Cosmic flows and the expansion of the Local Universe from nonlinear phase-space reconstructions

This work investigates the impact of cosmic flows and density perturbations on Hubble constant $H_0$ measurements using nonlinear phase-space reconstructions of the Local Universe (LU). In particular, we rely on 25 constrained N-body simulations of the LU using the 2MRS galaxy sample within distances of 90 Mpc/h. These have been randomly extended up to volumes enclosing 360 Mpc/h with augmented Lagrangian perturbation theory (750 simulations in total), accounting in this way for gravitational mode coupling from larger scales, correcting for periodic boundary effects, and estimating systematics of missing attractors ($\sigma_{\rm large}=134$ km/s). We report on Local Group speed reconstructions, which are compatible with CMB-dipole measurements: $|v_{\rm LG}|=685\pm137$ km/s. The direction $(l,b)=(260.5\pm 13.3,39.1\pm 10.4)^\circ$ is compatible with observations after considering the variance of large scales. Accounting for large scales, our local bulk flow estimations assuming a $\Lambda$CDM model are compatible with estimates based on velocity data derived from the Tully-Fisher relation. We focus on low redshift supernova measurements ($0.01<z<0.025$), which have been found to disagree with probes at larger distances. Our analysis indicates that there are two effects contributing to this tension. First, the anisotropic distribution of supernovae aligns with the velocity dipole and induces a systematic boost in $H_0$. Second, a divergent region surrounding the Virgo Supercluster is responsible for an additional positive bias in $H_0$. Taking these effects into account yields a correction of $\Delta H_0=-1.76 \pm 0.21 $ km/s/Mpc, thereby reducing the tension between local probes and more distant probes. Effectively $H_0$ is lower by about $2\%$.


INTRODUCTION
Measurements of the Hubble constant coming from different probes show high variations. In particular cosmic microwave background (CMB) measurements reported by the Planck Collaboration et al. (2013) and recession velocity measurements of Type Ia Supernovae (SNe Ia) (see e.g., Riess et al. 2009;Perlmutter et al. 1999) show a discrepancy of about 2.4σ. Jha et al. (2007) reported 6.5 ± 1.8% E-mail: hess.aip@gmail.com † E-mail: kitaura@aip.de, Karl-Schwarzschild fellow higher values for a local sample (z < 0.025 corresponding to about < 75 h −1 Mpc), as compared to the more distant set.
There are presumably two combined main effects which could explain this discrepancy: first the cosmic variance (CV), and second systematic errors in the standardisation of SNe Ia due to environmental dependencies (Rigault et al. 2013(Rigault et al. , 2014. A number of studies have focused on estimating the influence of CV from large sets of simulations (see e.g. Marra et al. (2013); Wojtak et al. (2014); Ben-Dayan et al. (2014)), as local probes are particularly sensitive to the local den-sity and velocity field. These theoretical studies analyse the impact of random seeded mock catalogues on the Hubble measurement, thereby studying cosmic variance in a probabilistic way (see also Hui & Greene 2006;Davis et al. 2011).
As an alternative to statistical CV estimates, one can use the actual velocity information of the Local Universe to reduce the influence of CV on H0 estimations. Interestingly, Wiltshire et al. (2013) find from radial velocity data, that the amplitude over the Hubble flow changes markedly over the range 32 to 60 h −1 Mpc. Neill et al. (2007) and Riess et al. (2009) managed to find moderate corrections to the Hubble constant estimate based on peculiar velocity corrections derived from galaxy redshift data using linear theory.
We want to extend these works by investigating the inferred nonlinear velocity field from high precision constrained N-body simulations based on Bayesian selfconsistent phase-space reconstructions of the Local Universe focusing on low redshift supernovae (z < 0.025) .
This paper is structured as follows. First, we study in §2 the qualitative impact of cosmic variance through peculiar motions on the estimation of the Hubble constant from a theoretical perspective. In §3 we compute the peculiar velocities from constrained simulations of phase-space reconstructions. The resulting shifts on the Hubble constant measurement are presented in §4. Then we discuss our results in §5. Finally, we present our conclusions.

