A & A 608, A 44 (2017)
D O I: 10.1051/0004-6361/201630276
© E S O 2017
Astronomy
&
Astrophysics
The VIMOS Public Extragalactic Redshift Survey (VIPERS)
G ravity test from the com bination of redshift-space distortions and galaxy-galaxy lensing at 0.5 < z < 1.2*
S. de la Torre1, E. Jullo1, C. Giocoli1, A. Pezzotta2,3, J. Bel4, B. R. Granett2, L. Guzzo2,5, B. Garilli6, M. Scodeggio6, M. Bolzonella7, U. Abbas8, C. Adami1, D. Bottini6, A. Cappi7,9, O. Cucciati10,7, I. Davidzon1,7, P. Franzetti6,
A. Fritz6, A. Iovino2, J. Krywult11, V. Le Brun1, O. Le Fevre1, D. Maccagni6, K. Małek12, F. Marulli10,13,7, M. Polletta6,14,15, A. Pollo12,16, L. A. M. Tasca1, R. Tojeiro17, D. Vergani18, A. Zanichelli19, S. Arnouts1, E. Branchini20,21,22, J. Coupon23, G. De Lucia24, O. Ilbert1, T. Moutard25,1, L. Moscardini10,13,7, J. A. Peacock26,
R. B. Metcalf10, F. Prada27,28,29, and G. Yepes30
(Affiliations can be found after the references) Received 17 D ecem ber 2016 / Accepted 2 A ugust 2017
ABSTRACT
We carry out a jo in t analysis of redshift-space distortions and galaxy-galaxy lensing, w ith the aim o f m easuring the growth rate o f structure; this is a key quantity for understanding the nature of gravity on cosm ological scales and late-tim e cosm ic acceleration. We make use of the final VIPERS redshift survey dataset, w hich maps a portion o f the Universe at a redshift of z - 0.8, and the lensing data from the CFHTLenS survey over the same area o f the sky. We build a consistent theoretical m odel that combines non-linear galaxy biasing and redshift-space distortion models, and confront it w ith observations. The two probes are combined in a Bayesian m axim um likelihood analysis to determ ine the growth rate of structure at two redshifts z = 0.6 and z = 0.86. We obtain measurements of fix 8(0.6) = 0.48 ± 0.12 and fix 8(0.86) = 0.48 ± 0.10. The additional galaxy-galaxy lensing constraint alleviates galaxy bias and ix8 degeneracies, providing direct m easurem ents o f f and ix8: [ f (0.6),ix8(0.6)] = [0.93 ± 0.22,0.52 ± 0.06] and [f(0.86), ix8(0.86)] = [0.99 ± 0.19,0.48 ± 0.04]. These m easurem ents are statistically consistent with a Universe where the gravitational interactions can be described by G eneral Relativity, although they are not yet accurate enough to rule out some comm only considered alternatives. Finally, as ^com plem entary test we m easure the gravitational slip parameter, E G, for the first time at z > 0.6. We find values of E G(0.6) = 0.16 ± 0.09 and E G(0.86) = 0.09 ± 0.07, when E G is averaged over scales above 3 h -1 Mpc. We find that our E G measurem ents exhibit slightly lower values than expected for standard relativistic gravity in a A CD M background, although the results are consistent within 1-2ix.
Key words. large-scale structure o f Universe - cosmology: observations - cosm ological param eters - dark energy - galaxies: high-redshift
1. Introduction
T he origin o f the late-tim e acceleration o f the universal expan
sion is a m ajo r question in cosm ology. T he source o f this acceler
ation and its associated energy d ensity are crucial in u nderstand
ing th e properties o f th e U niverse and its evolution and fate. In the standard cosm ological m odel, this cosm ic acceleration can be associated w ith the presence o f a d ark energy com ponent, a cosm ological fluid w ith negative pressure, w hich opposes the gravitational force on large scales. H ow ever, this app aren t a c celeration can conversely b e interpreted as a failure o f th e stan
dard relativistic theory o f gravity. A key goal fo r cosm ology is
* Based on observations collected at the European Southern O bser
vatory, Cerro Paranal, Chile, using the Very Large Telescope under programmes 182.A-0886 and partly 070.A-9007. Also based on obser
vations obtained with M egaPrime/M egaCam, a jo in t project of CFHT and CEA/DAPNIA, at the Canada-France-Haw aii Telescope (CFHT), which is operated by the N ational Research Council (NRC) o f Canada, the Institut N ational des Sciences de l’Univers of the Centre National de la Recherche Scientifique (CNRS) o f France, and the University of Hawaii. This w ork is based in part on data products produced at TER- APIX and the Canadian Astronomy D ata Centre as part o f the Canada- France-Hawaii Telescope Legacy Survey, a collaborative project of NRC and CNRS. The VIPERS web site is
h t t p : / / w w w . v i p e r s . i n a f . i t /
therefore to investigate the n ature o f gravity em pirically. To be clear, w h at can p otentially b e falsified is the v alidity o f E in ste in ’s field equations, rath er than G eneral R elativity itself; this sets a b ro ad er fram ew ork w ithin w hich E instein gravity o r m odified alternatives can operate.
T he large-scale structure o f the U niverse has proved to be very pow erful fo r testing the cosm ological m odel through the use o f various observables such as the tw o-point statistics o f the galaxy distribution and its features (e.g. P eacock e t al. 2 0 0 1 ; C o l e e ta l. 2 0 0 5 ; T e g m a rk e ta l. 2 0 0 4 ; E isenstein e t al. 2 0 0 5 ; G u z z o e ta l. 2 0 0 8 ; Percival et al. 2 0 1 0 ; B e u tle re ta l. 2 0 1 1 ; B lake e ta l. 2 0 1 2 ; A nderson et al. 2 0 1 4 ; A la m e ta l. 2 0 1 7 , and references therein). In this context, a unique p robe o f gravita
tional physics is the large-scale com ponent o f galaxy p eculiar velocities affecting the observed galaxy distribution in redshift surveys (G uzzo et al. 2008) , sensitive to the grow th rate o f struc
ture f d efined as dln D /d ln a, w here D and a are respectively the linear grow th factor and scale factor. In turn, the grow th rate o f structure tells us about the strength o f gravity acting on cosm o
logical scales and is a d irect p rediction o f gravity theories. The distortions induced by pecu liar velocities in the apparent galaxy clustering, the so-called redshift-space distortions (RSD), are a very im portant cosm ological p robe o f the n ature o f gravity. In the last decade, they have been studied in large galaxy redshift
Article published by EDP Sciences A44, page 1 of 21
surveys, show ing a b ro ad consistency w ith A -cold dark m a t
ter (A C D M ) and G eneral R elativity predictions (e.g. B lake et al.
2 0 1 2 ; B eutler e t al. 2 0 1 2 ; de la Torre e t al. 2 0 1 3 ; S am ushia et al.
2 0 1 4 ; G il-M arfn e t al. 2 0 1 6 ; C huang et al. 2016) .
A lthough galaxy red sh ift surveys are pow erful co sm ologi
cal tools for understanding the geom etry and the dynam ics o f the U niverse, they are fundam entally lim ited by the inherent u n certainty rela ted to the bias o f galaxies, the fact that these are n o t faithful tracers o f the underlying m atter distribution. G ravita
tional lensing represents a p ow erful p ro b e that is com plem entary to galaxy redshift-space clustering. In the w eak regim e in p ar
ticular, the statistical shape deform ations o f b ackground g alax ies p robe the relativistic gravitational deflection o f lig h t by the projected dark m atter fluctuations due to foreground large-scale structure. T here are several techniques associated w ith w eak gravitational lensing; one th at is p articularly useful for co m b in ing w ith galaxy clustering is galaxy-galaxy lensing. This tech nique consists o f studying the w eak deform ations o f background galaxies around foreground galaxies, w hose associated dark m a t
ter com ponent acts as a gravitational lens. This is particularly useful for probing the galaxy-m atter cross-correlation, w hich in turn p rovides insights on the bias o f foreground galaxies and the m atter energy density f l m, although th e pro jected n ature o f the statistic m akes it insensitive to R SD . T he com bination o f galaxy- galaxy lensing w ith red sh ift-sp ace galaxy correlations is th ere
fore a very prom ising w ay to study gravitational physics, given that both lensing inform ation on back g ro u n d sources an d spec
troscopic inform ation on foreground galaxies are available on the sam e field.
