# Testing Bayesian reconstruction methods from peculiar velocities

Aurélien Valade,1,2 Noam I Libeskind1,2 Yehuda Hoffman3 nd Simon
Pfeifer1^1$Leibniz-
Institutf\"{u}rAstrophysikPotsdam(AIP),AnderSternwarte16,14482Potsdam,Germany\\\
$^2$UnivLyon,UnivClaudeBernardLyon1,CNRS,IP2ILyon/IN2P3,IMR5822,F-69622,Villeurbanne,France\\\
$^3$RacahInstituteofPhysics,HebrewUniversity,Jerusalem91904,Israel$

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The reconstruction of the large scale density and velocity fields from surveys
of galaxy distances, is challenging. The data is very noisy, sparse,
inhomogeneously and anisotropically distributed. Estimated distances, and
thereby peculiar velocities, are strongly affected by the Malmquist-like
lognormal bias. Two algorithms have been recently introduced to perform
reconstructions from such data: the Bias Gaussian correction coupled with the
Wiener filter (BGc/WF) snd the Hamlet implementation of the Hamiltoniam Monte
Carlo forward modelling. The two methods are tested here against mock catalogs
that mimic the Cosmicflows-3 data. Considering the nearby ($d\lesssim
40h^{-1}\,{\rm Mpc}$), the intermediate (($40\lesssim d\lesssim
120h^{-1}\,{\rm Mpc}$) and the distant ($d\gtrsim 120h^{-1}\,{\rm Mpc}$)
regimes, and analyzing the residual between the reconstructed and target
density and velocity fields, the following is found. In the nearby regime the
two methods perform roughly equally well. Hamlet does a bit of a better job in
the intermediate regime. The main differences between the two appear in the
distant regime, close to the edge of the data. The Hamlet outperforms the
BGc/WF in terms of better and tighter correlation, yet in the distant regime
the Hamlet yields somewhat biased reconstruction. Such biases are missing from
the BGc/WF reconstruction. We consider such biases to be part of the
‘teething’ problem of the Hamlet algorithm to be solved. In sum both methods
perform well and create reliable reconstructions with significant differences
apparent when details are examined. Specifically the reconstructed cosmography
and moments of the velocity field (monopole, dipole) are examined. Lastly, a
comparisson is made to the “exact” wiener filter as well.

###### keywords:

keyword1 – keyword2 – keyword3

††pubyear: 2019††pagerange: Testing Bayesian reconstruction methods from
peculiar velocities–LABEL:lastpage

##  1 Introduction

Mapping the large scale distribution of matter and its corresponding three-
dimensional (3D) velocity field is of great interest. The motivation is
threefold. For one, mankind at large, and astronomers in particular, explore
their Universe by charting it. Making maps is a first step towards
understanding ones surroundings and ones place within a greater environment.
Mapping the large scale distribution of matter in the Universe is thus an end
unto itself. Furthermore, explaining the large scale velocity and density
fields within the context of the $\Lambda$CDM paradigm of structure formation,
allows for the estimation of cosmological parameters that are inherent to that
model XXX. A third motivation is to map these fields at high redshift, thereby
allowing for the reconstruction of the initial conditions of the Local
Universe (XXX).

We focus here on mapping the large scale structure (LSS) of the universe from
surveys of peculiar velocities, or rather from surveys of galaxies with
measured distances from which the radial component of the peculiar velocities
may be extracted. Measuring galaxy distances is a formidable challenge in
observational cosmology. All distance measures rely on comparing an observed
magnitude with an (inferred or assumed) absolute magnitude. There are many
ways to estimate a galaxy’s distance. For example, scaling relations tie the
size of an elliptical galaxy, or the angular velocity of disc galaxy to its
intrinsic luminosity. Other methods include resolving stars at the tip of the
red giant branch, measuring Supernovae light curves, Cepheid variables
pulsations or the scale of fluctuations in a galaxies surface brightness. Each
method has errors associated with it, a combination of instrumentational
errors, systematic errors, or calibration errors.

As such, compilations of peculiar velocities are difficult to analyze (for a
comprehensive review 1995PhR...261..271S) and are usually a patchwork of
various surveys and methods observed with different telescopes in different
locations on earth (or in space). The POTENT method was the first attempt to
produce continuous maps of the density and velocity fields based on peculiar
velocities surveys (1989ApJ...336L...5B). The main underlying assumption of
the POTENT method is that galaxy velocities are drawn from an irrotational,
potential flow. No further assumptions were made on the statistical nature of
the flow field. Therefore, its ability to handle the shortcomings of such
peculiar velocity surveys was limited. Subsequent approaches to the
reconstruction of the LSS from peculiar velocities have been formulated within
Bayesian frameworks - these include the Wiener filter (WF) and constrained
realizations (CRs) methodology (1993ApJ...415L...5G; 1999ApJ...520..413Z;
Tully2019) as well as Monte Carlo Markov Chain algorithms (MCMC Lavaux2016;
2019MNRAS.488.5438G). Often, these methods are tested on mock data in order to
gauge how reliable they are (XXX). They have been remarkably successful in
“mapping the invisible” and recovering the underlying cosmic fields.

Figure 1: The distance, in units of km/s, of the CF3 data points (magenta) and
mock data points (green) projected on the three supergalactic principal
planes. Note the ZOA is accurately reproduced in the mock catalogues. Figure
2: From left to right, the distribution SGB, SGL and $z$ of CF3 and mock data
points. Note that in panels (a) and (b) the two curves are on top of one an
other.

Beyond the issues of noisy, sparse data, plagued with inhomogenous errors,
there is one additional inherent conceptual problem common to all surveys of
peculiar velocities and that is that peculiar velocity itself is not observed
but is a derived quantity. Given the redshift of and a distance to a galaxy,
it is the radial component of its peculiar velocity that can be derived. But
only the redshift is observed; distances themselves are not directly observed.
What is measured is the distance modulus of a galaxy (cf. Tully2016). Because
this error of the measured distance modulus is assumed to be normally
distributed, the errors on the observed distances are thus lognormally
distributed (see equaition XXX). This leads to a biased estimate of the
distances and peculiar velocities with respect to the actual distances (see
2021MNRAS.505.3380H). Often this bias is treated as yet another manifestation
of the Malmquist bias (see 1995PhR...261..271S). Here we refer to it as the
lognormal bias. For the WF/CRs reconstruction algorithm the lognormal bias is
treated outside of the Bayesian framework in a separate parallel process
(2015MNRAS.450.2644S; Tully2014; 2016MNRAS.461.4176H; 2021MNRAS.505.3380H).
For Monte Carlo methodologies the lognormal bias is treated within a
comprehensive algorithm (Lavaux2016; 2019MNRAS.488.5438G).

The Constrained Local UniversE Simulations (CLUES) project focuses on the
reconstruction of LSS of our nearby cosmic neighbourhood from surveys of
galactic distances and thereby peculiar velocities, in particular the
Cosmicflows database (cf. Tully2016, and references therein). Two main
methodologies have been employed by the CLUES for the reconstruction of the
local LSS and the setting of initial conditions for constrained cosmological
simulations - one that is based on the WF/CRs methodology (Hoffman and Ribak,
1991, Zaroubi et al 2995) and the the other on MCMC and of Hamiltonian Monte
Carlo (HMC) sampling. In particular, within the WF/CRs framework the issue of
the lognormal bias has been handled by two independent alfgorithms, that of
Sorce (2015) and that of Hoffman et al (2021) and within the Monte Carlo
sampling approach by Graziani et al (2019) and Valade, Hoffman and Libeskind
(2022).

Our aim here to test the quality of two methods that aim to reconstruct the
LSS from peculiar velocities: the WF/CRs method with a lognormal bias
correction algorithm, known as the Bias Gaussian Correction (BGc;
2021MNRAS.505.3380H) and The HAmiltonian Monte carlo reconstruction of the
Local EnvironmenT (Hamlet for short) method XXX. These two methods are applied
to mock data catalogues drawn from a cosmological simulation designed to
imitate the CosmicFlows-3 data(Tully2016). The quality of the two
reconstructions are compared with the so-called “target” simulation to gauge
their fidelity. The New Multidark 2 dark matter (DM) only simulation
(Riebe2013) is used to construct the mock catalogue.

This paper is structured as follows. In Section XXX the Hamlet and WF/CR
reconstructions methods are briefly described. In section XXX the nature of
the input data as well as its biases and a bias correction scheme is
presented. In section XXX the algorithm for constructing halo catalogues that
mock the cosmic flows data is presented. The results of applying to the two
reconstruction methods to these mocks, as well as a comparison between them is
presented in section XXX. A summary and conclusion is offered in section XXX.

##  2 Methods

###  2.1 Mock Catalogue construction

ֿAureliean - please note that in Figs. 1 and 2 the heading uses CF3+ while the
captions refer to CF3. The heading of Fig. 3 uses CF3. In the text I couldn’t
find CF3+. Please clarify the matter and be consistent.
We wish to identify a set of dark matter (DM) haloes from a cosmological
simulations that statistically is as close to the grouped CF3 catalogue as
possible, since it is on this observational data to which the methods tested
here, will eventually be applied. To do so the MultiDark Planck 2
simulation111The MultiDark simulations are publicly available:
www.cosmosim.org (MDPL2, Riebe2013) is used, an $N$-body run of $N=3840^{3}$
particles in a periodic box of side length $L=1\,h^{-1}{\rm Gpc}$. The
cosmological parameters of the simulation are drawn from the 2nd Planck data
release XXX which suggests the “concordance cosmology” i.e. a flat
$\Lambda$CDM Universe ($\Omega_{\rm m}=0.307,~{}\Omega_{\rm
b}Ω0.048,~{}\Omega_{\Lambda}=0.693,~{}\sigma_{8}=0.8228,~{}n_{s}=0.96$ and a
dimensionless Hubble parameter $h=0.678$ where $H_{0}=100~{}h~{}$km/s/Mpc). A
Friend-Of-Friend’s (FOF) algorithm with a linking length of 0.2 times the mean
inter particle separation is used to identify haloes whose mass is roughly
$M_{200}$. It is appropriate to use a FOF halo infder in this case since it is
the grouped CF3 catalogue which is being mocked. Grouping the members of a
virialized object together averages out nonlinear motions implying that the
(e.g.) cluster’s peculiar velocity is a better traces of the flow field. Note
that the MDPL2 box size of $L=1\,h^{-1}{\rm Gpc}$ is large enough to embed the
CF3 catalogue, whose effective depth is roughly $160\,h^{-1}{\rm Mpc}$.

An “observer” is associated with a randomly selected halo of mass in the range
of $[0.9\,\text{---}\,2.0]\times 10^{12}h^{-1}M_{\sun}$. The simulation is
then re-centered on this halo and the simulation’s coordinate axes are then
arbitrarily labelled as Supergalactic (SGX, SGY, SGZ). Furthermore a mock “sky
projection” is made such that each halo is given a sky position (SGL, SGB).

