An Estimation of the Local Growth Rate from Cosmicflows-3 Peculiar

MNRAS 000, 000–000 (0000) Preprint 20 March 2019 Compiled using MNRAS LATEX style ﬁle v3.0

An estimation of the local growth rate from Cosmicﬂows-3 peculiar velocities

Alexandra Dupuy1 ?, Helene M. Courtois1, Bogna Kubik1 1University of Lyon, UCB Lyon 1, CNRS/IN2P3, IPN Lyon, 69622 Villeurbanne, France

20 March 2019

ABSTRACT This article explores three usual estimators, noted as v12 of the pairwise velocity, ψ1 and ψ2 of the observed two-point galaxy peculiar velocity correlation functions. These estimators are tested on mock samples of Cosmicflows-3 dataset (Tully et al. 2016) , derived from a numerical cosmological simulation, and also on a number of constrained realizations of this dataset. Observational measurements errors and cosmic variance are taken into consideration in the study. The result is a local measurement of f σ8 = 0.43 (±0.03)obs (±0.11)cosmic out to z = 0.05, in support of a ΛCDM cosmology. 1 INTRODUCTION in order to constrain the growth rate of large scale structures and related parameters. On the one hand, Hudson Since the late 70’s, several publications discussed the theory 0.55 & Turnbull(2012) measured f σ8 ≡ Ωm σ8 = 0.40 ± 0.07 of galaxy pairwise peculiar velocity statistics, such as the by comparing the observed peculiar velocities of 245 su- 2-point peculiar velocity correlation function (ψ1 and ψ2) or pernovae (extracted from a compilation dubbed the First the mean pairwise velocity (v12; Monin & Yaglom 1975; Amendment, A1) to the galaxy density field predicted by Davis & Peebles 1977; Peebles 1980, 1987; Gorski 1988). It the Point Source Catalogue Redshift Survey (PSCz, Saun- has been shown that such statistics can be measured di- ders et al. 2000). This method has been later applied by rectly from only the radial part of peculiar velocities. Since Carrick et al.(2015) on galaxies from the 2M++ redshift these statistics are related to the growth factor of large scale compilation (Lavaux & Hudson 2011), finding a much more structures f = Ωγ , where γ is the growth index (Lahav et al. m accurate estimate of the growth factor f σ8 = 0.401 ± 0.024. 1991), observed peculiar velocities can be used as cosmolog- On the other hand, Johnson et al.(2014) analyzed the two- ical probes to estimate the matter density parameter Ωm point statistics of the peculiar velocity field and obtained (Ferreira et al. 1999; Juszkiewicz et al. 1999). However, f f σ8 = 0.418 ± 0.065 from a sample gathering peculiar veloc- and σ8, the amplitude of the density fluctuations on 8 Mpc ities of ∼ 9,200 galaxies from the Six Degree Field Galaxy −1 h scales (where h = H0/100 and H0 is the Hubble con- Survey peculiar velocity catalog (6dFGS, Jones et al. 2004, stant), are degenerate and cannot be constrained separately 2006, 2009) and various supernovae distance measurements. when using only galaxy peculiar velocity data. Alternatively, using again the same two-point statistic (v12) The first attempts of constraining cosmological param- as Juszkiewicz et al.(2000) and Feldman et al.(2003), ap- eters such as the density parameter have been made by plied on the Cosmicflows-2 catalog containing ∼8,000 galaxy Peebles(1976), Kaiser(1990) and Hudson(1994). Later, distances (CF2, Tully et al. 2013), Ma et al.(2015) found Juszkiewicz et al.(2000) gave Ωm = 0.35 ± 0.15 with mea- Ω0.6 +0.384 +0.73 surprising results: m h = 0.102−0.044 and σ8 = 0.39−0.1 . surements of the mean pairwise velocity on the Mark III Moreover, by measuring the covariance of radial peculiar catalog of radial peculiar velocities of roughly 3,000 spi- velocities in two catalogs, a sample of 208 low redshift su- ral and elliptical galaxies (Willick et al. 1995, 1996, 1997). pernovae (named SuperCal) and a set of roughly 9,000 pe- arXiv:1901.03530v2 [astro-ph.CO] 19 Mar 2019 Then, Feldman et al.(2003) obtained a very similar value of culiar velocities from 6dFGS, Huterer et al.(2016) evalu- +0.17 +0.22 Ωm = 0.30 and also measured σ8 = 1.13 . This was ated +0.048 at . Finally, Adams & Blake −0.07 −0.23 f σ8 = 0.428−0.045 z = 0.02 done by using the same estimator of Juszkiewicz et al.(2000) (2017) measured +0.067 by modelling the cross- f σ8 = 0.424−0.064 , but on a much larger dataset combining peculiar velocities covariance of the galaxy overdensity and peculiar velocity of approximately 6,400 galaxies extracted from several cat- fields and applying their analysis to the observed peculiar alogs: Mark III, Spiral Field I-Band (Giovanelli et al. 1994, velocities from the 6dFGS data. 1997a,b; Haynes et al. 1999a,b), Nearby Early-type Galaxies Survey (da Costa et al. 2000) and the Revised Flat Galaxy Despite the increase in number of measurements and in Catalog (Karachentsev et al. 2000). redshift coverage, peculiar velocity catalogs are still not large A decade later, datasets improved and methods of an- enough and remain noticeably sparse at large distances. alyzing peculiar velocity to constrain cosmology evolved. Hence, growth rate estimations are affected by uncertainties Some authors proposed new statistical methodologies using introduced by cosmic variance as they are obtained from lo- observed peculiar velocities, which differs from v12 and ψ1,2, cal observations. Hellwing et al.(2016) discussed the effect