B eyond the d eterm ination o f th e grow th rate o f structure, one can define consistency tests o f gravity that are sensitive to both the N ew tonian and curvature gravitational potentials, Y and $ respectively (e.g. Sim pson et al. 2013) . O ne is the gravitational slip, E g , w hich w as originally pro p o sed by Z hang e t al. (2007) and im plem ented b y R eyes et al. (2010) in term s o f the ratio b e tw een th e galaxy-galaxy lensing signal and the R SD param eter P = f /b tim es the g alaxy clustering signal o f th e lenses. H ere b is the galaxy linear bias. E g effectively tests w hether the L aplacian o f Y + $ , to w hich gravitational lensing is sensitive, and that o f Y, to w hich galaxy pecu liar velocities are sensitive, are consistent w ith standard gravity predictions. In the standard cosm ological m odel, E g asym ptotes to Q m/ f on large linear scales. A failure o f this test w ould either im ply an incorrect m atter energy d en sity o r a d eparture from standard gravity. This te st has been p er
form ed a t low red sh ift in th e SDSS survey by R eyes e t al. (2010) and m o re recently at redshifts up to z = 0.57 by B lake et al.
(2016) and Pullen e t al. (2016) .
T he E g statistic is form ally defined as E g = Ygm/(P Y gg), w here Ygm and Ygg are filtered versions o f th e real-space p ro je c te d g alaxy-m atter an d galaxy-galaxy correlation functions r e spectively, a n d P is the R SD param eter. In practice, its im plem en
tation involves m e asu rin g P and the ratio Ygm/Y gg separately, to finally com bine them . B u t sin c eP and Ygg are extracted from the sam e observable, nam ely th e anisotropic tw o-point correlation function o f lens galaxies, this is suboptim al and does n o t a c count fo r the covariance betw een them . In this analysis, w e fo l
low a different approach. W e com bine the galaxy-galaxy lensing quantity Ygm and the redshift-space anisotropic correlation fu n c
tion m onopole and quadrupole m om ents £0 and (from w hich P can be estim ated) in a jo in t likelihood analysis, to p rovide co n
straints on f and gravity a t redshifts above z = 0.6. W e n ote that w e do n o t include Ygg b ecause o f the red u n d a n t cosm ological inform ation shared w ith £0 and £2.
T he V IM O S P ublic E x tragalactic R ed sh ift S urvey (V IPER S) is a large galaxy red sh ift survey probing th e z - 0.8 U n i
verse w ith an unprecedented density o f spectroscopic galaxies o f 5 x 10-3 h3 M p c-3 and covering an overall area o f 23.5 deg2 on the sky. T he prim e goal o f V IPER S is an accurate m easu re
m en t o f the grow th ra te o f structure a t red sh ift around unity.
A first m easurem ent has been perfo rm ed using the P ublic D ata R elease 1 (PD R -1), setting a reference m easurem ent o f f< r8 at z = 0.8 (de la Torre et al. 2013) . T he survey is now com plete and several analyses including this one are using the final dataset to produce the V IPER S definitive grow th rate o f structure m easu re
m ents, b u t follow ing a variety o f approaches. T he present an al
ysis aim s a t m axim izing th e cosm ological inform ation available and takes advantage o f the overlapping lensing inform ation p ro vided by C F H T LenS lensing survey, to provide a p recise gravity test at redshifts 0.5 < z < 1.2 b y com bining R SD and galaxy- galaxy lensing.
T he p ap e r is organized as follow s. T he data are described in Sect. 2; Sect. 3 describes o ur m ethods for estim ating galaxy clustering and galaxy-galaxy lensing; Sect. 4 describes the th e oretical m odelling th at is tested in Sect. 5; Sect. 6 presents how the likelihood analysis is constructed; Sect. 7 describes the cosm ological results, and Sect. 8 sum m arizes ou r findings and concludes.
T hroughout this analysis and if n o t stated o ther
w ise, w e assum e a flat A C D M cosm ological m odel w ith (Q m, O b, n s) = (0 .3 ,0 .0 4 5 ,0 .9 6 ) and a H ubble co nstant o f H 0 = 100 h km s-1 M p c-1 .
2. Data
2.1. C o m b in e d V IP E R S -C F H T L e n S d a ta s e t
T he V IPER S galaxy target sam ple was selected from the optical p hotom etric catalogues o f the C anada-F rance-H aw aii Telescope L egacy Survey W ide (C FH T L S-W ide, G oranova et al. 2009).
V IPER S covers 23.5 deg2 on the sky, divided over tw o areas w ithin the W 1 an d W 4 C FH TLS fields. G alaxies are selected to a lim it o f iAB < 22.5, applying a sim ple and ro b u st g ri colour p re selection to efficiently rem ove galaxies at z < 0.5. C oupled w ith a highly optim ized observing strategy (S codeggio et al. 2009), this allow s us to double the galaxy sam pling ra te in the red- shift ran g e o f interest, w ith resp e ct to a p u re m agnitude-lim ited sam ple. A t the sam e tim e, th e area and depth o f the survey r e sult in a relatively large volum e, 5 x 107 h -3 M p c3, analogous to that o f the Two D egree F ield G alaxy R edshift S urvey (2dF- G R S) a t z - 0.1 (C olless e t a l. 20 0 1 , 2003) . Such a com bina
tion o f sam pling rate and depth is unique am ongst current red- shift surveys at z > 0.5. V IPER S spectra are collected w ith the V IM O S m ulti-o b ject spectrograph (L e F ev re e t al. 20 0 3 ) at m o derate resolution (R = 220) using th e L R R ed grism , p ro v id ing a w avelength coverage o f 5 5 0 0 -9 5 0 0 A and a red sh ift er
ro r corresponding to a galaxy pecu liar velocity error a t any red- shift o f ^ vel = 163 k m s-1 . T he full V IPER S area o f 23.5 deg2 is covered through a m osaic o f 288 V IM O S pointings (192 in the W 1 area, and 96 in the W 4 area). A discussion o f th e sur
vey d ata reduction and m an ag em en t infrastructure is p resen ted in G arilli et al. (2 014) . A com plete description o f the survey co n struction, from the definition o f the target sam ple to the actual spectra and red sh ift m easurem ents, is given in the survey d e
scription p ap e r (G uzzo et al. 2014) .
T he data u sed here correspond to the publicly r e leased P D R -2 catalogue (S codeggio et al. 2017) th at includes 8 6 7 7 5 galaxy spectra, w ith the exception o f a sm all sub-set o f
S. de la Torre et al.: Gravity test from RSD and galaxy-galaxy lensing in VIPERS
redshifts (340 galaxies m issing in the range 0.6 < s < 1.1), for w hich the redshift and quality flags w ere revised close to the release date. C oncerning the analysis p resented here, this has no effect. A quality flag has been assigned to each object in the process o f d eten n in in g their redshift from the spectrum , w hich quantifies the reliability o f the m easu red redshifts. In this an aly sis (as w ith all statistical analyses p resen ted in the p arallel papers o f the final science release), w e use only galaxies w ith flags 2 to 9 inclusive, corresponding to objects w ith a red sh ift confi
dence level o f 96.1% o r larger. This has been estim ated from repeated spectroscopic observations in the V IPER S fields (see Scodeggio et al. 2017). The catalogue used here, w hich w e w ill refer to ju st as the V IPER S sam ple in the follow ing, includes 76 584 galaxies w ith reliable redshift m easurem ents.
