https://orcid.org/0000-0001-9005-2792
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 data set (Tully, Courtois & Sorce), derived from a numerical cosmological simulation, and also on a number of constrained realizations of this data set. Observational measurements errors and cosmic variance are taken into consideration in the study. The result is a local measurement of fσ8 = 0.43(± 0.03)obs(± 0.11)cosmic out to z = 0.05, in support of a ΛCDM cosmology.
1 INTRODUCTION
Since the late 70’s, several publications discussed the theory of galaxy pairwise peculiar velocity statistics, such as the two-point peculiar velocity correlation function (ψ1 and ψ2) or the mean pairwise velocity (v12; Monin & Yaglom 1975; Davis & Peebles 1977; Peebles 1980, 1987; Gorski 1988). It has been shown that such statistics can be measured directly from only the radial part of peculiar velocities. Since these statistics are related to the growth factor of large-scale structures |$f=\Omega _m^\gamma$|, where γ is the growth index (Lahav et al. 1991), observed peculiar velocities can be used as cosmological probes to estimate the matter density parameter Ωm (Ferreira et al. 1999; Juszkiewicz, Springel & Durrer 1999). However, f and σ8, the amplitude of the density fluctuations on 8 Mpc h−1 scales (where h = H0/100 and H0 is the Hubble constant), are degenerate and cannot be constrained separately when using only galaxy peculiar velocity data.
The first attempts of constraining cosmological parameters such as the density parameter have been made by Peebles (1976), Kaiser (1990), and Hudson (1994). Later, Juszkiewicz et al. (2000) gave Ωm = 0.35 ± 0.15 with measurements of the mean pairwise velocity on the Mark III catalogue of radial peculiar velocities of roughly 3000 spiral and elliptical galaxies (Willick et al. 1995, 1996, 1997). Then, Feldman et al. (2003) obtained a very similar value of |$\Omega _{\rm m} = 0.30 ^{+ 0.17} _{- 0.07}$| and also measured |$\sigma _8 = 1.13 ^{+0.22} _{- 0.23}$|. This was done by using the same estimator of Juszkiewicz et al. (2000), but on a much larger data set combining peculiar velocities of approximately 6400 galaxies extracted from several catalogues: Mark III, Spiral Field I-Band (Giovanelli et al. 1994, 1997a,b; Haynes et al. 1999a,b), Nearby Early-type Galaxies Survey (da Costa et al. 2000), and the Revised Flat Galaxy Catalog (Karachentsev et al. 2000).
A decade later, data sets improved and methods of analysing peculiar velocity to constrain cosmology evolved. Some authors proposed new statistical methodologies using observed peculiar velocities, which differs from v12 and ψ1, 2, in order to constrain the growth rate of large-scale structures and related parameters. On the one hand, Hudson & Turnbull (2012) measured |$f\sigma _8 \equiv \Omega _{\rm m}^{0.55} \sigma _8 = 0.40 \pm 0.07$| by comparing the observed peculiar velocities of 245 supernovae (extracted from a compilation dubbed the First Amendment, A1) to the galaxy density field predicted by the Point Source Catalogue Redshift Survey (PSCz; Saunders et al. 2000). This method has been later applied by Carrick et al. (2015) on galaxies from the 2M++ redshift compilation (Lavaux & Hudson 2011), finding a much more accurate estimate of the growth factor fσ8 = 0.401 ± 0.024. On the other hand, Johnson et al. (2014) analysed the two-point statistics of the peculiar velocity field and obtained fσ8 = 0.418 ± 0.065 from a sample gathering peculiar velocities of 9200 galaxies from the Six Degree Field Galaxy Survey peculiar velocity catalogue (6dFGS, Jones et al. 2004, 2006, 2009) and various supernovae distance measurements. Alternatively, using again the same two-point statistic (v12) as Juszkiewicz et al. (2000) and Feldman et al. (2003), applied on the Cosmicflows-2 catalogue containing 8000 galaxy distances (CF2; Tully et al. 2013), Ma, Li & He (2015) found surprising results: |$\Omega _{\rm m}^{0.6} h = 0.102 ^{+ 0.384} _{- 0.044}$| and |$\sigma _8 = 0.39 ^{+0.73} _{- 0.1}$|. Moreover, by measuring the covariance of radial peculiar velocities in two catalogues, a sample of 208 low-redshift supernovae (named SuperCal) and a set of roughly 9000 peculiar velocities from 6dFGS, Huterer et al. (2016) evaluated |$f \sigma _8 = 0.428 ^{+0.048} _{-0.045}$| at z = 0.02. Finally, Adams & Blake (2017) measured |$f\sigma _8 = 0.424 ^{+0.067} _{-0.064}$| by modelling the cross-covariance of the galaxy overdensity and peculiar velocity fields and applying their analysis to the observed peculiar velocities from the 6dFGS data.
