Nonlinear Effects in the Amplitude of Cosmological Density Fluctuations
Roman Juszkiewicz, Hume A. Feldman, J. N. Fry, Andrew H. Jaffe
aa r X i v : . [ a s t r o - ph . C O ] F e b Weakly nonlinear dynamics and the σ parameter Roman Juszkiewicz a,b , Hume A. Feldman c,d,e , J. N. Fry f , Andrew H. Jaffe e a Institute of Astronomy, 65-516 Zielona G´ora b Copernicus Astronomical Center, 00-716 Warsaw, Poland c Department of Physics and Astronomy, University of Kansas, Lawrence KS 66045, USA d Department of Physics and Astronomy, University College London, London, WC1E 6BT, UK e Astrophysics, Blackett Laboratory, Imperial College, London SW7 2AZ, UK f Department of Physics, University of Florida, Gainesville FL 32611-8440, USA
Email: [email protected], [email protected], [email protected], [email protected]
ABSTRACT
The amplitude of cosmological density fluctuations, σ , has been studied and estimated byanalysing many cosmological observations. The values of the estimates vary considerably betweenthe various probes. However, different estimators probe the value of σ in different cosmologicalscales and do not take into account the nonlinear evolution of the parameter at late times. Weshow that estimates of the amplitude of cosmological density fluctuations derived from cosmicflows are systematically higher than those inferred at early epochs from the CMB because ofnonlinear evolution at later times. We discuss the past and future evolution of linear and nonlinearperturbations, derive corrections to the value of σ and compare amplitudes after accounting forthese differences. Subject headings: cosmic flows, cosmological parameters from LSS, cosmological parameters from CMBR,galaxy dynamics
1. Introduction
The σ parameter is one of the most importantand least well known parameters in cosmology. Itis a convenient measure of the degree of the inho-mogeneity of the Universe. It is the rms matterdensity contrast in a sphere with a comoving ra-dius of 8 h − Mpc at present, where h is the usualdimensionless Hubble constant in units of 100 kms − Mpc − . The original motivation for the useof this parameter was the need to define the clus-tering amplitude: the variance in the number den-sity of optical galaxies in 8 h − Mpc spheres, σ ,was observed to be unity (Davis & Peebles 1983;Strauss & Willick 1995). This parameter is sim-ply related to scale-independent linear bias (Kaiser1988), defined as the square root of the ratio ofthe number density variance to the mass variance b ≡ σ gal /σ mass = 1 /σ . Later, σ provided thestandard way to normalize the theoretical powerspectra of density fluctuations, determined from first principles only up to an arbitrary multiplica-tive constant. A theory can be tested against theempirical data by comparing its assumed value of σ with a value implied by observations. A com-parison of values of σ , estimated from various ob-servations can provide an important measure ofsystematic errors introduced by different estima-tors, based on different physical effects, e.g., bulkflows, cosmic microwave background (CMB) fluc-tuations or gravitational lensing. Such a compar-ison also provides an internal consistency test forthe gravitational instability theory.To compare the σ ( z ), estimated at high red-shift, z , with its current value, estimated from ob-servations at z = 0, it is necessary to take thegrowth of the fluctuations into account. Since allknown estimates of this parameter were of the or-der of unity or less, until recently the linear per-1urbation theory was used for this purpose: σ (0) ≈ σ ( z ) D (0) D ( z ) , (1)where D is the linear growth factor (Peebles 1980).As we show below, ignoring higher order terms inthe above expression introduces a systematic errorof order of 10 to 15%. With the latest improve-ments in the quality of observations, such efectsshould not be neglected and we provide a simplerecipe how to take them into account. We focus onthe comparison of σ estimated from the peculiarvelocity field at an effective redshift z = 0, to σ implied by the CMB (effective redshift z ≈ σ and Ω m , the den-sity of the nonrelativistic matter. In particular,from pairwise velocities we found in Feldman et al.(2003) σ = 1 . +0 . − . . (2)From CMB temperature fluctuations, the WMAPcollaboration (Dunkley et al. 2009) found σ = 0 . ± . , (3)described as the “linear theory amplitude” (seetheir Table 1). The above two estimates differ onlyslightly, at the level of 1.5- σ , and it is a success ofthe model that inferences by such different meth-ods applied at two greatly different epochs are ingood agreement.In this paper we bring the two estimates evencloser by taking nonlinear dynamics into account.We find that the nonlinear correction is modestlysignificant, given the improving accuracy in cos-mology: the linear value of σ , hereafter denoted σ L , can be smaller by ∼
10% than the nonlin-ear value, a systematic difference that is compara-ble to the current statistical uncertainties. In sec-tion 2 we describe the spatial window functionsused to define σ . In section 3 we provide therecipe for recovering σ L from σ . In section 4 wecompare our revised σ parameters to other ob-servational estimates of σ . We summarize ourresults in section 5.