The (proper) distance $d$ of each halo from the center, is used to compute a
cosmological redshift $\bar{z}$ by numerically integrating:

$d=cH{{}^{-1}_{0}}\int_{0}^{\bar{z}}{1\over\sqrt{\Omega_{m}(1+\mathcal{Z})^{3}+(1-\Omega_{m})}}{\rm
d}{\mathcal{Z}}$ (1)

where $\Omega_{m}$ is the cosmological matter density parameter. The (proper)
distance $d$ is also turned into a luminosity distances ${d}_{\text{\tiny L}}$
by

${d}_{\text{\tiny L}}=d\times(1+\bar{z})$ (2)

which is used to compute the halo’s distance modulus:

$\mu=5\log{\bigg{(}\frac{{d}_{\text{\tiny L}}}{\rm Mpc}\bigg{)}}+25$ (3)

The radial peculiar velocity $v_{r}$ is combined with the cosmological
redshift to obtain the full redshift (TamaraDavis)

$z+1=(\bar{z}+1)\bigg{(}\frac{v_{r}}{c}+1\bigg{)}$ (4)

At this point each halo’s position relative to the observer has been
transformed into two “obserrvable” quantities: 1. a redshift $\bar{z}$ (which
includes a contribution from the radial peculiar velocity $v_{r}$) and 2. a
distance modulus $\mu$.

The mock catalogue aims to have the same distribution of $P({\rm SGB})$,
$P({\rm SGL})$ and $P(z)$ as the $N_{CF3}$ data points in the CF3 data. This
is accomplished with a monte-carlo style algorithm in the following way:
$N_{CF3}$ haloes are drawn at random from the simulation, within a sphere of
around 300Mpc$/h$. A merit is assigned to this initial set of haloes by
computing the absolute difference between its distribution (histogram) of
$P({\rm SGB})$, $P({\rm SGL})$ and $P(z)$ and that of CF3. Iterations proceed
by adding and subtracting one halo at a time and evaluating the merit of the
new $P({\rm SGB})$, $P({\rm SGL})$ and $P(z)$, compared to CF3’s. If a new
potential halo improves the merit of the distributions, it is kept; otherwise
it is rejected. In this way, the process converges halo by halo, towards
reproducing the distribution CF3’s $P({\rm SGB})$, $P({\rm SGL})$ and $P(z)$.

Once the process has converged and a suitable mock catalogue has been
constructed, the observational errors from the CF3 catalogue can be added to
the mock. Namely, the redshift and distance modulus of each CF3 data point is
given as $z+\sigma_{z}$ and $\mu+\sigma_{\mu}$ where $\sigma_{z}$ and
$\sigma_{\mu}$ denote the errors associated with each measurement. The error
on $z$ is assumed to be entirely due to the precision of the instrument and
taken as a constant $c\sigma_{z}=50$km/s for every data point while
$\sigma_{\mu}$ depends on which standard candle is used and may range from 5%
for Supernova to 20% for scaling relations. For each CF3 data point a random
number is drawn from a Gaussian of width $c\sigma_{z}=50$km/s and
$\sigma_{\mu}$. These are then added, on a halo by halo basis, to the values
of $\bar{z}$ and $\mu$ assigned to the haloes that constitute the mock. The
final mock catalogue is a list of $N_{CF3}$ halos each assigned a $\bar{z}$
and a $\mu$ from the equations above that have been perturbed according to the
distribution of errors given by CF3.

Figure 3: The distance of a galaxy as a function of its peculiar velocity is
shown for the grouped CF3 data (magenta) as well as the mock catalogue
(green). The lognormal bias is evident here in the lack of symmetry about
$v_{r}=0$; beyond around 70Mpc$/h$ the universe appears to be systematically
collapsing, in a so-called “breathing mode”. Bottom panel: after application
of the BGc correction, symmetry is reestablished. Figure 4: From left to
right: (a) The distributions of the radial peculiar velocity, (b) the distance
and (c) the redshift for the target (black solid lines), the BGc/WF (red
dotted) and Hamlet (Blue dash dotted) reconstruction methods. The Ex/WF method
is shown in green dashed.

The fidelity of the mock to the CF3 catalogue, is shown in Figs. 1 and 2. Fig.
1 shows the three supergalactic projections with the mock data points in green
and the CF3 constraints in purple. The reader will note that the two sets of
points are overplotted, occasionally obscuring the CF3 population. The Zone of
Avoidance (ZOA) and visual distribution of the catalogues are well recovered.
Quantitatively this is shown by looking at the distributions of $P({\rm
SGB})$, $P({\rm SGL})$ and $P(z)$ themselves shown in Fig. 2a, b, and c,
respectively. The distribution of SGB, SGL and $cz$ for the mock galaxies and
CF3 constraints, are largely indistinguishable from each other. Fig. 2c shows
that within $\sim 20$Mpc$/h$, the number of CF3 constraints is much greater
than the mock catalogues presented here. This is because of there are too many
CF3 constraints in this region, with respect to the resolution of our
simulation.

In principle the original unperturbed $d$, $d_{L}$, $\bar{z}$ and $v_{r}$ for
each halo that in the mock can be “forgotten” and new values can be computed
using the values of $\bar{z}$ and $\mu$ that include the observational errors.
These new values should exhibit similar biases to the observational data by
construction. This is seen in LABEL:fig:res:obs_rpv_d, where the radial
velocity as a function of distance is plotted. The diagonal cut in this plot
is indicative of the log normal bias discussed below in section XXX. Namely,
at a given distance there is an unequal number of galaxies moving towards and
away from the observer, making it appear that the universe is contracting. The
scheme employed to correct this bias is presented below and in XXX.

###  2.2 The Lognormal bias and the Bias Gaussian correction (BGc)

A brief note on the so-called lognormal bias. One of the main purposes of
constructing such a detailed mock catalogue as described above, is to ensure
that the lognormal bias is reproduced, thereby allowing us to gauge the
ability of the two reconstruction methods to handle this bias. Much hand
wringing and literature has been devoted to the handling of biases in peculiar
velocity surveys, and we refer the reader to Strauss and Willick (1995)  for a
comprehensive explanation. Here we briefly explain what the lognormal bias is
and how it is handled in the context of the BGc and WF/CR as proposed by
2021MNRAS.505.3380H. We refer the reader to that work for a comprehensive
description of the de-biasing techinque lognormal bias and its correction by
the BGc algorithm.

The reader will note that a Gaussian error on the distance modulus transforms
into a log-normal error on the luminosity distance (e.g. the inverse of Eq.
3). In other words, if the same galaxy is observed many times, the mean of the
different distance measures will not coincide with its actual value. This bias
changes the spatial distribution of the galaxies as well as their inferred
peculiar velocities. The log-normal bias can be seen in Fig. 3 where the CF3
and mock catalogue peculiar velocity $v_{r}$ is plotted as a function of
distance. Beyond around $\sim 70{\rm Mpc}/h$, there is no longer symmetry in
the distribution of $v_{r}$ about zero: more galaxies have negative $v_{r}$
and the universe naively appears to be collapsing, a so-called “breathing”
mode. In theory it can be corrected since the standard $\Lambda$CDM model
makes an explicit prediction that the expected scatter for the radial
component of the velocity is roughly $\sigma_{vp}\sim 275\ {\rm s^{-1}km}$.

The log-normal bias is treated by a correction scheme and the WF/CRs procedure
is applied to the bias-corrected data. The essence of the scheme is to map the
log-normal distribution of the inferred distances around their respective
redshift distances into a normal distribution around the median of the log-
normal one. The width of that normal distribution is treated as a free
parameter set to be $\sim 2h^{-1}\,{\rm Mpc}$, in agreement with the
$\Lambda$CDM prediction that the intrinsic scatter of the radial velocities is
$\sigma_{vp}\sim 275{\rm s^{-1}km}$. The same procedure is applied to the
observed radial velocities, retaining the median of the distribution of the
radial velocities of data points in a given redshift bin. Yet for the
velocities, unlike the inferred distances, the variance of the distribution is
preserved as well. the peculiar velocity in cosmological shells to have
$\sigma_{vp}\sim 275{\rm s^{-1}km}$. Namely the log-normal distribution of the
observed distances is mapped to a Gaussian distribution, while preserving the
median of the log-normal distribution. It is the invariance of the median
under the normal - log-normal transformation which constitutes the backbone of
the BGc scheme. After the application of the BGc scheme to the data, the
breathing mode dissapears and the radial peculiar velocities scatter normally
about 0, as can be seen in Fig. 3, bottom panel.

###  2.3 Wiener Filter and constrained realisation

The WF/CR method is tried and tested algorithm for reconstructing the large
scale density distribution of the universe, based on a limited number of
peculiar velocity measurements. Being roughly 3 decades old, and we refer the
reader to the voluminous literature Hoffman and Ribak (1991), Zaroubi et al
(1995) and Zaroubi, Hoffman and Dekel (1999) for a thorough description,
reviewing just the bare essentials here.

The WF is a Bayesian estimator that finds the most probable 3D velocity field
(and associated density) given a set of observed radial velocities and a given
assumed prior model of the distribution of the peculiar velocities. In
cosmological applications of the WF the $\Lambda$CDM concordance cosmological
model is taken to be the prior model. The WF finds the 3 dimensional
continuous velocity field (and associated density) which is most consistent
with the discrete radial peculiar velocity sample that is its input. The
algorithm is based on a minimal variance method, that assumes a Bayesian
framework and the concordance cosmology. Accordingly: a. The WF provides the
most probable continuous density and 3D velocity fields given a finite number
of observed ’noisy’ radial velocities and the assumed $\Lambda$CDM model. b.
The CRs sample the constrained residual around the WF field. c. The WF and CRs
act so as to interpolate between data points and then extrapolate beyond it.

The WF/CRs methodology recovers the linear density and 3D velocity fields.
Linear theory is clearly violated on small scales, and the density field is
more susceptible to non-linear dynamics than the velocity field. The linear
WF/CRs constitute a reasonable proxy to the actual velocity field down to the
scale of roughly $5-10h^{-1}\,{\rm Mpc}$ (1999ApJ...520..413Z). The WF
estimator is the outcome of the ‘tug-of-war’ between the data and the assumed
prior, $\Lambda$CDM in the present case. Where the data is ‘good and strong’
the estimated WF field is ‘close’ to the input data point. Otherwise, where
the data is weak the WF solution is dominated by the prior model, namely the
solution tends to the null density and velocity fields. Consequently the
constrained variance, i.e variance spanned by the CRs, is small in the good
data regime and converges towards the cosmic variance as the data
deteriorates.

One last remark regarding the BGc/WF method. Because there exist no
possibility to homogenize the sampling (namely the Zone of Avoidance will
always inhibit full sky coverage), the only source of uncertainty that could,
one day, be mitigated, is the observational uncertainty in the distance
measurement. In order to test the BGc/WF’s inherent ability to reconstruct the
underlying fields, an additional “method” is compared: the exact WF (hereafter
labeled Ex/WF). This is the WF applied to a mock where the error on each data
point has been artificially set to zero (and thus no BGc scheme is applied).
This serves the purpose of testing the WF in the case where the only source of
uncertainty is the sampling. In other words in the results section XXX, the
reconstruction based on the BGc/WF, the Ex/WF and the Hamlet method are
presented.

###  2.4 Hamlet

The second approach examined here utilizes a Hamiltonian Monte Carlo sampling
of the posterior PDF via the HAmiltonian Monte carlo reconstruction of the
Local EnvironmenT (Hamlet ) code, which is described in full in an
accompanying paper Valade et al (2022).

Unlike the WF/CRs formalism, the Hamlet algorithm treats the real distances of
the data as unknown dynamical variables that need to be estimated, much in the
same way as the density and velocity field. It is Bayesian in nature because
an _ab initio_ PDF of the variables that are to be estimated can be specified.