In addition to the V IPER S spectroscopic sam ple, w e m ake use o f the public lensing data from the C anada-F rance-H aw aii L ensing S urvey (H eym ans et al. 2012), h ereafter referred to as C F H T LenS. The C FH TLenS survey analysis com bined w eak lensing data processing w ith THELI (E rben et al. 2013), shear m easurem ent w ith LENSFIT (M iller et al. 2013), and p h o to m etric redshift m easurem ent w ith P S F -m atched photom etry (H ildebrandt et al. 2012). A full system atic error analysis o f the shear m easurem ents in com bination w ith the photom etric re d shifts is p resen ted in H eym ans et al. (2012), w ith additional er
ror analyses o f the photom etric redshift m easurem ents p resented in B enjam in et al. (2013).
2.2. S a m p le se le c tio n
F or this analysis, w e define tw o redshift intervals covering the full volum e o f the V IPER S survey: 0.5 < c < 0.7 and 0.7 <
c < 1.2. The num ber density o f galaxies in the com bined W1 and W 4 fields is p resented in F ig. 1, after correction w ith sur
vey incom pleteness w eights u f (see Sect. 3.1). It is w orth e m phasizing that after application o f survey incom pleteness cor
rections, the V IPER S spectroscopic sam ple represents a statisti
cally unbiased subset o f the parent /Ai; < 22.5 photom etric c a t
alogue (G uzzo et al. 2014; G arilli et al. 2014; S codeggio et al.
2017). The redshift distribution is m od elled using the Vmax m ethod (C ole 2011; de la Torre et al. 2013) and show n w ith the solid curve in the figure. In this m ethod, w e random ly sam ple 500 tim es the Vmax o f each galaxy, defined as the com oving v o l
um e betw een the m inim um an d m axim um redshifts w here the galaxy is observable given its apparent m agnitude and the m a g nitude lim it o f V IPE R S, Iab = 22.5. The redshift distribution thus obtained is regular and can be straightforw ardly interpolated w ith a sm ooth function, show ed w ith the solid curve in F ig. 1.
In addition to V IPER S spectroscopic galaxies, photom etric galaxies from the C F H T LenS survey on the overlapping areas w ith V IPER S survey, have b een used for the galaxy-galaxy len s
ing. The lens sam ple satisfies the V IPER S selection /Ai; < 22.5 and uses V IPER S spectroscopic redshifts w hen available (i.e.
for about 30% o f objects) or C FH TLenS m axim um likelihood photom etric redshifts otherw ise. The sources have been selected to have Iab < 24.1 and thus have a higher surface density.
Sources inside the m ask delim iting b ad p hotom etric areas in the C FH TLenS catalogue have been discarded. W e also m ake use o f the individual source redshift p robability distribution fu n c
tion estim ates obtained from b p z (H ildebrandt et al. 2012) as d e
scribed in Sect. 3.2. Source galaxies extend above cph0t = 1.4 and their num ber density is represented w ith the unfilled histogram in F ig. 1.
3. Galaxy clustering and galaxy-galaxy lensing estimation
3.1. A n iso tro p ic g a la x y clu ste rin g e stim a tio n
We estim ate the redshift-space galaxy clustering b y m easuring the tw o-point statistics o f the spatial distribution o f galaxies in configuration space. F or this w e infer the anisotropic tw o-point correlation function £ ( s ,g ) using the L andy & Szalay (1993) estim ator:
„ , G G ( s , p ) - 2 G R ( s , p ) + R R ( s , p )
= M i J ) 1 (1)
w here G G (s ,p ), G R (s ,p ), and R R (s ,p ) are respectively the norm alized galaxy-galaxy, galaxy-random , and random -random num ber o f pairs w ith separation ( s ,p ) . Since w e are interested in quantifying R SD effects, w e have decom posed the three- dim ensional galaxy separation vector s into p o la r coordinates ( s ,p ), w here s is the norm o f the separation vector an d p is the cosine o f the angle betw een the line-of-sight and separa
tion vector directions. This estim ator m inim izes the estim ation variance and circum vents discreteness and finite volum e effects (Landy & Szalay 1993; H am ilton 1993). A ran d o m catalogue needs to be constructed, w hose aim is to accurately estim ate the num ber density o f objects in the sam ple. It m ust be an u n clustered p opulation o f objects w ith the sam e radial and angular selection functions as the data. In this analysis, w e use random sam ples w ith 20 tim es m ore objects than in the data to m inim ize the shot noise contribution in the estim ated correlation functions, and the redshifts o f random points are draw n random ly from the m odel n(z) p resented in Fig. 1.
In o rder to study R SD , w e further extract the m ultipole m o m ents o f the anisotropic correlation function £ ( s ,p ) . This ap proach has the m ain advantage o f reducing the num ber o f o b servables, com pressing the cosm ological inform ation contained A44. page 3 of 21 Fig. 1. N um ber densities o f VIPERS galaxies in the individual W 1 and W 4 fields and of CFHTLenS/VIPERS photom etric redshift galaxies, as a function o f redshift. The num ber densities o f VIPERS galaxies are corrected for the survey incom pleteness by weighting each galaxy in the counts by its associated inverse completeness w eight u f. The solid curve corresponds to the m odel n(z) used in the analysis. It was obtained by random ly sampling galaxy redshifts within their Vnax (see text for details).
in the correlation function. This eases th e estim ation o f the covariance m atrices associated w ith th e data. W e ado p t this m ethodology in this analysis and use the tw o first non-null m o m ents £0(s) and £2(s), w here m o st o f the relevant inform ation is contained, and ignore the contributions o f the m ore noisy sub
sequent orders. T he m u ltipole m om ents are related to £ ( s ,u ) as
B y applying these w eights w e effectively u p-w eight g alax ies in th e pair counts. It is im portant to note th at the spatial distribution o f the random objects is kept consistently uniform across the survey volum e. T he final w eights assigned to G G , GR, and RR pairs com bine the survey com pleteness an d angular p air w eights as
(2)
w here L is the L egendre polynom ial o f order I. In practice the integration o f Eq. (2 ) is approxim ated b y a R iem ann sum over the b in n e d £ (s ,u ). W e use a logarithm ic binning in s w ith A log(s) = 0.1 and a linear binning in u w ith Ap = 0.02.
V IPER S has a com plex angular selection function w hich has to b e taken into account carefully w hen estim ating the co rrela
tion function. This has been studied in detail for th e V IPERS Public D ata R elease 1 (PD R -1; G uzzo et al. 2 0 1 4 ; G arilli et al.
2014) an d p articularly for th e galaxy clustering estim ation in de la Torre et al. (2013) and M arulli e t al. (2 013) . W e follow the sam e m ethodology to account for it in this analysis w ith only sm all im provem ents. W e sum m arize it in the follow ing and refer the read er to the com panion paper, P ezzotta e t al. (2017), for fur
ther details and tests o f the m ethod w hen applied to the V IPERS final dataset.
T he m ain source o f incom pleteness in the survey is in tro duced by the V IM O S slit positioner, SSPO C , and th e V IPERS one-pass observational strategy. This results in an incom plete and uneven spectroscopic sam pling, d escribed in detail in G uzzo e t al. (2014) and G arilli et al. (2014) . In term s o f galaxy clustering, th e effect is to introduce an underestim ation in the am plitude o f the m easured galaxy correlation function, w hich becom es scale-dependent on the sm allest scales. W e dem onstrate in de la Torre et al. (2013) th at this can b e corrected by w eig h t
ing each galaxy in the estim ation o f the correlation function. F or this w e define a survey com pleteness w eight, wC, w hich is d e
fined for each spectroscopic galaxy as w ell as an angular pair w eight, wA, w hich is applied only to G G pair counts. T he la t
ter is obtained from the ratio o f one plus th e angular correlation functions o f targeted and spectroscopic galaxies, as described in de la Torre et al. (2013) .
T he im provem ents com pared to the PD R -1 analysis only concern the estim ation o f survey com pleteness w eights wC.