© 0000 The Authors 2 Dupuy et al. of the observer location in the universe on the derivation in Gaussian distributed errors on peculiar velocities: of two-point peculiar velocity statistics (v , ψ and ψ ) by 12 1 2 cz considering two sets of randomly chosen observers and Lo- u = cz ln . (1) cal Group -like observers. The authors showed that the local H0d environment, especially the Virgo cluster, systematically in- Equation1 will be used throughout this paper to derive troduces deviations from predictions. radial peculiar velocities from observed distances. More recently, Nusser(2017) measured f σ8 = 0.40±0.08 This article will only focus on the CF3 distances cata- by measuring velocity - density correlations on the largest log. Two radial peculiar velocity samples will be considered: and most recent catalog of ∼ 18,000 accurate galaxy dis- the ungrouped sample and the grouped sample, containing tances, Cosmicflows-3 (CF3, Tully et al. 2016). And, last galaxies and groups of galaxies respectively. Groups are fre- but not least, Wang et al.(2018) analyzed the peculiar ve- quently used by authors√ because they allow to reduce un- locity correlation functions through the estimators ψ1 and certainties with an N improvement on observed distances, ψ2 applied to the Cosmicflows catalogs (CF2 and CF3) to and thus on radial peculiar velocities. These uncertainties constrain cosmological parameters: Ω +0.205 and m = 0.315−0.135 are due to the virial motions of group members. However, +0.440. σ8 = 0.92−0.295 the methodology presented in this article is valid for pairs On the grounds of the previous literature works intro- of galaxies, and not for pairs of groups of galaxies. The CF3 duced above, this article studies two classical two-point pe- grouped catalog is tested in this article since recent studies culiar velocity statistics, using radial peculiar velocities pro- (Ma et al. 2015; Nusser 2017, both introduced above in Sec- vided by the Cosmicflows-3 catalog, to constrain the local tion1) made use of the grouped versions of the Cosmicflows value of the growth rate factor f σ8. Its structure is orga- catalogs to derive f σ8. However, it will be seen in the discus- nized as follows. Section 2 provides details on the peculiar sion that using grouped data to constrain the growth rate velocity data used for the analysis. The methodology of two- leads to incoherent results. point correlation functions of peculiar velocities is described Tully et al.(2016) shows that the most consistent value in Section 3 and is tested and validated on mocks in Sec- of the Hubble constant with CF3 distances when computing −1 −1 tion 4. Section 5 shows the velocity statistics measured on radial peculiar velocities is H0 = 75 ± 2 km s Mpc . This observed peculiar velocities. The main result of this article, value is preferred as it minimizes the monopole term with the estimate of the growth rate from Cosmicflows radial pe- CF3 distances and results in a tiny global radial infall and culiar velocities, is discussed in Section 6. outflow in the peculiar velocity field. A larger value of H0 would give a large overall radial infall towards the position of the observer, while choosing a smaller H0 would yield a large radial outflow (cf. Figure 21 in Tully et al. 2016). For these reasons, in this article the value H0 = 75 km 2 DATA s−1 Mpc−1 is used to compute radial peculiar velocities of 2.1 Observed peculiar velocities: Cosmicflows-3 CF3 galaxies and groups. We note that this high value of H0 is consistent with other values of the Hubble constant The latest CF3 catalog (Tully et al. 2016) provides dis- measured in the local universe. tances for 17,648 galaxies which can be redistributed within 11,936 groups, up to 150 Mpc h−1. It is an expansion of the previous CF2 catalog (Tully et al. 2013). It contains 2.2 Cosmicflows mock catalogs 8,188 galaxy distances with a homogeneous volume coverage up to 80 Mpc h−1, mostly derived with the Tully-Fisher A three-dimensional peculiar velocity field computed with (TF) relation (Tully & Fisher 1977), linking the luminosity the Constrained Realization (CR) methodology (Hoffman & to the HI line width for spiral galaxies, and the Fundamen- Ribak 1991) is considered to construct mock catalogs. Us- tal Plane (FP) relation (Djorgovski & Davis 1987; Dressler ing the CF3 grouped dataset and assuming a ΛCDM cos- 1987) for elliptical galaxies. The main additions to the CF3 mological model (Ωm = 0.3, dark energy density parameter −1 −1 catalog are new distances, obtained with the FP relation ΩΛ = 0.7 and H0 = 70 km s Mpc ), this velocity field is from 6dFGS, and distances computed with the TF relation. composed of a velocity field obtained with the Wiener-Filter About 60 percent of CF3 distances are therefore measured technique (Zaroubi et al. 1995, 1999, WF) and a random with the FP method and mostly located in the Southern ce- component derived from the Random Realizations method. lestial hemisphere, while around 40 percent of distances are The velocity field used in this article is reconstructed in a obtained with the TF relation. Few distance measurements box 2000 Mpc wide, in Cartesian Supergalactic coordinates are obtained from various methods where applicable such as and centered on the Milky Way, with 1283 cells. Cepheids, Tip of the Red Giant Branch, type Ia Supernovae, In order to test the various estimators of the peculiar and surface brightness fluctuations. velocity correlation function, mock catalogs are prepared as From the distance d of a galaxy and its redshift z, it explained hereafter. Radial peculiar velocities, predicted by is possible to derive the radial component of its peculiar the three-dimensional velocity field of a CR, are assigned to velocity, u = cz − H0d, where c is the speed of light in galaxies or groups from the CF3 data. The mock catalogs vacuum, and H0 is the Hubble constant. However, as dis- are prepared with the following method. Considering galax- tance moduli have Gaussian distributed errors, the peculiar ies and groups of the CF3 catalog, their predicted three- velocities computed with this equation have non-Gaussian dimensional peculiar velocities are extracted from the CRs’ (skewed) distributed errors. To solve this problem, Watkins peculiar velocity field at the redshift positions of the galax- & Feldman(2015) introduced a new estimator which results ies. The radial part of the peculiar velocity, which is the