2. Window functions
Like many cosmological experiments, measure-ments of cosmic flows are sensitive to a windowedintegral of the matter power spectrum. In general,such an observable can be characterized as Q i = Z ∞ dk W ( k ) P p ( k ) T i ( k ) (4)where P p ( k ) gives the primordial power spectrum, k is the comoving wavenumber, T i ( k ) gives thetransfer function which contains the physics of theevolution of the particular observable from the pri-mordial spectrum. The window function, W ( k ),describes the experimental setup (sky coverage,depth, errors etc.). This formalism describesstraightforward measurements of the galaxy powerspectrum, in which case Q i = P gal ( k i ), the CMBspectrum for which Q i = C ℓ i , and the ampli-tude of the cosmological velocity field where Q i = P v ( k i ), the velocity power spectrum. It is cru-cial to note that the transfer function depends im-plicitly upon the other cosmological parametersand hence any lack of knowledge thereof will (orat least should) translate to increased uncertaintyupon the amplitude.Bulk flow and shear measure the velocity-velocity power spectrum [or covariance, Watkins et al.(2009); Feldman et al. (2009)], whereas pair-wise velocities measure the density-velocity cross-spectrum (Feldman et al. 2003). Under linear evolution, the density contrast is proportionalto the divergence of the peculiar velocity in realspace, or v ∝ k ρ in Fourier space, so these powerspectra differ by powers of wavenumber k from thedensity power spectrum, which can be absorbedinto the appropriate transfer function.An amplitude parameter such as σ is in essencea spectral observable as well. We define the win-dow function W R ( x , x ′ ) = W R ( x − x ′ ), normal-ized so that R d x W R ( x ) = 1. For our sphericaltop hat, W R ( r ) = 1 /V where V = 4 πR /
3, when r = | x ′ − x | ≤ R , 0 otherwise. Hence, the densitycontrast, spatially averaged over a sphere arounda particular point x is simply δ R ( x ) = Z d x ′ W R ( x − x ′ ) δ ( x ′ ) . (5)The ensemble avarage of δ R at redshift z is given2y σ R ( z ) ≡ h δ R i = Z ∞ dkk ∆( k, z ) | ˜ W ( kR ) | , (6)where ˜ W ( kR ) = (3 /kR ) j ( kR ) (7)is the Fourier transform of the window function W R ( r ), and j is a spherical Bessel function of thefirst kind, while∆ ≡ πP ( k, z ) k / (2 π ) (8)is the dimensionless power per wavenumber oc-tave and P ( k, z ) is the power spectrum of the massdensity fluctuations. The mean square fluctuationthat we would measure from the actual densitycontrast depends upon the actual nonlinear powerspectrum, but we can analogously define the linearvariance by replacing the power spectrum in theabove expression with the linear spectrum, con-strained to have the same amplitude for k → σ R ( z ) = 1 V Z V d x d x ′ ξ ( | x − x ′ | , z ) , (9)where ξ is the two-point correlation function,and V = 4 πR /
3. In Feldman et al. (2003)we have used the above equation to estimatethe true present-day value of the σ parame-ter from the PSC z survey correlation function(Hamilton & Tegmark 2002). We have also usedother empirical correlation functions, derived fromdifferent surveys, as a template and we found thatthe resulting value of σ was not sensitive to suchvariations; see Feldman et al. (2003).
