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Eric ]. z etn eea eaiiy eas on u h interpret the out point ΛCD also We with consistency relativity. demonstrate general results testing The model. energy ASnmes 98.80.-k,95.36.+x numbers: PACS nlsso aaycutrn aa tigi substantial a in fitting distance ga data, diameter Moreover, clustering galaxy times. of distortions. different space analysis from redshift giving coming redshift, whi sight their epoch change of same the line from the coming sight to of line the to transverse eff u bevtoso h nvreaefnaetlyanisotr fundamentally are Universe the of observations Our n t omlgclipc o etgnrto surveys. generation next for impact cosmological its and , .INTRODUCTION I. 3 nttt o h al nvreWU waWmn Universit Womans Ewha WCU, Universe Early the for Institute 5 oe nttt o dacdSuy ogamng,Sol1 Seoul Dongdaemun-gu, Study, Advanced for Institute Korea 2 oe srnm n pc cec nttt,Deen305-3 Daejeon Institute, Science Space and Astronomy Korea 1 4 , nvriyo cec n ehooy ajo 0-3,Kor 305-333, Daejeon Technology, and Science of University ; 2 , [email protected] 3 1 21 iJ Oh MinJi , 16 eklyLbadBree etrfrCsooia Physics, Cosmological for Center Berkeley and Lab Berkeley D A , – ubeparameter Hubble , 18 22 nvriyo aiona ekly A970 USA 94720, CA Berkeley, California, of University .Asmn h in- the Assuming ]. ]. 19 2 , 4 , epiOkumura Teppei , 20 Dtd etme 2 2018) 12, September (Dated: ,btthe but ], H n rwhrate growth and , nlz h DSD9glxe nteBS MS [ CMASS BOSS the clustering in We galaxies power. the DR9 of SDSS of ring the analyze BAO distortions the distinct including isocontours, of give power Variation these statistics. of clustering each 2D the from taneously l parameter ble tion. u suigLD rayohrdr nrymodel. distance energy angular dark the for other fit any directly we or with- Instead manner, LCDM independent assuming model out substantially a in out in ooti siae of estimates func- correlation obtain quadrupole to and tions monopole the of [ shape by confirmed were surements analyses, spectra power where and functions correlation typical Following [ distribution. in lowest tested clustering the method a the the or of on ratio, multipoles (AP) relying few squashing than a average, rather spherical analysis, (transverse-radial) a olwdb oedtie td hc on the found which study detailed ratio more distance a by followed was “ring (BAO) [ oscillation power” acoustic of baryon the particular paying to plane, attention transverse-radial the in function apea neetv esitof redshift effective an at sample ih osrit nteaglrdaee itneand distance diameter constant: angular placed Hubble the the wedges” on “clustering constraints tight using analysis Anisotropic and be a hnue opaetgtcntanso h cos- the on constraints [ tight parameters place mological to used then was ables rate A eeto nD9cmn rm[ from coming DR9 in first the detection distances, BAO cosmological measuring in results cant 3 epromafl w-iesoa anisotropy two-dimensional full a perform We nte datg fti nlssi hti scarried is it that is analysis this of advantage Another nual-vrgdsaitc,sc stemultipoles the as such statistics, Angularly-averaged hsdt a led ie iet eea signifi- several to rise given already has data This rsin .Sabiu G. 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Due to a strong correlation between density mological parameters. However, such statistics become and velocity fields, the mapping between real and red- complicated when one considers excluding data along the shift space is intrinsically non–linear [36]. In general, line-of-sight, e.g. that are much noisier than the data per- it appears as a not–closed iteration of polynomials for pendicular to it [32] or are difficult to model accurately which a more elaborate description than simple factor- because of velocity or nonlinear effects. It is thus mean- ized formulation needs to be used. However at large ingful to present the analysis of the correlation function separation several leading polynomials dominate. In ad- in the transverse-radial plane, without angle averaging, dition, we apply the non–linear correction terms using as a complementary method. the resummed perturbation theory called RegPT [37, 38]. Another advantage of using the full 2D correlation When restricting analysis to the quasi-linear regime, the function is that one can easily distinguish between the result is the non–linear portions of the power spectra are geometric and velocity (RSD) effects, clarifying the phys- better