A Monte-Carlo technique is used to sample the various posterior PDFs that are
under consideration: the cosmic density field, the velocity field as well as
the distances of the constraining data. The technical challenges of the Monte
Carlo approach is twofold. First the (assumed) prior and posterior PDFs of the
density and velocity field need to be extended to also include the
distribution of datum distances. Then an efficient sampling of the posterior
PDF needs to devised. Given the extremely high dimensionality of the problem,
the Hamiltonian Monte Carlo algorithm is the tool of choice to perform that
sampling, outperforming Markov Chain Monte-Carlo algorithms by many orders of
magnitude (Valade et al 2022).

Beyond its inhomogeneous distribution (e.g. the ZOA) the CF3 catalogue has a
fairly sharp cut off at around  a redshift distance of 150$h^{-1}$Mpc (see
Fig. 2c). In practice this defines and limits the extent within which th
Hamlet reconstruction method is valid. Hinton2017 investigated the problem
that occurs when sampling from a distribution that has an abrupt cut off. They
show that the reconstructed fields (an inferred variables) close to the cut
off will be biased, effectively nullifying the applicability of the
reconstruction beyond a small amount closer than the cut off (say $\sim
90\%$). It is therefore expected that any reconstruction based on these data
will not be valid beyond around 150$h^{-1}$Mpc.

##  3 Results

YH: paragraph slightly revised. The results are presented in three sub-
sections where we 1. compare the predicted constraints themselves differ from
their real values (Section 3.1); 2. examine the accuracy of the reconstruction
of the cosmic fields (Section 3.2); and 3. compare the reconstructed monopole
and dipole (i.e. bulk flow) multipoles with their target counterparts (Section
3.3).

###  3.1 Reconstructed data

After applying the BGc/WF and Hamlet methods to the mock catalogues (as well
as the WF to the exact, no error mocks), the first things to check is how well
the distributions of radial peculiar velocities, distances and redshifts of
the data points, match the input sample their input values. This is shown in
Fig. 4a,b and c, respectively, where the “target” curve (black) represents the
true distributions of the mock catalogue; in otherwords namely, the closer the
BGc/WF (red dotted) or the Hamlet (blue dot-dashed) curve is to the black, the
more accurate the reconstructions. The values of the reconstructed RPVs of the
mock data points are obtained by interpolation over the grid points. Aurelien:
I don’t understand tour Fig. 4b. It shows the histograms of the distances. The
case of the Ex/WF is of distances identical to the target. Yet they are
slightly different. WHat is happening here?  We remind the reader that the
Exact WF (green dashed) represents the limits of the WF method. Fig.4(a) shows
that the Hamlet reconstruction method does an exceptional job at recovering
the distribution of radial peculiar velocities. Note also that the WF in its
purest form too recovers the target distribution. The BGc/WF struggles
slightly by narrowing the data’s distribution with a slight over emphasis on
smaller values of the peculiar velocity at the expense of the large values. We
note, as an aside, that the fact that the target (and hence the
reconstructions) are not centered at $v_{r}=0$ is due to the specific nature
of the mock observer chosen (i.e. cosmic variance). In conclusion, we expect
that BGc/WF reconstructed data to have slightly smaller peculiar velocities
than reality. The BGc/WF suppression of the reconstructed radial velocities of
the data points relative to the target is inherent in the WF algorithm, where
the estimated signal is the weighted ’compromise’ between the data and the
prior model. Where the data is not very strong the WF estimator is biased
towards the null field predicted by the prior.

In Figs. 4(b) and (c) the distance (See my remark above) and redshift
distributions are examined. For both of these quantities the two
reconstructions do a remarkably good job at matching the target, rendering
their curves practically indistinguishable from the target. Note however that
the BGc/WF method tends to “exagerate” some of the peaks and valleys in the
distance distributions (Fig. 4b). All models reliably follow the input’s form.
As a final point note that the Ex/WF is quite literally indistinguishable from
the target. This fact hints that in the absence of errors222An unrealistic but
scientist’s dream. this method perfectly recovers the target: discrepancies
seen are therefore entirely due to the observational errors. In the absence of
errors, i.e. the Ex/WF case, the reconstructed RPVs of the data points should
be equal to the input constraints taken from thhe tharget simulation (Hoffman
and Ribak, 1991). The slight mismatch between the Ex/WF reconstructed values
and the input constraints is due to the interpolation across the coarse grid
and the small but not negligible observational errors on the redshifts.

It is important to understand by how much each constraint shifts during the
de-biasing and reconstruction procedures. In Fig. 5 the difference between the
reconstructed $v_{r}$ and the input $v_{r}$ is compared on a constraint by
constraint basis and as function of distance. From top to bottom this
difference is shown for show the BGc/WF (5a), Hamlet (Fig. 5b), and Ex/WF
(Fig. 5c). The difference between the two main reconstrcution methods (BGc/WF
and Hamlet ) is shown in the final panel, Fig. 5d. In these plots each
constraint is a dot. The median Aurelien: median pr mean? value of the
difference is shown as a black line and the standard deviation of the
distribution is designated with error bars. An examination of Fig. 5, reveals
that the methods based on the WF, tend to underestimate the $v_{r}$ in the
inner most distance shells (below $\leq 60$Mpc$/h$) while overestimating it in
the outer shells. This is ture even for the ideal case of the Ex/WF. The mean
of Hamlet method, however (Fig. 5b) indicates the constraints are not
systematically shifted in the region $\sim 40-110h^{-1}\,{\rm Mpc}$, but
underestimate $v_{r}$ outside this range.

Figure 5: RPV as a function of the comobile distance for the same sets of
constraints. The exact source catalogue, the noisy constraints as well as they
corrected equivalent are presented here. The BGc only source (orange) shows
the constraints before optimization by the WF, while the BGc/WF shows the
results at the end of the full pipeline. Scatter plots of the residual of the
BGc/WF (panel a), of the Hamlet (panel b) and of the WX/WF (panel c)
reconstructed RPVs evaluated at the data points. The residual of the BGc/WF
from the Hamlet reconstructed RPVs at the data points is shown as well (panel
d).

###  3.2 Reconstructed Cosmic Fields

In this section the reconstructed density and velocity fields are examined and
compared with the target.

####  3.2.1 Reconstructed Density maps

Aurelien: I assume that by the density of the target simulation you mean
$\delta_{\rm lin}=-\nabla\cdot\vec{v}/H_{0}f(\Omega)$. If so please correct in
Figs. 6 and 7 to $\delta_{\rm lin}$ where the target density field is
considered.
In order to visually inspect the reconstructed density distribution, a
4Mpc$/h$ thick smoothed slab at the super galactic plane (SGZ=0) is chosen.
This is not an arbitrary choice: given that the largest numbers of constraints
are expected to lie in or close to SGZ=0, we expect this slab to be the most
important (and possibly the most accurate) for reconstructions.

Figure 6: Fields, uncertainties and pool maps $\delta$ for the ${\rm SGZ}=0$
plan. From top to bottom : BGc/WF, Ex/WF, HMC. A comparison of the of the
BGc/WF (left column), Ex/WF (central column) and the Hamlet (right column)
reconstructed (over)density fields with the target simulation. The top panel
presents the linearized density, $\delta_{\rm lin}$, of the target simulation.
The middle panels present the reconstructed $\delta$ and the bottom ones show
the constrained variance normalized by the cosmic variance,
$\Sigma_{\delta}/\sigma_{\delta}$. All plots refer to the $SGZ=0$ plane of the
target simulation and all fields are Gaussian smoothed with a $5h^{-1}\,{\rm
Mpc}$ kernel.

Fig. 6 examines the density distribution in this slab. The different plots
show the fractional overdensity, $\delta=(\rho/\bar{\rho}-1$, where
$\bar{\rho}$ is the cosmological mean density. For the target non-linear
density field the fractional over density is replaced by $\delta_{\rm
lin}=-\nabla\cdot\vec{v}/H_{0}f(\Omega)$ which coincides with $\delta$ in the
linear regime, where $f(\Omega)$ is the linear growth factor. The $\delta$
fields are Gaussian smoothed with a kernel of $5h^{-1}\,{\rm Mpc}$. The top
panel (a) is the target density distribution. The column below it (namely the
middle column, Fig. 6c,f) shows the Ex/WF results, while the left column (Fig.
6b,e) shows the BGc/WF results and the right column (Fig. 6,d,g) shows the
Hamlet Hamlet results. The middle row (panels b,c,d) shows the reconstructed
density distribution. Some conclusions may be drawn from a visual examination
of Fig. 6b,c,d): The Ex/WF generally recovers the features of the local
cosmography at all distances. The reconstruction is not exact; given that
there are no “observational” errors here, this implies that the mismatch
between the Ex/WF and the target (i.e. between Fig. 6c and Fig. 6a), are
entirely due to the finite, inhomogenous and anisotropic sampling. Comparing
the BGc/WF (Fig. 6b) with the target indicates a steady deterioration of the
accuracy of the cosmographic reconstructions with distance decline of power o
the reconstructed density field with the distance from the observer, yet the
general structure of the cosmic web of over- and under-dense regions is
recovered. The HAMLET reconstructed $\delta$ field does not exhibit the same
loss of power as in the BGc/WF case but it suffers from a loss of spatial
resolution with distance (Fig. 6d). The more distant structures become more
fuzzy and diffuse with distance. The same deterioration is not as severe (but
nevertheless present) in the HAMLET method (Fig. 6d), which fairly accurately
reproduces the main features of the target at greater distances than the
BGc/WF method.

The bottom panels of Fig. 6 present the constrained variance, $\delta$ fields,
$\Sigma^{2}_{\delta}$, of the three reconstructed. It is defined as the local,
cell by cell, variance calculated over an ensemble of CRs for the Ex/WF and
BGc/WF case and over a set of independent states of the Monte Carlo chain in
the HAMLET case. The panels show the square root of constrained variance
normalized by the cosmic variance, $\Sigma_{\delta}/\sigma_{\delta}$. The
cosmic variance is calculated by calculating the variance over all CIC cells
in the target simulation. The value of $\Sigma_{\delta}/\sigma_{\delta}$
gauges the constraining power of the constraints and the assumed prior model.
One expects it to be small at small distances and to approach unity
asymptotically with distance. The bottom row of Fig. 6, (namely Fig. 6e,f,g)
quantifies the how well constrained a region is by comparing the variance in
the reconstructions ($\Sigma_{\delta}$) with cosmic variance
($\sigma_{\delta}$). What is plotted here is their ratio
$\Sigma_{\delta}/\sigma_{\delta}$. The constrained variance is determined, on
a cell by cell basis, by averaging over all the constrained realizations of
the density field. Each cell thus has a mean value of the reconstructed
density and a standard deviation about this value. The standard drviation is
what is meant here by “constrained variance” (i.e.$\Sigma_{\delta}$) The
cosmic variance is just the standard deviation of the density field of a
$\Lambda$ CDM universe at $z$=0. This can be computed in many ways. In this
case, it is found by simply compting the standard deviation of all the cells
in the target simulation and is thus a fixed number. When
$\Sigma_{\delta}=\sigma_{\delta}$, the cell is essentially unconstrained,
since the variance in the density field across all constrained realizations is
equal to the cosmic variance. Thus when $\Sigma_{\delta}/\sigma_{\delta}=1$
the region is fully unconstrained while when
$\Sigma_{\delta}/\sigma_{\delta}=0$ the region is extremely well constrained.

Fig. 6e,f,g) quantifies what is visually apparent from Fig. 6b,c,d) namely
that the inner regions are well constrained but that this fades with
increasing distance. The reconstruction methods that include errors (i.e. Fig.
6e,g) are never “perfect”, while the Ex/WF method Fig. 6f), does obtain values
of $\Sigma_{\delta}/\sigma_{\delta}$ close to 0. Interestingly, the impact of
the Zone of Avoidance (ZoA) on the reconstruction method is glaringly obvious
in Fig. 6(f). Here the simple effect of being unable to sample the ZoA, causes
a very clear limitation of the expected ability to reconstruct the density
field.