Despite the increase in number of measurements and in redshift coverage, peculiar velocity catalogues are still not large enough and remain noticeably sparse at large distances. Hence, growth rate estimations are affected by uncertainties introduced by cosmic variance as they are obtained from local observations. Hellwing et al. (2016) discussed the effect of the observer location in the Universe on the derivation of two-point peculiar velocity statistics (v12, ψ1 and ψ2) by considering two sets of randomly chosen observers and Local Group-like observers. The authors showed that the local environment, especially the Virgo cluster, systematically introduces deviations from predictions.
More recently, Nusser (2017) measured fσ8 = 0.40 ± 0.08 by measuring velocity–density correlations on the largest and most recent catalogue of 18 000 accurate galaxy distances, Cosmicflows-3 (CF3, Tully, Courtois & Sorce 2016). And, last but not least, Wang et al. (2018) analysed the peculiar velocity correlation functions through the estimators ψ1 and ψ2 applied to the Cosmicflows catalogues (CF2 and CF3) to constrain cosmological parameters: |$\Omega _{\rm m} = 0.315^{+0.205}_{-0.135}$| and |$\sigma _8 = 0.92^{+0.440}_{-0.295}$|.
On the grounds of the previous literature works introduced above, this article studies two classical two-point peculiar velocity statistics, using radial peculiar velocities provided by the Cosmicflows-3 catalogue, to constrain the local value of the growth rate factor fσ8. Its structure is organized as follows. Section 2 provides details on the peculiar velocity data used for the analysis. The methodology of two-point correlation functions of peculiar velocities is described in Section 3 and is tested and validated on mocks in Section 4. Section 5 shows the velocity statistics measured on observed peculiar velocities. The main result of this article, the estimate of the growth rate from Cosmicflows radial peculiar velocities, is discussed in Section 6.
2 DATA
2.1 Observed peculiar velocities: Cosmicflows-3
The latest CF3 catalogue (Tully et al. 2016) provides distances 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 catalogue (Tully et al. 2013). It contains 8188 galaxy distances with a homogeneous volume coverage up to 80 Mpc h−1, mostly derived with the Tully–Fisher (TF) relation (Tully & Fisher 1977), linking the luminosity to the H i line width for spiral galaxies, and the fundamental plane (FP) relation (Djorgovski & Davis 1987; Dressler 1987) for elliptical galaxies. The main additions to the CF3 catalogue are new distances, obtained with the FP relation from 6dFGS, and distances computed with the TF relation. About 60 per cent of CF3 distances are therefore measured with the FP method and mostly located in the Southern celestial hemisphere, while around 40 per cent of distances are obtained with the TF relation. Few distance measurements are obtained from various methods where applicable such as Cepheids, Tip of the Red Giant Branch, type Ia Supernovae, and surface brightness fluctuations.
Equation (1) will be used throughout this paper to derive radial peculiar velocities from observed distances.