3. Linear and nonlinear amplitudes
We use two different methods to estimate thenonlinear corrections for σ , one based in pertur-bation theory, which allows us to express the cor-rection as a simple analytical expression, and oneusing a phenomenological mapping based on con-servation of pair counts and calibrated using nu-merical simulations, which allows us to explore theeffects of changing many parameters individually. Under linear evolution, the spatial and tempo-ral dependence of clustering separates. The den-sity perturbation δ ( x , a ) = δρ/ρ can be describedas (Peebles 1980) δ ( x , a ) = δ (1) ( x ) D ( a ) , (10)where δ (1) gives the linear density perturbationfield as a function of comoving spatial coordinates x at some fiducial time and D is the growth func-tion, here parameterized by the scale factor a asa time coordinate (here and below we keep onlythe fastest-growing modes). In flat ΛCDM modelsthe growing mode is given by (Heath 1977; Peebles1993) D ( a ) = 5Ω m E ( a )2 Z a duu E ( u ) , (11)where E ( a ) ≡ (cid:2) Ω m a − + 1 − Ω m (cid:3) / , (12)In the early Universe, when the scale factor issmall, a →
0, equation (11) is well approximatedby the expression D ( a ) = a , (13)as in Einstein-de Sitter Universe. In the oppo-site limit, the cosmological constant becomes dy-namically dominant and the linear growth factorwill saturate at a maximum value, as gravitationalclustering is balanced by the effective force of ac-celerated expansion. It is easy to show that in thelimit a → ∞ , the growth factor is given by theexpression D ∞ = 2Γ( )Γ( ) √ π (cid:18) Ω m − Ω m (cid:19) / . (14)Using the above two asymptotic expressions wehave found a new fitting formula for the growthfactor, valid for all flat ΛCDM cosmological mod-els: D ( a ) = a h a/D ∞ ) . i / . . (15)In Figure 1 we show that equation (15) remainswithin two-percent level of the exact solution (11)in both the past and the future. Note that someexpressions for D ( a ) and its logarithmic deriva-tive, d log D/d log a , frequently quoted in the lit-erature (Lahav et al. 1991; Carroll et al. 1992) ap-ply only to the past and fail for a > D , plotted as a func-tion of the scale factor a for three different valuesof Ω m . The exact solution (11) is represented bythe solid red curve, while the dashed blue curvewas derived from the approximate formula (15).All models assume zero spatial curvature, Ω Λ =1 − Ω m . The scale factor at present equals unity.Hence the past corresponds to a/a <
1, while thefuture to a/a > a ≡ a ( z = 0) = 1). The perturbative solution of the equations ofmotion of the cosmic fluid can be written as δ ( x , a ) = ∞ X J =1 δ ( J ) ( x ) D J ( a ) , (16)where δ ( J ) D J is the solution of order J and weassume that the linear solution is a Gaussian ran-dom field, so all its odd-order moments vanish.The mean value of the square of the above expan-sion is therefore given by a series of even powersof D ( a ), h δ i = h [ δ (1) ] i D + h δ (2) +2 δ (1) δ (3) i D + . . . (17)One-loop perturbative corrections to the lead-ing order variance σ L ( r ) for power-law powerspectra are given by Lokas et al. (1996) andScoccimarro & Frieman (1996) σ = σ L + β σ L , (18)where the factor β is related to the logarithmicslope of the two-point correlation function γ ( r ) = − d ln ξ/d ln r by the following relation: β = 1 . − . γ. (19)The above equations and more generally, theweakly nonlinear gravitational instability theorywere confirmed by N-body simulations and bymeasurements of the galaxy skewness and bispec-trum in redshift surveys (see e.g. Juszkiewicz et al.1993; Scoccimarro et al. 2001; Feldman et al.2001; Verde et al. 2002, and references therein).For non-power-law spectra, γ is a slowly varyingfunction of scale. A convenient representation ofthe correlation function over scales of interest hastwo power laws (Hamilton & Tegmark 2002), ξ ( r ) = q (cid:0) x − γ + x − γ (cid:1) , (20)where x j = r/r j , r = 2 . h − Mpc, r =3 . h − Mpc, γ = 1 . γ = 1 .