The accuracy of the density field reconstructions - specifically their
accuracy as a function of distance \- is shown in Fig. 7. These are scatter
plots which compare, on a cell by cell basis, the density of the target with
the BGc/WF (top row), Ex/WF (middle row) and HAMLET (bottom row). The line,
$y=ax+b$, which best describes the correlation is shown in red; its slope and
the Pearson correlation coefficient is given in each sub-panel. In the ideal
case where a reconstruction method perfectly matches the target this would
simply be the $y=x$ case a slope of $a=1$ and an offset or bias of $b=0$ line
with zero scatter (shown in black), namely a Pearson correlation coefficient
of unity. The columns in this figure denote different 40 Mpc$/h$ thick radial
shells under consideration. Note that a slope less than unity indicates that
the reconstruction under estimates the over dense regions and over estimates
the under dense regions. A slope greater than unity represents the opposite
(exaggerates over and under dense regions). An offset of $b\neq 0$ means a
biased reconstruction.

There are a number of important features of this Fig. 7. First, considering
the inner most bin (leftmost column) the Ex/WF reconstruction recovers very
well the density of the target. A slope of unity  and practically null offset
of $b=0.01$ and a correlation coefficient of 0.92 indicates that in general in
this region the Ex/WF reconstructions is  very well recovered. This implies
that the nearby sampling of the CF3 catalog is almost optimal. Obviously, the
HAMLET and the BGc/WF methods do worse in recovering the density field. Moving
to the outer shells all three density reconstruction systematically degrade
with slopes decreasing and correlation coefficients increasing. The reader
will note that the slop in all cases is less than unity, indicating that the
reconstructions suppress high density regions and enhance low density ones the
power of the recovered density field. This effect increases with increasing
distance fro the observer diminishing power increases with the distance from
the observer. This is because in the limit of no data, the reconstruction
methods recover the mean field. This effect is acutely visible in the
rightmost column where the slope of the BGc/WF method approached 0. The
conclusion drawn here is that given the enormous errors at these distances,
the reconstruction methods essentially recover the mean field and are no
longer good realizations of field.  The BGc/WF suffers from that lose of
correlation with distance than the HAMLET. Yet, the latter reconstruction is
quite biased at large distances with $b=0.12$ for the distance range of $120\
\leq\ d\ \leq\ 160\ hmpc$. The BGC/WF behaves, on the other hand, by the
’Bayesian book’ - where the sampling is very sparse and the errors are much
larger than the signal, the unbiased prior is recovered.

I suggest to remove the following paragraph
Looking at Fig. 7 more quantitatively, specifically comparing the middle row
to the top row, shows the effect of errors on the BGc/WF reconstruction method
compared to the limitations of the sampling. Namely even with no errors, we
expect inaccuracies at large distances simply via the sampling issues
previously elucidated. The bottom row indicates that the HAMLET is remarkably
successful at reconstructing the density field even at large distances. Its
slope and correlation coefficient is closer to unity than the BGc/WF method at
all distances (albeit not as close as the Ex/WF method).

Figure 7: Density scatter plots of $\delta$ reconstructed versus $\delta$
target. Rows from top to bottom : HMC, Ex/WF BGc/WF. Columns from left to
right: within spheres of 50, 100, 150, 200 $h^{-1}{\rm Mpc}$. The red line
represents the best fitted line whose line equation is $y=ax+b$. The
parameters of the line and the Pearson correlation coefficient are given in
the insert. The black line of a slope of a slope of unity and no offset is
shown for reference.
Aurelien: please change the xtitle from ’$\delta$ target’ to ’$\delta_{\rm
lin}$ target’.

The correlation between the reconstructed mean field and the target is shown
as a function of distance in Fig. 8. Essentially these are just the
correlation coefficients from the scatter plots (Fig. 7) plotted as a function
of distance in order to gauge the degradation of the reconstruction methods as
data becomes more sparse and volumes become large. The binning is also finer
(20 vs 40 Mpc$/h/$ shell thickness, hence the exact values of the correlation
coefficient are not identical). The solid lines in Fig. 8 represent the mean
correlation coefficient between the reconstruction and the target; the error
corridor represents the 2$\sigma$ variance about this mean. Two main
conclusions can be drawn here: as expected the Ex/WF is always a superior to
the BGc/WF and the HAMLET method. With the exception of the inner most bin,
the Hamlet method achieved higher correlation coefficients than the BGc/WF
method. At the edge of the data, no method achieves a correlation coefficient
of greater than 0.5

Figure 8: Statistics per shell of distance. Absolute value of the coefficient
of correlation for $\delta$ between the different reconstructions and the
target field. The error bars represent the envelope represents the $2\sigma$
variation on of the ensemble of realizations.

###  3.3 Reconstructed radial velocity maps

In this section we examine the reconstructed radial peculiar velocity field in
the same way that the density field was examined in the previous section The
examination of the radial component of the velocity field follows here that of
the density field (§3.2.1).  The same SGZ=0, smoothed, 4Mpc$/h$ thick slab of
the supergalactic plane is shown in Fig. 9. Again, the top panel is the target
(mock) radial peculiar velocity field while the left column shows the BGc/WF
reconstruction, the middle column the Ex/WF reconstruction and the right most
column, the Hamlet reconstruction.

By visual inspection the reader will note that the radial velocity field (Fig.
9b,c,d) is much more accurately reconstructed than the density field. The same
outflows and inflows are generally visible and the cosmographic landscape is
recognisable in all three cases. Although the reader will note that the
accuracy of the velocity reconstruction, like the density field, deteriorates
at larger distances. Features are recognisable but distorted and smoothed out.

As above, in Fig. 9(e,f,g) the constrained variance of the radial peculiar
velocity field is compared to the cosmic variance of the same quantity. As
stated in section Section 3.2.1, the quantification of how well constrained a
region of the reconstruction is, is computed by the ratio of
$\Sigma_{v_{r}}/\sigma_{v_{r}}$. Here $\Sigma_{v_{r}}$ is the standard
deviation, for a given cell, of the reconstructed peculiar velocity, averaged
over all constrained realizations and $\sigma_{v_{r}}$ is the cosmic variance
of this quantity in a $\Lambda$CDM Universe obtained by simply computing the
mean radial peculiar velocity of each cell (in a reconstruction) and the
corresponding standard deviation of all cells. As such the ratio
$\Sigma_{v_{r}}/\sigma_{v_{r}}$=1 when the peculiar velocity is fully
unconstrained, and thus varies as much as a random cosmography, and equal to
zero when fully constrained with no differences across the constrained
realizations. The reader will note that these plots are considerably “darker”
than their density field counterparts in Fig. 6, indicating that the radial
peculiar velocity field is much better constrained than the density field. Our
inability to reconstruct within the ZoA is still visible in the radial
peculiar velocity albeit much less prominent than as in the density maps. What
is very different between the reconstructed density and peculiar velocity
field is the accuracy at the edge of the data: whereas the density field
reconstructions become more or less random or at least statistically
indistinguishable from random, the reconstructed peculiar velocity fields
barely obtain such large errors. Fig. 9(g) for instance never gets to
$\Sigma_{v_{r}}/\sigma_{v_{r}}$=1. The constrained and cosmic variances of the
radial velocity, $\Sigma_{v_{r}}$ and $\sigma_{v_{r}}$ respectively, are
calculated much in the same way as in for the density field (§3.2.1). The
imprint of the ZOA is clearly seen in the $\Sigma_{v_{r}}/\sigma_{v_{r}}$ map
of the Ex/WF map. Yet, in all cases considered here the constrained variance,
normalized by the cosmic variance, is much smaller than in density case.
Namely, the velocity field is much more constrained by the CF3 data than the
density field. Also, the Hamlet outperforms the BGc/WF reconstruction.

A close inspection of Fig. 9g uncovers one troubling feature. At the edge of
the reconstructed volume, at distances close to $150\ h^{-1}\,{\rm Mpc}$ the
reconstructed is ’bluer’ than in the corresponding target and the Ex/WF maps.
Namely the Hamlet reconstructed velocity field has a spurious negative infall.

Again, we turn to a scatter plot, on a cell by cell basis to quantify the
quality of the reconstruction as a function of distance in Fig. 10, which Fig.
10 is structured identically to Fig. 7 \- namely radial extent increasing
column wise from left to right, while the rows from top to bottom being
BGc/WF, Ex/WF and Hamlet . This figure is qualitatively also identical to its
density field counter part (Fig. 7) in that the same behavioural trends
between the different reconstructions methods and as a function of distance
exist. The main difference however is the quality of the radial velocity field
reconstructions which are superior to the density field in almost every way.
In most cases the slope and Pearson correlation coefficient is much closer to
unity. The degradation as a function of distance is not as rapid. Of specific
remarkable interest is that fact that the Hamlet method has a strong
correlation coefficient of above 0.8 even 80 to 100 Mpc$/h$ away. These plots
indicate that when it comes to examining radial velocity field, the methods
presented here do an excellent job, despite the limitations of sampling and
observational errors. The correlation analysis of the Ex/WF and BGc/WF cases
behaves much in the same way as for the density field - a degradation of the
correlation with distance, a slope ($a$) that is close to unity nearby and
diminishes with distance, and essentially with zero offset ($b\sim 0$). Yet,
the quality of the reconstruction of the radial velocity is much better than
that of the density. The Hamlet reconstruction shows a somewhat unexpected
behaviour. The slope of the best fit line for the distance range of $40\ \leq\
d\ \leq\ 80\ h^{-1}\,{\rm Mpc}$ exceeds unity, $a=1.22$, i.e. there is an
excess of power compared with the target and the Ex/WF cases. This is
unexpected for a Bayesian algorithm. The best linear fit for the range of
$120\ \leq\ d\ \leq\ 160\ h^{-1}\,{\rm Mpc}$ yields a significant negative
offset of $b=-151{\rm s^{-1}km}$, in agreement with the visual inspection of
Fig. 9g.

The correlation between the reconstructed radial velocity field and the target
is shown as a function of distance in Fig. 11, similar to Fig. 8. Essentially
these are just the correlation coefficients from the scatter plots
(LABEL:fig:scatters_vr) plotted as a function of distance. The green dashed,
red dotted and blue dot dashed lines in Fig. 8 represent the mean correlation
coefficient between the BGc/WF, Ex/WF and Hamlet reconstruction methods
(respectively) and the target; the error corridor represents the 2$\sigma$
variance about this mean. As expected, and similar to the behaviour of the
density field in Fig. 8, the Ex/WF is always a superior to the BGc/WF and the
HAMLET method. In fact, the radial peculiar velocity field is extraordinarily
well reconstructed by the Ex/WF, out to the edge of the data: even at $\sim
150$Mpc$/h$, the Ex/WF achieves a correlation with the target of $\sim 80\%$.
The fact that at these distances the BGc/WF drops to correlations of just
$\sim 20\%$, indicates how errors are critical in these reconstructions.
Again, as is with the case of the density field, the HAMLET method achieves
higher correlation coefficients than the BGc/WF method everywhere except the
inner most bin. Note that insofar as reconstructing the velocity field goes
the Hamlet and BGc/WF method are roughly equal out to $\sim 70$Mpc$/h$.