T hese in fact correspond to the inverse effective sam pling rate, E SR , an d are defined for each galaxy as
Ng Ng
G G (s ,u ) = X Ś w fw jw A(6 ;j)0 ;j ( s ,u ) i=1 j=i+1
Ng Nr
G R (s ,u ) = ^ ^ wC0 ; j ( s , u ) i=1 j=1
Nr Nr
R R (s ,u ) = E E 0 ; j ( s , u ) ,
;=1 j=;+1
(4)
(5)
(6)
w here 0; j( s ,p ) is equal to unity for lo g (s; j) in [log(s) - A lo g (s)/2 , lo g (s)+ A lo g (s)/2 ] a n d p; j in [ u - A p /2 ,p + A p / 2 ] , and n ull otherw ise. W e define th e separation associated w ith each logarithm ic bin as th e m edian p air separation inside the bin. This definition is m o re accurate than using the b in centre, particularly at large s w hen the bin size is large.
O ne can also extract real-sp ace clustering inform ation from the anisotropic red sh ift-sp ace correlation function. This can be done b y m easuring the latter w ith the estim ator o f Eq. ( 1), but w here the redshift-space g alaxy separation vector is decom posed in tw o com ponents, rp and n, respectively p erpendicular and p ar
allel to th e line-of-sight (F ish er et al. 1994) . This decom position allow s the isolation o f the effect o f pecu liar velocities as these m odify only the com ponent p arallel to th e line-of-sight. This way, R SD can then b e m itigated b y integrating £ (rp,n ) over n, thus defining the pro jected correlation function
(7)
W e m easure wp(rp) using an optim al value o f n max = 50 h -1 M pc, allow ing us to red u ce the underestim ation o f the am plitude o f wp(rp) on large scales and at the sam e tim e to avoid in cluding noise from uncorrelated pairs w ith separations o f n >
50 h -1 M pc. F ro m the p rojected correlation function, one can derive the follow ing quantity
(8)
(3) w here SSR, T S R are respectively the spectroscopic and target sam pling rates (for details, see G uzzo e t al. 2014) . A significant effort has been invested in im proving the estim ation o f th e SSR and TSR . In p articular the SSR, w hich characterizes our ab il
ity o f m easuring the redshifts from observed galaxy spectra, has been refined an d now accounts for new galaxy property d epen
dencies, as described in S codeggio et al. (2017) . T he TSR , d e
fined as th e fraction o f spectroscopically observed galaxies in the paren t target catalogue, has been recom puted w ith b etter an gular resolution, on rectangular apertures o f 60 by 100 arcsec2 around spectroscopic galaxies. In o rder to m itigate the shot noise contribution in the galaxy counts in such sm all apertures, w e use the D elaunay tesselation th at n aturally adapts to local d en sity o f points (for details, see P ezzotta et al. 2017). T he accuracy o f this new set o f w eights is tested in the nex t section and in P ezzotta et al. (2 017) .
w here r0 is a cut-off radius, pc = 3 H2/(8 n G ) is the critical d en sity, H (a ) = a / a is th e H ubble param eter, and G is the grav
itational constant. This quantity is equivalent to Yg m, w hich is m easurable from galaxy-galaxy lensing (see n ex t section), but for galaxy-galaxy correlations instead o f g alaxy-m atter ones. It enters th e definition o f the gravitational slip p aram eter EG. In order to m easure it in practice, since th e logarithm ic binning in rp is rath er large in ou r analysis, w e interpolate wp(rp) using cu bic spline interpolation b efore evaluating the integral in Eq. (8) num erically. W e find that Ygg is m o re accurately m easured w ith this technique than by m odelling wp(rp) as a pow er law to p er
form the integral, as is often done (e.g. M andelbaum e t al. 2013) .
3.2. G a la xy -g a la xy le n sin g e stim a tio n
W e use in this analysis the w eak lensing technique usually referred to as galaxy-galaxy lensing, in w hich one infers the
2t + 1 r 1
& (s) = ~ 2 ~
J
0«)dju,X
^maxŹ(rp, n) dn.
nmax
2 f rp r2 '
Ygg(rp, ro) = pc — rwp(r) d r - Wp(rp) + - 2Wp(ro) ,
< rp J -o rp >
wC = E S R -1 = (SSR X T S R )-1 ,
S. de la Torre et al.: Gravity test from RSD and galaxy-galaxy tensing in VIPERS
tangential shear o f back g ro u n d sources yt around foreground objects (lenses) induced by the pro jected m atter distribution in betw een. This quantity is sensitive to the p rojected cro ss
correlation betw een lens galaxies and the underlying m atter d is
tribution. Since the shear signal is w eak and the intrinsic ellip- ticity o f galaxies is unknow n, one has to average the form er over a large num ber o f foreground sources. The quantity that is effec
tively m easured is the differential excess surface density
J r»00
dzsP sizs^ ^ z^ zs),
Zi.
w hich leads to the follow ing estim ator (e.g. M iyatake et al.
2015; B lake et al. 2016):
(9)
(10)
In the above equations, rp is the com oving transverse distance betw een lens and source galaxies, D s , D LS, D L are the a n g u lar diam eter observer-source, lens-source, an d observer-lens d is
tances, and c is the speed o f light in the vacuum .
W e use the inverse variance-w eighted estim ator for the d if
ferential excess surface density (e.g. M andelbaum et al. 2013):
(11)
w here the i and j indices run over source and lens galaxies r e spectively, N s an d N \. are respectively the num ber o f source and lens galaxies, e ,j is the tangential ellipticity for each lens-source pair, ws are statistical w eights accounting for b iases in the deter
m ination o f b ackground source ellipticities, and ©,y(rp) is equal to u nity for rPt ,7 in [rp - A r p/2 , r + A r p/2 ] and null otherw ise. The p rojected separation rp is calculated as rp = PpL, w here 9 andqy.
are respectively the angular distance betw een the lens and the source, and the radial com oving distance o f the lens. This e sti
m ato r includes an inverse-variance w eight fo r each lens-source p air I - 2,, w hich dow nw eights the pairs at close redshifts that contribute little to the w eak lensing signal (M andelbaum et al.
2013).
This estim ator is u nbiased if the redshifts o f the sources are perfectly know n, but here w e have only photom etric redshift e s
tim ates: the m axim um likelihood photom etric redshift and the norm alized red sh ift probability distribution function for each source p s(z). U sing the m axim um likelihood p hotom etric re d shift o f sources in E q. (11) and restricting the sum to pairs w ith cs > c l can possibly lead to a dilution o f the signal induced by the non-negligible p robability that cs < .7 .. T his effect can be m itigated b y replacing in Eq. (11) b y its average over the source red sh ift p robability distribution function p s
(12)
(13)
In principle, those estim ators hold in the lim it w here the lens red sh ift distribution is narrow and lens redshifts accurate (N akajim a et al. 2012). To b etter u nderstand the im portance o f the effects introduced by an im perfect know ledge o f the source
Fig. 2. Relative difference between various estim ates o f AEgm, based on different assumptions for source and lens redshifts. and the fiducial es
timate in the data at 0.5 < z < 0.7. The quantity shown in the figure is AEgm/AE^ - 1 as a function o f the projected separation rp. The fiducial estimate AE“ is that obtained by using Eq. (13). which includes the individual redshift probability distribution function p s(z) of the sources, and for the lenses, the VIPERS spectroscopic redshift Uspec) when avail
able or the CFHTLenS m axim um likelihood photom etric redshift Gphot) otherwise (see text). It corresponds to the adopted estim ate for the anal
ysis. The grey shaded area represents the relative statistical error ex
pected in the survey.
and lens redshifts in the data, w e p erform a com parison o f differ
ent estim ates using E q. (11) o r Eq. (13), an d various assum ptions on the source an d lens redshifts. This is p resented in term s o f the relative difference w ith respect to a fiducial estim ate in Fig. 2.
The fiducial estim ate is that obtained b y using Eq. (13), w hich includes the individual red sh ift probability distribution function p s(z) o f the sources, an d for the lenses, the V IPER S spectro
scopic red sh ift w hen available o r the C FH TLenS m axim um lik e
lihood p hotom etric red sh ift otherw ise.