RELATION BETWEEN THE HUBBLE CONSTANT AND THE VELOCITY FIELD
Let us consider the Local Volume, where the redshift evolution is negligible. The radial recession or Hubble flow is then given by vH ≡ H0r, with r being the line-of-sight distance vector, and H0 the Hubble constant at redshift zero. Hence, the Hubble constant H0 measuring the expansion speed of the Universe is directly related to the divergence of the Hubble flow by The observed Hubble constant H0 obs is in general measured from an average of N discrete distance tracers where each tracer i contributes to the average with H i 0 ,r is the unitary line of sight vector, and the velocities are given by the sum of the Hubble flow and the peculiar motion term with vp being the peculiar velocity of each tracer (we refer to § 4.2 for a relativistic treatment). Each tracer i at a distance ri approximately measures the Hubble constant within the volume Vi = 4π 3 r 3 i . However, this will in general only be true under the assumption that the line of sight projected peculiar velocities are isotropic, as can be seen from applying Gauss theorem to the volume averaged Hubble constant Hence, an anisotropic line of sight peculiar velocity field will introduce a systematic bias in the Hubble constant measurement. This effect is amplified by an anisotropic tracer distribution in the sum of Eq. 2, as we will show below.
Unfortunately, supernovae and their host galaxies, are not evenly distributed in space, but are affected by various radial selection effects and a complex biasing w.r.t. the underlying dark matter distribution. Let us therefore consider a simple model, including a radial selection function depending on the distance r and a bias as a function of the local density field ρ = ρ(r) (for more complex non-local biasing relations see, e.g., McDonald & Roy 2009) We can now obtain a more realistic model of the observed Hubble constant by introducing the selection biased Hubble constant H where we have introduced the effective volume V f = dV f (r) and the average f = 1 Furthermore we used Eq. 3 and have applied Gauss theorem in the reversed order to Eq. 4.
We find the deviation between the true value of the Hubble constant and the one obtained from a selection biased tracer is given by It is obvious from this equation that in the limiting case of negligible peculiar motions the difference vanishes. However, interestingly, there are more contributions in the general case, beyond the effective divergent flow term (first term in Eq. 9). Local peculiar velocities modulate terms dominated by radial selection effects and density perturbations (second and third terms, respectively).
We conclude from this analysis that a proper measurement of the Hubble constant should include accurate peculiar velocity flow corrections under the consideration of the distribution of matter beyond divergent bulk flow corrections, as we will show in this work.

COSMIC FLOWS RECONSTRUCTION
The supernova data permit one to estimate the Hubble constant by linear regression of their redshifts against their luminosity distances. Their measured redshift positions are however affected by peculiar motions of the large scale structure. Therefore a correction of the redshift space distortions is necessary to improve the Hubble constant estimation, as discussed in the previous section.
In this section we describe our nonlinear peculiar halo velocity field reconstruction, followed by the study of the influence of larger scales beyond the reconstructed volume and the dependence on cosmological parameters. Finally we show the supernova peculiar motion correction based on simulations using the halo phase-space reconstruction.