###  3.4 Multipole moments of the reconstructed velocity field

Aurelien: are you showing here $-\nabla\cdot\vec{v}/H_{0}f(\Omega)$ or
$-\nabla\cdot\vec{v}/H_{0}$ as the monoploe term?
The first two moments of the velocity field, the monopole and dipole, are
examined here. The effect of errors and sampling on the fidelity of these two
physical quantities is of particular interest since the monopole and dipole
are often used as probes of the scale of homogeneity and can effect probes of
the cosmological model in particular.

Fig. 12(a) shows the target and reconstructed velocity monopole as a function
of distance. The same colouring and line style convention used in Fig. 8 and
Fig. 11 is adopted here too, with the moments of the target simulation plotted
in black. Note that the monopole - the mean infall or outflow of matter, is
the zeroth order moment of the velocity field. Namely it is simply the mean of
the divergence of the velocity field in spheres of radius $d$ and as such is
called the “breathing mode” of the velocity field. In the linear theory of the
cosmological gravitational instability the density and velocity fields are
related by $\delta=-\nabla\cdot\vec{v}/H_{0}f(\Omega)$, hence we opted here to
present the monopole term by means of
$\delta=-\nabla\cdot\vec{v}/H_{0}f(\Omega)$. Thereby, Fig. 12a effectively
presents the mean linear density with spheres of radius $d$.  (divided by the
growth factor times the Hubble constant) and as such is often called the
“breathing mode”. The Ex/WF is nearly indistinguishable from the target here:
the error corridor (which corresponds to variance across all the constrained
realisations) is tiny and the black and green dashed line are practically on
top of each other. As for the more realistic cases of the Hamlet and BGc/WF
curves, they succeed and fail in different ways. The BGc/WF curve over
estimates the monopole in the inner parts (within $\sim 50\ h^{-1}\,{\rm
Mpc}$) while underestimating it outside that range. This increased monopole
implies an overestimation of the density in the inner parts of the mock
universe - this is confirmed by examining the equation of the best fit line to
the scatter plot Fig. 7 (upper row, left column, $d<40h^{-1}\,{\rm Mpc}$). The
best fit line has a large $y$-intercept (0.11) has an offset of $b=0.11$,
meaning that there is a systematic increase in the estimated densities,
consistent with the artificially higher monopole. The fact that the BGc/WF
method and the target both approach zero monopole at around the 100
$h^{-1}\,{\rm Mpc}$ mark, is a result of the $\Lambda$CDM Copernican nature of
the universe on these scales (Noam - this has nothing to do with Copernican
nature). Both the reconstructions and the target tend to zero infall at these
large scales. The Hamlet method on the other hand behaves inversely to the
BGc/WF method, underestimating the target monopole at small scales and over
estimating it at large scales. Similar to the BGc/WF over-estimation at small
scales, the Hamlet method too has a high $y$-intercept for the most distnt
regions of the reconstruction (ie in the bottom right panel of
LABEL:fig:scatters_divv). The Hamlet ’s monopole term at the edge of the data
reveals an excess of density at $d\sim(120-150)\ h^{-1}\,{\rm Mpc}$, in
agreement with LABEL:fig:scatters_divv (lower/right panel). Otherwise, the
Hamlet method succeeds in tracking the target monopole over a large range from
$\sim 20$ to $\sim 100\ h^{-1}\,{\rm Mpc}$.

Fig. 12(b,c) refer to the second moment of the velocity field, namely the
dipole or the bulk flow. Fig. 12(b) refers to the magnitude of the bulk flow,
while Fig. 12(c) refers to its direction. Accordingly all methods do a fine
job of recovering the magnitude of the bulk flow beyond around $\sim
30$Mpc$/h$. The Ex/WF has, predictably, a smaller error corridor than the
other two methods, which are roughly similar in size. With respect to
direction, Fig. 12(c0 shows the dot product between the target bulk flow
direction and the reconstructed one (hence in this plot there is no black
target line). The bulk flow directions for the Hamlet and BGc/WF method are
aligned to within $\sim 15$ deg of the target out to a distance of $\sim
50$Mpc$\h$, while the Ex/WF is well aligned to greater distances. Note that
however, even the Ex/WF curve begins to deviate significantly at the
reconstructed edge. This indicates that even in the best case scenario of zero
errors, sampling at these great distances is a limiting factor in terms of
recovering the cosmic dipole. The problem is (obviously) exacerbated when
examining the BGc/WF and Hamlet curves at large distances. Taken together Fig.
12 indicates that although the monopole and dipole are well recovered across a
large range, the direction of the reconstructed dipole begins to deteriorate
when the sampling drops.

Figure 9: Fields, uncertainties and pool maps $v_{r}$ for the ${\rm SGZ}=0$
plan. From top to bottom : BGc/WF, Ex/WF, HMC.
Figure 10: Scatter plots of the target $v_{r}$ versus the reconstructed
$v_{r}$. Rows from top to bottom : HMC, Ex/WF BGc/WF. Columns from left to
right indicate 40 $h^{-1}{\rm Mpc}$ shells with cinreasing distance from the
observer. The red line represents the best fitted line whose line equation is
$y=ax+b$. The parameters of the line and the Pearson correlation coefficient
are given in the insert. The black line of a slope of a slope of unity and no
offset is shown for reference. Figure 11: Statistics per shell of distance.
Absolute value of the coefficient of correlation for $v_{r}$ between the
different reconstructions and the target field. The error bars represent the
variation on realizations. Figure 12: The monopole moment (upper panel), the
amplitude of the dipole moment (i.e. the bulk velocity; middle panel), and the
cosine of the angle of alignment between the reconstructed and target bulk
velocities (lower panel) are shown. The profiles present the mean and
‘$1\sigma$’ scatter of the mean profile in spheres of radius $d$. The
reconstructions correspond here to the Ex/WF (green dashed line), the BGc/WF
(red dotted line) and the Hamlet (blue dot-dashed line) case. The scatter is
the constrained variance of the different reconstruction and the target
simulation is presented by the black solid line (middle and upper panels).
Aurelien - note that I ask you to redraw the plot ad use cosine angle.

##  4 Conclusion Summary

The reconstruction of the large scale density and velocity fields from
Cosmicflows-like databases of galaxy distances, and hence peculiar radial
velocities, is challenging. The data is sparse, extremely noisy with
Noise/Signal ratio larger than a few for the majority of the data, non-
uniformly and anisotropically distributed. Furthermore that data suffers from
the lognormal bias, which leads to a non-linear bias in the estimated
distances and velocities. The CLUES project focuses on the reconstruction of
the local LSS and the setting of constrained initial conditions for
cosmological simulations designed to reproduce our local patch of the
Universe. Three main algorithms are used within the CLUES to ‘unbias’ the data
and perform the reconstruction: the bias minimization of Sorce (2015), the
BGc/WF of Hoffman et al (2021) and the Hamlet of Valade, Hoffman and Libeskind
(2022). The present paper compare the BGc/WF and the Hamlet algorithms by
testing them against mock CF3-like surveys draw from one the MultiDark
cosmological simulations.

The quality of the reconstruction is gauged by studying the residual between
the reconstructed and target density and velocity fields. The residual is
mostly analyzed by quadratic measures and as such it is characterized by the
mean and variance of the distribution. An optimal reconstruction should make
the mean of the residual to be as close as possible to the null field and aim
at minimizing its variance. A related measure is the linear correlation
analysis which yields the best ’line’, $y=ax+b$, that fits the linear
dependence of reconstructed field on the target one, and the Pearson
correlation coefficient. The values of the offset, $b$, for the case of the
linear over-density and for the radial velocity are consistent with zero fro
the BGc/WF, in agreement with the theoretical expectations. The distant data
points are extremely noisy and very sparsely distributed, hence the WF
reconstruction is domiated by the $\Lambda$CDM prior model. The Hamlet ’s
significant offset is inconsistent with the prior model.

We define here three different regions: the nearby ($d\lesssim 40h^{-1}\,{\rm
Mpc}$), the intermediate (($40\lesssim d\lesssim 120h^{-1}\,{\rm Mpc}$) and
the distant one ($d\gtrsim 120h^{-1}\,{\rm Mpc}$). Based on the above criteria
we conclude that nearby the BGc/WF and the Hamlet methods are doing roughly
equally well. The methods diverge at large distance - with the Hamlet
outperforming the BGc/WF with a tighter correlation and smaller variance but
underperforming in terms of the bias. This is most noticeable for distant
region (the right columns of Figs. 7 and 11).

The three panels of Fig. 12 deserve a special attention here. The upper panel
shows the radial profile of the monopole moment. The four profiles shown there
- target, Ex?WF, BGc/WF and Hamlet \- are all constructed under the assumption
of $\Lambda$CDM value of $H_{0}=67.7\,\rm km\,s^{-1}Mpc$. Yet, the negative
offset of the monopole moment at the edge of the data implies that the local
value of $H_{0}$ is somewhat smaller than it is global value. A phenomenon
expected for any finite volume realization in the $\Lambda$CDM cosmology (see
Hoffman et al, 2021 for a quantitative assessment). A proper adjustment of the
local value of $H_{0}$ would bring the target and Ex/WF profiles to converge
to zero at the edge of the data, together with the BGc/WF asymptotic value.
This would leave the Hamlet positive offset standing out with a systematic
bias. The amplitude of the dipole moment, namely the bulk velocity, is
recovered equally well by the three reconstruction in very good agreement with
the target. The bottom panel shows the cosine of the angle between the
reconstructed and the target bulk velocities. The BGc/WF behaves as expected -
the mean misalignment is consistent with the full alignment to within one
$\sigma$ of the constrained variance. This is not the case with the Hamlet
reconstruction, where the misalignment is more than 1 $\sigma$ away from the
expected alignment.
Aurelien - please note. I’m suggesting to present the alignment of the vectors
by means of the cosnine of the angle. For obvious reasons.

Our overall assessment of the Hamlet and the BGc/WF reconstructions is that
the former outperforms the latter one in terms of reduced scatter and tighter
correlation between the reconstructed and the target density and velocity
fields. Yet, the Hamlet suffers from biases in the reconstructed LSS at the
distant regime - ones that do not appear in the BGc/WF reconstruction. It
follows that the Hamlet should be the method of choice for the reconstruction
of the LSS and the study of the cosmography of our local patch of the
Universe. The BGc/WF reconstruction is the preferred tool for performing
quantitative analysis and parameters estimation and possibly also for setting
initial conditions for constrained cosmological simulations. One last comment
is due here. The WF/CRs is a very well tested approach that is based on a
solid theoretical foundations (Hoffman and Ribak, 1991, Zaroubi et al, 1995,
Zaroubi, Hoffman and Dekel 1999). As such it provides an attractive framework
for perfoming Bayesian reconstruction of the nearby LSS. Yet, any bias in the
observational data and in particular the lognormal one needs to be addressed
and apply outside that framework in some ad-hoc and approximate way. The HMC
methodology, and in particular its Hamlet implementation, still suffer from
some teething problems that need to be overcome. The ability of the MCMCM
methodology in general and the HMC in particular to address the issue of
reconstruction of the LSS, the handling of observational biases and the
estimation of cosmological parameters withing one computational self-
consistent framework make the Hamlet a very attractive tool in the CLUES’
toolbox. The incredible improvement in the CPU efficiency of the Hamlet
compared with previous implementation of MCMC algorithms (Valade et al 2022)
makes it even more promising for future implementations within the CLUES
project.

Noam, Aurelien - this Conclusion reads more like an Introduction. I suggest to
replace it entirely.