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where uìi and δi correspond respectively to the peculiar velocity and the density contrast at the location of the galaxies i = A, B, and < ... >ρ speciﬁes a pair-weighted average with −1 (1 + δA)(1 + δB)(1 + ξ(r)) as the weighting factor. In the non-linear regime, i.e for pairs of very close galaxies, ξ(r) 1, and the solution of the pair conservation equation is v12(r) = −H0r. In the case of the linear regime, i.e for large separation distances, ξ(r) 1 and the solution of the conservation equation for v12 is then given by the pertur- bative expansion of ξ(r). In order to measure cosmological parameters such as the total matter density parameter Ωm, Juszkiewicz et al.(1999) introduce a solution for v12 valid in both regimes, linear and non-linear, by interpolating the linear and non-linear solutions:

Figure 1. Pair of galaxies considered throughout this article. 2 h i v (r) ≈ − Hr f ξ¯(r) 1 + αξ¯(r) , (3) 12 3

¯( ) ¯( )/[ ( )] ¯( ) −3 ∫ r ( ) 2 only observable component, is then derived from the three- where ξ r = ξ r 1 + ξ r and ξ r = 3r 0 ξ x x dx is the dimensional velocity. two-point correlation function averaged in a sphere of radius Throughout this article, parameters that are not fixed r. The parameter α = 1.2−0.65γ depends on the logarithmic by the cosmology of the CR are set to their Planck 2015 slope of ξ(r) denoted by the quantity 0 < γ < 3 given by: values (Planck Collaboration et al. 2016, ΩΛ = 0.69, Ωm = 0.31, σ = 0.82). 8 d ln ξ(r) γ = − . (4) d ln r ξ=1 3 METHODOLOGY From the approximate solution for v12 (equation3), it We consider in this article a pair of galaxies A and B lo- is possible to recover the linear solution if ξ → 0, and the cated at the positions rìA and rìB respectively. The spatial solution valid in the non-linear regime if x → 0. Equation3 separation of these two galaxies is given by rì = rìA − rìB. has been tested and validated by Juszkiewicz et al.(1999) Their peculiar velocities are vìA and vìB and their radial on N-body simulations for 0.1 < ξ(r) < 1000. ˆ ˆ ˆ components are given by uìA = uArìA = vìA · rìA rìA and ˆ ˆ ˆ ˆ uìB = uBrìB = vìB · rìB rìB, where rìA,B are the unit direction vectors of the galaxies. The cosines of the angles between the ˆ ˆ ˆ ˆ different directions are given by cos θ A = rìA · rì, cos θB = rìB · rì 3.1.2 Estimator and cos θ = rìˆ · rìˆ . Figure1 illustrates the geometry and AB A B Equation3 shows that the amplitude of the mean pairwise quantities defined above. velocity v is related to the growth rate f , as shown in To partially avoid the Malmquist bias, the galaxies (or 12 the left panel of Figure2. The statistic v can therefore groups) are located at their redshift positions, as position- 12 be used to constrain this parameter. However, observations ing objects at their observed distance leads to much larger give access only to the radial part of the peculiar velocities errors, especially for the most distant ones. of galaxies. Therefore, one cannot use equation3 to compute v12 directly from observed data. An estimator that can 3.1 Mean pairwise velocity be used to compute v12(r) directly from observed radial peculiar velocities has to be considered. Ferreira et al.(1999) 3.1.1 Model introduced such an estimator to determine the mean pairwise velocity directly for observational catalogs of peculiar The mean pairwise velocity v was introduced for the 12 velocities: first time in the context of the Bogolyubov-Born-Green- Kirkwood-Yvon theory (BBGKY, Yvon 1935; N. N. Bogoli- ubov 1946; Kirkwood 1946, 1947; Born & Green 1946). In Í 2 (uA − uB)(cos θ A + cos θB) this theory, the conservation equation of pairs of galaxies v12(r) = , (5) Í (cos θ + cos θ )2 links the two-point correlation function ξ(r) to the growth A B rate of large scale structures f and the mean pairwise veloc- where the sums are computed for all pairs separated by a ì/ ity v12r r (Davis & Peebles 1977; Peebles 1980). For a pair distance r. of galaxies separated by a distance r, the mean pairwise ve- A new estimator which relies on the transverse com- locity is given by (Juszkiewicz et al. 1998): ponent of peculiar velocities (instead of the radial one) has been introduced by Yasini et al.(2018). This estimator will ì − ì ( )( ) D Ê uA uB 1 + δA 1 + δB allow to analyze pairwise velocities derived from upcoming v12(r) = uìA − uìB · rì = , (2) ρ 1 + ξ(r) transverse peculiar velocity surveys such as Gaia (Hall 2018).