separated from the linear spectra, for which the ical interpretation. The 2D correlation function including assumption of perfect cross–correlation between density the BAO scale was first analyzed by [33] but the analysis and velocity fields is verified. relied on linear theory [34]. In [23] we developed a for- In brief, we adopt the redshift-space power spectrum, malism that predicts the correlation function in the 2D P˜(k,µ), given in Ref. [36], which can be recast as plane with nonlinear perturbation theory. Following the 4 method tested in [23], we here fit the clustering correla- ˜ 2n FoG tion function in the transverse-radial plane to data. P (k,µ)= Q2n(k)µ G (kµσp) , (1) nX=0 The outline of this paper proceeds as follows. In Sec. II we briefly review the analysis method and treatment of where σp is a free parameter representing small scale ve- nonlinearities and velocity effects. Section III details the locity effects. Our previous analysis suggests that as long measurement procedure including estimation of the co- as we consider the weakly nonlinear scales, cosmologi- variance matrix. The results are presented in Sec. IV and cal analysis can be made independently of the functional the implications for cosmological models are discussed in form of FoG effect. The functions Q2n are given in [39]. Sec. V. We conclude in Sec. VI, with Appendix A ex- From the power spectrum one can compute the corre- ploring cautions regarding interpretation of zeff at the lation function by Fourier transform. The redshift-space accuracy of next generation surveys. correlation function ξ(σ, µ) is generally expanded as
3 d k k s ξs(σ, π)= P˜(k,µ)ei · II. THEORETICAL MODEL Z (2π)3 = ξ (s) (ν) , (2) ℓ Pℓ The two–point correlation function of galaxy cluster- ℓXeven ing, ξs, is described as a function of σ and π in the with being the Legendre polynomials. Here, we define distant-observer limit, where σ and π are the transverse P 2 2 1/2 and the radial directions with respect to the observer. ν = π/s and s = (σ + π ) . The moments of cor- As mentioned in the Introduction, several effects give relation function are given in [23]. Here we include the rise to anisotropy between these directions. In the linear moments up to ℓ = 6, since the higher-order moments ℓ 8 are shown to contribute negligibly. density perturbation regime, RSD causes the clustering ≥ pattern to be squeezed along the line of sight (i.e. the π-direction), leading to an apparent enhancement of the amplitude of the observed correlation function. This is III. MEASUREMENTS known as Kaiser effect [1]. On the other hand, in the non– linear regime, the random virial motions of galaxies resid- A. Data Sample ing in halos introduce elongated clustering along the line of sight, which is dubbed the Finger of God effect (FoG). We use data from the Sloan Digital Sky Survey [SDSS; This dispersion effect has significant impact, and even 40], Data Release 9 (DR9). SDSS has mapped over one on large scales (in linear theory), a simple description of quarter of the sky in five photometric bands down to a ξs(σ, π) using the Kaiser formula alone may not be ade- limiting magnitude of r 22.5. The photometric data quate along the π direction (e.g., [32]). In our previous is reduced and from it are∼ selected targets for followup paper, we combined this dispersion effect with the Kaiser spectroscopy. The spectroscopic survey, known as the formula to analyze two–dimensional anisotropy structure Baryon Oscillation Spectroscopic Survey (BOSS), is de- of DR7 catalogue [35]. signed to obtain spectra for 1.5 million galaxies over a The precision of the updated DR9 clustering catalog is 10,000 square degree footprint.∼ greatly improved. Due to this improvement, systematic In an effort to control the evolution of galaxy bias over uncertainties in accounting for the anisotropic clustering large redshift ranges the BOSS targets are selected in effects gain greater influence. Therefore we here employ such a way as to have approximately constant stellar improved distortion models to analyze the better preci- mass (CMASS). This is obtained using colour selections 3
FIG. 1. The measured 2D clustering correlation function ξ(σ, π) is plotted, adopting early universe priors from WMAP9 (left) or Planck (right). based on the passive galaxy template of [41]. The ma- is assigned a weight according to jority of CMASS galaxies are bright, central galaxies (in the halo model framework) and are thus highly biased i 1 wF KP = , (5) (b 2) [42]. 1+ n (z)P0 ∼ i The CMASS sample [43] is defined by
