Observing Baryonic Acoustic Oscillations in tomographic cosmic shear surveys
YYITP-20-15
Observing Baryonic Acoustic Oscillations in tomographic cosmic shear surveys
Francis Bernardeau,
1, 2
Takahiro Nishimichi,
3, 4 and Atsushi Taruya
3, 4 Institut d’Astrophysique de Paris & Sorbonne Universit´e and CNRS,98 bis boulevard Arago, 75014, Paris, France. Universit´e Paris-Saclay, CNRS, CEA, Institut de physique th´eorique, 91191, Gif-sur-Yvette, France Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan Kavli Institute for the Physics and Mathematics of the Universe (WPI),UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan (Dated: April 8, 2020)We show that it is possible to build effective matter density power spectra in tomographic cosmicshear observations that exhibit the Baryonic Acoustic Oscillations (BAO) features once a nullingtransformation has been applied to the data. The precision with which the amplitude and positionof these features can be reconstructed is quantified in terms of sky coverage, intrinsic shape noise,median source redshift and number density of sources. BAO detection in Euclid or LSST like widesurveys will be possible with a modest signal-to-noise ratio. It would improve dramatically forslightly deeper surveys.
The shape deformation of background galaxies inducedby the gravitational lensing of cosmological matter den-sity fluctuations along the line-of-sight – cosmic shear –has been detected for the first time in the early 2000,[1–3]. Results obtained in more recent surveys (such asthe CFHTLS survey [4], DES , Subaru HSC [5, 6], KiDS(+VIKING) survey [7]) have confirmed that it is possi-ble to build dedicated cosmic shear wide surveys in whichthis effect can be measured with exquisite precision of-fering new means to constrain fundamental cosmologicalparameters. This is at the heart of large projects such asthe LSST or EUCLID .More precisely, on cosmological scale, cosmic shear sig-nals depend on the geometry of Universe through a com-bination of angular diameter distances, mass density ofthe universe and amplitude of the density fluctuations(see for instance [9]). With tomographic information,correlation functions in general give access to precise con-straint on the expansion history of the Universe [10]. An-other solid probe of the background expansion of the uni-verse has been put forward in this context, namely thesignature of sound waves in the photon-baryon plasmain the early Universe. It has now been measured withhigh precision in Cosmic Microwave Background (CMB)data . These Baryon Acoustic Oscillations (BAO) arehowever not limited to CMB observations: they havebeen seen in the galaxy distribution as a preferred co-moving separation of galaxies of 150 Mpc [14], or, equiv-alently, as a series of oscillations in the galaxy power see [8]. See for instance [11, 12] on how they have been used to constraincosmological models and in particular dark energy parametersexploiting the fact that the mechanisms at play are fully under-stood, e.g. [13]. spectrum [15].The purpose of the letter is to demonstrate that BAOfeatures can also be detected in intrinsic cosmic shearobservations, despite the fact that in essence weak lens-ing effects collect projected information of the large-scalemass distribution making it hard to detect the acousticsignature from the resultant featureless spectra. Here,we propose to use the nulling technique developed in [16],which can substantially mitigate the mixing of small- andlarge-scale modes due to projection effects, hence enablesto make BAO features visible. We shall below examinein detail the feasibility of BAO detection based on thismethod.In the context of this study we simply assume that cos-mic shear observations lead to the construction of multi-ple convergence maps κ (ˆ n ) where ˆ n is a unit vector point-ing in the celestial direction ˆ n . With detailed determi-nation of the (photometric-)redshifts of the backgroundgalaxies, it is furthermore possible to define vectors ofcosmic-shear observations, κ i (ˆ n ) corresponding to differ-ent populations of sources. In particular it is in principlepossible to split the source populations in redshift binsto create tomographic data sets as exemplified in [10].The data vectors we can manipulate are then conver-gence maps related to the mass density contrast, δ matter ,along the line of sight : κ i (ˆ n ) = ˆ d χ w i ( χ ) δ matter ( χ, ˆ n ) (1)where w i ( χ ) is the resulting radial selection function forgalaxies selected in the i -th redshift bin (written here as This relation is however an approximation as it makes use ofthe Born approximation, ignores lens couplings