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Figure 2. ΛCDM models of the three usual peculiar velocity statistics v12 (left), ψ1 (middle) and ψ2 (right) as a function of the pair separation r. The three statistics are computed for several values of f σ8 (see colobar).

3.2 Velocity correlation function and ψ2 as: In the linear regime, the two-point correlation tensor of a homogeneous and random velocity field vì(rì) is defined as Í ˆ ˆ uìA · rì uìB · rì Í (Monin & Yaglom 1975; Strauss & Willick 1995): uAuB cos θ A cos θB ψ2(r) = = Í , Í ˆ ˆ ˆ ˆ ˆ ˆ cos θ AB cos θ A cos θB rìA · rìB rìA · rì rìB · rì Ψij (r) ≡ vi(rìA)vj (rìB) = Ψk(r) − Ψ⊥(r) rîrˆj + Ψ⊥(r)δij, (6) (11) where i and j are the Cartesian coordinates. The quantities The sums in equations 10 and 11 are performed over all Ψk(r) and Ψ⊥(r) are the radial (i.e along rì) and transverse pairs with fixed separation r. The denominators normalize velocity correlation functions respectively. The spectral rep- the sums in order to preserve the norm of the velocity field. resentations of Ψk(r) and Ψ⊥(r) are given by (Gorski 1988): 3.2.2 Model 2( )2 ¹ H0 f σ8 j1(kr) Ψk(r) = P0(k) j0(kr) − 2 dk (7) The correlation function of radial peculiar velocities can be 2π2 kr derived from the two-point velocity correlation tensor: and

u (rì )u (rì ) = rˆ rˆ Ψ (r)rˆ rˆ , (12) 2( )2 ¹ m A n B Am Bn ij Ai Bj H0 f σ8 j1(kr) Ψ⊥(r) = P0(k) dk (8) where i, j, m and n are the Cartesian coordinates. Inserting 2π2 kr equation 12 into equations 10 and 11, the quantities ψ1(r) where j (x) and j (x) are the spherical Bessel functions of 0 1 and ψ2(r) can be written as functions of Ψk(r) and Ψ⊥(r) the ﬁrst kind: (Gorski et al. 1989):