W e find that the estim ate b ased on Eq. ( 1 1), w hich only uses m axim um likelihood photom etric redshifts for b oth lenses and sources, underestim ate the signal on all p robed scales b y about 15% w ith respect to the fiducial case. H ere, w e im pose cs >
0.1 + cl, including the additive term o f 0.1 to account fo r ty p i
cal p hotom etric red sh ift errors (e.g. C oupon et al. 2015). F urther including the source redshift p robability distribution function through the estim ator o f Eq. (13) allow s a slight im provem ent, reaching an underestim ation o f about 10% w ith respect to the fiducial case. The tw o previous estim ates are still affected b y the uncertainty on the lens redshifts, w hich effectively tends to dilute the overall signal. If w e now use as lenses only V IPER S spec
troscopic galaxies, w hich represents about 30% o f all galaxies w ith ;AB < 22.5, w e find a rem arkably good agreem ent w ith the fiducial estim ate. In principle, this estim ate m ay be considered as the reference u nbiased estim ate, how ever on the largest scales p robed b y the data, i.e. at rp = 10 - 2 0 / r 1 M pc, the signal drops significantly. This can be im puted to the lack o f source-lens pairs induced by the red u ced num ber o f lenses, directly affecting our ability to probe the largest scales signal. H owever, w e find that this effect can be m itigated by adding photom etric lenses from
A44. page 5 of 21 AUgmC^p) — ^crit
w here
y _ g2 D s cnt 4 nG Dl sDl '
, T , ,
z ą z f c .f t .J & e M r ,)
5 ' " '
E &
^ " e m V p j — o
2 S 2 ? i « f f e y ) “©y(/-p)
the C F H T LenS catalogue, taking the m axim um likelihood p h o tom etric redshifts: this corresponds to the fiducial estim ate. W e n o te th at the expected statistical uncertainty, w hich is show n in Fig. 2 w ith the grey shaded area, is n o t negligible particularly above rp = 10 h -1 M pc, and higher than any residual system atic effect. This test m akes us confident that our fiducial estim ate o f A 2gm(rp) is robust, given the expected level o f statistical error in the data. S im ilar results are fo u n d at 0.7 < z < 1.2, leading to the sam e conclusions.
A non-negligible source o f system atics in w eak lensing m e a
surem ents is related to the m easurem ent o f b ackground galaxy shapes. This can lead to system atic biases in the lensing m e a
surem ents. T he C FH TLenS collaboration has studied these ex tensively in M iller et al. (2013) and H eym ans et al. (2012) , and w e follow their m eth o d to correct o ur m easurem ents. W e used the additive and m ultiplicative shear calibration corrections c and m, as w ell as th e o ptim al w eights wS provided b y LENSFIT, w hich are available in th e C F H T LenS catalogue. In particular, to correct for the m ultiplicative bias w e applied the correction fa c tor (1 + K (rp)) 1 to A 2gm(rp) as d escribed in M iller et al. (2013) and V elander et al. (2014) . W e found this correction to b o o st the galaxy-galaxy lensing signal by about 5% independently o f the scale.
F or the purpose o f constraining the cosm ological m odel, it can b e difficult to use A 2gm as its m odelling is non-linear. O ne o f the difficulties is to m odel the non-linear scales and the intrinsic m ixing o f sm all-scale non-linear and large-scale linear in fo rm a
tion (B a ld a u f e t al. 2010) . This is achievable b u t at th e expense o f introducing additional n uisance p aram eters in th e m odel (e.g.
C acciato e t al. 2 0 1 3 ; M o re et al. 2015) . A n alternative approach, w hich w e use in this analysis, consists o f using a derived statis
tic th at allow s the m itigation o f non-linearities: the annular d if
ferential surface density Ygm, w hich is defined as (B a ld a u f et al.
2010)
6g and m a tte r overdensity 6 as:
6g(x) = M ( x ) + 2 b2 [62(x) - a 2] + 1 b s2 [s2(x) - <s2>]
+ O (s3(x)), (15)
w here b 1 and b2 are the linear and second-order non-linear bias term s, bs2 the non-local bias term , s is the tidal tensor term from w hich n on-locality originates. T he a 2 and (s 2> term s ensure the condition (6g> = 0.
4.2. A n n u la r differential e x c e s s su rfa c e d e n s ity
T he galaxy-galaxy lensing quantity th at w e observe is the differ
ential excess surface density. It is defined as A^gm(rp) = 2 gm(rp) ^gm(rp),
w here
- 2 r rp
^gm(rp) = - p i ^gm(r) r d r rp 0
and 2 gm(rp) is the p ro jected surface density defined as
^gm(rp) = ^m Pc ^1 + ^gm( -\Jrp + dX.
(16)
(17)
(18)
In th e above equation, Q m is m a tte r energy density an d x is the radial com oving coordinate. Ygm can be p redicted from A 2gm by using Eq. ( 14) or d irectly from the galaxy-m atter cro ss
correlation function as (B a ld a u f e t al. 2010)
Ygm(rp) = £gm(x)WY(x, rp, r0)dx, (19)
0
w here W Y (x, rp, r0) is the w indow function (B a ld a u f et al. 2010) :
WY( x, rp, r0) = ^ ( ^ X ^ 0 ( x - o ) - ^ x 2 - rp 0 ( x - rp) j
(14)
This statistic rem oves the sm all-scale n o n-linear contribution o f A 2gm below a cut-off radius r 0. W e use this quantity in our an al
ysis and study the im p act o f the choice o f r 0 in Sect. 5 .1 .
4. Theoretical modelling 4.1. G a la xy b ia sin g
G alaxies are n o t faithful tracers o f th e underlying m atter d is
tribution and this has to b e taken into account in co sm ologi
cal analyses, since cosm ological m odels p rim arily p red ict m atter observables. T he m odelling o f galaxy b iasing is sim plified w hen focusing on large scales, w here bias can be considered as linear and sim ply b e represented as a constant m ultiplicative factor in front o f th e m atter pow er spectrum . This is a com m on assu m p tion in R SD analyses. In our case, however, the relatively sm all survey volum e m eans th at m uch o f ou r inform ation lies below fully lin e ar scales; for this reason, an d because o f the in trin sic non-linearities in the excess surface density A 2gm, additional care m u st b e taken to m odel galaxy biasing. W e use a n on-linear prescription for galaxy bias b ased on the cosm ological p ertu r
bation theory th at allow s describing it m o re accurately dow n to translinear scales. W e adopt the n o n-linear n on-local bias m odel o f M cD onald & R oy (2009) that relates the galaxy overdensity
(20)
w here 0 ( x ) is th e H eaviside step function.
F ro m these equations one can see explicitly th at Yg m is related to th e g alaxy-m atter cross-correlation function £g m or cross-pow er spectrum Pg m. I f w e assum e the biasing m o d el o f Eq. ( 15), Pg m can be w ritten as (M cD onald & R oy 2009) P g m(k) = b1 P6 6(k) + b2 P b 2 , 6(k) + bs 2 P b s 2 , 6 (k)
+ b3 n la3(k)Pl i n(k), (21)
w here P6 6 is the non-linear m atter density-density pow er spec
trum , b3n l is a third-order non-local bias term , pl i n is the lin ear m atter pow er spectrum , and Pb 2 , 6, Pb s 2 , 6 are 1-loop integrals given in A ppendix A . In the local L agrangian p icture w here one assum es no in itial non-local bias, one can p red ict th at the non-local bias term s at la ter tim e are related to b1 such that (C han et al. 2 0 1 2 ; Saito e t al. 2014)
4
bs 2 = - 7<b1 - 1) (22)
32
b3 n l = 3 1 5 (b1 - 1). (23)
W e adopt these relations and o ur m odel has finally tw o galaxy biasing param eters: b1 an d b2, b1being th e standard lin e ar bias param eter.
r 2
Ygm(rp, ro) = A2gm(rp) - A2gm(ro).