The halo peculiar velocity field from nonlinear phase-space reconstructions
We aim at obtaining a full nonlinear reconstruction of the peculiar velocity field. To solve this problem we rely on the kigen-code (Kitaura 2013;, updated with a number of improvements reported in Heß et al. (2013) and Nuza et al. (2014). This method is based on a Bayesian networks machine learning algorithm, which iteratively samples Gaussian fields, whose phases are constrained by the distribution of observed tracers given a structure formation and a cosmological model. In particular, it employs two Gibbs-sampling steps. In the first step, the phases of the Gaussian fields are sampled given a distribution of matter tracers of proto-haloes at Lagrangian initial conditions, for which a lognormal-Poisson distribution (Kitaura et al. 2010) including a linear Lagrangian bias (power law bias within the lognormal framework) is assumed (see also Kitaura et al. 2014). The log-normal model is a fair assumption in a Lagrangian co-moving description when shell-crossing is negligible (see Coles & Jones 1991;. In the second step, the positions of the proto-halo tracers are obtained given the initial Gaussian phases from the previous step. Here a Chi-squared likelihood comparison minimising the quadratic distance between the final mod-  (Kogut et al. 1993), from the constrained N -body simulations within 2.3 h −1 Mpc distance v LG , and including large scale modes uncertainty estimates: v large LG .
elled matter positions and the observed positions of galaxies is used. The structure formation model including redshift space distortions connecting the initial Gaussian field with the observations is based on augmented Lagrangian perturbation theory (ALPT, ). An uncertainty variance in the Chi-square of 1 h −1 Mpc accounts for the inaccuracy of the ALPT approximation and the probability that a galaxy is associated to a particular halo. This scheme is iterated until it reaches convergence of the power spectra of the initial Gaussian fields and the crosspower spectra between the reconstructed dark matter field in redshift space and the observed galaxy field. Then a couple of thousand additional iterations are run to produce an ensemble of initial conditions compatible with the observations. The Lagrangian bias is self-consistently constrained to yield unbiased power spectra with respect to the theoretical linear power spectrum model. Based on the Two-Micron Redshift Survey (2MRS) galaxy catalogue (Huchra et al. 2012) we have selected the set with the highest correlation with the galaxy field to perform 25 constrained N -body simulations ). These comprise a cubic volume of 180 h −1 Mpc and simulate the nonlinear structure formation assuming WMAP7 (Komatsu et al. 2011) cosmology. These simulations resemble the Local universe on scales larger than 2 h −1 Mpc and resolve spherical overdensity halos with masses m > 1.6 × 10 11 h −1 M . Therefore most of the heavily star-forming and hence supernovaforming objects are resolved.
This permits us to study the full nonlinear peculiar velocity field based on the halo population, and supposes a considerable improvement with respect to the dark matter peculiar velocity field based on second order Lagrangian perturbation theory presented in .
We identify in our simulations an under-density of δ = −.02 (δ = −.01) within cz < 7000 s −1 km (within cz < 7400 s −1 km), indicating that we have a closely fair sample on those scales. This is in agreement with the cosmic web study performed with the same simulations in Nuza et al. (2014), and with a number of other studies of the Local Universe (see e.g. Branchini et al. 1999;Giovanelli et al. 1999). We note that other recent works claim to have found evidence for an under-density on distances of about 300 Mpc Keenan et al. (2013); Whitbourn & Shanks (2014), which are not included in our simulations. We need to consider scales well beyond our reconstructed volumes to thoroughly analyse the uncertainties in our cosmic flows measurements.

