Uncertainty (1 σ ) Uncertainty (1 σ ) Uncertainty (1 σ ) Bias/ σ Bias/ σ Bias/ σ Ω m b h n s σ w w a (cid:96) -modes up to 5000 are included, and no k -cut is made, while the ‘maximum k -cut’ columns denote the situation where (cid:96) -modesup to 5000 are included, but a k -cut is taken at k = 5 . h Mpc − , as this is the maximum k -cut to achieve non-significantbiases. Finally, the ‘pessimistic’ case is when only (cid:96) -modes up to 1500 are included, and no k -cut is taken. The ‘maximum k -cut’ option is able to suppress the biases to the point of not being significant, while still achieving more precise constraintsthan the ‘pessimistic’ option. Here σ denotes the 1 σ uncertainty.Cosmological Optimistic ( (cid:96) max = 5000) Maximum k -cut Pessimistic ( (cid:96) max = 1500) Optimistic Maximum k -cut PessimisticParameter Uncertainty (1 σ ) Uncertainty (1 σ ) Uncertainty (1 σ ) Bias/ σ Bias/ σ Bias/ σ Ω m b h n s σ w w a B. k -cut Cosmic Shear and Reduced Shear forStage IV Surveys We calculated the cosmological parameter constraints,and the biases resulting from neglecting the reducedshear approximation, for a range of k -cut values. Thechanging constraints are shown in Figure 2, whilst thebiases are shown in Figure 3. As expected, taking lower k -cuts results in weaker constraints. In general, biases re-duce as a lower k -cut is taken. The behaviour of the biasin Ω b is non-trivial, due to the complex way in which thisparameter interacts with the non-linear component of thematter power spectrum. A bias is considered significantif its magnitude is greater than 0.25 σ , as beyond this theconfidence contours of the biased and unbiased parameterestimates overlap by less than 90% [41]. The maximum k -cut required in order to ensure that no parameter bi-ases are significant is 5.37 h Mpc − . Table IV shows thebiases and constraints at this k -cut, as well as the bi-ases and constraints when no k -cut is taken for both the‘optimistic’ ( (cid:96) max =5000), and ‘pessimistic’ ( (cid:96) max =1500)scenarios for a Euclid -like survey. From this, we see thatthe optimum k -cut increases the size of all of the pa- rameter constraints by less than 10%. This is a markedimprovement over the ‘pessimistic’ case in which all buttwo of the parameters have their constraints increased bymore than 10% compared to the ‘optimistic’ case.These findings support the idea that k -cut cosmic shearcan be successfully used to access smaller angular scalesfor upcoming Stage IV weak lensing surveys. It has al-ready been shown that this technique can bypass the needto model baryonic physics [13], while allowing access tosmall physical scales. Now, these results indicate that k -cut cosmic shear can also address the impact of thereduced shear approximation. While explicit calculationof the reduced shear correction yields the most precisecosmological parameter constraints, it is prohibitivelycomputationally expensive [14]. The k -cut approachesbypasses this cost while only marginally weakening theconstraints.We note that if the photometric redshifts are system-atically mis-calibrated, the BNT transform we computewould be inaccurate. In fact, given that the lensing ker-nels have some width, using the peak of the kernel as arepresentative comoving distance value for the k -cut isalready technically inaccurate. Despite this, the k -cuttechnique proves successful [13]. Given that we wouldexpect any biases in the photometric redshifts to be nar-rower than the width of the kernel, we do not anticipatethat these biases would significantly affect the validityof the k -cut method. In addition, if there is no mis-calibration, the BNT transformed cross-spectra shouldbe small, and dominated by shot-noise, which is wellknown and cosmology-independent. If there is significantphotometric redshift calibration bias, these cross-spectrawill no longer be small. Accordingly, the BNT transformcan also serve as a null-test for mis-calibration.Furthermore, another consideration is our choice of IAmodel. The NLA model used here can be overly restric-tive, and artificially improve constraining power. Thiscould lead to an overestimate of the biases, and accord-ingly the determination of a lower than needed k -cut.However, in any case the limiting k -cut value will be thatnecessitated by baryonic physics. C. k -cut Cosmic Shear and Reduced Shear forKinematic Weak Lensing Surveys The predicted cosmological parameter constraints fora hypothetical kinematic lensing survey which includes (cid:96) -modes up to 5000, together with the expected biasesin those constraints from neglecting the reduced shearapproximation, are stated in Table V. From this we seethat the reduced