and assumes thereduced shear can be approximated by the shear itself. We do notthink though that these approximations have significant impacton the developments presented in this paper. a r X i v : . [ a s t r o - ph . C O ] A p r Ω i (cid:72) z (cid:76) Figure 1: The nulling radial kernel functions. The thin linesare the standard tomographic weak-lensing kernel functions.The thick lines are those obtained after the nulling procedurehas been implemented. The results are presented here for 8equally populated tomographic bins. a function of the comoving distance χ ), that is, in flatcosmology, w i ( χ ) = 3Ω H c ˆ bin i d z n ( z ) ( χ ( z ) − χ ) χχ ( z ) a ( χ ) (2)where n ( z ) is the source redshift distribution functionand the integration of z is restricted to the bin i . Thequantities Ω , H and c are respectively the mass densityparameter, Hubble parameter at present time, and thespeed of light. The core observables are then the autoand cross-spectra of such maps given as a function ofmultipole (cid:96) , C ij ( (cid:96) ), defined as (cid:10) a ( i ) (cid:96)m a ( j ) ∗ (cid:96) (cid:48) m (cid:48) (cid:11) = δ (cid:96)(cid:96) (cid:48) δ mm (cid:48) C ij ( (cid:96) ) (3)where a ( i ) (cid:96)m are the spherical harmonics coefficients of themaps κ i (ˆ n ). The cosmic shear spectra are then related tothe (redshift dependent) matter power spectrum P m ( k, z )with C ij ( (cid:96) ) = ˆ χ (cid:48) ( z )d zχ ( z ) w i ( χ ( z )) w j ( χ ( z )) P m (cid:18) (cid:96) + 1 / χ ( z ) , z (cid:19) . (4)The fundamental property we want to take advantageof is called nulling . It has indeed been shown in [16] thatone can define a transformation matrix p ij such that thetransformed radial selection function, ˆ w i = (cid:80) j p ij w j , arefully localised in redshift. This is illustrated on Fig. 1.where the “nulled” radial selection function is shown asthick solid lines. We recall here how this transformationis built. Let us first define the following quantities, n (0) i = ˆ bin i d z n ( z ) , n (1) i = ˆ bin i d z n ( z ) 1 χ ( z ) , (5) Notions of nulling in this context were first introduced in [17].We refer here to a more recent implementation of it.
10 50 100 50010001.02.03.01.5 multipole (cid:123) (cid:123) C ii (cid:72) (cid:123) (cid:76) (cid:144) (cid:72) Π (cid:76) (cid:137)
10 50 100 50010000.20.51.02.05.0 multipole (cid:123) (cid:123) C ii (cid:72) (cid:123) (cid:76) (cid:144) (cid:72) Π (cid:76) (cid:137) Figure 2: Cosmic shear tomographic (top panel) and nulled(bottom) spectra. The spectra are color-coded with the sameconvention as in Fig. 1. The color lines correspond to thehalofit power spectra. The solid and dashed grey lines cor-respond to linear and 2-loop standard perturbation theoryprediction respectively. Most of the oscillatory features ofinterest here are well described by the 2-loop perturbationtheory calculations which furthermore capture the departure,in amplitude and shape, from linear theory. We show hereonly the last three nulled bins for clarity. then, the two conditions (to be implemented for i > i (cid:88) j = i − p ij n (0) j = 0 , i (cid:88) j = i − p ij n (1) j = 0 (6)together with p ij = 0 for j < i − w i ( z )vanishes for redshifts in the 1 , . . . , i − n ( z ) ∝ z e − / ( zz m ) / (7)with z m is the median redshift of the sources which isabout 0.9 for or the Euclid wide survey, [8]. As stressedin [16], this transformation allows to separate the contri-butions of different scales. It was pointed in particularthat it permits the application of the Perturbation The-ory for the calculations of power spectra (see below). Weshow explicitly here that, by the same token, it allows toactually locate features in power spectra, such as BAO.More precisely the observables we want to exploit aredefined as ˆ C ij ( (cid:96) ) = p ia p jb C ab ( (cid:96) ) (8)and are built from the original spectra with the helpof the nulling transformation coefficients. The expectedspectra ˆ C ii , for i = 6 , k which is spanned for a given (cid:96) , the BAO featuresare made visible (bottom panel), and this is in markedcontrast to what is obtained from standard tomographicspectra (top panel).One can be more precise on the conditions for whichsuch observations are made possible: for each (cid:96) and eachbin i , the range in k which is spanned for the computationof C (cid:96) is now bounded. Its range is typically about∆ k eff . ( i ; (cid:96) ) = (cid:96)χ eff . ( i ) ∆ χ eff . ( i ) χ eff . ( i ) , (9)where χ eff . ( i ) is the average angular distance of the bin i and ∆ χ eff . ( i ) its width. One can check that for (cid:96) about200, ∆ k eff . ( i ) varies from 0 .