rp 2x rp© (x - rp) r2© (x - ro)
r p , V x2 - rp2 V x2 - r2 >
S. de la Torre et al.: Gravity test from RSD and galaxy-galaxy lensing in VIPERS
4.3. R e d s h ift-s p a c e distortions
T he m o st general form alism describing the redshift-space anisotropies in the pow er spectrum derives from w riting the m a t
ter density conservation in real and red sh ift space (K aiser 1987) . In p articular, in the p lane-parallel approxim ation th at is assum ed in this analysis, the anisotropic pow er spectrum o f m a tte r has the general com pact form (S coccim arro e t al. 1999)
P s(k ,v ) = f e -ik r ( e -ik/vA“''x J (2 n )3 '
[6(x) + /3,, u, (x )][6 (x ') + /3,, u, ( x ')]) (24) w here v = k\\/k, u\\(r) = - v \\(r ) /( /a H (a ) ), v,(r) is the line-of- sight com ponent o f the pecu liar velocity, 6 is the m atter density field, Auy = u||(x) - uy(x') and r = x - x '. It is w orth noting that in F ourier space, for an irrotational velocity field, 3,, u„ is related to the divergence o f th e velocity field 9 v ia 3,, u„ (k ) = v29(k). A l
though exact, Eq. (2 4 ) is im practical an d w e use the app ro x i
m ation pro p o sed b y Taruya et al. (2 0 1 0 ) . In th e case o f perfect m atter tracers, the latter m odel takes the form
P s(k, v) = D(kvcrv) [P66(k) + 2v2/P 6 9 (k) + v4/ 2P99(k)
+CA(k,v, / ) + CB(k ,v , / ) ] , (25) w here D(kv<rv) is a dam ping function, P 66, P 69, P 99 are r e spectively th e n o n-linear m atter density-density, density-velocity divergence, and velocity divergence-velocity divergence pow er spectra, and ^ v is an effective pairw ise velocity dispersion that w e can fit for an d then treat as a nuisance param eter. T he ex pressions for CA(k, v, / ) and C B(k, v, / ) are given in Taruya et al.
(2010) and de la Torre & G uzzo (2012) . This phenom enological m odel can b e seen in configuration space as a convolution o f a p airw ise velocity distribution, th e dam ping function D(kp<rv) that w e assum e to b e L orentzian in F ourier space, i.e.
D ( k w v) = (1 + k2v2 ^ 2 ) - 1 , (26)
and a term involving th e density and velocity divergence co rre
lation functions an d their spherical B essel transform s.
This m odel can b e generalized to the case o f biased trac
ers by including a biasing m odel. B y introducing th at o f Eq. ( 15) , one obtains for the redshift-space galaxy pow er spec
trum (B eutler et al. 2 0 1 4 ; G il-M arfn e t al. 2014)
Pg(k, v) = D(kv<rv) [P gg(k) + 2v2/ P g9(k) + v4/ 2P ^ k ) + C A (k,v, / , b{) + CB(k ,v , / , b1)] (27) w here
Pgg(k) = b \P 66(k) + 2b2b1 Pb2,6 (k) + 2 b s2 b P b ^ / k ) + b2 Pb22(k) + 2b2bs2 P b2s2 (k) + b 2s2 Pbs22(k)
+ 2b1b3nl^3(k)Plin(k) + N, (28)
P g9(k) = b 1P 69 (k) + b2 P *2,9 (k) + b s2 P bs2,9(k)
+ b3nl^3(k)Plin(k). (29)
In the above equations P69 is the n o n-linear m atter density- velocity divergence pow er spectrum , Plin is the m atter linear p ow er spectrum , and Pb2,6, Pbs2,6, Pb2,9, Pbs2,9, Pb22, Pb2s2, Pbs22,
are 1-loop integrals given in A ppendix A .
T he final m odel for £g(s) is obtained from its F ourier c o u n terpart P*(k) defined as
2* + 1 p1
P* (k) = — ^t J Pg(k, v ) L (v) dv, (30)
w here
P k 2
£ ( s ) = i j ^ P * ( k ) j * ( k s ) dk. (31) In the above equation, j* denotes the spherical B essel functions.
T he ingredients o f the m odel are the non-linear pow er spec
tra o f density and velocity divergence at the effective redshift o f the sam ple. T hese pow er spectra can be pred icted from p er
turbation theory or sim ulations for different cosm ological m o d els. T he n o n-linear m a tte r pow er spectrum can also b e o b tained to a great accuracy from sem i-analytical prescriptions such as H A L O F IT (Sm ith e ta l. 2003), for various co sm o lo gies. In particular, H A L O F IT allow s the p rediction o f P 66 from the linear m atter pow er spectrum and the know ledge o f the scale o f n on-linearity at the red sh ift o f interest, knl(z). W e note th at at fixed linear m atter pow er spectrum shape, variations o f <r8 (z) can b e straightforw ardly m apped into variations o f knl(z) (see Sm ith et al. 2003) . In this analysis, the linear m a t
ter pow er spectrum is pred icted using the C LA SS B oltzm ann code (L esgourgues 2011) , and w e use th e latest calibration o f H A L O F IT by Takahashi e t al. (2012) to obtain P 66. To p redict P99 and P69, w e use the n early universal fitting functions o f Bel et al. (in prep.) th at dep en d on the linear pow er spectrum and
(z) as
P 99 (z) = Plin(z)e-ta1< 2(z) (32)
P69(z) = (P66(z)Plin(z)e-kB1^ 2(z)) 1/2 , (33) w here Plin is the linear pow er spectrum and (m1, m2, n1, n2) are free param eters calibrated on sim ulations. W e ad o p t here the v al
ues (m 1, m2, n 1, « 2) = (1 .9 0 6 ,2 .1 6 3 ,2 .9 7 2 ,2 .0 3 4 ). T hese p red ic
tions for P99 and P69 are accurate at the few p ercen t level up to k - 0.7 (B el et al., in prep.). Therefore, the overall degree o f n on-linearity in P 66, P 69 an d P 99 is solely co ntrolled by ^ 8(z), w hich is left free w hen fitting the m odel to observations.
In the m odel, the linear bias and grow th rate param eters, b1 and / , are degenerate w ith the n orm alization o f th e m atter pow er spectrum p aram eter ^ 8. G enerally w ith R SD , only th e com bina
tion o f b 1^ 8 and / ^ 8 can b e constrained if n o assum ption is m ade on th e actual value o f ^ 8. H ow ever in the Taruya e ta l.
(2010) m odel, b^/ ^ 8 , b 1 / 2^ j , and / V g term s appear in the correction term CA (see Taruya et al. 2 0 1 0 ; de la Torre & G uzzo 2012) . A ccordingly, in the general case, ( / , b 1, b 2, ^ 8, ^ v) are treated as separate param eters in the fit and w e provide m arg in al
ized constraints on th e derived / ^ 8.
4.4. R e d s h ift errors
R edshift errors can p o tentially affect th e anisotropic RSD sig
nal. In the anisotropic correlation function they have a sim ilar effect as galaxy ran d o m m otions in virialized objects: they in troduce a sm earing o f th e correlation function along th e line o f sight at sm all transverse separations. If the probability distribu
tion function o f red sh ift errors is know n, their effect can b e for
w ard m odelled b y adding another m ultiplicative dam ping fu n c
tion in the redshift-space galaxy pow er spectrum o f Eq. (19). In th at case, th e dam ping function should b e the F ourier transform o f the error p robability distribution function. W e follow this ap proach and the final m o d el is obtained by m ultiplying Eq. (19) b y a G aussian w ith standard deviation set to the estim ated p air
w ise red sh ift dispersion o f V IP E R S galaxies such th at the final R SD m odel Pg reads
Pg (k, v) = G (k v ^ ) P g (k, v), (34)
A44, page 7 of 21
geom etry w hereas R SD are sensitive to the grow th o f cosm o
logical perturbations.