sin x sin x cos x j0(x) = , j1(x) = − , (9) ψ (r) = A (r)Ψ (r) + 1 − A (r) Ψ (r), (13) x x2 x 1,2 1,2 k 1,2 ⊥ and P0(k) is the non-normalized linear matter power spec- where trum measured today. In the rest of this article P0(k) is computed with CAMB in the Planck 2015 cosmology. Í cos θ cos θ cos θ A (r) = A B AB , (14) The quantities Ψ (r) and Ψ (r) both depend on the pa- 1 Í 2 k ⊥ cos θ AB ( )2 rameter f σ8 . These correlation functions can therefore and be used to constraint the combined cosmological parameter f σ , deﬁned as the normalized growth rate of large scale 8 Í 2 2 structures. cos θ A cos θB A2(r) = Í . (15) cos θ A cos θB cos θ AB The functions A ( ) and A ( ) contain information 3.2.1 Estimator 1 r 2 r about the geometry of the sample, and measure the contri- Gorski et al.(1989) introduced two velocity statistics, noted butions of Ψk(r) and Ψ⊥(r) to the functions ψ1(r) and ψ2(r). here as ψ1 and ψ2, which depend only on radial peculiar As one can see on the middle and right panels of Figure velocities. The statistic ψ1 is deﬁned as: 2, the amplitude of the two statistics ψ1 and ψ2 depends on the growth factor f σ8, allowing to constrain this cosmolog- Í uì · uì Í u u cos θ ical parameter. The higher f σ8 is, the higher the amplitude ψ (r) = A B = A B AB , (10) 1 2 Í 2 of ψ1 or ψ2 is: a universe with a large f σ8 appears more Í ˆ ˆ cos θ AB rìA · rìB compact and peculiar velocities get larger. However, in the

MNRAS 000, 000–000 (0000) Growth rate 5 case of ψ2, the curves corresponding to different f σ8 get Results obtained from these tests are shown in section closer and closer as r increases (see right panel of Figure 4. 2). It is difficult to constrain cosmological models for sepa- As different bins of separation distances share the same ration distances higher than 60 Mpc/h. The statistic ψ2 is galaxies (or groups of galaxies), errors between bins are cor- therefore not robust enough to estimate the growth rate on related and thus the covariance matrix needs to be consid- peculiar velocity catalogs such as CF3, and especially on the ered when fitting the measured statistics to extract the nor- upcoming big surveys. Therefore, this statistic will not be malized growth rate f σ8. considered for the rest of this paper. From these Monte-Carlo realizations or cosmic variance mocks, the covariance matrix C between bins of separation distances of galaxy pairs can be computed. The covariance 3.3 Observational errors and cosmic variance between bins rm and rn is computed as: N Two kinds of uncertainties on peculiar velocity statistics are 1 Õk considered in this article: measurement (or observational) Cmn = Sk (rm) − Sm Sk (rn) − Sn (16) N − 1 error and cosmic variance. k k=1 Observational errors on peculiar velocity statistics v12, where S denotes the statistic considered and Sm,n = ψ1 and ψ2, i.e errors due to the uncertainty in distance mea- 1 ÍNk Sk (rm,n) is defined as the mean of all realizations surement, are derived by Monte-Carlo synthetic realizations. Nk k=1 for the bins r and r respectively and N = 100 is the num- These realizations are constructed by adding a random error m n k ber of mocks (realizations). to the radial peculiar velocity. The random error is extracted from a normal distribution with a standard deviation equal to the measurement error on the peculiar velocity. This measurement error is derived from the uncertainty on the dis- 4 DERIVING THE LOCAL GROWTH RATE tance (or distance modulus). In this article, 100 realizations 4.1 Verification of estimators on mocks have been computed for each sample (mocks and observed data). The normalized growth rate f σ8 is estimated by fitting the In addition to measurements uncertainties, one ought theoretical models of the statistics v12 and ψ1, noted Smod to take into account the impact of cosmic variance when es- and computed with equation3 and 13, to the quantity Smeas timating cosmological parameters. However, computing un- measured with the radial estimators of the two statistics certainties caused by cosmic variance cannot be done on defined in equations5 and 10 respectively. The value of the a Constrained Realization of CF3, described in section2, growth rate f σ8 is obtained by minimizing the following chi- which represents our local universe. Therefore, a ΛCDM square function: dark matter only N-body simulation is considered in this paper in order to estimate the impact of the cosmic variance on Nbins peculiar velocity statistics and growth rate measurements. 2 Õ h −1 χ ( f σ8) = Smeas(ri) − Smod(ri; f σ8) Cij Two tests, whose results are presented later in this paper, i, j=0 (17) on dark matter halos extracted from the MultiDark Planck i 2 simulation (MDPL2, Prada et al. 2012) of size 1 Gpc h−1 Smeas(rj ) − Smod(rj ; f σ8) are carried out. The underlying cosmology of the simulation is the Planck 2015 cosmology. Only mock galaxies with halo The minimization is conducted with MINUIT (Function Min- 11 12 imization and Error Analysis software, James & Roos 1975). mass between 10 M and 10 M are taken into account to construct these mocks. Data samples for the tests are The error on the fitted parameter is also given by MINUIT as prepared as follows: the second derivative of the chi-square. Peculiar velocity statistics v12 and ψ1 have been mea- • as the CF3 catalog can (mostly) be contained in a sured on mock and observed radial peculiar velocities. sphere of a 250 Mpc h−1 radius, one can place 8 of such Throughout this article, statistics are computed out to a dis- independent spheres in a cube of side 1 Gpc h−1. There- tance of 100 Mpc h−1 in 20 equal bins of 5 Mpc h−1. In all fore 8 CF3-like samples are generated for each octant cube figures of this article displaying velocity statistics computed of the simulation. The observer of each sample is placed at on mocks or observed data, scattered points are located at the center of its associated octant. Then radial components the middle of the bins. of peculiar velocities are extracted at the position of CF3 The statistics v12 and ψ1 have been tested and validated galaxies with respect to the observer. These mocks are com- by the authors who introduced them. They allow one to re- pletely independent from each other as they do not share cover the underlying cosmology from a homogeneous and any halos. spherical universe. But CF3 is very sparse and asymmet- • As only 8 samples is not high enough to get a robust rical. Before constraining the growth rate, one must check result, a total of 100 more mocks are generated. Instead of if the spatial distribution of the CF3 catalog alone inhibits positioning observers such that samples do not share halos, such statistics from accurately recovering the underlying cos- a total of 100 observers are placed at a random locations mology. This is done on the 100 CF3 mocks generated from within the simulation box. Then radial components of pe- a Constrained Realization as described in section2. culiar velocities are extracted at the position of CF3 galax- Figure3 shows as solid lines the ΛCDM models for ies with respect to the observers. In this case, the samples galaxies in the CF3 ungrouped (red) and CF3 grouped (blue) are not independent as a single halo can belong to several mocks. For the statistic v12, errors bars of the two mocks do spheres, so results will be correlated. not include the ΛCDM model. This means that due to the