This article will only focus on the CF3 distances catalogue. Two radial peculiar velocity samples will be considered: the ungrouped sample and the grouped sample, containing galaxies and groups of galaxies, respectively. Groups are frequently used by authors because they allow to reduce uncertainties with an |$\sqrt{N}$| improvement on observed distances, and thus on radial peculiar velocities. These uncertainties are due to the virial motions of group members. However, the methodology presented in this article is valid for pairs of galaxies, and not for pairs of groups of galaxies. The CF3 grouped catalogue is tested in this article since recent studies (both introduced above in Section 1 Ma et al. 2015; Nusser 2017) made use of the grouped versions of the Cosmicflows catalogues to derive fσ8. However, it will be seen in the discussion that using grouped data to constrain the growth rate leads to incoherent results.
Tully et al. (2016) shows that the most consistent value of the Hubble constant with CF3 distances when computing radial peculiar velocities is H0 = 75 ± 2 km s−1 Mpc−1. This value is preferred as it minimizes the monopole term with CF3 distances and results in a tiny global radial infall and 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. fig. 21 in Tully et al. 2016).
For these reasons, in this article the value H0 = 75 km s−1 Mpc−1 is used to compute radial peculiar velocities of CF3 galaxies and groups. We note that this high value of H0 is consistent with other values of the Hubble constant measured in the local Universe.
2.2 Cosmicflows mock catalogues
A three-dimensional peculiar velocity field computed with the constrained realization (CR) methodology (Hoffman & Ribak 1991) is considered to construct mock catalogues. Using the CF3 grouped data set and assuming a ΛCDM cosmological model (Ωm = 0.3, dark energy density parameter |$\Omega _\Lambda =0.7$| and H0 = 70 km s−1 Mpc−1), this velocity field is composed of a velocity field obtained with the Wiener–Filter technique (WF; Zaroubi et al. 1995; Zaroubi, Hoffman & Dekel 1999) and a random component derived from the random realizations method. The velocity field used in this article is reconstructed in a box 2000 Mpc wide, in Cartesian Supergalactic coordinates and centred on the Milky Way, with 1283 cells.
In order to test the various estimators of the peculiar velocity correlation function, mock catalogues are prepared as explained hereafter. Radial peculiar velocities, predicted by the three-dimensional velocity field of a CR, are assigned to galaxies or groups from the CF3 data. The mock catalogues are prepared with the following method. Considering galaxies and groups of the CF3 catalogue, their predicted three-dimensional peculiar velocities are extracted from the CRs’ peculiar velocity field at the redshift positions of the galaxies. The radial part of the peculiar velocity, which is the only observable component, is then derived from the three-dimensional velocity.
Throughout this article, parameters that are not fixed by the cosmology of the CR are set to their Planck 2015 values (|$\Omega _\Lambda = 0.69$|, Ωm = 0.31, σ8 = 0.82; Planck Collaboration XIII 2016).
3 METHODOLOGY
We consider in this article a pair of galaxies A and B located at the positions |$\vec{r}_{\rm A}$| and |$\vec{r}_{\rm B}$|, respectively. The spatial separation of these two galaxies is given by |$\vec{r} = \vec{r}_{\rm A} - \vec{r}_{\rm B}$|. Their peculiar velocities are |$\vec{v}_{\rm A}$| and |$\vec{v}_{\rm B}$| and their radial components are given by |$\vec{u}_{\rm A} = u_{\rm A} \hat{\vec{r}}_{\rm A} = \left(\vec{v}_{\rm A} \cdot \hat{\vec{r}}_{\rm A}\right) \hat{\vec{r}}_{\rm A}$| and |$\vec{u}_{\rm B} = u_{\rm B}\hat{\vec{r}}_{\rm B} = \left(\vec{v}_{\rm B} \cdot \hat{\vec{r}}_{\rm B}\right) \hat{\vec{r}}_{\rm B}$|, where |$\hat{\vec{r}}_{\rm A,B}$| are the unit direction vectors of the galaxies. The cosines of the angles between the different directions are given by |$\cos \theta _{\rm A} = \hat{\vec{r}}_{\rm A} \cdot \hat{\vec{r}}$|, |$\cos \theta _{\rm B} = \hat{\vec{r}}_{\rm B} \cdot \hat{\vec{r}}$| and |$\cos \theta _{\rm AB} = \hat{\vec{r}}_{\rm A} \cdot \hat{\vec{r}}_{\rm B}$|. Fig. 1 illustrates the geometry and quantities defined above.