where ni(z) is the comoving number density of galaxy z > 0.4 (3) population i at redshift z and one conventionally eval- 17.5
d⊥ = r i (g r )/8.0 . (4) range 0.43 Ngal Each spectroscopically observed galaxy is weighted to i wFKP,i zi zeff = , (6) account for three distinct observational effects: redshift P Ngal w failure, w ; minimum variance, w ; and angular i FKP,i fail F KP P variation, wsys. These weights are described in more de- giving the value zeff =0.57. The effective volume tail in [45] and [46]. Firstly, galaxies that lack a redshift due to fiber collisions or inadequate spectral information 2 are accounted for by reweighting the nearest galaxy by a n(zi)P0 Veff = ∆V (zi) , (7) weight wfail = (1+ N), where N is the number of close X 1+ n(zi)P0 neighbours without an estimated redshift. Secondly, the finite sampling of the density field leads to use of the min- where ∆V (z) is the volume of a shell at redshift z, is 3 imum variance FKP prescription [47] where each galaxy Veff 2.2Gpc . ∼ 4 B. Measuring the correlation function where Nmocks = 611, ξk(ri) represents the value of the correlation function in the ith bin of (σ, π) in the kth r r We compute the redshift-space 2-dimensional correla- realization, and ξ( i) is the mean value of ξk( i) over tion function ξ(σ, π) using the BOSS DR9 galaxy cat- all the realizations. We can then obtain the normalized alog [45]. We perform the coordinate transforms for covariance matrix as two fiducial spatially-flat cosmological models: WMAP9 ˆ Cov(ξi, ξj ) (ωb = 0.02264, ωc = 0.1138, h = 0.70) and PLANCK Cij = . (10) Cov(ξi, ξi)Cov(ξj , ξj ) (ωb = 0.022068, ωc = 0.12029, h = 0.67). Although the parameter fitting procedure should be insensitive to the p In order to reduce the statistical noise in our covari- choice of fiducial model, we perform this check for con- ance matrix, we perform a singular value decomposition sistency. (SVD) of the matrix as done in [35, 50], We estimate the correlation function using the stan- dard Landy-Szalay estimator [48], ˆ † Cij = UikDklVlj , (11) DD 2DR + RR ξ(σ, π)= − , (8) where U and V are orthogonal matrices that span the RR 2 range and the null space of Cˆij , and Dkl = λ δkl, a diag- where DD is the number of galaxy–galaxy pairs, DR the onal matrix with the singular values along the diagonal. number of galaxy-random pairs, and RR is the number of When using SVD the χ2 value becomes more difficult random–random pairs, all separated by a distance σ ∆σ to interpret as it changes as one cuts the noisiest eigenval- and π ∆π. Each pair is weighted by the product of± the ues. However we establish that the reduced χ2 converges individual± weightings of each point. to a constant value above 250 modes. To be conservative The random point catalogue constitutes an unclustered we use 350 out of 400 available modes. but observationally representative sample of the BOSS The estimate of the covariance matrix obtained from CMASS survey. The points are chosen to reside within a finite number of realizations is necessarily biased ([51], the survey geometry and the redshifts are obtained via see also [29]). To obtain the unbiased covariance matrix the random shuffle method of [46]. The randoms are also C, we multiply the original covariance Cˆ by a correction assigned completeness weights, just as for the galaxies. factor To reduce the statistical variance of the estimator we use 20 times as many randoms as we have galaxies. −1 Nmocks Nbins 2 ˆ−1 ∼ C = − − C , (12) We calculate the correlation function in 15 bins of di- Nmocks 1 − mension 10 h−1 Mpc, linearly spaced in the range 0 < σ,π < 150 h−1 Mpc. The resulting two point correlation where Nbins is the number of bins of ξ(σ, π) used for the function in Fig. 1 shows the typical Kaiser [1] compres- analysis. sion at small σ (near to the line of sight) and the emer- gence of the 2D BAO ring at √σ2 + π2 100 h−1 Mpc. ≈ IV. RESULTS OF 2D ANISOTROPY ANALYSIS C. Covariance matrix A. Fitting method In addition to the correlation function we need to know To fit the correlation function with as model indepen- the errors on it. Because different bins of the correlation dent cosmology inputs as possible, we assume that the function are strongly correlated, it is necessary to esti- shape of the power spectra is given by the early universe mate a covariance matrix to give correct constraints on conditions measured by CMB experiments. We denote cosmological parameters. As