Influence of large scales
The constrained simulations only probe distances of ∼ 90 h −1 Mpc along the SG axis (and of ∼ 127 h −1 Mpc along the diagonal within the SG XY plane). In this way For visualisation purposes of the shell crossing regions of the peculiar velocity field coinciding with high density regions, the dark-blue, the reddish, and the light-yellow colour-codes indicate low, close to mean, and high densities, respectively. The length of the arrows is proportional to the average speed at that location.
the Shapley concentration (SC), with its centre located at distances of about 130-140 Mpc/h (in the diagonal direction within the SG plane), is mainly excluded from our volume (see for instance Fig. 19 in Erdoǧdu et al. (2006) including the SC). We note that the reconstructed volume was limited to distances where uncertainties due to the selection function, especially the Kaiser-rocket effect, are negligible (Nusser et al. 2014). Therefore missing attractors, such as the SC, and larger modes are expected to have an effect on our reconstructions (see e.g., Nusser et al. 2014). Another source of systematic errors comes from the periodicity assumption within the reconstruction process. Let us represent the resulting halo peculiar velocity field from our constrained N -body simulations by v h (x), where x represents Eulerian real-space. To asses the influence of the above mentioned effects we embed the constrained initial conditions in a bigger volume of (720 h −1 Mpc) 3 , effectively including distances of up to 360 h −1 Mpc with respect to the Local Group. We exploit the advantage of having reconstructed the initial conditions, partially following the methods by Tormen & Bertschinger (1996) and Schneider et al. (2011). In particular we compute the white noise (initial density field divided by the square root of the power spectrum) of the set corresponding to the 25 best reconstructed fields and augment it by adding random phases of unity variance beyond the reconstructed volume to boxes of 720 h −1 Mpc side (similar to Hoffman et al. (2001)). We finally multiply the augmented white noise with the square root of the power spectrum in Fourier space to produce the primordial Gaussian density field from which we compute the peculiar velocity field with ALPT.
To accurately estimate the effects of mode-coupling beyond the reconstructed volume, we compute the differences between the constrained and augmented boxes in Lagrangian space (we denote Lagrangian coordinates with q). For definiteness we compute difference using ALPT for both boxes. We can therefore define the correction, as the difference between the long range (enclosing the big volume) and the constrained component (enclosing the small volume): We then use the constrained displacement field to evaluate the correction in Eulerian-space with x = q + Ψ, where Ψ is given by the smaller constrained volume. In this way we do not alter the displacement fields, which have been accurately constrained within the self-consistent reconstruction process. We can finally compute the long range corrected halo peculiar motions by adding the correction to the full nonlinear constrained component: For each of the 25 constrained small boxes we compute the mean and variance of 30 augmented ALPT simulations comprising 750 realisations in total. Fig. 1 shows the influence of modes beyond the reconstructed volume on the divergence of the velocity field in one of these augmented ALPT simulations.
From this analysis we get the following results.
(i) By inspecting the ensemble mean of the realisations we find v large h (x) , a systematic deviation of 39 s −1 km in the direction of (l, b) = (37, 19) • respectively, caused by the periodicity assumption or the missing modes, since the attractors beyond the reconstructed volume cancel out in the ensemble average. An illustration of the cosmic velocity field in the Local Universe derived from our calculations is shown in Fig. 2. The caustics of the peculiar velocity field show a remarkable correlation with well-known structures like the Local Super-cluster, the Great Attractor, the Coma, and the Perseus-Pisces clusters, indicating the accuracy of the peculiar velocity field. Let us report, as part of the results of this work, the peculiar motion of the Local Group with a speed of |vLG| = 685 ± 36 s −1 km and pointing towards galactic longitude and latitude of (l, b) = (260.5 ± 2.5, 39.2 ± 1.7) • , respectively, after taking into account the mean correction. We have considered the mean peculiar velocity of LG like haloes contained within 2.3 h −1 Mpc distance to the location of the observer. This distance accounts for the 2−3 h −1 Mpc uncertainty in the location of haloes . One should note that the error bars indicated here represent the error of the mean of the ensemble of constrained halo catalogues, and a number of systematic effects are not included, as we will do below. We find however, already an interesting result with respect to previous works based on linear or even second order Lagrangian perturbation theory, as the speed is considerably larger than in those previous studies (see e.g. ). This speed is compatible with independent CMB-dipole measurements (|vLG| = 627 ± 22 s −1 km (Kogut et al. 1993)).
(ii) We find by inspecting the variance of the realisa- By including these uncertainty estimates we find compatible results with the CMB-dipole local group velocity measurement as summarised in Tab. 1. However, our analysis does not specify the direction of the missing bulk flow motion. To assess the direction we can use the CMB-dipole information ( |vLG| = 627 ± 22 s −1 km towards (l, b) = (276 ± 3, 30 ± 3) • Kogut et al. (1993)). To reconcile the halo velocity in the centre of the box with the Local Group velocity we need to assume a bulk motion of the simulation box of |v large | = 187 s −1 km towards (l, b) = (354, −39) • which is consistent with our analysis, corresponding to 1.4σ large . Interestingly, this missing component is closely perpendicular to the constrained flow speed, which was already compatible with the CMB measurement. . Average bulk flow within radius R of potential SN hosting halos from the ensemble of constrained simulations. Shown are estimates without (blue) and with restframe correction of the simulation box (red). The blue shaded region marks the uncertainty containing the error of the ensemble mean and the standard deviation of the bulk flow estimate. The dashed lines extend the uncertainty due to differences in cosmology. As comparison serves the Local Group speed (Kogut et al. 1993) in yellow.