28, and q = σ (8 h − Mpc) / . σ from the pairwise peculiar motions ofgalaxies (Feldman et al. 2003) . We have also con-sidered other observational estimates of ξ ( r ) andfound that the resulting values of σ and Ω m were4naffected. The effective slope of the correlationfunction is then γ ( r ) ≡ − d ln ξ ( r ) d ln r = γ x − γ + γ x − γ x − γ + x − γ , (21)independent of the amplitude q . At r = 8 h − Mpc,the effective slope is γ = 1 . β = 0 . σ to σ L , we invertequation (18): σ L ( r ) = p βσ ( r ) − β . (22)Note that for β →
0, the above expression gives σ L = σ , as it should. In Feldman et al. (2003)we obtained σ = 1 .
13. Using this value with β =0 .
216 in equation (22), we obtain for the centralvalue σ L = 1 . , (23)a modest but significant decrease. Our other method of relating linear and nonlin-ear variance is non-perturbative. It uses the map-ping of scale proposed in Hamilton et al. (1991),for which conservation of mass or pair counts re-lates a scale in the linear regime to a nonlinear“collapsed” scale r L = Z R d ( r ) [1 + ξ ( r )] = R [1 + σ ( R )] . (24)The variance σ is then a (nearly) universal func-tion σ ( R ) = g [ σ L ( r L )] . (25)The relation is verified and the function g iden-tified in numerical simulations (Hamilton et al.1991; Peacock & Dodds 1996; Smith et al. 2003).There is an important difference between theperturbative and the phenomenological calcula-tions. The non-perturbative mapping, given byequation (25), explicitly uses the linear expressionfor D ( a ) and therefore in order to derive σ L from σ , we have to choose specific values of the cos-mological parameters; for technical details, see,for example Peacock & Dodds (1996). This is notnecessary for the perturbative formula (22): to de-rive σ L we only need to know σ and γ , the slopeof the two-point correlation function at present. Fig. 2.— Fully evolved variance σ plotted vs. lin-ear variance σ L . For comparison, the isolated redsolid line shows σ = σ L . Second order perturba-tion theory, based on Eq. (18) with β = 0 .
216 givesthe long-dashed black line. The remaining curvesare derived from the phenomenological mapping,given by Eq. (25). The differences between themshow how sensitive they are to different assump-tions about the cause of change in the amplitudeof density perturbations. The growth induced bygravitational instability acting on density pertur-bations with the ‘standard’ ΛCDM initial condi-tions (Ω m = 0 . , Ω Λ = 0 . , h = 0 .
7) and the ap-propriate D ( a ) factor, is shown by the blue short-dashed line. The solid red line differs in redshiftor amount of evolution (and hence saturates at σ L ≈ . h = 0 . − .
7) or scale ( R = 5 −
19 Mpc)which changes n eff and thus γ and so tracks a lit-tle differently from the others. We also show theWMAP (Dunkley et al. 2009) result ( σ L = 0 . σ = 0 . σ = 1 .
13 correspond-ing to σ L = 1 .