in our previous paper [23], this primordial spectrum convolved with the transfer we use the mock galaxy catalog created by [49]. This function as Dm(k). This then evolves coherently through catalog has the same survey geometry and number den- all scales from the last scattering surface. That is, the sity as the CMASS galaxy sample that was used in our growth occurs through a time-varying, scale-independent analysis and 611 mock realizations were created using amplitude growth factor (this assumption breaks down second-order Lagrangian perturbation theory (2LPT) for in theories that introduce significant scale dependence, the galaxy clustering. such as some modified gravity theories). Propagating For each realization we compute the correlation func- this through to cosmological parameters requires assump- tion as we did for the observed catalog in section IIIB tions on the cosmological model, e.g. the nature of dark and obtain a covariance matrix by energy. To remain substantially model independent we use the growth rate itself as our variable. Cov(ξi, ξj )= (9) The power spectra are then given by N 1 mocks 2 r r r r Pbb(k,a)= Dm(k)Gb (a), [ξk( i) ξ( i)][ξk( j ) ξ( j )] , 2 Nmocks 1 − − PΘΘ(k,a)= D (k)G (a) , (13) − Xk=1 m Θ 5 FIG. 2. The measured, best fit, and LCDM-predicted versions of ξ(σ, π) are plotted, using the Planck early universe prior. The blue filled contours represent the measured ξ(σ, π), and the black unfilled contours represent the best fit ξ(σ, π) with two −1 −1 different σcut. The left panel uses σcut = 20 h Mpc in the fit, and the right panel uses σcut = 40 h Mpc. The quarter rings denote the 2D BAO ring, for the fit (solid) and Planck LCDM prediction (dotted). −1 fid where Gb and GΘ denote the growth functions of density and H is fitted to the observed ξ using the rescaling and peculiar velocity. We define here Gb = b Gδm where in the above equations. b is the standard linear bias parameter between galaxy Given the early universe prior on the power spectrum tracers and the underlying dark matter density. The ex- shape, both distances and growth functions are measured pression of Dm(k) is available in [35], and assumed to be simultaneously to high precision [52]. This holds even given precisely by CMB experiments, such as WMAP9 without assuming the FRW integral relation between and Planck experiments. We refer to this as an early −1 H and DA. Thus we do not have to assume any par- universe prior. We incorporate the uncertainty from the ticular cosmological model or restrict to zero curvature CMB anisotropy data in the amplitude determination of 2 or LCDM. the initial spectra, AS, into the growth function GX , i.e. ∗ ∗ ∗ Finally, we introduce a parameter representing non– GX = GX AS /AS where GX is the intrinsic growth func- tion. linear contamination to the power spectra of the density The clustering correlation function ξ(σ, π) is measured and velocity fields. Even on linear scales the damping in comoving distances, while galaxy locations use angu- effect on the power spectrum amplitude caused by ran- lar coordinates and redshift in galaxy redshift surveys. dom galaxy motions still remains. This is described by A fiducial cosmology is required for conversion into co- the Gaussian model for the FoG effect in Eq. 1, with σp moving space. We use the best fit LCDM universe to a free parameter giving the velocity dispersion. How- WMAP9 or Planck. The observed anisotropy correlation ever, the non–perturbative damping effects are not fully function using this model is transformed into true comov- understood, and the Gaussian model may be insuficient ing coordinates using the transverse and radial distances, on non–linear scales. We therefore do not use the mea- −1 sured ξ(σ, π) for bins in which this breakdown is likely. involving DA and H , respectively. The approximate fiducial maps are created by rescaling the transverse and Two cut–off’s are used: 1) scut represents the scales on radial distances, using which non–linear description of ∆PXY is uncertain, and 2) σcut represents the scales on which Gaussian FoG func- fid fid D true tional form may not be appropriate. These are set to A −1 −1 σ = true σ be scut = 50 h Mpc and σcut = 40 h Mpc (although DA −1 we also consider σcut = 20 h Mpc). This strategy was H−1fid πfid = πtrue , (14) tested and proved valid using simulations in our previous H−1 true work. We follow the same method as presented in Song, where “fid” and “true” denote the fiducial and true dis- Okumura