Cosmology dependence
The reconstruction scheme as well as the constrained simulations have been performed assuming WMAP7 cosmology. However both the reconstruction and the simulations have been performed in units of h −1 , without choosing a particular H0, and are hence suitable for this study. Other cosmological parameters have an influence on the peculiar velocity field. To gauge this effect we have performed a constrained simulation with Planck cosmology (Planck Collaboration et al. 2013) using existing constrained initial conditions. We find that halo velocities tend to be slightly higher especially at small scales. In Fig. 3 we indicate the uncertainty introduced by the difference of the two cosmologies (dashed lines). We note that this study tends to overestimate the effects of the cosmological parameters. The reason being that the initial conditions have been found with WMAP7 cosmology. A self-consistent reconstruction of the initial conditions with PLANCK cosmology would certainly yield closer results. Nevertheless, the bulk flows remain very similar at radii larger than about 10 h −1 Mpc, demonstrating the robustness of our results.

Supernova peculiar motion correction
In the previous sections we have described the method used to obtain an ensemble of halo distributions with the corresponding peculiar motions constrained on the Local Universe, including an accurate assessment of uncertainties due to systematics (periodic boundary conditions and missing modes from larger scales) and cosmic variance (missing attractors beyond the reconstructed volume). The ensemble of catalogues permits us to account for uncertainties derived from the reconstruction method itself. We note that the phase space reconstructions reproduce information of the large-scale structure down to about 2 h −1 Mpc, meaning that structures below these scales are essentially random (see Heß et al. (2013)). This is reflected in the lack of correlation on these scales between realisations within the ensemble.
To each SN we assign the mean of peculiar velocities of constrained haloes (v large h ) which are nearby in redshift space (see Nuza et al. 2014).
Let us call the resulting estimated supernova peculiar motion vSN. In this way the information of the ensemble of solutions is used to obtain a single and conservative correction. We note that we are neglecting the peculiar motion of supernovae within galaxies.
We find that the amplitudes of the peculiar velocities tend to get slightly underestimated due to the conservative nature of the ensemble mean. The distribution of radial velocities for the reference haloes have σvH = 282 s −1 km, whereas the radial velocities assigned to the supernovae have σvSN = 224 s −1 km. Given the additional uncertainty of the assignment itself, of σvassign ∼ 20 s −1 km this leaves a residual uncertainty in the peculiar velocities of σvres ∼ 80 s −1 km, indicating that we reach about a factor of two more accurate estimates than previous works (see e.g. Neill et al. 2007, based on linear theory).
We note that the peculiar velocity term in Eq. 3 is weighted with 1 + z (see e.g., Davis et al. (2011)) first within the reconstruction process ( §3.1) and in the Hubble fit (see next section).

HUBBLE CONSTANT MEASUREMENTS
In this section we start presenting the input supernova sample used in our study, followed by our results on Hubble constant measurements applying the peculiar velocity correction shown in the previous section. We finally present our cosmic flows reconstructions and in particular our estimates on bulk flows and the Local Group motion.