02, with the vertical and horizon-tal intervals shown in long dashed black lines therange in Eqs. (26) and (27)).5n Figure 2 we show nonlinear σ plotted againstthe inferred linear σ (bundle of curves); for com-parison the isolated solid red line shows σ L . Theperturbation theory result of equation (22) isshown by the long-dashed black line. To verifythat perturbative results still make sense even atthe threshold of the validity of perturbation the-ory, when σ L ≈
1, we also plot the phenomeno-logical mapping results, based on equation (25)for a variety of parameters. In this mapping wemust identify why σ L has changed, which may befrom a change of fluctuation amplitude inducedby the standard gravitational instability, evolu-tion epoch, or scale; each has a slightly differenteffect. In general, the perturbative curve agreeswell with the phenomenological results, based onthe transfer function from Bardeen et al. (1986)and the Peacock & Dodds (1996) fitting. Usingthe Eisenstein & Hu (1998) transfer function andthe Smith et al. (2003) fitting make very littledifference in the results except when the Hubbleconstant or scale changes. This occurs because achange in scale substantially changes the value of γ and so of β . This is yet another confirmation of thereliability of our perturbative calculations. It alsoshows that the PSCz ξ ( r ), assumed in the pertur-bative calculation, agrees well with the ‘standard’ΛCDM ξ ( r ), assumed in the phenomenologicalmapping.In the phenomenological ΛCDM relation, thenonlinear signal range σ = 1 . +0 . − . , (26)maps to σ L = 1 . +0 . − . . (27)Note that since σ is steeper than σ L , the range in σ L is somewhat narrower than the one in σ . Thesevalues and the resulting limits are also shown inFig. 2.More generally, the existing descriptions andansatzen of nonlinear clustering were created todescribe the evolution of clustering from the pastthrough to the present-day; they are not necessar-ily adequate representations of future clustering,even in the very near future. In Figure 3 we showthe value of ∆ ( k, z ), the mean square of the di-mensionless density fluctuation at various epochsusing both the Peacock & Dodds (1996) mappingand the Smith et al. (2003) formula; the former becomes time independent whereas the actual evo-lution will concentrate more and more matter insmaller haloes over time. We also see this in Fig-ure 4 which shows the value of both σ and σ asa function of expansion factor. The latter scaleof 1 h − Mpc is nonlinear today and therefore al-ready shows a difference between linear evolutionand the different nonlinear formulae, whereas thecanonical 8 h − Mpc scale is just going nonlineartoday.Fig. 3.— The linear (short-dashed blacklines) and nonlinear values of the dimensionless∆ ( k ) as obtained from the nonlinear ansatz ofPeacock & Dodds (1996, long-dashed red lines)and from Smith et al. (2003, solid blue lines), forexpansion factors a = 1 /
4, 1/2, 1, 2, 4, and 8(where the present is a = 1).These differences arise partly from the factthe various fitting formulae did not attempt toreproduce future clustering, but moreover, fromthe well-known fact that we live at a specialtime: in ΛCDM, linear clustering will “saturate”soon. Hence, any prediction such as that ofPeacock & Dodds (1996) based on a mapping ofthe linear spectrum will also necessarily satu-rate. Moreover, the various densities are cur-rently evolving very rapidly. For example, at a = 1 /
2, Ω m was approximately what Ω Λ is to-day; at a = 2, Ω m will be approximately whatΩ b is today. This also gives us some insight into6ig. 4.— The value of σ (lower set of curves)and σ (upper set of curves) as a function ofexpansion factor, for linear theory (short-dashedblack lines), Peacock & Dodds (1996, long-dashedred lines) and from Smith et al. (2003, solid bluelines).the underlying (rather than just practical) lim-itations of these methods. The mapping pro-posed by Peacock & Dodds (1996) essentially ap-plies equally to all scales. However, in an evolvedΛCDM universe, large–scale dynamics are dom-inated by the accelerated expansion and smallscales by highly nonlinear stable clustering, whichis captured somewhat better in the halo model.
4. Discussion
We will now compare different estimates of σ . CMB . The σ is estimated from the CMBflutuations, observed at z ≈ σ from the CMBdepends on details of the data-analysis proce-dure, in particular on the assumed Bayesianpriors on the cosmological parameters consid-ered. The “recommended” value for a flat ΛCDM model ( http://lambda.gsfc.nasa.gov ) is σ =0 . ± .
04. However, considering different modelsand priors on those parameters can give variationsover 0 . . σ . . Cosmic flows . Observations of peculiar ve-locities of galaxies provide an estimate of σ at the present, nonlinear regime. Masters et al.(2006) estimated σ Ω . m = 0 . ± .