Mapping the large scale distribution of matter in the Universe from scant,
sparse observational data is an attractive problem for a multitude of reasons.
For one, mapping ones environment is a way to understand it, and thus one’s
place in it. More importantly, depending on the nature of the tracers, the
estimation of a continuous distribution from a limited discrete sample of
tracers, is a well formulated mathematical problem.

There are different ways to estimate the cosmic density field (XXXX). Many of
these methods, justifiably, rely on measuring the density of light, typically
by using large spectroscopic surveys. Although such methods have the advantage
of hundreds of thousands, if not tens of millions (e.g. the EUCLID
spectroscopic survey) of tracers, they suffer from some drawbacks: red shift
space distortions and unobserved regions like the ZoA limit the ability to map
the entire universe. Galaxy peculiar velocities on the other hand suffer from
neither of these draw backs since a galaxy’s peculiar velocity is a result of
the entire mass distribution in the universe. Thus a relatively small number
of tracers are needed. Not only that, but structures can be inferred where
they cant be seen by looking at the peculiar velocity field and there is a
long history in cosmology of such discoveries (from the great attractor XXX to
the vela supercluster Kran-Korteweg)

But how accurate are these methods? Measuring a galaxy’s peculiar velocity is
a challenge that requires both spectroscopy and photometry thus cosmic
velocity sample sizes are small (CF3 has $\sim 10^{4}$ data points compared
with two orders of magnitude more data in spectrosopic surveys); whats more:
errors on the distances can be very large (i.e up to 20% with standard scaling
relation). Given these limitations how well do such methods do in
reconstructing the underlying cosmic fields?

In this work we focus on two methods that claim to reconstruct the comic
distribution of matter, and thus the full velocity field by using measurements
of galaxy distances: the Hamlet method (Valade et al 2022) and the Bias
Gaussian Corrected Wiener Filter method (Hoffman et al 2021). Details on how
these methods work are presented first and foremost in those two paper but
also in sections 2.2, 2.3 and 2.4.

The basis of this paper is to apply these two methods to a mockj cataloge
drawn from a numerical simulations such that we know the underlying density
and velocity field and we can compare the reconstruced one rto the true one
(which we call the “target”). A mock catalogue is constructed by “observing"
haloes in a simulation in a way which mimics the real CosmicFlows data. The
methods are then applied to the mock data and the resulting reconstructions
are then gauged and checked against the simulation’s density and velocity
field. The existence of the underlying mock universe allows us to extrapolate
how these methods perform on real data.

Although many of our results employ the human eye as judge (i.e. Fig 6 and 9)
the fidelity is quantified as well. In general the main conclusion is that
these two method do better in recovering the radial velocity field than the
density field. This is perhaps no surprising since the radial velocity field
is what is measured, however the measurements are error prone. In the
unrealistic case of no observational errors, the Wiener Filter radial velocity
field and the input data are nearly indistinguishable. When errors are
conisdered the two methods recover the input very well within $\sim
70h^{-1}{\rm Mpc}$. It turns out that the density field too is well recovered
in a region thats slightly smaller - around 50h^-1Mpc.

extra ordinarly well in reconstructing the density and velocity field where
there is good data \- good data here meaning a well sampled field namely in
the inner parts. The main features are well recovered

In this paper we have tested in great detail two methods that reconstruct the
cosmic density and velocity field from measurements of pecukiar velocities,
similar to the cosmic flows project.

## Acknowledgements

Useful discussions with Tamara Davis, concerning the Hinton2017 paper, are
acknowledged. This work has been done within the framework of the Constrained
Local UniversE Simulations (CLUES) simulations. AV and NIL acknowledge
financial support from the Project IDEXLYON at the University of Lyon under
the Investments for the Future Program (ANR-16-IDEX-0005). YH has been
partially supported by the Israel Science Foundation grant ISF 1358/18.

## Data availability

The data underlying this article will be shared on reasonable request to the
corresponding author.


Tsemo Aristide

PKFokam Institute of Excellence,

Yaounde, Cameroon.

<EMAIL_ADDRESS>

Globally linearizable actions on topological manifolds.

Abstract.

Let $M$ be a finite dimensional topological aspherical manifold whose
universal cover is ${\bf R}^{n}$. In this paper, we study $Aff(M)$, the
subgroup of the group of homeomorphisms of $M$, whose elements can be lifted
to affine transformations of ${\bf R}^{n}$. We show that if $M$ is closed, the
connected component $Aff(M)_{0}$ of $Aff(M)$ acts locally freely on $M$. We
deduce that $Aff(M)_{0}$ is a solvable Lie group, and is nilpotent if $M$ is a
polynomial manifold. We study the foliation defined by the orbits of
$Aff(M)_{0}$ if $dim(Aff(M)_{0})=dim(M)-1$.

I. Introduction.

Let $M$ be a topological $n$-dimensional manifold, and $G$ a topological group
which acts on $M$. The action of $G$ on $M$ is linearizable at $x$, if there
exists a neighbourhood $U$ of $x$ such that $U$ is stable by $G$, and the
restriction of $G$ to $U$ is conjugated to a linear action. If $M$ is the
Euclidean space ${\bf R}^{n}$, we say that the action is globally linearizable
if we can choose $U={\bf R}^{n}$. Remark that an infinitesimal version of this
problem has been studied by Poincare in his thesis. He has given conditions
which ensure that an ordinary nonlinear differential equation is equivalent to
a linear differential equation. A generalization of this Poincare’s problem is
given by a finite set $L$ of differential equations which form a Lie algebra
${\cal L}$. Suppose that a group $G$ acts freely, properly and discontinuously
by diffeomorphisms on ${\bf R}^{n}$, and the equations of $L$ are invariant by
$G$. This implies that the quotient of ${\bf R}^{n}$ by $G$ is a manifold $M$,
and there exists a Lie algebra ${\cal L}_{G}$ of $M$ whose lifts by the
universal covering map of $M$ is ${\cal L}$. If ${\cal L}$ is conjugated by a
diffeomorphism of ${\bf R}^{n}$ to an affine system, or equivalently to a Lie
algebra contained in the Lie algebra of the group of affine transformations of
${\bf R}^{n}$, we say that ${\cal L}$ is globally linearizable. Remark that,
if $M$ is compact, ${\cal L}_{G}$ is the image of an infinitesimal action of a
simply connected Lie group on $M$, whose action can be lifted to ${\bf
R}^{n}$.

In this paper, we study a topological version of global linearizable actions.
Let $M$ be an $n$-dimensional topological manifold whose universal cover is
homeomorphic to ${\bf R}^{n}$. We denote by $\pi_{1}(M)$ the fundamental group
of $M$. We consider the group $Aff(M)$ of homeomorphisms of $M$ which can be
lifted to affine maps of ${\bf R}^{n}$. We characterize the subgroup
$N(\pi_{1}(M))$ of homeomorphisms of ${\bf R}^{n}$ which are lifts of elements
of $Aff(M)$ by showing that it is generated by its normal subgroup
$\pi_{1}(M)$ and a subgroup of the group of affine transformations of ${\bf
R}^{n}$. We adapt a result of Fried, Goldman and Hirsch by showing that if $M$
is closed, the action of $\pi_{1}(M)$ on ${\bf R}^{n}$ is irreducible, that
is, does not preserve any proper non empty affine subset. This enables to show
that the action of the connected component $Aff(M)_{0}$ of $Aff(M)$ on $M$ is
locally free. We deduce that if $M$ is a compact $Aff(M)_{0}$ is solvable, and
if $M$ is a compact polynomial manifold, $Aff(M)_{0}$ is nilpotent. We remark
that if $Aff(M)_{0}$ acts transitively on $M$, then $M$ is a solvmanifold
endowed with an affine structure. Finally, we determine the topology of $M$,
if $M$ is differentiable and $dim(Aff(M)_{0})=dim(M)-1$.

2\. General properties of the group of affine transformations of a topological
manifold.

In this section, we are going to define affine transformations of topological
manifolds and study some of their general properties.

Definition 2.1. A group of homeomorphisms $G$ of the Euclidean space ${\bf
R}^{n}$ acts freely and properly discontinuously on ${\bf R}^{n}$ if and only
if:

For every elements $x$ of ${\bf R}^{n}$, and $\gamma$ of $G$, $\gamma(x)=x$
implies that $\gamma=Id_{{\bf R}^{n}}$.

For every element $x$ of ${\bf R}^{n}$, there exists an open subset $U$
containing $x$ such that for every $\gamma$ in $G$, $\gamma(U)\cap U$ is not
empty implies that $\gamma=Id_{{\bf R}^{n}}$.

For such an action, the quotient of ${\bf R}^{n}$ by $G$ is a topological
manifold $M$. We denote by $p_{M}:{\bf R}^{n}\rightarrow M$ the quotient map.
It is a covering map. We can define:

Definition 2.2. An ${\bf R}^{n}$-aspherical manifold $M$, is a topological
manifold $M$, such that there exists a group $G$ which acts continuously,
freely, and properly discontinuously on ${\bf R}^{n}$, and the quotient of
${\bf R}^{n}$ by $G$ is $M$.

We can identify the fundamental group $\pi_{1}(M)$, of $M$, with $G$.

Definitions 2.3. Let $G$ be a group which acts on ${\bf R}^{n}$ by
homeomorphisms. An affine subset $V$ of ${\bf R}^{n}$ is stable by $G$ if for
every $x$ in $V$, and every $g$ in $G$, $g(x)$ is an element of $V$.

The action of $G$ on ${\bf R}^{n}$ is irreducible if the only affine subsets
of ${\bf R}^{n}$ stable by $G$ are the empty subset and ${\bf R}^{n}$.

The following fundamental proposition for this paper is adapted from [3],
theorem 2.2 p. 496.

Proposition 2.1. Let $X$ be an $n$-dimensional contractible topological
manifold, suppose that a group $G$ acts freely and properly discontinuously on
$X$ and the quotient of $X$ by $G$ is compact. Let $Y$ be a non empty
topological contractible submanifold of $X$. Suppose that for every element
$g\in G,g(Y)\subset Y$, then $dim(Y)=n$.

Proof. The quotient $M$ of $X$ by $G$ (resp. $N$ of $Y$ by $G$) are Eilenberg-
McLane $K(G,1)$-spaces. Up to a $2$-cover, we can assume that $M$ and $N$ are
oriented, this implies that the singular homology group $H_{n}(M,{\bf Z})$ is
not zero since $M$ is compact (see [1] p. 346). we deduce that $H_{n}(N,{\bf
Z})$ is not zero. This implies that $dim(Y)=n$.

Corollary 2.1. Let $M$ be a compact ${\bf R}^{n}$-aspherical manifold, the
action of $\pi_{1}(M)$ on ${\bf R}^{n}$ is irreducible.

Proof. Let $V$ be a non empty affine subspace such that for every
$\gamma\in\pi_{1}(M)$, $\gamma(V)=V$. The proposition 2.1 implies that the
dimension of the quotient of $V$ by $G$ is $n$. We deduce that $V={\bf
R}^{n}$.

Let $M$ be an ${\bf R}^{n}$-aspherical manifold, we can lift every
homeomorphism $f$ of $M$ to an homeomorphism $\hat{f}$ of ${\bf R}^{n}$ such
that the following diagram is commutative (see [1] Theorem 4.1):

$\matrix{{\bf R}^{n}&{\buildrel\hat{f}\over{\longrightarrow}}&{\bf R}^{n}\cr
p_{M}\downarrow&&\downarrow p_{M}\cr M&{\buildrel f\over{\longrightarrow}}&M}$

Definition 2.4. An homeomorphism $f$ of the ${\bf R}^{n}$-aspherical manifold
$M$ is an affine transformation, if there exists a lift $\hat{f}$ of $f$ which
is an element of $Aff({\bf R}^{n})$, the group of affine transformations of
${\bf R}^{n}$. We denote by $Aff(M)$ the set of affine transformations of $M$.