Pair of galaxies considered throughout this article.
To partially avoid the Malmquist bias, the galaxies (or groups) are located at their redshift positions, as positioning objects at their observed distance leads to much larger errors, especially for the most distant ones.
3.1 Mean pairwise velocity
3.1.1 Model
From the approximate solution for v12 (equation 3), it is possible to recover the linear solution if ξ → 0, and the solution valid in the non-linear regime if x → 0. Equation (3) has been tested and validated by Juszkiewicz et al. (1999) on N-body simulations for 0.1 < ξ(r) < 1000.
3.1.2 Estimator
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 fσ8 (see colour bar).
A new estimator which relies on the transverse component of peculiar velocities (instead of the radial one) has been introduced by Yasini, Mirzatuny & Pierpaoli (2018). This estimator will allow to analyse pairwise velocities derived from upcoming transverse peculiar velocity surveys such as Gaia (Hall 2019).
3.2 Velocity correlation function
The quantities Ψ∥(r) and Ψ⊥(r) both depend on the parameter (fσ8)2. These correlation functions can therefore be used to constraint the combined cosmological parameter fσ8, defined as the normalized growth rate of large scale structures.
3.2.1 Estimator
3.2.2 Model
The functions |$\mathcal {A}_1(r)$| and |$\mathcal {A}_2(r)$| contain information about the geometry of the sample, and measure the contributions of Ψ∥(r) and Ψ⊥(r) to the functions ψ1(r) and ψ2(r).
As one can see on the middle and right panels of Fig. 2, the amplitude of the two statistics ψ1 and ψ2 depends on the growth factor fσ8, allowing to constrain this cosmological parameter. The higher fσ8 is, the higher the amplitude of ψ1 or ψ2 is: a universe with a large fσ8 appears more compact and peculiar velocities get larger. However, in the case of ψ2, the curves corresponding to different fσ8 get closer and closer as r increases (see right-hand panel of Fig. 2). It is difficult to constrain cosmological models for separation distances higher than 60 Mpc h−1. The statistic ψ2 is therefore not robust enough to estimate the growth rate on peculiar velocity catalogs such as CF3, and especially on the upcoming big surveys. Therefore, this statistic will not be considered for the rest of this paper.
3.3 Observational errors and cosmic variance
Two kinds of uncertainties on peculiar velocity statistics are considered in this article: measurement (or observational) error and cosmic variance.
Observational errors on peculiar velocity statistics v12, ψ1, and ψ2, i.e. errors due to the uncertainty in distance measurement, are derived by Monte Carlo synthetic realizations. These realizations are constructed by adding a random error 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 distance (or distance modulus). In this article, 100 realizations have been computed for each sample (mocks and observed data).
In addition to measurements uncertainties, one ought to take into account the impact of cosmic variance when estimating cosmological parameters. However, computing uncertainties caused by cosmic variance cannot be done on a constrained realization of CF3, described in Section 2, which represents our local Universe. Therefore, a ΛCDM dark matter only N-body simulation is considered in this paper in order to estimate the impact of the cosmic variance on peculiar velocity statistics and growth rate measurements. Two tests, whose results are presented later in this paper, on dark matter haloes extracted from the MultiDark Planck 2 simulation (MDPL2; Prada et al. 2012) of size 1 Gpc h−1 are carried out. The underlying cosmology of the simulation is the Planck 2015 cosmology. Only mock galaxies with halo mass between 1011 M⊙ and 1012 M⊙ are taken into account to construct these mocks. Data samples for the tests are prepared as follows:
As the CF3 catalogue can (mostly) be contained in a sphere of a 250 Mpc h−1 radius, one can place eight of such independent spheres in a cube of side 1 Gpc h−1. Therefore, eight CF3-like samples are generated for each octant cube of the simulation. The observer of each sample is placed at the centre of its associated octant. Then radial components of peculiar velocities are extracted at the position of CF3 galaxies with respect to the observer. These mocks are completely independent from each other as they do not share any haloes.