06 using thepeculiar velocities from a sample of clusters inthe SFI++ survey; Zaroubi et al. (2002) usingdensity-density and velocity-velocity comparisonsfound σ Ω . m = 0 . ± .
05 both of which translateto σ &
1. At scales of 100 h − Mpc, Lavaux et al.(2008) found σ = 1 . ± .
56 from 2MASS, al-though they only directly measure the velocityfield within 30 h − Mpc, on which scales they es-timate a somewhat lower fluctuation amplitude.As mentioned above, pairwise velocity analysis(Feldman et al. 2003) estimates a present daynonlinear value of σ = 1 . +0 . − . . Recent bulkflow measurements using the best available pecu-liar velocity surveys (Watkins et al. 2009) requirethe nonlinear σ > .
11 (0 .
88) at a 95% (99%)confidence level. Thus, flow measurements givehigher values of σ than the CMB, as expectedfrom the ideas discussed above. We expect moreprecise measurements of σ from cosmic flows inthe near future. The new surveys are deeper,denser, have better sky coverage (Masters et al.2006; Springob et al. 2007, 2009), and we haveimproved our understanding of the distance in-dicators needed to extract the peculiar veloci-ties (Feldman et al. 2003; Radburn-Smith et al.2004; Pike & Hudson 2005; Sarkar et al. 2007;Watkins & Feldman 2007; Feldman & Watkins2008). Other cosmological probes . For compar-ison, we provide a table, where apart from theabove estimates of σ , we also show the val-ues, derived from Ly α observations (Tytler et al.2004); cluster number density measurements(Vikhlinin et al. 2009); weak lensing measure-ments of cosmic shear (Benjamin et al. 2007);the Sunyaev-Zeldovich effect using the ACBARreceiver (Reichardt et al. 2009), and from thegalaxy clustering power spectrum (Cole et al.2005; Tegmark et al. 2004; Eisenstein et al. 2005).All of these measurements fall in the range0 . < σ < . . < σ < . σ from various estimatorsMethod Parameter valueCMB σ L . ± . α σ L . ± . σ L . ± . σ L . ± . σ . +0 . − . Galaxies σ gal8 . ± . σ . +0 . − . Bulk flow σ > .
11 (0 .
88) at 95% (99%) CL Dunkley et al. (2009); Tytler et al. (2004); Benjamin et al. (2007); Vikhlinin et al. (2009); Reichardt et al. (2009); Cole et al. (2005); Tegmark et al. (2004); Eisenstein et al. (2005); Feldman et al.(2003); Watkins et al. (2009)
5. Conclusions
We have presented a formalism to calculate σ ,the amplitude of cosmological density fluctuationson scales of 8 h − Mpc, that incorporates the non-linear evolution of the parameter. Estimates of σ depend directly on the epoch and scale of thesurveys used. When using deep, high-redshift sur-veys (CMB) that estimate σ in the linear regime,the results suggest systematically lower values of σ . When analyzing shallow, local data (pecu-liar velocities) which are affected by nonlinearities,we get higher σ . The results from other cosmo-logical probes, shown in Table 1, show a similartrend. When results from deep surveys are beingcorrected for this effect, most estimates from vari-ous independent surveys on all scales agree betterwith each other: the systematic differences are re-duced.Quantitatively, our main result here is the re-duction in the value of σ , derived from the ob-served mean pairwise velocity of galaxies. Theoriginal estimate, as we have discussed earlier, is σ = 1 . +0 . − . . (28)After the correction based on the second order per-turbation theory, this becomes σ L = 1 . +0 . − . . (29)Somewhat more cumbersome non-perturbativemethods give identical results, bringing late-time estimates of σ close to high-redsift measurements,which appear in Table 1. Acknowlegements:
We would like to thank theanonymous referee for her/his constructive criticalremarks. RJ was supported by the Polish Min-istry of Science grant NN203 2942 34 and an IN-TAS grant No. 06-1000017-9258. HAF has beensupported in part by a grant from the ResearchCorporation, by an NSF grant AST-0807326 andby the National Science Foundation through Ter-aGrid resources provided by the NCSA. AHJ wassupported by STFC in the UK.