The following proposition can be proved straightforward by following the
definitions:

Proposition 2.2. The set of affine transformations $Aff(M)$ is a group.

Let $M$ be an ${\bf R}^{n}$-aspherical manifold, consider $N(\pi_{1}(M))$, the
set of elements of the group of homeomorphisms of ${\bf R}^{n}$ such that, for
every element $g\in N((\pi_{1}(M))$, there exists an element $f$ of $Aff(M)$
whose lift is $g$. Remark that $g$ is not necessarily an affine transformation
of ${\bf R}^{n}$. For example, every element of $\pi_{1}(M)$ is an element of
$N(\pi_{1}(M))$ since it is the lift of $Id_{M}$, the identity of $M$, and
$Id_{{\bf R}^{n}}$ is the lift of $Id_{M}$. We have the following proposition:

Proposition 2.3. Let $g$ be an element of $N(\pi_{1}(M))$, there exists an
element $\hat{f}$ of $Aff({\bf R}^{n})$, and an element $\gamma$ of
$\pi_{1}(M)$ such that $g=\gamma\circ\hat{f}$.

Proof. There exists an element $f$ of $Aff(M)$ such that $g$ is the lift of
$f$. Since $f$ is an element of $Aff(M)$, there exists $\hat{f}$ in $Aff({\bf
R}^{n})$ such that $\hat{f}$ is the lift of $f$. Let $x$ be an element of
${\bf R}^{n}$, since $p_{M}(g(x))=p_{M}(\hat{f}(x))$, there exists $\gamma$ in
$\pi_{1}(M)$ such that $\gamma(\hat{f}(x))=g(x)$. We are going to show that
$g=\gamma\circ\hat{f}$.

The subset $V$ of elements of ${\bf R}^{n}$ on which $g$ and
$\gamma\circ\hat{f}$ coincide is closed since $g$ and $\gamma\circ\hat{f}$ are
continuous. Let $x$ be an element of $V$, there exists an open subset $U$ of
${\bf R}^{n}$ containing $g(x)$ such that, for every element $\alpha$ of
$\pi_{1}(M)$, $\alpha(U)\cap U$ is not empty implies that $\alpha=Id_{{\bf
R}^{n}}$. The open subset $g^{-1}(U)\cap(\gamma\circ\hat{f})^{-1}(U)$ is not
empty, since it contains $x$. For every element $y$ of
$W=g^{-1}(U)\cap(\gamma\circ\hat{f})^{-1}(U)$, $p_{M}(g(y))$ and
$p_{M}(\hat{f}(x)))$ coincide. This implies that there exists $\alpha\in G$
such that $g(y)=\alpha(\hat{f}(y))$. We deduce that $\alpha(\hat{f}(y))$ and
$\gamma(\hat{f}(y))$ are elements of $U$. We deduce that
$\alpha\circ\gamma^{-1}(U)\cap U$ is not empty, since
$(\alpha\circ\gamma^{-1})(\gamma(\hat{f}(y)))=\alpha(\hat{f}(y))$. This
implies that $\alpha\circ\gamma^{-1}=Id_{{\bf R}^{n}}$, henceforth
$\alpha=\gamma$. We deduce that $\gamma\circ\hat{f}$ and $g$ coincide on $W$.
This implies that $V$ is open and closed. We deduce that $V$ is ${\bf R}^{n}$
since ${\bf R}^{n}$ is connected.

Let $G$ be a group of homeomorphisms of the topological space $X$. For every
compact subset $K$ of $X$, and any open subset $U$ of $X$, we denote by
$C_{K,U}$ the subset of $G$ such that for every $g\in C_{K,U}$, $g(K)$ is
contained in $U$. The family $C_{K,U}$ is a basis of a topology defined on $G$
called the compact-open topology. We suppose that $Aff(M)$ and $N(\pi_{1}(M))$
are endowed with the compact-open topology. Remark that the compact-topology
endows $\pi_{1}(M)$ with the structure of a discrete set:

Proposition 2.4. The compact-open topology endows $\pi_{1}(M)$ with the
structure of a discrete set.

Proof. Let $x$ be an element of ${\bf R}^{n}$, there exists an open subset $U$
of ${\bf R}^{n}$ containing $x$ such that for every $\gamma\in\pi_{1}(M)$,
$\gamma(U)\cap U$ is not empty if and only if $\gamma$ is the identity of
${\bf R}^{n}$. Let $\gamma$ be an element of $\pi_{1}(M)$. Let
$C_{x,\gamma(U)}$ the subset of elements of $\pi_{1}(M)$ such that for every
$\alpha$ of $C_{x,\gamma(U)}$, $\alpha(x)\subset\gamma(U)$. The subset
$C_{x,\gamma(U)}$ is a neighbourhood of $\gamma$ for the compact-open topology
defined on $\pi_{1}(M)$. If $\alpha$ is an element of $C_{x,\gamma(U)}$, the
fact that $\alpha(x)\in\gamma(U)$, implies that $\gamma^{-1}(\alpha(x))$ is an
element of $U$. We deduce that $(\gamma^{-1}\circ\alpha)(U)\cap U$ is not
empty. The definition of $U$ implies that $\gamma^{-1}\circ\alpha=Id_{{\bf
R}^{n}}$, and $\gamma=\alpha$. We deduce that $C_{x,\gamma(U)}$ contains only
$\gamma$, and $\pi_{1}(M)$ is a discrete subgroup for the compact-open
topology.

Proposition 2.5. An element of $N(\pi_{1}(M))$ which lifts the identity is
contained in $\pi_{1}(M)$.

Proof. Let $f$ be an element of $N(\pi_{1}(M))$ above the identity, and $x$ an
element of ${\bf R}^{n}$. There exists an element $\gamma$ of $\pi_{1}(M)$
such that $\gamma(x)=f(x)$, since $p_{M}(x)=p_{M}(f(x))$. Let $V$ be the
subset of ${\bf R}^{n}$, such that for every $y\in V$, $f(x)=\gamma(x)$. The
subset $V$ is closed since $\gamma$ and $f$ are continuous. Since the action
of $\pi_{1}(M)$ is free and properly discontinuous, there exists an open
subset $U$ of ${\bf R}^{n}$ containing $f(x)$ such that for every element
$\alpha$ in $\pi_{1}(M)$, $\alpha(U)\cap U$ is not empty implies that $\alpha$
is the identity. The open subset $W=\gamma^{-1}(U)\cap f^{-1}(U)$ is not empty
since it contains $x$. Let $y\in W$, there exists $\alpha\in\pi_{1}(M)$ such
that $f(y)=\alpha(y)$. This implies that, $\alpha^{-1}(f(y))=y$, and
$\gamma(\alpha^{-1}(f(y)))=\gamma(y)$ is an element of $U$. We deduce that
$(\gamma\circ\alpha^{-1})(U)\cap U$ is not empty. This implies that
$\alpha=\gamma$, and henceforth $f=\gamma$.

Proposition 2.6. The group $\pi_{1}(M)$ is a normal subgroup of
$N(\pi_{1}(M))$, and $Aff(M)$ is isomorphic to the quotient of $N(\pi_{1}(M))$
by $\pi_{1}(M)$.

Proof. Let $\gamma$ be an element of $\pi_{1}(M)$, and $f$ an element of
$N(\pi_{1}(M))$, $f\circ\gamma\circ f^{-1}$ lifts the identity. We deduce that
it is an element of $\pi_{1}(M)$, and $\pi_{1}(M)$ is normal in
$N(\pi_{1}(M))$.

Let $P_{M}:N(\pi_{1}(M))\rightarrow Aff(M)$ the map which assigns to the
element $\hat{f}$ of $N(\pi_{1}(M)))$, and element $f$ of $Aff(M)$ such that
$\hat{f}$ lifts $f$. The map $f$ is well-defined. Suppose that $\hat{f}$ is
the lift of the elements $f$ and $g$ of $Aff(M)$. Let $x$ an element of $M$,
and $\hat{x}$ an element of $p^{-1}_{M}(x)$, we have
$f(x)=g(x)=p_{M}(\hat{f}(\hat{x}))$.

The map $P_{M}$ is surjective, since every element of $Aff(M)$ can be lifted
to $\hat{M}$.

The map $P_{M}$ is a morphism of groups. The proposition 2.5 implies that
$P_{M}(\hat{f})$ is the identity if and only if $\hat{f}$ is contained in
$\pi_{1}(M)$. This is equivalent to saying that the kernel of $P_{M}$ is
$\pi_{1}(M)$.

3\. The commutator of $\pi_{1}(M)$ and the action of $Aff(M)_{0}$.

Let $Aff(M)_{0}$ be the connected component of $Aff(M)$, and
$N(\pi_{1}(M))_{0}$ the connected component of $N(\pi_{1}(M))$. We denote by
$Com(\pi_{1}(M))$, the subgroup of the group of homeomorphisms of ${\bf
R}^{n}$ which commute with $\pi_{1}(M)$. We endow $N(\pi_{1}(M))_{0}$ and
$Aff(M)_{0}$ with the compact-open topology.

We have the following result:

Proposition 3.1. The group $N(\pi_{1}(M))_{0}$ is contained in
$Com(\pi_{1}(M))$. The image of the restriction of $P_{M}$ to
$N(\pi_{1}(M))_{0}$ is $Aff(M)_{0}$.

Proof. Let $g$ an element of $N(\pi_{1}(M))_{0}$, and $I=[0,1]$ the closed
interval of ${\bf R}$, and $f:I\rightarrow N(\pi_{1}(M))_{0}$ a continuous map
such that $f(0)=Id_{{\bf R}^{n}}$ and $f(1)=g$. For every element $\gamma$ of
$\pi_{1}(M)$, the map defined by $g_{\gamma}(t)=f(t)\circ\gamma\circ
f(t)^{-1}$ is a continuous map contained in $\pi_{1}(M)$ since $\pi_{1}(M)$ is
a normal subgroup of $N(\pi_{1}(M))_{0}$. Since $\pi_{1}(M)$ is discrete for
the compact open topology, we deduce that $g_{\gamma}$ is constant and for
every $t\in[0,1]$, $g_{\gamma}(t)=g_{\gamma}(0)=\gamma$. We deduce that every
element of $N(\pi_{1}(M))_{0}$ commutes with $\pi_{1}(M)$.

The quotient map $P_{M}:N((\pi_{1}(M))\rightarrow Aff(M)$ is a covering map
since its kernel is discrete. We deduce that we can lift any path through the
identity of $M$ contained in $Aff(M)_{0}$ to a path through the identity of
${\bf R}^{n}$ contained in $N(\pi_{1}(M))$. This implies that
$P_{M}(N(\pi_{1}(M))_{0})=Aff(M)_{0}$.