As only eight samples is not high enough to get a robust result, a total of 100 more mocks are generated. Instead of positioning observers such that samples do not share haloes, a total of 100 observers are placed at a random locations within the simulation box. Then radial components of peculiar velocities are extracted at the position of CF3 galaxies with respect to the observers. In this case, the samples are not independent as a single halo can belong to several spheres, so results will be correlated.
Results obtained from these tests are shown in Section 4.
As different bins of separation distances share the same galaxies (or groups of galaxies), errors between bins are correlated and thus the covariance matrix needs to be considered when fitting the measured statistics to extract the normalized growth rate fσ8.
4 DERIVING THE LOCAL GROWTH RATE
4.1 Verification of estimators on mocks
Peculiar velocity statistics v12 and ψ1 have been measured on mock and observed radial peculiar velocities. Throughout this article, statistics are computed out to a distance of 100 Mpc h−1 in 20 equal bins of 5 Mpc h−1. In all figures of this article displaying velocity statistics computed on mocks or observed data, scattered points are located at the middle of the bins.
The statistics v12 and ψ1 have been tested and validated by the authors who introduced them. They allow one to recover the underlying cosmology from a homogeneous and spherical universe. But CF3 is very sparse and asymmetrical. Before constraining the growth rate, one must check if the spatial distribution of the CF3 catalogue alone inhibits such statistics from accurately recovering the underlying cosmology. This is done on the 100 CF3 mocks generated from a constrained realization as described in Section 2.
Fig. 3 shows as solid lines the ΛCDM models for galaxies in the CF3 ungrouped (red) and CF3 grouped (blue) mocks. For the statistic v12, errors bars of the two mocks do not include the ΛCDM model. This means that due to the unique CF3 geometry and selection function, this estimator can not recover the underlying ΛCDM cosmology. Therefore, we stress that it cannot be used to estimate to local growth rate with the real CF3 catalogue, 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 data set, whose footprint is similarly 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.
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 fσ8 can be fitted, see the limitations described in the text.
Furthermore, one can see in Fig. 3 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 except for small separations bins which contain galaxies close to each other (i.e. within clusters). When fitting fσ8 to this statistic, the effect of non-linearity will not be taken into account: bins corresponding to separations lower than 20 Mpc h−1 will be omitted.
Tests on mocks reported in Fig. 3 show that the underlying value of fσ8 in the CR is recovered by the estimator ψ1 in an interval of robustness 20–60 Mpc h−1. However, considering the depth of the CF3 catalogue and the size of the constrained realization, the measured value of the growth rate with CF3 data sets 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 Universe.
Fig. 4 shows peculiar velocity statistics obtained with the eight independent mocks extracted from the MDPL2 simulation (see Section 3.3) as red triangles with error bars. The underlying cosmology of the simulation, shown by the black line, is well recovered by the estimator ψ1(r). Results obtained with the 100 CF3-like mocks generated considering random observers in the MDPL simulation is shown in Fig. 4 as green triangles with error bars. The underlying cosmology of the simulation is once again well recovered. This shows that despite its depth, the CF3 catalogue may allow measuring the growth rate of large-scale structures. Nevertheless, errors bars due to cosmic variance and the extent of the sample are large and cannot be ignored. They must be taken into account when constraining fσ8 with CF3 data.
Peculiar velocity statistic ψ1 as a function of the pair separation r computed in the large MDPL2 simulation box. The simulation’s underlying ΛCDM model is shown as a black solid line. Red scattered triangles with error bars represent results obtained from the 8 CF3-like independent mocks. Green scattered triangles with error bars represent results obtained from the 100 CF3-like mocks constructed with randomly positioned observers. Vertical dashed lines show the region where where fσ8 can be fitted, see the limitations described in the text.