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Figure 3. The two peculiar velocity statistics v12 (left) and ψ1 (right) as a function of the pair separation r. The ΛCDM model (Ωm = 0.3, γ = 0.55, σ8 = 0.82) is shown as a black solid line. Scattered points with error bars represent results obtained from mock peculiar velocities constructed from a ΛCDM constrained realization of CF3. The CF3 ungrouped and grouped mocks are shown as red triangles and blue squares respectively. Vertical dashed lines show the region where f σ8 can be ﬁtted, see the limitations described in the text.

unique CF3 geometry and selection function, this estimator can not recover the underlying ΛCDM cosmology. There- fore we stress that it cannot be used to estimate to local growth rate with the real CF3 catalog, and will not be considered in the analysis that follows. Also, this explains why Ma et al.(2015) obtained incoherent values for Ωm and σ8 by applying v12 on the CF2 dataset, whose footprint is sim- ilarly inhomogeneous. For the estimator ψ1, the amplitude of the CF3 mocks (red and blue dots with error bars for the ungrouped and grouped samples respectively) is slightly lower but error bars are consistent with the models up to 60 Mpc h−1. This shows that the geometry and the sparseness of the current survey prevent from deriving any growth rate for separation distances larger than 60 Mpc h−1. Furthermore, one can see in Figure3 that for both statistics the overall differences between the CF3 grouped and ungrouped samples are very small. This shows that non- linearity does not have any impact on the ψ1 estimator ex- cept for small separations bins which contain galaxies close Figure 4. Peculiar velocity statistic ψ1 as a function of the pair to each other (i.e within clusters). When fitting f σ8 to this separation r computed in the large MDPL2 simulation box. The statistic, the effect of non-linearity will not be taken into ac- simulation’s underlying ΛCDM model is shown as a black solid count: bins corresponding to separations lower than 20 Mpc line. Red scattered triangles with error bars represent results ob- h−1 will be omitted. tained from the 8 CF3-like independent mocks. Green scattered Tests on mocks reported in Figure3 show that the un- triangles with error bars represent results obtained from the 100 CF3-like mocks constructed with randomly positioned observers. derlying value of f σ8 in the CR is recovered by the estimator ψ in an interval of robustness 20 – 60 Mpc/h. However, Vertical dashed lines show the region where where f σ8 can be 1 fitted, see the limitations described in the text. considering the depth of the CF3 catalog and the size of the constrained realization, the measured value of the growth rate with CF3 datasets gives only its local value. Due to cosmic variance, this local value may not represent the global value of the growth rate of large scale structures of the entire tained with the 100 CF3-like mocks generated considering universe. random observers in the MDPL simulation is shown in Fig- Figure4 shows peculiar velocity statistics obtained with ure4 as green triangles with error bars. The underlying cos- the 8 independent mocks extracted from the MDPL2 simu- mology of the simulation is once again well recovered. This lation (see section 3.3) as red triangles with error bars. The shows that despite its depth, the CF3 catalog may allow underlying cosmology of the simulation, shown by the black measuring the growth rate of large scale structures. Never- line, is well recovered by the estimator ψ1(r). Results ob- theless, errors bars due to cosmic variance and the extent of

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Table 1. Constraints of the normalized growth rate f σ8 obtained

from measurements of the peculiar velocity statistic ψ1, within 0 its interval of robustness as described in the text, on the Cos- micﬂows datasets of observed peculiar velocities. The two last ∆ obs ∆ cosmic columns f σ8 and f σ8 correspond to observational and cosmic variance uncertainties on the constrained parameter respectively.