W e follow X u e t a l . (2013) and m odel A P distortions u s
ing th e a an d e param eters, w hich characterize respectively the isotropic and anisotropic distortion com ponents associated w ith AP. T hese are given by
(36)
(37)
w here quantities calculated in the fiducial cosm ology are d e
n oted w ith prim es. T hose param eters m odify the scales a t w hich the correlation function is m easured such that
(38) (39)
Fig. 3. Probability distribution function of redshift errors at 0.5 < z <
0.7 and 0.7 < z < 1.2 in the VIPERS data. This is obtained from the redshift differences of reobserved galaxies, for w hich there are two in
dependent redshift m easurem ents. The dotted and dashed curves are best-fitting Gaussians for the redshift intervals 0.5 < z < 0.7 and 0.7 < z < 1 .2 respectively.
w here Pg(k, v) is taken from Eq. (2 7 ), G is the F ourier transform o f th e G aussian kernel
Therefore, for the m odel correlation function m onopole and quadrupole in a tested cosm ology, the corresponding quantities in the fiducial cosm ology are obtained as (X u e t al. 2013)
(40)
(41)
(35)
and is the pairw ise standard deviation associated w ith th e red- shift error probability distribution function.
T he G aussian form is m otivated b y th e data them selves as show n in Fig. 3 . In this figure are show n the distributions o f re d shift differences at 0.5 < z < 0.7 and 0.7 < z < 1.2 in V IPERS reobservations (1061 at 0.5 < z < 0.7 and 1086 at 0.7 < z < 1.2), for w hich w e have tw o independent red sh ift m easurem ents for the sam e galaxies (see S codeggio e t al. 2017) . T hese distribu
tions can b e rath er w ell m odelled by G aussians, and by doing so, w e obtain values o f ^ z = 1 3 1 x 10-3 and ^ z = 1 3 6 X 10-3 for the pairw ise red sh ift standard deviations at 0.5 < z < 0.7 and 0.7 < z < 1.2 respectively. T hese are fu rth er converted in com oving length assum ing the fiducial cosm ology to enter the m odel in Eq. (34) .
4.5. A lc o c k -P a c z y n sk i e ffe c t
A dditional distortions can arise in galaxy clustering b ecause o f the n ee d to assum e a fiducial cosm ology to convert red sh ift and angular positions into com oving distances, and th e fact that this fiducial cosm ology is n o t n ecessarily the true one. This is the A lcock & P aczynski ( 1979) effect
(Ap).
M ore specifically, since the line-of-sight separations req u ire the know ledge o f the H ubble param eter, H (z), an d transverse separations that o f the angular d iam eter distance, D A(z), any difference in H (z) and DA(z) betw een the fiducial and true cosm ologies, translates into an anisotropic clustering, independently o f RSD . A lthough AP and R SD anisotropies are degenerate to som e extent in th e o b servables (B allinger e t al. 1996; M atsu b ara & Suto 1996) , they have a fundam entally different origin: A P is sensitive to theIn the case o f the galaxy-galaxy lensing statistic th at w e are co n sidering, since it is a function o f th e transverse separation rp, the corresponding Yg m in the fiducial cosm ology is sim ply given by
Ygm(rp) = Ygm (a (1 + e)-1 rp) . (42)
4.6. C o sm o lo g ica l in sig h ts from g a la x y clu sterin g a n d g a la x y -g a la x y le n sin g
G ravitational physics on cosm ological scales can b e tested from m easurem ents o f th e grow th rate o f structure, w hich is w ell m e a
sured from R SD in the galaxy clustering pattern. W e have seen th at in practice, the correlation function m u ltipole m om ents d e
p en d n o t o nly on the grow th rate o f structure f , b u t also on the shape and am plitude ^ 8 o f the m atter pow er spectrum , the galaxy bias param eters b 1 and b2, an d the p airw ise velocity d is
persion ^ v. To derive the grow th rate o f structure, one then needs to m arginalise over those nuisances. This is o f course a source o f uncertainty in the determ ination o f the grow th ra te o f structure.
M oreover, since there is a degeneracy betw een the am plitude o f the m atter pow er spectrum ^ 8, the grow th rate o f structure f , and the linear bias p aram eter b 1, R SD alone are sensitive to the f ^ 8 and b 1^ 8 p aram eter com binations.
O n the o ther hand, galaxy-galaxy lensing probes the real- space galaxy-m atter correlations th at are described by the shape and am plitude ^ 8 o f the m atter p ow er spectrum , the galaxy bias param eters b 1 an d b 2, and the m atter density p aram eter Q m. P ro je c te d galaxy-galaxy correlations are also sensitive to ^ 8, b 1, and b 2. B u t b y looking in detail a t those dependencies, w e can see th at in the linear regim e Ygm ^ O mb 1 ^ , w hile Ygg ^ b ^ , such th at b y com bining the tw o w e can b reak the degeneracy betw een b 1 and ^ 8. W e no te th at £ (s ,u ), from w hich £0 and £2 are derived, has th e sam e p aram eter dependences as Ygg, except
r ' = a (1 + e)2ri r± = a ( 1 + e)-1 ^ .
DA H i f a = — ^ —
D '2 H ' A / . = ( DA H I 1' 3 - 1,
I D a H ) '
C f '\ £ f \ , 2 \ , d^2(a s )
& (s ) = & ( a s ) + 5 e 3^2( a s ) + d ln (s)
H l ^ o d ^ o (as) , / , , 6 \ 4 d & ( a s ) (s > = 2^ - d w S 7 + ( 1 + 7 ^ + 7 e "dlnTST"
+ 4 . [5^4( a s ) + ^ ( ^ ■
7 L dln(s) .
( k2 v2 a ? \ G (k v a z) = exp I ---2— I ,
S. de la Torre et al.: Gravity test from RSD and galaxy-galaxy lensing in VIPERS
for the additional f dependence. Therefore, additional galaxy- galaxy lensing inform ation brings an independent handle on the bias param eters b 1 and b2, and the pow er spectrum am plitude ^ 8, reducing the uncertainties on th e grow th rate o f structure induced by th e lack o f know ledge on th e bias o f galaxies, as w ell as a supplem entary sensitivity to Q m.
5. Tests on simulated data 5.1. S im u la te d data
To test the robustness o f redshift-space galaxy clustering, galaxy-galaxy lensing, and associated error estim ates, w e m ake use o f a large n um ber o f m o c k galaxy sam ples, w hich are d e
signed to b e a realistic m atch to the V IP E R S final dataset.
W e u sed the m o c k lensing lightcones p resented in G iocoli et al.
(2 016) . T hese have been b u ilt upon th e Big M ultiD ark dark m atter N -b o d y sim ulation (K ly p in e ta l. 2016) , w hich assum es a flat A C D M cosm ology w ith (Q m, O a , Q b, h, n, ^ 8) = ( 0 .3 0 7 ,0 .6 9 3 ,0 .0 4 8 2 ,0 .6 7 8 ,0 .9 6 0 ,0 .8 2 3 ) and covers a v o l
um e o f 15.625 h -3 G pc3. T hese lightcones contain the shear inform ation associated w ith sim ulated b ackground galaxies distributed uniform ly on the sky but follow ing the redshift distribution o f C F H T LenS galaxies. M ore specifically, the light- cones have been b u ilt to m atch the effective n um ber d en sity and red sh ift distribution o f the C FH TLenS lensing ca ta
logue. W e added G aussian ran d o m errors w ith standard deviation
= ( ^ 2 + ^ 2) 1/2 = 0.38 to th e ellipticities to m im ic those in the C F H T LenS data. T he size o f the sim ulation allow ed us to create 54 independent lightcones for W 1 and W 4, spanning the red sh ift range 0 < z < 2.3 (for details, see G iocoli e t al. 2016).
W e populate these lightcones w ith foreground galaxies using the halo occupation distribution (H O D ) technique and apply the detailed V IP E R S selection function and observational strategy.