4.2 Estimation of fσ8 on data
Fig. 5 shows the results of the computation of ψ1 on the CF3 observations, for the ungrouped and grouped samples represented, respectively, as red triangles and blue squares. Errors bars include both, observational and cosmic variance uncertainties. Predictions of statistic ψ1 for different cosmological models are shown as solid (ΛCDM, fσ8 = 0.4) and dotted (fσ8 = 0.2, 0.3, 0.5, 0.6) lines. The two estimators obtained for CF3 peculiar velocities are consistent with ΛCDM within their interval of robustness, as predicted by tests on mocks shown in Fig. 3. Besides, one can observe that results obtained from CF3 groups have a slightly larger amplitude than results derived from CF3 galaxies.
Observed peculiar velocity statistic ψ1 as a function of the pair separation r. The ΛCDM model, fσ8 = 0.4, predicted for CF3 galaxies is shown as a black solid line. Peculiar velocity statistics predicted for other cosmological models, fσ8 = 0.2, 0.3, 0.5, 0.6, are shown as black dashed and dotted lines. Scattered points with error bars represent results obtained from observed peculiar velocities. The ungrouped sample of CF3 is shown as red triangles and the grouped CF3 sample is shown as blue squares. Vertical dashed lines show the region where fσ8 can be fitted, see the limitations described in the text.
For completeness, ψ1 has also been applied to the previous CF2 catalogue (not shown). The results are similar to CF3 and also in agreement with ΛCDM.
The data in Fig. 5 are fitted by minimizing the χ2 defined in equation (17). The fits are done within the interval [20, 60] Mpc h−1 of pair separation distances. The results are displayed in Table 1 which shows the values of fσ8 fitted directly from observational data. Both observational and cosmic variance uncertainties on the constrained growth rate are reported.
Constraints of the normalized growth rate fσ8 obtained from measurements of the peculiar velocity statistic ψ1, within its interval of robustness as described in the text, on the Cosmicflows data sets of observed peculiar velocities. The two last columns |$\Delta f\sigma _8^\mathrm{obs}$| and |$\Delta f\sigma _8^\mathrm{cosmic}$| correspond to observational and cosmic variance uncertainties on the constrained parameter, respectively.
| CF3 sample – statistic | fσ8 | |$\Delta f\sigma _8^\mathrm{obs}$| | |$\Delta f\sigma _8^\mathrm{cosmic}$| | ||
|---|---|---|---|---|---|
| CF3 ungrouped | |$\boldsymbol{0.43}$| | |$\boldsymbol{\pm }$| | |$\boldsymbol{0.03}$| | |$\boldsymbol{\pm }$| | |$\boldsymbol{0.11}$| |
| CF3 grouped | 0.45 | ± | 0.04 | ± | 0.11 |
| CF3 sample – statistic | fσ8 | |$\Delta f\sigma _8^\mathrm{obs}$| | |$\Delta f\sigma _8^\mathrm{cosmic}$| | ||
|---|---|---|---|---|---|
| CF3 ungrouped | |$\boldsymbol{0.43}$| | |$\boldsymbol{\pm }$| | |$\boldsymbol{0.03}$| | |$\boldsymbol{\pm }$| | |$\boldsymbol{0.11}$| |
| CF3 grouped | 0.45 | ± | 0.04 | ± | 0.11 |
Constraints of the normalized growth rate fσ8 obtained from measurements of the peculiar velocity statistic ψ1, within its interval of robustness as described in the text, on the Cosmicflows data sets of observed peculiar velocities. The two last columns |$\Delta f\sigma _8^\mathrm{obs}$| and |$\Delta f\sigma _8^\mathrm{cosmic}$| correspond to observational and cosmic variance uncertainties on the constrained parameter, respectively.