obs cosmic ∆ ∆ CF3 sample — statistic f σ8 f σ8 f σ8 CF3 ungrouped 0.43 ± 0.03 ± 0.11 CF3 grouped 0.45 ± 0.04 ± 0.11 00

0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0

Figure 5. Observed peculiar velocity statistic ψ1 as a function of the pair separation r. The ΛCDM model, f σ = 0.4, pre- 8 0 dicted for CF3 galaxies is shown as a black solid line. Pecu- liar velocity statistics predicted for other cosmological models, f σ8 = 0.2, 0.3, 0.5, 0.6, are shown as black dashed and dotted lines. 00 Scattered points with error bars represent results obtained from observed peculiar velocities. The ungrouped sample of CF3 is 0 shown as red triangles and the grouped CF3 sample is shown as blue squares. Vertical dashed lines show the region where f σ8 can be ﬁtted, see the limitations described in the text.

0 0 0 0 0 the sample are large and cannot be ignored. They must be taken into account when constraining f σ8 with CF3 data.

Figure 6. Observed peculiar velocity statistic ψ1 as a function 4.2 Estimation of f σ8 on data of the pair separation r. The ΛCDM model, f σ8 = 0.4, predicted for CF3 galaxies is shown as a black solid line. Scattered points Figure5 shows the results of the computation of ψ on the 1 with error bars represent results obtained from observed peculiar CF3 observations, for the ungrouped and grouped samples velocities of the CF3 galaxies, derived by considering different val- represented respectively as red triangles and blue squares. ues of the Hubble constant H0 = [66, 68, 70, 72, 73, 74, 75, 76, 77, 79] Errors bars include both, observational and cosmic variance km/s/Mpc. uncertainties. Predictions of statistic ψ1 for different cosmological models are shown as solid (ΛCDM, f σ8 = 0.4) and dotted ( f σ8 = 0.2, 0.3, 0.5, 0.6) lines. The two estimators obtained for CF3 peculiar velocities are consistent with cosmic variance uncertainties on the constrained growth rate ΛCDM within their interval of robustness, as predicted by are reported. tests on mocks shown in Figure3. Besides, one can observe Results shown in Figure5 and Table1 are obtained that results obtained from CF3 groups have a slightly larger using the radial peculiar velocities computed from CF3 dis- −1 −1 amplitude than results derived from CF3 galaxies. tances with equation1 and H0 = 75 km s Mpc . However, For completeness, ψ1 has also been applied to the pre- Tully et al.(2016) suggested that the range H0 = 75 ± 2 km vious CF2 catalog (not shown). The results are similar to s−1 Mpc−1 gives a reasonable value for the monopole flow. CF3 and also in agreement with ΛCDM. This uncertainty on the Hubble constant may affect the un- The data in Figure5 are fitted by minimizing the χ2 certainty on the growth rate measurements, as the amplitude defined in equation 17. The fits are done within the interval of ψ1 depends on the chosen value of H0 when deriving ra- [20, 60] Mpc h−1 of pair separation distances. The results dial peculiar velocities, as shown in Figure6. Therefore, we are displayed in Table1 which shows the values of f σ8 fit- considered other values of H0 within the error range given in −1 −1 ted directly from observational data. Both observational and Tully et al.(2016). Taking H0 = 73 km s Mpc to com-