T he haloes w ere identified in th e sim ulation using a friends-of- friends algorithm w ith a relative linking length o f b = 0.17 tim es the inter-particle separation. T he m ass lim it to w hich th e halo catalogues are com plete is 101195 h -1 M 0 . B ecause this lim it
ing m ass is too large to h o st th e faintest galaxies observed w ith V IPE R S, w e use the m eth o d o f de la Torre & P eacock (2013) to reco n stru ct haloes below the resolution lim it. This m ethod is b ased on stochastically resam pling the halo n um ber density field using constraints from the conditional halo m ass function.
F or this, one needs to assum e the shapes o f the halo bias fa c tor and halo m ass function at m asses below the resolution lim it and use th e analytical form ulae obtained b y Tinker et al. (20 0 8 , 2010) . W ith this m eth o d w e are able to popu late th e sim ulation w ith low -m ass haloes w ith a sufficient accuracy to have u n b i
ased galaxy tw o-point statistics in the sim ulated catalogues (for details, see de la Torre et al. 2013) . T he m inim um reconstructed halo m ass w e consider for the p urpose o f creating V IPERS m ocks is 1010 h -1 M 0 .
In this process, w e populate each halo w ith galaxies acco rd ing to its m ass, the m ean nu m b er o f galaxies in a halo o f a given m ass being given by the H O D . It is com m on usage to differen
tiate betw een central an d satellite galaxies in haloes. W hile the form er are p u t at rest a t halo centres, the latter are random ly distributed w ithin each halo according to a N F W radial profile (N avarro et al. 1996, 1997) . T he halo occupation function and its dependence on red sh ift and lu m inosity/stellar m ass m u st be p recisely chosen in order to obtain m o ck catalogues w ith rea l
istic galaxy clustering properties. W e calibrated th e halo o cc u p ation function directly on the V IPER S data, as p resented in de la Torre et al. (2013) . W e add velocities to the galaxies and
m easure their redshift-space positions. W hile the central g alax ies are assigned th e velocity o f their h o st halo, satellite galaxies have an additional random com ponent for w hich each C artesian velocity com ponent is draw n from a G aussian distribution w ith a standard deviation that depends on the m ass o f th e h o st halo.
D etails ab o u t the galaxy m o c k catalogue construction technique are given in A ppendix A of de la Torre et al. (2013) .
T he final step in obtaining fully realistic V IPER S m ocks is to ad d the detailed survey selection function. W e start by applying the m agnitude cut iAB < 22.5 and the effect o f the colour selection on th e radial distribution o f th e m ocks. This is achieved by depleting th e m ocks a t z < 0.6 so as to rep ro duce the V IPER S colour sam pling rate (see G uzzo et al. 20 1 4 , for detail). T he m o c k catalogues that w e obtain are then sim i
lar to th e paren t p hotom etric sam ple in th e data. W e n ex t apply the slit-positioning algorithm w ith the sam e setting as for the data. This allow s us to reproduce the V IPER S footprint on the sky, the sm all-scale angular incom pleteness and the variation o f T SR across the fields. Finally, a ran d o m red sh ift error is added to the redshifts as in the data. W e are thus able to produce realistic m o ck galaxy catalogues th at contain the detailed survey co m pleteness function and observational biases o f V IPE R S, w hich w e refer to as the ‘o bserved’ m o c k catalogues in the follow ing.
W e note that another set o f V IPER S m ock catalogues span
ning the red sh ift ran g e o f 0.4 < z < 1.2 have been co n structed. This set, w hich com prises 306 and 549 lightcones o f W 1 an d W 4 fields respectively, has n o t been explicitly used in this analysis, b u t in accom panying V IPER S P D R -2 an aly ses (e.g. H aw ken et al. 2 0 1 7 ; P ezzo tta e t al. 2 0 1 7 ; W ilson et al., in prep.; R ota e t al. 2017) .
5.2. S y s te m a tic s o n th e correlation function m o n o p o le a n d q u a d ru p o le
T he m o ck sam ples are crucial for testing the redshift-space clu s
tering estim ation in V IPE R S, w hich is n o t trivial given the co m plex selection function o f the survey. W e first study the im p act o f th e survey selection function on the m easurem ent o f the m onopole and quadrupole correlation functions. W e m easured these quantities in the observed m ocks, applying the different w eights defined in Sect. 3.1, and com pare them to th e refer
ence m easurem ents obtained from the paren t m ocks, including V IPER S typical spectroscopic red sh ift errors. T he relative differ
ences in and as a function o f separation and averaged over the m ocks are show n in Figs. 4 and 5 , respectively for the tw o sam ples a t 0.5 < z < 0.7 and 0.7 < z < 1.2. F irst o f all, it is clear from these figures that w ith o u t any correction the spectroscopic strategy introduces biases in the estim ation o f the galaxy clu s
tering. B u t w hen applying the survey com pleteness w eights wC, one can recover w ithin a few p erc en t the correct am plitude o f the correlation functions on scales above 5 h -1 M pc. B y further ap plying th e angular w eights wA, w e obtain an alm ost u nbiased e s
tim ate o f the m onopole an d q uadrupole dow n to a few h -1 M pc.
T he statistical relative error induced b y sam ple variance an d e s
tim ated from the dispersion am ong the m ock sam ples, is show n w ith the shaded area in these figures. It is im portant to note that it is m uch larger than any residual system atics over th e range o f scales considered. Finally, it is w orth m entioning th at in the quadrupole, th e apparent higher level o f system atics at around s = 10 h -1 M p c is an artefact due to the zero crossing o f the functions at slightly different separations.
A44, page 9 of 21
Fig. 4. Relative systematic errors on the correlation function m onopole (top panel) and quadrupole (bottom panel) at 0.5 < z < 0.7 and ef
fects of target sampling rate (TSR) and angular pair w eighting (wA) corrections. The grey shaded areas represent the relative statistical error expected in the survey, w hile light grey band m ark ±1% relative uncer
tainties for reference.
5.3. S y s te m a tic s o n th e grow th rate o f stru ctu re
W e further study ou r ability to determ ine f a 8 w hen com bining RSD and galaxy-galaxy lensing m easurem ents in a m axim um likelihood analysis. F or this p urpose w e p erform several an aly ses o f the m ean RSD an d galaxy-galaxy lensing m easurem ents in the observed m ocks, for different m in im u m separations sm i n in th e correlation functions and different cut-off scale r0 in the annular differential excess surface density. T hese analyses are perform ed on the m ean quantities to reduce th e im pact o f statis
tical errors and concentrate on system atics. T he p recision m atrix is estim ated from th e m ocks as explained in Sect. 6 , except that each elem ent is further divided b y the nu m b er o f m ocks to char
acterize the error on the m ean. As an illustration, w e p rese n t in this section only the case o f th e sam ple a t 0.5 < z < 0.7. The sam ple at 0.7 < z < 1.2 provides very sim ilar system atic levels.
F igure 6 presents the system atic errors on f a8, i.e. the re l
ative difference o f recovered values w ith resp ect to the fidu
cial value o f the m ocks, as a function o f sm i n and for r0 = (1 h -1 M pc, 1.5 h -1 M pc). W e consider rath er sm all m inim um scales and cut-off rad ii to explore the extent to w hich o ur m o d elling is robust in the translinear regim e. W e can see in this figure th at o ur m odel allow s the recovery o f the fiducial value o f f a8 dow n to sm i n = 6.3 h-1 M pc, w ith system atic errors below 5% , independently o f the choice o f r0. In principle, val
ues o f r0 sm aller than th e typical radius o f haloes hosting these
Fig. 5. Same as in Fig. 4 but for the redshift interval 0.7 < z < 1.2.
Fig. 6. Relative systematic error on f a8 at 0.5 < z < 0.7 as a function of sm in, for different values o f r0 (r0 = 1 h-1 M pc and r0 = 1.5 h-1 Mpc) and when including or not redshift error. The error bars represent the relative statistical error associated to analysing the m ean m ock predic
tions. The shaded area shows the 1 a confidence region associated with the relative statistical error expected in VIPERS. The squares and trian
gles are artificially shifted along smin axis to improve the clarity of the figure.