| CF3 sample – statistic | fσ8 | |$\Delta f\sigma _8^\mathrm{obs}$| | |$\Delta f\sigma _8^\mathrm{cosmic}$| | ||
|---|---|---|---|---|---|
| CF3 ungrouped | |$\boldsymbol{0.43}$| | |$\boldsymbol{\pm }$| | |$\boldsymbol{0.03}$| | |$\boldsymbol{\pm }$| | |$\boldsymbol{0.11}$| |
| CF3 grouped | 0.45 | ± | 0.04 | ± | 0.11 |
| CF3 sample – statistic | fσ8 | |$\Delta f\sigma _8^\mathrm{obs}$| | |$\Delta f\sigma _8^\mathrm{cosmic}$| | ||
|---|---|---|---|---|---|
| CF3 ungrouped | |$\boldsymbol{0.43}$| | |$\boldsymbol{\pm }$| | |$\boldsymbol{0.03}$| | |$\boldsymbol{\pm }$| | |$\boldsymbol{0.11}$| |
| CF3 grouped | 0.45 | ± | 0.04 | ± | 0.11 |
Results shown in Fig. 5 and Table 1 are obtained using the radial peculiar velocities computed from CF3 distances with equation (1) and H0 = 75 km s−1 Mpc−1. However, Tully et al. (2016) suggested that the range H0 = 75 ± 2 km s−1 Mpc−1 gives a reasonable value for the monopole flow. This uncertainty on the Hubble constant may affect the uncertainty on the growth rate measurements, as the amplitude of ψ1 depends on the chosen value of H0 when deriving radial peculiar velocities, as shown in Fig. 6. Therefore, we considered other values of H0 within the error range given in Tully et al. (2016). Taking H0 = 73 km s−1 Mpc−1 to compute peculiar velocity of CF3 galaxies gives fσ8 = 0.58 ± 0.03, and taking H0 = 77 km s−1 Mpc−1 gives fσ8 = 0.54 ± 0.03.
![Observed peculiar velocity statistic ψ1 as a function of the pair separation r. The ΛCDM model, fσ8 = 0.4, predicted for CF3 galaxies is shown as a black solid line. Scattered points with error bars represent results obtained from observed peculiar velocities of the CF3 galaxies, derived by considering different values of the Hubble constant H0 = [66, 68, 70, 72, 73, 74, 75, 76, 77, 79] km s−1 Mpc−1.](https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/mnras/486/1/10.1093_mnras_stz901/1/m_stz901fig6.jpeg?Expires=1749449610&Signature=mxaWWxQHH~1SipxGMIceuDMoTV9FUmfnKBgnlIiOolD-5~kYGNfTuTh1pOJpvXfVxqJC-ATEDazHbIOFw9Jy2uBBdHA053~t7L65GGnxkkS6eDWObTzHZjAKTSrDwOHhcoFZiTEXWUTC1hraY5bdoHvjbsDDIylzAo2rOrakg80g82cM-x2wVXeniN6k5x3xKQbGchwhNy2hVRO5fT5fL5w-1AsI24IAHcYBg9nCjONvo0FOO3gp~mMNv1aVY6NT0UIaCfy6CZlBko-vh94O-2XnCLXeyTn99G6SrgEFukSHXBsH8hKBOtnAKK6ploLT9WVTeCf3V5E52VDPRq2H4A__&Key-Pair-Id=APKAIE5G5CRDK6RD3PGA)
Observed peculiar velocity statistic ψ1 as a function of the pair separation r. The ΛCDM model, fσ8 = 0.4, predicted for CF3 galaxies is shown as a black solid line. Scattered points with error bars represent results obtained from observed peculiar velocities of the CF3 galaxies, derived by considering different values of the Hubble constant H0 = [66, 68, 70, 72, 73, 74, 75, 76, 77, 79] km s−1 Mpc−1.
Wang et al. (2018) show a similar study in the fig. 7 of their paper. They published error bars that are twice smaller than the ones we derived for ψ1 (see fig. 8 of Wang et al. 2018).