MNRAS 000, 000–000 (0000) 8 Dupuy et al. pute peculiar velocity of CF3 galaxies gives f σ8 = 0.58±0.03, ACKNOWLEDGMENTS and taking H = 77 km s−1 Mpc−1 gives f σ = 0.54 ± 0.03. 0 8 Noam Libeskind, Carlo Schimd, Romain Graziani, Yannick Wang et al.(2018) show a similar study in the Fig7 of Copin and Mickael Rigault are gratefully thanked for scien- their paper. They published error bars that are twice smaller tific discussions. Support has been provided by the Institut than the ones we derived for ψ (see Fig8 of Wang et al. 1 Universitaire de France and the CNES. (2018)). The CosmoSim database used in this article is a service by Figure7 shows the normalized growth rate f σ as a 8 the Leibniz-Institute for Astrophysics Potsdam (AIP). The function of redshift z. The black solid line corresponds to the MultiDark database was developed in cooperation with the ΛCDM model (γ = 0.55). Other cosmological models (modi- Spanish MultiDark Consolider Project CSD2009-00064. fied gravity, γ , 0.55) are represented by dotted and dashed black lines. Scattered points with error bars show results taken from literature summarized in Table2. Colors depend on the cosmological probe used to constraint the growth REFERENCES rate: SN Ia in yellow (Turnbull et al. 2012), galaxy peculiar velocities in blue (Johnson et al. 2014), Baryon Acoustic Os- Adams C., Blake C., 2017, MNRAS, 471, 839 cillations in green (Blake et al. 2011; Reid et al. 2012) and Amendola L., et al., 2016, preprint, (arXiv:1606.00180) Beutler F., et al., 2012, MNRAS, 423, 3430 Redshift Space Distortions in pink (Hawkins et al. 2003; Blake C., et al., 2011, MNRAS, 418, 1707 Samushia et al. 2012; Beutler et al. 2012; de la Torre et al. Born M., Green H. S., 1946, Proceedings of the Royal Society of 2013). Growth rate measurements predictions for future cos- London Series A, 188, 10 mological surveys are shown in red, with a predicted error Carrick J., Turnbull S. J., Lavaux G., Hudson M. J., 2015, MN- bar of 10 per cent (McConnachie et al. 2016; Amendola et al. RAS, 450, 317 2016; da Cunha et al. 2017). Davis M., Peebles P. J. E., 1977, ApJS, 34, 425 Local growth rate constraint obtained from radial pecu- Djorgovski S., Davis M., 1987, ApJ, 313, 59 liar velocities of CF3 galaxies is shown in Figure7 as a blue Dressler A., 1987, ApJ, 317, 1 point with error bars. Errors bars include the observational Feldman H., et al., 2003, ApJ, 596, L131 uncertainties only, in coherence with other data points. Ferreira P. G., Juszkiewicz R., Feldman H. A., Davis M., Jaffe A. H., 1999, ApJ, 515, L1 Giovanelli R., Haynes M. P., Salzer J. J., Wegner G., da Costa L. N., Freudling W., 1994, AJ, 107, 2036 Giovanelli R., Haynes M. P., Herter T., Vogt N. P., Wegner G., Salzer J. J., da Costa L. 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N., finding f σ8 = 0.488. Freudling W., Salzer J. J., Wegner G., 1999b, AJ, 117, 2039 Currently, as seen in Figure7 some local cosmological Hellwing W. A., Nusser A., Feix M., Bilicki M., 2016, preprint, probes like BAO and RSD do not have error bars that could (arXiv:1609.07120) constrain the growth rate of large scale structures. Even Hoffman Y., Ribak E., 1991, ApJ, 380, L5 large redshift surveys such as SDSS, VIPERS or WiggleZ Hudson M. J., 1994, MNRAS, 266, 475 Hudson M. J., Turnbull S. J., 2012, ApJ, 751, L30 have a too large observational error budget to discriminate Huterer D., Shafer D., Scolnic D., Schmidt F., 2016, preprint, a cosmological model. In contrast, one can see that using (arXiv:1611.09862) only a few thousands of peculiar velocities of galaxies as James F., Roos M., 1975, Computer Physics Communications, cosmological probes to measure f σ8 allows to restrict the 10, 343 range of cosmological models. 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MNRAS 000, 000–000 (0000) Growth rate 9

Table 2. Measurements of normalized growth rate f σ8 at redshift z taken from literature. These constraints are obtained by using various cosmological probes: peculiar velocities (Vpec), Type Ia Supernovae (SN Ia), Redshift Space Distortions (RSD) and Baryon Acoustic Oscillations (BAO).

Redshift z Normalized growth rate f σ8 Publication Cosmological probe +0.079 6dFGS 0.05 0.428−0.068 Johnson et al.(2014) Vpec A1 0.03 0.40 ± 0.07 Turnbull et al.(2012) SN Ia

2dFGRS 0.20 0.46 ± 0.07 Hawkins et al.(2003) RSD

6dFGRS 0.067 0.423 ± 0.055 Beutler et al.(2012) RSD

WiggleZ 0.22 0.42 ± 0.07 Blake et al.(2011) BAO 0.41 0.45 ± 0.04 0.60 0.43 ± 0.04 0.78 0.38 ± 0.04

SDSS-LRG 0.25 0.3512 ± 0.0583 Samushia et al.(2012) RSD 0.37 0.4602 ± 0.0378

BOSS 0.57 0.451 ± 0.025 Reid et al.(2012) BAO

VIPERS 0.80 0.47 ± 0.08 de la Torre et al.(2013) RSD

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0 0 0

00 0 0 0 0 0 .

Figure 7. Normalized growth rate f σ8 as a function of the redshift z. The ΛCDM model (γ = 0.55) is represented by a black solid line. Dotted and dashed black lines correspond to other cosmological models. Results taken from literature and displayed in Table2 are shown by scattered points with error bars. Colors depend on the cosmological probe used to constraint the growth rate: SN Ia in yellow, galaxy peculiar velocities in light blue, Baryon Acoustic Oscillations in green and Redshift Space Distortions in pink. Growth rate measurements predictions for future cosmological surveys are shown in red. Result of this article obtained from observed peculiar velocities of CF3 galaxies is represented by a blue point with error bars. Errors bars include observational uncertainties only.

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