and Taruya (2013) [23]. tances. Thus the theoretical ξ with potentially true DA In summary, we have Gb and GΘ to describe growth 6 FIG. 3. As Fig. 2, but using WMAP9 instead of Planck. −1 functions, DA and H to fit distance measures, and σp shape of an object is a priori known, can provide a mea- to model the FoG effect. The form of the FoG is taken sure of HDA (AP test). to be Gaussian and the shape of the linear spectra is The outer measured ξ(σ, π) contours are too vague assumed to be given as an early universe prior by CMB to reveal detailed BAO peak structure, but those peak experiments. points can define the measured 2D BAO ring. Fig- ure 2 shows the 2D correlation function contours, and the best fit 2D BAO rings. The left and right panels use −1 −1 B. Cut–off scales and 2D BAO ring σcut = 20 h Mpc and 40 h Mpc, respectively. If the correlation function model is accurate down to σcut = 20 h−1 Mpc then the two rings should be consistent. First, we investigate the appropriate cut–off scales. −1 However, the 2D BAO ring using σcut = 20 h Mpc not The scut is introduced due to the uncertainty of the re- −1 only disagrees with that using σcut = 40 h Mpc but summed perturbation theory RegPT at smaller scales. It −1 also from the measured circle. is conservatively set to be scut = 50 h Mpc which al- lows the perfect cross–correlation between density and Basically the small, nonlinear scales where the model velocity fields. In addition σcut is used because the is imperfect are distorting the results at all scales. improved ξ(σ, π) in Eq. 2 is not applicable at bins in This can be seen by looking at several inner con- which the higher order terms of non–perturbative effect tours at small scales, those corresponding to ξ = are dominant along the line of sight. It was set to be (0.2, 0.06, 0.016, 0.005). In the left panel the solid curves −1 σcut = 20 h Mpc in [23], and reproduced the true val- attempt to fit tightly the small scale contours very close ues successfully. But we find that it may be too ambitious to the line of sight, at the price of a poor fit to the large for the actual DR9 CMASS catalogue. scale, linear contours. By contrast, in the right panel −1 When the broadband shape of spectra and the distance with σcut = 40 h Mpc the residual non–perturbative measures are known, the 2D BAO ring is invariant to the effects are observed clearly in the inner contours, but the changes of the coherent galaxy bias and coherent motion linear contours are better behaved. This problem with an overambitious use of small scales is seen as well in growth function. When Gb increases/decreases, the BAO tip points coherently move counter–clockwise/clockwise. Fig. 3 using the WMAP9 early universe prior instead. When GΘ increases/decreases, the BAO tip points move Therefore we use more conservative bound at σcut = −1 toward/away from the pivot point (equal radial and 40 h Mpc. We tested our final results using different −1 transverse separation). If the correct distance model σcut at 20, 30, 40, and 50 h Mpc and found they con- −1 verged for σcut 40 h Mpc. The effect on cosmology is known, the tip points of BAO peaks form an invari- ≥ ant ring regardless of galaxy bias and coherent motion. of using a cut allowing more of the non–linear regime is The ratio between the observed transverse and radial dis- discussed in Sec. V. tances varies with the assumed cosmology and, if the The dashed contours in Fig. 2 represent the ξ(σ, π) of 7 −1 FIG. 4. The 68% and 95% confidence contours from the galaxy clustering data are plotted in the DA − H plane, with the best fit denoted by the large X. The values predicted from the CMB within LCDM cosmology are shown by the blue square. The left panel uses the WMAP9 early universe prior on the power spectrum shape, while the right panel uses the Planck prior. Overlaid are theory curves giving the relation between the two cosmological quantities within certain cosmologies; note that all standard cosmologies lie in a restricted band. In addition to LCDM (black solid), we show wCDM with w = −0.8 (blue) or w = −1.2 (magenta), and their generalizations to include spatial curvature (dotted). Each curve covers the range of Ωm = 0.2 at their upper ends to Ωm = 0.35 at their lower ends, with large dots showing the Ωm = 0.3 case. the Planck LCDM concordance model. They are derived Parameters Fiducial values Measurements −1 using the fiducial (DA,H , GΘ), and best fit (Gb, σp). With WMAP9 prior −1 +27.2 The right panel of Fig. 