Input supernova data
We use the data from the Extragalactic Distance Database (EDD) 1 (Tully et al. 2009) comprising five sources (Prieto et al. 2006;Jha et al. 2007;Hicken et al. 2009;Amanullah et al. 2010;Folatelli et al. 2010), which was compiled by . It consists of 308 supernovae within 0 < z < 0.12. In particular we focus in this work on the local sample z < 0.025, which is contained inside our simulated box. The density of supernovae at increasing distance is shown in Fig. 4.

Distance calculation
We proceed now to perform a linear regression of the supernova data based on their spectroscopic redshifts cz and the corresponding inferred luminosity distances dL. Let us consider the cases with and without redshift space distortions,  i.e., f (H0dL) = H0dL − vSN ·r and f (H0dL) = H0dL, respectively, and their relation to the redshift distance g(cz): g(cz) = f (H0dL). In particular we consider following three approximations to study the robustness of our results: linear relation g(cz) = cz, relativistic relation g(cz) = c 2z+z 2 2+2z+z 2 (for a discussion see Davis & Lineweaver 2004), and approximate integral relation g(cz) = cz + (1 − 3 4 Ω)cz 2 + (9Ω − 10) Ω 8 cz 3 (see Pen 1999). The first two are approximations for low redshifts (for a discussion see Davis & Lineweaver 2004) independent of cosmology. The latter depends on the matter density ΩM and we fix ΩMh 2 according to WMAP7 (Komatsu et al. 2011). In the case without peculiar motions corrections we assume a radial velocity dispersion of σv = 282 s −1 km, as extracted from our analysis in §3.4. This accounts for the uncertainty due to peculiar velocities within the Chi-square fit. These error bars are considerably reduced when adding the actual information of the peculiar motions according to our analysis. We find slightly different Hubble constant estimates depending on the choice of approximation (g(cz)), as we show in Fig. 5. We should also note that the calibration of the supernovae have a considerable impact on the absolute value ranging from about 74 to 76 s −1 km Mpc −1 (see e.g. . We note that recent environmental studies indicate that the impact could be even larger (Rigault et al. 2013(Rigault et al. , 2014. However the relative shift (using the same approximation) due to the peculiar velocity correction remains unchanged within error bars. Therefore we will focus in this work on the relative corrections and not on the absolute estimates. In fact we find, that taking peculiar motions corrections into account yields a correction of ∆H0 = −1.76 s −1 km Mpc −1 for z < 0.025. Reducing the range to 0.01 < z < 0.025, and hence excluding the local super-cluster, as is done in several studies (see e.g. Riess et al. 2009;Ben-Dayan et al. 2014), we find a correction of ∆H0 = −2.13 ± 0.27 s −1 km Mpc −1 (Jackknife Error Estimates). Hence, these results reduce the tension between local and more distant probes of H0 by about 3%. Riess et al. (2009)

DISCUSSION
In §2 we discussed different possible sources of systematic deviations in the measurement of the Hubble constant, which originate in the presence of peculiar motions (see Eq. 9).
Let us analyse our results in this section and the causes of the biases in the Hubble constant measurement.
The cumulative distribution of supernovae shows a deviation towards low distances from a flux limited sample following a one over squared distance law (see Fig. 4). Beyond the radial selection effect and inhomogeneity, there is an apparent anisotropic distribution of supernovae. These effects have an impact on the Hubble measurement through the second and third terms in Eq. 9. Fig. 6 indicates the residuals of a linear fit. There is a correspondence between the deviations and the corrections for distances that are well inside the reconstructed volume.
From these two figures it is apparent that there is a supernova under-density at radii smaller than ∼ 60 h −1 Mpc leading to a radially diverging flow. According to Eq. 9 this will then lead to positively biased Hubble constant estimations.
Let us now analyse in detail the first term of Eq. 9 dominated by the velocity divergence. Fig. 7 shows slices through the velocity divergence field. Here the constrained simulations have been augmented with random large scale modes and the average of the resulting velocity divergence field is shown.
It appears that in a shell between 27 h −1 Mpc and 47 h −1 Mpc there is an expanding region (enclosed by black circles in Fig. 7). The velocity divergence field as a function of radius r in Fig. 8 clearly demonstrates the expansion at distances above 27 h −1 Mpc just beyond the Virgo Supercluster (VSC) (enclosed by green lines in Fig. 8). Under-densities in that region have been noted before, as it comprises e.g. the local void (Tully et al. 2008). Moreover, within supernova data a signal has been detected at 4800 s −1 km by Jha et al. (2007). However we want to stress that this region is not a void by any standard definition since it is not a convex volume and contains considerable overdensities including part of the great attractor. Nevertheless it is a spherical shell that is on average diverging.
The bottom panel of Fig. 8 indicates the impact of the peculiar velocity field on each tracer, solely due to the velocity divergence term. We find that the velocity divergence is indeed the dominating correction by comparing the cumulative velocity divergence (bottom panel of Fig. 8)  absolute peculiar velocity correction ∆vp (bottom panel of Fig. 6).