REFERENCES
Bardeen, J. M., Bond, J. R., Kaiser, N., & Szalay,A. S. 1986, ApJ, 304, 15Benjamin, J., et al. 2007, MNRAS, 381, 702Carroll, S. M., Press, W. H., & Turner, E. L. 1992,ARAA, 30, 499Cole, S., et al. 2005, MNRAS, 362, 505Davis, M., & Peebles, P. J. E. 1983, ApJ, 267, 465Dunkley, J., et al. 2009, ApJS, 701, 1804Eisenstein, D. J., & Hu, W. 1998, ApJ, 496, 605Eisenstein, D. J., et al. 2005, ApJ, 633, 560Feldman, H., et al. 2003, ApJ, 596, L1318eldman, H. A., Frieman, J. A., Fry, J. N., & Scoc-cimarro, R. 2001, PRL, 86, 1434Feldman, H. A., & Watkins, R. 2008, MNRAS,387, 825Feldman, H. A., Watkins, R., & Hudson, M. J.2009, ArXiv e-printsHamilton, A. J. S., Kumar, P., Lu, E., &Matthews, A. 1991, ApJ, 374, L1Hamilton, A. J. S., & Tegmark, M. 2002, MNRAS,330, 506Hamilton, A. J. S., & Tegmark, M. 2002, MNRAS,330, 506Heath, D. J. 1977, MNRAS, 179, 351Juszkiewicz, R., Bouchet, F. R., & Colombi, S.1993, ApJ, 412, L9Kaiser, N. 1988, MNRAS, 231, 149Lahav, O., Lilje, P. B., Primack, J. R., & Rees,M. J. 1991, MNRAS, 251, 128Lavaux, G., Tully, R. B., Mohayaee, R., &Colombi, S. 2008, ArXiv e-printsLokas, E., Juszkiewicz, R., Bouchet, F. R., &Hivon, E. 1996, Astrophys. J., 467, 1Masters, K. L., Springob, C. M., Haynes, M. P.,& Giovanelli, R. 2006, ApJ, 653, 861Peacock, J. A., & Dodds, S. J. 1996, MNRAS, 280,L19Peebles, P. J. E. 1980, The large-scale structure ofthe universe (Princeton, N.J., Princeton Uni-versity Press, 1980)—. 1993, Principles of physical cosmology (Prince-ton, N.J., Princeton University Press, 1980)Pike, R. W., & Hudson, M. J. 2005, ApJ, 635, 11Radburn-Smith, D. J., Lucey, J. R., & Hudson,M. J. 2004, MNRAS, 355, 1378Reichardt, C. L., et al. 2009, ApJ, 694, 1200Sarkar, D., Feldman, H. A., & Watkins, R. 2007,MNRAS, 375, 69 Scoccimarro, R., Feldman, H. A., Fry, J., & Frie-man, J. 2001, ApJ, 546, 652Scoccimarro, R., & Frieman, J. A. 1996, ApJS,105, 37Smith, R. E., et al. 2003, MNRAS, 341, 1311Springob, C. M., Masters, K. L., Haynes, M. P.,Giovanelli, R., & Marinoni, C. 2007, ApJS, 172,599—. 2009, ApJS, 182, 474Strauss, M. A., & Willick, J. A. 1995, Phys. Rep.,261, 271Tegmark, M., et al. 2004, ApJ, 606, 702Tytler, D., et al. 2004, ApJ, 617, 1Verde, L., et al. 2002, MNRAS, 335, 432Vikhlinin, A., et al. 2009, ApJ, 692, 1033Watkins, R., & Feldman, H. A. 2007, MNRAS,379, 343Watkins, R., Feldman, H. A., & Hudson, M. J.2009, MNRAS, 392, 743Zaroubi, S., Branchini, E., Hoffman, Y., & daCosta, L. N. 2002, MNRAS, 336, 1234