2 shows strong agreement between DA ( h Mpc) 946.0 916.2−25.4 −1 −1 +102.0 the derived best fit model and the theoretical Planck H ( h Mpc) 2241.5 2163.1−85.8 − +0.07 LCDM concordance model. Gb 1.07−0.09 +0.09 For Fig. 3 using the WMAP9 early universe prior, GΘ 0.44 0.51−0.08 −1 − +4.6 while the estimated 2D BAO ring agrees approximately σp ( h Mpc) 1.0 with the measured 2D BAO ring, peak points along the Parameters Fiducial values Measurements ring do not well match to each other. The dashed con- With Planck prior −1 +26.7 DA ( h Mpc) 932.6 939.7−32.6 tours here represent ξ(σ, π) of the WMAP9 LCDM con- −1 −1 +82.3 cordance model. Unlike the Planck case, the measured H ( h Mpc) 2177.5 2120.5−100.6 − +0.07 Gb 1.11−0.10 peak points shift toward the pivot point for the outer con- +0.10 GΘ 0.46 0.47−0.07 tour, less so for the inner contours. As discussed above, −1 +4.0 σp ( h Mpc) − 1.2 this is a signature of an increased velocity growth func- tion; we expect the measured GΘ to be higher than fidu- cial in this case. TABLE I. We summarize the values predicted by the CMB data and the values measured from the BOSS data of the −1 distance quantities DA and H and the growth quantities C. The measured distances and growth functions Gb and GΘ, as well as the velocity damping scale σp, with 68% confidence level errors. (The CMB data does not predict We present the results for the measured distances and values of the astrophysical parameters Gb and σp.) growth functions in Table I. Our baseline value of σcut = 40 h−1 Mpc is used throughout this section. The angular diameter distance DA, related to trans- distortions are relevant to the radial direction, and verse separations, is measured to be consistent with the it is expected that DA is not biased much. With LCDM predictions. Most uncertainties of anisotropic the Planck early universe prior, DA is measured to 8 FIG. 5. As Fig. 4 but for the Gθ − DA plane. Here the allowed cosmology band is wider (we do not plot the owCDM models). +26.7 −1 be 939.7−32.6 h Mpc, in excellent agreement with the we show this explicitly in Sec. V. Planck LCDM best fit prediction. Using the WMAP9 The galaxy bias is estimated from the Gb measure- early universe prior, the measured DA is 1 σ from the ment. The bias b is measured to be 1.9and 1.8 for Planck − WMAP9 LCDM prediction. and WMAP9 respectively. Those values are consistent 1 The line of sight distance quantity H− , related to ra- with CMASS catalogues [49]. The velocity dispersion σp dial separations, is also measured to be consistent with indicates the level of the FoG effect. For both Planck LCDM predictions. For either the Planck or WMAP9 and WMAP9 cases, it is observed to be small, about −1 early universe priors the agreement is within 1 σ. σp =1 h Mpc, but with significant uncertainty. −1 − Greater tension is seen if one uses σcut = 20 h Mpc, which lowers the measured H−1. As mentioned earlier, the growth functions influence V. TESTING COSMOLOGY the location of peaks along the rings of power (see [23] for illustrations). For the Planck early universe prior, Our analysis approach has been model independent, −1 the best fit peak structure is nearly identical to that pre- obtaining constraints on the distances DA and H – dicted by the Planck LCDM model. The measured co- without even assuming a Friedmann integral relation be- +0.10 herent growth function has GΘ = 0.47−0.07, while the tween them – and on the velocity growth factor GΘ. fiducial value is 0.46. This measurement can be con- While we have so far compared the values individually to verted to a value at zeff =0.57 of the standard parame- the best fit LCDM predictions from the CMB, we should ter fσ8 =0.48, which is very close to the fiducial model also look at the joint probabilities. We can test for consis- value of 0.47. When the WMAP9 early universe prior is tency with the LCDM model by examining whether the used, the measured GΘ becomes bigger than LCDM pre- fixed relations between these quantities in LCDM, i.e. the −1 −1 diction. Like the distance measurements, the measured 1D curves in the DA H , DA GΘ, and H GΘ GΘ is offset by 1 σ. − − − ∼ − planes, all intersect the measured confidence contours. Note that GΘ has a relatively large error, about Furthermore, we can generalize the test by allowing for 15 20%. This is partly caused by floating σ as a free spatial curvature or non-Λ dark energy. For the growth − p parameter. In the linear regime, when the first order factor GΘ this comparison also