shear correction is also necessary forpotential future kinematic lensing surveys, as the bias in n s is significant. This is due to the fact that constrainton n s is improved, compared to the standard Stage IVcase. The spectral index is most sensitive to high- (cid:96) modes[42], and this is where the hypothetical kinematic surveyperforms better than the standard survey. The kinematicsurvey has a higher signal-to-noise ratio at high- (cid:96) , and alower signal-to-noise ratio at low- (cid:96) , as the shot-noise islow by construction, and because it covers a smaller areathan the Stage IV survey which means sample varianceis relatively more important.For such a survey, we find that the maximum k -cutrequired for the biases from the reduced shear correc-tion to no longer be significant is 5.82 h Mpc − . Thisis higher than the value in the Stage IV survey case, be-cause the kinematic survey is less deep in redshift. Conse-quently, the same (cid:96) -mode corresponds to a higher k -modefor the kinematic survey than in the Stage IV experimentcase. Since the the reduced shear correction is only non-negligible at the highest (cid:96) -modes, this is where a cut willalleviate biases, and shallower surveys can include higher k -modes before reaching this regime. Table V shows thepredicted parameter constraints and reduced shear biasesat this k -cut. For comparison, the constraints and biasesfor the pessimistic case of the kinematic survey, whereonly (cid:96) -modes up to 1500 are probed, are also shown here.As with the Stage IV cosmic shear survey, the k -cut tech-nique degrades the predicted cosmological constraints for a kinematic lensing survey less than the exclusion of (cid:96) -modes above 1500. With the k -cut, the largest increaseis on the constraint on h , which increases by 27%. Incomparison, in the pessimistic case, the lowest increasein constraints is of 44%, for Ω b . V. CONCLUSIONS
In this paper, we have examined the validity of the re-duced shear approximation when applying k -cut cosmicshear to Stage IV cosmic shear experiments, and a hypo-thetical kinematic lensing survey. We first compared thereduced shear correction calculated using two differentmodels for the matter bispectrum: the fitting formulaeof [27], and the BiHalofit model [29]. Despite the dif-ferences between the two approaches, we found that theirresulting reduced shear corrections were not significantlydifferent, and that accordingly the reduced shear correc-tion was robust to the choice of bispectrum model.The k -cut cosmic shear technique is used to removesensitivity to baryonic physics, while allowing access tosmall physical scales. We examined whether it would alsoaffect the impact of the reduced shear approximation. Avariety of k -cuts were applied to the BNT transformedtheoretical shear power spectra and reduced shear correc-tions for the ‘optimistic’ case of a Euclid -like survey. Thisscenario assumes (cid:96) -modes up to 5000 are probed. Wedemonstrated that, in this case, k -cut cosmic shear pref-erentially removes scales sensitive to the reduced shearapproximation, reducing it’s importance. This techniquemakes this ‘optimistic’ scenario more achievable, whilebypassing the significant computational expense posedby having to explicitly calculate the reduced shear correc-tion. The disadvantage is that the inferred cosmologicalparameter constraints are weakened. However, with k -cut cosmic shear applied to the ‘optimistic’ case, the pa-rameters constraints are weakened significantly less thanthose found in the ‘pessimistic’ case for such a survey;where only (cid:96) -modes up to 1500 are included. We alsorepeated this analysis for a theoretical kinematic lensingsurvey; finding similarly that the k -cut technique reducedsensitivity to the reduced shear approximation. ACKNOWLEDGMENTS
The authors would like to thank Eric Huff for provid-ing the theoretical galaxy distribution for a hypotheti-cal TF-Stage III kinematic lensing survey. ACD wishesto acknowledge the support of the Royal Society. PLTacknowledges support for this work from a NASA Post-doctoral Program Fellowship. Part of the research wascarried out at the Jet Propulsion Laboratory, CaliforniaInstitute of Technology, under a contract with the Na-tional Aeronautics and Space Administration.0 [1] Euclid Collaboration, A. Blanchard, S. Camera, C. Car-bone, V. F. Cardone, S. Casas, S. Ili´c, M. Kil-binger, T. Kitching, M. Kunz, et al. , arXiv e-prints, arXiv:1910.09273 (2019), arXiv:1910.09273 [astro-ph.CO].[2] A. Albrecht, G. Bernstein, R. Cahn, W. L. Freedman,J. Hewitt, W. Hu, J. Huth, M. Kamionkowski, E. W.Kolb, L. 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