Fig. 7 shows the normalized growth rate fσ8 as a function of redshift z. The black solid line corresponds to the ΛCDM model (γ = 0.55). Other cosmological models (modified gravity, γ ≠ 0.55) are represented by dotted and dashed black lines. Scattered points with error bars show results taken from literature summarized in Table 2. Colours depend on the cosmological probe used to constraint the growth rate: SN Ia in yellow (Turnbull et al. 2012), galaxy peculiar velocities in blue (Johnson et al. 2014), Baryon Acoustic Oscillations in green (Blake et al. 2011; Reid et al. 2012) and Redshift Space Distortions in pink (Hawkins et al. 2003; Beutler et al. 2012; Samushia, Percival & Raccanelli 2012; de la Torre et al. 2013). Growth rate measurements predictions for future cosmological surveys are shown in red, with a predicted error bar of 10 per cent (Amendola et al. 2016; McConnachie et al. 2016; da Cunha et al. 2017).
Normalized growth rate fσ8 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 Table 2 are shown by scattered points with error bars. Colours 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.
Measurements of normalized growth rate fσ8 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 fσ8 | Publication | Cosmological probe | |
|---|---|---|---|---|
| 6dFGS | 0.05 | |$0.428^{+0.079}_{-0.068}$| | 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 |
| Redshift z | Normalized growth rate fσ8 | Publication | Cosmological probe | |
|---|---|---|---|---|
| 6dFGS | 0.05 | |$0.428^{+0.079}_{-0.068}$| | 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 |
Measurements of normalized growth rate fσ8 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 fσ8 | Publication | Cosmological probe | |
|---|---|---|---|---|
| 6dFGS | 0.05 | |$0.428^{+0.079}_{-0.068}$| | 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 |
| Redshift z | Normalized growth rate fσ8 | Publication | Cosmological probe | |
|---|---|---|---|---|
| 6dFGS | 0.05 | |$0.428^{+0.079}_{-0.068}$| | 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 |
Local growth rate constraint obtained from radial peculiar velocities of CF3 galaxies is shown in Fig. 7 as a blue point with error bars. Errors bars include the observational uncertainties only, in coherence with other data points.
5 CONCLUSION
This article presents a measurement of the local value of fσ8 in the nearby Universe by means of a peculiar velocity survey: Cosmicflows-3. We obtained a measurement of fσ8 = 0.43(± 0.03)obs(± 0.11)cosmic out to z = 0.05. Uncertainties correspond to observational 'obs’ and cosmic variance ’cosmic’ uncertainties, respectively.
This is in complete agreement with the measurement made by Wang et al. (2018), finding fσ8 = 0.488.
Currently, as seen in Fig. 7 some local cosmological probes like BAO and RSD do not have error bars that could constrain the growth rate of large scale structures. Even large redshift surveys such as SDSS, VIPERS, or WiggleZ have a too large observational error budget to discriminate a cosmological model. In contrast, one can see that using only a few thousands of peculiar velocities of galaxies as cosmological probes to measure fσ8 allows to restrict the range of cosmological models. However, due to the unique geometry of the CF3 survey as well as its sparseness, we have found that we are unable to use the full arsenal of velocity field statistics to probe fσ8. We look forward to increased coverage to improve on this situation.
Upcoming redshift and peculiar velocity surveys (such as TAIPAN, MeerKAT, WALLABY, Euclid and MSE) are expected to improve uncertainties in a large range of redshifts from the very local Universe up to z = 1.7. Hopefully, this will allow to constrain the results also in cosmic variance allowing finally to discriminate a single cosmological model using galaxy two-point statistics of peculiar velocities.
ACKNOWLEDGEMENTS
Noam Libeskind, Carlo Schimd, Romain Graziani, Yannick Copin and Mickael Rigault are gratefully thanked for scientific discussions. Support has been provided by the Institut Universitaire de France and the CNES.
The CosmoSim data base used in this article is a service by the Leibniz-Institute for Astrophysics Potsdam (AIP). The MultiDark data base was developed in cooperation with the Spanish MultiDark Consolider Project CSD2009-00064.