allows a test of general contribution of the Gaussian FoG function dominates, relativity since within this theory the distance quantities this factor is nearly featureless and becomes significantly (measuring the cosmic expansion) have a definite relation degenerate with coherent growth function. Using more to the growth quantity GΘ. non-linear scales (smaller σcut) would break this degen- Figures 4, 5, 6 show the three planes of pairs of the eracy, reducing the error contour but introducing bias; cosmological quantities and their joint measurement con- 9 −1 FIG. 6. As Fig. 4 but for the Gθ − H plane. Here the allowed cosmology band is wider (we do not plot the owCDM models). tours, overlaid with the allowed theory curves of LCDM, Figure 7 shows what occurs in the cosmology param- −1 oLCDM (with spatial curvature), wCDM (dark energy eters if data down to σcut = 20 h Mpc is used. The with constant equation of state ratio w), and owCDM. shifting of the best fit values, and the reduction in the Each one is shown for a WMAP9 (left panels) or Planck uncertainty on GΘ, clearly indicate that substantial in- (right panels) early universe prior. formation to fit cosmology is coming from small scales, −1 not just the BAO ring scales. Unfortunately, the sensi- In the DA H space, the cosmological models all − tivity of the results to low σcut (as opposed to the conver- lie within a narrow swath, somewhat separated from the −1 best fit point in the Planck prior case. However, the gence found when σcut & 40 h Mpc) indicates that the 68% confidence level contour of the measurements over- modeling of the 2D correlation function on these scales is −1 inadequate. Further improvements are necessary before laps the LCDM model. In the DA GΘ or H GΘ planes, the standard cosmologies span− a wider range− of these scales can be used to provide robust results. the space. In both planes the measurements are consis- In terms of Fourier wavenumber, note that tent with LCDM at the 68% confidenece level. There 2π 40 h−1 Mpc is no sign of significant deviation from LCDM in either k =0.16 h Mpc−1 . (15) σcut σcut distances or growth, and hence no sign of deviation from ≈ general relativity either. In comparisons to simulations the 2D anisotropic clus- Note, however, that if we attempt to push the data tering model (not simply the 1D real space power spec- by using data to smaller, non-linear scales, then we do trum or angle averaged correlation function) performed −1 find deviations. In particular, GΘ rapidly becomes un- well down to 40 h Mpc [36]. Another way to spuriously derestimated, with values of 0.42 for a cutoff at σcut = produce a shift outside the swath of standard cosmolo- −1 −1 30 h Mpc and 0.34 for σcut = 20 h Mpc. However gies, and hence possibly conclude there is a violation of −1 increasing σcut above 40 h Mpc does not change the general relativity, is to misestimate zeff . In fact, as we result, indicating the value has converged. Had we in- discuss in Appendix A, zeff is itself anisotropic and will cluded the smaller scales, we would have found that no differ for different cosmological quantities but not at a cosmology (LCDM, oLCDM, wCDM) would have given level significant with current data. good fits to the measurement contours. Moreover, we would have apparent evidence for a violation of general relativity. The apparent strong growth suppression in VI. CONCLUSIONS the measured growth rate GΘ = dδ/d ln a would yield an apparent gravitational growth index γ [53] of γ & 0.7, in We have used the BOSS CMASS DR9 galaxies to per- contrast to the value 0.55 for general relativity. form a cosmology model independent, fully 2D anistropic 10 conservative results to those from a multipole analysis. Another aspect is that we find that the results from the real, observed, data are more contaminated with the small scale velocity and non-linear effects than those from the mock catalogues. In the simulations, σcut = 20 h−1 Mpc is acceptable to measure observables using the improved perturbation theory model. However, in the real dataset, the cut–off scale must be extended to −1 σcut = 40 h Mpc to obtain convergent results (insensi- tive to the exact choice of σcut). This can also be seen by comparing the 2D BAO ring with the measured BAO peak structure. The second caution comes from the interpretation de- pendence of the effective redshift zeff . Since it involves the galaxy power spectrum (or correlation function) it is intrinsically anisotropic and will take