17 to 0 . h/ Mpc which, forthe furthest bins, is indeed much smaller than the BAOwavelength in Fourier space (that is about 0 . h /Mpc).Furthermore, comparing the linear and the SPT or halofitpredictions shows that the BAO features are slightlysmeared out because of non-linear couplings but are stillclearly present. To be noted also is that although SPThas a limited validity range, it provides us with an ac-curate description of the BAO in the quasi-linear regimeshowing such features can then be captured from firstprinciple calculations. Incidentally note that the rangein which standard tomographic spectra can be accuratelycomputed from first principle is much more limited.One can take advantage of these quantities to constructeffective power-spectrum estimators as a function of thewavenumber kP ( i )eff . ( k ) = (cid:18) ˆ d χχ w i ( χ ) D ( χ ) (cid:19) − ˆ C ii ( kχ eff . ( i )) (10)where D + ( χ ) is the linear growth rate of the fluctuations.Such power spectra can then be aggregated together.With such a construction it is then possible to coher-ently add the observed spectra and preserve the infor-mation on the BAO positions. In Fig. 3 we present theresult of such an exercise. The adopted value for z m ishere 1.3. We used 10 bins to ensure the BAO featuresare sharp enough and use the bins 4 to 10 to do the re-construction. Closer bins are found to slightly smear the (cid:72) h (cid:144) Mpc (cid:76) k P e ff . (cid:72) k (cid:76) Figure 3: The reconstructed matter power spectra for eachbin and from a combination of observed cosmic shear spectraaccording to Eq. (10). The thin coloured lines are the powerspectra from individual bins (where the nonlinear effects areall the more important that the bins are closer). Bins 4 to 9are shown assuming a 10 equi-populated binning. The greydashed line is the prediction from the linear theory and thethick solid line is the reconstructed power-spectrum from theaggregation of bins 4 to 10. oscillatory features. The end result of the summationis an effective power spectrum whose shape is interme-diate between the linear and nonlinear predictions as itaggregates observations obtained at different redshifts.In the following we explore the precision with whichone could constrain the amplitude and position of theBAO in realistic observational scenarios. Rather than P eff . ( k ) the starting point of the computation is to usethe nulling power spectra and cross-spectra as a datavector. The full data vector can then be defined as thecollection of spectra ˆ V ij ( (cid:96) ) for i ∈ (1 , n b ), j ≥ i , and wethen assume to haveˆ V ij ( (cid:96) ) = ˆ C ij + ˆ S ij (11)for the nulled channel neglecting other sources of system-atics such as intrinsic alignment. The shape noise of thenulled spectra is thenˆ S ij ( (cid:96) ) = p it p jt σ s n g ( t ) , (12)where σ s is the intrinsic shape r.m.s. – and we adopt σ s = 0 . n g ( t ) is the number density of galaxiesin bin t . The data covariance is then computed assumingthe field obeys Gaussian statistics so that all (cid:96) are inde-pendent and that each bin is obtained from the measure-ment of all available modes. More precisely, for a surveythat covers a fraction f sky of the sky the number of avail-able modes can be estimated to be about (2 (cid:96) + 1) f sky as pointed in [23]. The covariance coefficients, c ij ; kl ( (cid:96) ),between data vector elements ˆ V ij and ˆ V kl are then c ij ; kl ( (cid:96) ) = 1(2 (cid:96) + 1) f sky (cid:104) ˆ V ik ( (cid:96) ) ˆ V jl ( (cid:96) ) + ˆ V il ( (cid:96) ) ˆ V jk ( (cid:96) ) (cid:105) . (13)The models we consider are simple single-parametermodels for the linear density power spectra that ei-ther linearly interpolate between oscillatory and non-oscillatory models or for two different positions of theoscillations. More specifically we define the α − model, C α ( (cid:96) ) = C ( (cid:96) ; ω = − α [ C nw ( (cid:96) ; ω = − − C ( (cid:96) ; ω = − β − model, C β ( (cid:96) ) = C ( (cid:96) ; ω = −