CONCLUSIONS
We have presented a detailed analysis of the Hubble constant measurement corrections in the Local Universe taking into account the impact of cosmic flows and density perturbations. Our findings indicate that low redshift supernova samples (0.01 < z < 0.025) overestimate the Hubble constant by about 3%. By making the appropriate peculiar motion corrections we find that the Hubble constant is reduced by ∆H0 = −2.13 ± 0.27 s −1 km Mpc −1 (∆H0 = −1.76 s −1 km Mpc −1 for z < 0.025). This correction should be considered in addition to environmental dependences which affect the systematic errors in the standardization of SNe Ia. Due to the small enclosed volume and the small distances, local observations of recession velocities are easily influenced by inhomogeneities, anisotropies and peculiar velocity. Our analysis relies on precise constrained N -body simulations which are based on unprecedented Bayesian self-consistent phase-space reconstructions of the initial conditions of the Local Universe using the 2MRS data. This ensemble of constrained nonlinear velocity fields permits us to deal with cosmic variance, which is paramount to unlock the full potential of the abundant supernovae distance measurements in the Local Universe. We have furthermore accounted for periodicity effects and missing attractors by increasing the simulations volume up to distances of 360 h −1 Mpc, running 750 simulations with augmented Lagrangian perturbation theory. The cosmic flows reconstructions presented in this work show an extraordinary resemblance with independent observations. In particular we obtain consistent results to observed bulk flows on scales up to 40 h −1 Mpc based on the Tully-Fisher relation with the 2MTF galaxies. Furthermore, our calculations yield a Local Group speed of |vLG| = 685 ± 137 s −1 km (l = 260.5 ± 13.3, b = 39.1 ± 10.4) compatible with the observed CMB dipole velocity. Our analysis suggests that there is a missing component of about 130 s −1 km, which, however, has mostly an impact on the direction. All these results show a remarkable agreement with ΛCDM.
We investigate the origin of our peculiar velocity correction further and find two main components. The first is the correction due to the alignment of the large scale bulk flow and the anisotropy of the supernovae distribution. The second contribution is due to a diverging shell at distances from 27 h −1 Mpc to 47 h −1 Mpc . Despite the very mild deviation from average density within a sphere of z < 0.025, this shell is below average density. It is located beyond the Virgo Super-cluster and limited by well known overdensities such as e.g. the Perseus-Pisces or the Coma clusters.
Still a number of aspects could be further improved in this work. Constrained simulations with even higher precision would be able to reduce the remaining small scale uncertainties further. This involves the challenges of improving the structure formation model and the redshift space distortion treatment, including N -body solutions within the reconstruction process.
Reconstructions on larger volumes would allow us to pick up more extended inhomogeneities and bulk flows that we included as uncertainties. However this requires a treatment of the Kaiser rocket effect and ever more sparse spectroscopic redshift observations on the full sky awaiting for new data.
This work represents a first attempt to make a full nonlinear analysis of the local cosmic flows and the Hubble constant measurement from galaxy redshift data.