on different values depending on what quantity is being measured. That is, D −1 H one formally has DA(zeff ), H (zeff ), etc. We estimate the magnitude of this effect and show that it could be- come relevant for next generation redshift surveys such as DESI or Euclid. −1 FIG. 7. Using small scale information, σcut = 20 h Mpc (in ACKNOWLEDGMENTS −1 dark orange) rather than our standard σcut = 40 h Mpc (in light grey), shifts the results into a region corresponding to no Several authors thank KASI for hospitality during re- reasonable cosmology within general relativity. Here we show −1 search visits. We especially thank Atsushi Taruya for the GΘ − H plane, with the Planck early universe prior, as many inputs and comments, and we thank Seokcheon Lee an example. for reading the manuscript. This work was supported in part by the US DOE grant DE-SC-0007867 and Contract No. DE-AC02-05CH11231, and Korea WCU grant R32- clustering analysis. Using an early universe prior from 10130 and Ewha Womans University research fund 1- CMB experiments, from the clustering correlation func- 2008-2935-001-2. Numerical calculations were performed tion we can extract the angular diameter distance DA, by using a high performance computing cluster in the Ko- −1 Hubble scale H , and growth rate GΘ at the effective rea Astronomy and Space Science Institute and we also survey redshift zeff =0.57. These are found to be consis- thank the Korea Institute for Advanced Study for pro- tent with LCDM, and by comparing expansion of cosmic viding computing resources (KIAS Center for Advanced distances with growth of cosmic structure we also test Computation Linux Cluster System). general relativity, again finding consistency. Two cautions are relevant to such an analysis, one im- portant already to current data and one entering for fu- Appendix A: Effective redshift variation ture, high precision surveys. Use of small scale measure- ments of the correlation functions, which can be signifi- The transverse and radial distances extracted from the cantly contaminated by non–linear gravitational physics, galaxy data do not in fact have the same zeff , as the opti- is fraught with peril. We find this can distort the cos- mal weighting depends on the strength of clustering [47], mological results, moving them wholly outside the range enhanced along the line of sight by redshift space distor- of standard cosmology and give a spurious signature of tions [e.g. the usual Kaiser factor (b + fµ2)2]. This is breakdown of general relativity. Insidiously, the extra most familiar perhaps in the power spectrum, where the data also helps shrink the contours, so the cosmological weighting 1/[1 + n(z) P (k,µ,z)] shows that the higher quantities appear well determined. power along the line of sight further deweights lower red- We employ the improved redshift distortion model of shift galaxies where clustering has grown. [36], but this is still limited in accuracy to scales where This is a small effect, negligible for previous red- higher order terms of the FoG effect are negligible. To shift surveys, but will become increasingly important for prevent bias we cut most of the measured ξ(σ, π) along larger, more precise surveys. Figure 8 calculates zeff as a the line of sight out from this analysis. This conserva- function of k and µ, using the power spectrum computed tive treatment is well defined in the full 2D anisotropy from mock simulations relevant to BOSS [54]. Since most analysis but could be problematic when using a multipole of the information for determining H−1 comes from ra- expansion instead. It will be interesting to compare our dial modes µ 1 and for determining D comes from ≈ A 11 transverse modes µ 0, we see that the fit quantities are D ≈ −1 H H D really DA(zeff ) and H (zeff ) where zeff zeff 0.004. This in turn would affect cosmological parameter− ≈ estima- tion. Around zeff 0.57, the Hubble parameter scales in LCDM as H(z)≈ 1+z so that dz/(1+z) dH−1/H−1. ∝ ≈− Note that a survey that should use zeff =0.58 rather than −1 0.57, say, due to the anisotropy of zeff , would bias H by 0.7%. This could be relevant for next generation sur- − veys. Since DA is extracted mostly from the transverse modes where the observed clustering is equal to the real space clustering, no shift should be needed in the conven- tional zeff estimation. (For completeness we note that if zeff = 0.58 rather than 0.57 then DA is biased high by 0.8%.) 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