1) + β dd ω [ C ( (cid:96) ; ω ) − C nw ( (cid:96) ; ω )](15)where C nw ( (cid:96) ; ω ) is the no-wiggle version of the spectraand where the projection effects are computed for a darkenergy component of constant equation of state P = ω ρ .The first model tests the possibility of detecting the os-cillatory features at a given angular position ; the secondtests the possibility of determining their angular positionthrough the dependence of the angular distance with theDark Energy equation of state. In our case, the posteriordistribution of α (or β ) behaves like, p ( α ) ∼ exp (cid:32) − α (cid:88) (cid:96) I α ( (cid:96) ) (cid:33) (16)where I α ( (cid:96) ) is a rate function for each (cid:96) obtained as I α ( (cid:96) ) = c − · (∆ ˆ V ) · (∆ ˆ V ) (17)where c − is the inverse matrix of c ij ; kl and ∆ ˆ V is thedifference between α = 0 and = 1, (or β ). The resultingr.m.s in the measurement of α (or β ) is then σ α = (cid:34) (cid:96) max (cid:88) (cid:96) =10 I α ( (cid:96) ) (cid:35) − / (18)and we define the signal-to-noise ratio in the following asits inverse, i.e., ( S/N ) ≡ /σ α (or 1 /σ β ).The signal-to-noise results are computed for a Euclidlike mission with a sky coverage of 15.000 deg , assum-ing the source distribution of z m = 0 .
9. The maximummultipole used for data analysis is set to (cid:96) max = 850although the results depend very weakly on its choiceand the adopted number of bins, n b = 8, correspondsto a case where the results have reasonably converged.No theoretical nuisance parameters are introduced as weare in a regime where power spectra can be computedfrom first principle. The dependence of the signal-to-noise ratio with respect to the number density of galaxiesor rather the depth of the survey is then explored. It isobviously not straightforward to predict how the numberdensity of galaxies – and their z -distribution – will evolvewhen one improves upon the sensitivity of the detection.For simplicity, while changing z m we assume that theform (7) generally holds and that the low z amplitude of m S i g n a l (cid:45) t o (cid:45) n o i s e Figure 4: The dependence of the signal-to-noise ratio on ob-servables α (solid blue lines) and β (dashed red lines) onthe median redshift when the number density of sources ischanged accordingly. All but the first two bins are taken intoaccount. The points are exact numerical results. The linesare fitted forms presented in Eq. (19). n ( z ) is left unchanged naturally leading to a higher num-ber density of galaxies for deeper surveys. The resultsare presented in Fig. 4 showing that the signal-to-noiseratio regularly increases with z m due to the combina-tion of two effects: as the number of tracers increasesthe shape noise diminishes making the number of use-ful modes larger and the depth of the survey increasesmaking the signal larger. These two effects are of similaramplitude. The resulting scaling of the signal-to-noiseratio is encapsulated in the form, (cid:18) SN (cid:19) BAO = 3 . (cid:18) f sky . (cid:19) . (cid:16) σ (cid:15) . (cid:17) − (cid:18) z m − . . (cid:19) (19)for the α -model (and 3.2 is replaced by 2.85 for the β -model). It shows that a significant signal-to-noise ratiocan be obtained if the number density of galaxies is largeenough. In practice a non-ambiguous detection of theBAO would be secured if one pushes the median redshiftto z m = 1 . in a wide survey.The performances of the Euclid mission as it is designed[8], with about 30 gal/arcmin for a median redshift of0.9, offer then only a marginal detection. Ground basedobservations such as provided by the LSST project mightperform slightly better with a median redshift of 1.2 butwith only 30 gal/arcmin available for cosmic-shear mea-surement, [24], relying on an efficient de-blending strat-egy [25]. But relation (19) clearly shows that observingBAO in cosmic shear is within reach of dedicated widecosmic shear surveys. Acknowledgments
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