Let $Aff({\bf R}^{n},M)$ be the intersection $N(\pi_{1}(M))\cap Aff({\bf
R}^{n})$. The restriction of $P^{A}_{M}$ of $P_{M}$ to $Aff({\bf R}^{n},M)$ is
surjective, and its kernel is $Aff({\bf R}^{n},M)\cap\pi_{1}(M)$. This implies
that $P^{A}_{M}$ is a covering map since $\pi_{1}(M)\cap Aff({\bf R}^{n},M)$
is discrete. Let $Aff({\bf R}^{n},M)_{0}$ be the connected component of
$Aff({\bf R}^{n},M)$. Remark that since $Aff({\bf R}^{n},M)_{0}$ is contained
in $N(\pi_{1}(M))_{0}$, its elements commute with the elements of
$\pi_{1}(M)$. We have the following result:

Proposition 3.2. Suppose that $M$ is compact, then the group $Aff({\bf
R}^{n},M)_{0}$ acts freely on ${\bf R}^{n}$.

Proof. Let $f$ be an element of $Aff({\bf R}^{n},M)_{0}$. Consider $Fix_{Aff}$
the set of affine subspaces of ${\bf R}^{n}$, such that for every element $V$
of $Fix_{Aff}$ and every $x$ in $V$, $f(x)=x$. Let $V$ be an element of
$Fix_{Aff}$ which has a maximal dimension. Suppose that $dim(V)=p$ with $p<n$
and $V$ is not empty. The corollary 2.1 implies that there exist an element
$x$ of $V$, and an element $\gamma$ of $\pi_{1}(M)$ such that $\gamma(x)$ is
not contained in $V$. Let $U$ be the direction of $V$, that is the vector
subspace of dimension $p$ of ${\bf R}^{n}$ such that for every $y\in V$, there
exists an element $u$ of $U$ such that $y=x+u$. We can write $f=(L,l)$ where
$L$ is a linear map of ${\bf R}^{n}$, $l$ an element of ${\bf R}^{n}$, and for
every $z$ in ${\bf R}^{n}$, $f(z)=L(z)+l$.

Write $\gamma(x)=x+v$. We have $f(\gamma(x))=\gamma(f(x))=\gamma(x)$. This
implies that $f(\gamma(x))=L(x+v)+l=f(x)+L(v)=x+v$. Since $f(x)=x$, we deduce
that $L(v)=v$. All the points of the affine subspace which contains $x$, and
whose direction is the direct sum $U\oplus{\bf R}v$ are fixed by $f$. This
implies that de dimension of $V$ is not maximal in $Fix_{Aff}$, contradiction.
We deduce that either $V$ is ${\bf R}^{n}$ or $V$ is empty. This is equivalent
to saying that $Aff({\bf R}^{n},M)_{0}$ acts freely on ${\bf R}^{n}$.

Remark that if $f$ is an element of $Aff({\bf R}^{n})$ which commutes with
$\pi_{1}(M)$, it is the lift of an element of $Aff(M)$. The subgroup $Com({\bf
R}^{n},M)$ of $Aff({\bf R}^{n})$ whose elements commute with the elements of
$\pi_{1}(M)$ is a Lie group since it is a closed subgroup of $Aff({\bf
R}^{n})$. Since it contains $Aff({\bf R}^{n},M)_{0}$, its connected component
is the connected of $Aff({\bf R}^{n},M)$. We deduce that $Aff({\bf
R}^{n},M)_{0}$ is a Lie group. We denote by $aff({\bf R}^{n},M)$ the Lie
algebra of $Aff({\bf R}^{n},M)$.

Corollary 3.1. Suppose that $M$ is compact, then $Aff({\bf R}^{n},M)_{0}$ and
$Aff(M)_{0}$ are solvable Lie groups.

Proof. Let $KAN$ be the Iwasawa decomposition of $Aff({\bf R}^{n},M)_{0}$. The
group $K$ is a compact Lie group which acts freely on ${\bf R}^{n}$, we deduce
that it is trivial. This implies that $Aff({\bf R}^{n},M)_{0}$ is a solvable
group. The restriction of $P_{M}$ to $Aff({\bf R}^{n},M)_{0}$ is a covering
map whose image is $Aff(M)_{0}$. This implies that $Aff(M)_{0}$ is solvable.

Corollary 3.2. Suppose that $M$ is compact, then group $Aff(M)_{0}$ is a Lie
group whose action on $M$ is locally free.

Proof. Let $x$ be an element of $M$, $\hat{x}$ an element of $p_{M}^{-1}(x)$.
Consider an open subset $V$ of the identity of $Aff({\bf R}^{n},M)_{0}$ such
that the restriction of $P_{M}$ to $V$ is an homeomorphism onto its image. We
can shrink $V$ and find a neighbourhood $U$ of $\hat{x}$ such that: the
restriction of $p_{M}$ to $U$ is an homeomorphism onto its image, and for
every element $\hat{f}$ of $V$, $\hat{f}(\hat{x})$ is contained in $U$. Let
$\hat{f}$ be an element of $V$, write $f=P_{M}(\hat{f})$. Suppose that
$f(x)=x$, this is equivalent to saying that $p_{M}(\hat{f}(\hat{x}))=x$. Since
$\hat{f}(\hat{x})$ is contained in $U$, we deduce that $\hat{f}(x)=\hat{x}$,
and $\hat{f}=Id_{{\bf R}^{n}}$ since $Aff({\bf R}^{n},M)_{0}$ acts freely on
${\bf R}^{n}$. This implies that $f$ is the identity of $M$, and the action of
$Aff(M)_{0}$ on $M$ is locally free.

We suppose now that the action of $\pi_{1}(M)$ on ${\bf R}^{n}$ is a
differentiable action. This implies that $M$ is a differentiable manifold, we
have the following result:

Corollary 3.3. The orbits of $Aff(M)_{0}$ are the leaves of a foliation.

We can also show:

Corollary 3.4. Suppose that $Aff({\bf R}^{n},M)_{0}=Com({\bf R}^{n},M)_{0}$ is
an algebraic group, $Aff(M)_{0}$ is a nilpotent group.

Proof. Let $U$ be the unipotent radical of $Aff({\bf R}^{n},M)_{0}$, (see [6),
there exists a reductive subgroup $R$ such that $Aff({\bf R}^{n},M)_{0}$ is
the semi-direct product of $U$ and $R$. Since any representation of a
reductive subgroup has a fixed point, we deduce that $R$ is trivial and
$Aff({\bf R}^{n},M)_{0}$ is unipotent, and henceforth $Aff(M)_{0}$ is
nilpotent.

A compact and complete polynomial manifold of dimension $n$ is a compact
differentiable manifold which is the quotient of ${\bf R}^{n}$ by a subgroup
of polynomial automorphisms which act freely and properly discontinuously on
${\bf R}^{n}$. Since $M$ is compact, its fundamental group is generated by a
finite set $S$ of polynomial automorphisms. This implies that $Com({\bf
R}^{n},M)$ is defined by a finite set of algebraic equations, and is an
algebraic group. We have:

Corollary 3.5. Suppose that $M$ is a compact and complete polynomial manifold,
then $Aff(M)_{0}$ is nilpotent.

Corollary 3.6. Suppose that $M$ is a compact differentiable manifold, and
$Aff(M)_{0}$ is a compact Lie group of dimension $l$. Then $M$ is the total
space of a bundle whose typical fibre is the torus ${\bf T}^{l}$ over an
orbifold.

Proof. The fact that $Aff(M)_{0}$ is compact implies that it is commutative,
since it is a solvable Lie group. We deduce that $M$ is endowed with the
locally free action of ${\bf T}^{l}$. This implies that that $M$ is the total
space of a fibre bundle whose typical fibre is ${\bf T}^{l}$, and whose base
space is an orbifold.

Proposition 3.3. Suppose that $M$ is a compact differentiable manifold, which
is the quotient of ${\bf R}^{n}$ by a free and properly discontinuous action
of its fundamental group. Moreover, suppose that the subset of elements of
$Aff({\bf R}^{n})$ which commute with $\pi_{1}(M)$ is not discrete, then the
Euler number of $M$ is zero.

Proof. Under the hypothesis of the proposition, the Lie algebra $aff(M)$ of
$Aff(M)$ is not zero, since it is isomorphic to the Lie algebra of $Com({\bf
R}^{n},M)$ which is not zero. Since $Aff(M)_{0}$ acts locally freely on $M$,
any non zero element of $aff(M)$ does not vanish on $M$. We deduce that the
Euler number of $M$ is zero.

Definitions 3.1. An affine manifold $(M,\nabla_{M})$ is a manifold endowed
with a connection whose curvature and torsion forms vanish identically. This
is equivalent to endow $M$ with an atlas whose coordinates change are affine
transformations. We will say that $\nabla_{M}$ is an affine connection.

Let $G$ be a Lie group, and $\hat{G}$ its universal cover, a left invariant
affine structure defined on $G$, is an affine connection $\nabla_{G}$
invariant by the left translations of $G$.

The restriction of the connection to ${\cal G}$ defines a left-symmetric
product, that is a product such that:

$ab-ba=[a,b].$

$a(bc)-(ab)c=b(ac)-(ba)c.$

Let $a$ be an element of ${\cal G}$, we denote by $L_{a}$, the linear map of
${\cal G}$ defined by $L_{a}(b)=ab$. We denote by $h(a)=(L_{a},a)$. The
following proposition has been shown by Koszul:

Proposition 3.4. Let $G$ be a Lie group endowed with a left-symmetric algebra.
The map $h(a)$ is an affine representation of the Lie algebra ${\cal G}$. Let
$H:\hat{G}\rightarrow Aff({\cal G})$ be the representation whose differential
at the origin is $h$. The orbit at the origin of $H$ is open. Conversely, let
$\hat{G}$ be an $n$-dimensional simply connected Lie group, suppose that there
exists an $n$-dimensional affine representation $H$ of $G$ which has an open
orbit, then ${\cal G}$ can be endowed with the structure of a left symmetric
algebra which induces $H$.

From this proposition, we deduce:

Proposition 3.5. Suppose that $M$ is compact, and $Aff(M)_{0}$ acts
transitively on $M$, then $M$ is a solvmanifold endowed with the structure of
an affine manifold.

Proof. The corollary 3.1 shows that $Aff({\bf R}^{n},M)_{0}$ is solvable.
Since $Aff({\bf R}^{n},M)_{0}$ acts transitively and freely on ${\bf R}^{n}$,
it is endowed with the structure of a left-invariant affine structure. We can
identify $\pi_{1}(M)$ with a discrete subgroup of $Aff({\bf R}^{n},M)_{0}$,
since it action on $M$ is transitive and locally free. This implies that $M$
is an affine solvmanifold.

4\. $dim(Aff(M)_{0})=dim(M)-1$.

In this section, we are going to study compact ${\bf R}^{n}$-aspherical
manifolds such that $dim(Aff(M)_{0})=n-1$. In this case, the orbit of
$Aff(M)_{0}$ are leaves of an $n-1$-dimensional foliations defined on $M$.
Such foliations have been studied by many authors. Suppose that $Aff(M)_{0}$
is commutative, the results of [2] imply:

Proposition 4.1. Suppose that $M$ is an $n$-aspherical compact differentiable
manifold such that $Aff(M)_{0}$ is commutative and has a compact orbit. Then
$M$ is the total space of a fibre bundle over the circle $S^{1}$ with fibre
the torus ${\bf T}^{n-1}$.

In [4], Hector, Ghys and Moriyama have studied closed manifolds endowed with
locally free actions of nilpotent Lie groups. The theorem $B$ of their paper
implies that if $Aff(M)_{0}$ is nilpotent, there exists a nilmanifold $N$ of
dimension superior or equal to $1$ which is the fibre space of a locally
trivial fibration whose total space is $M$.

The corollary 3.4 implies:

Proposition 4.2. Suppose that $M$ is a compact differentiable $n$-aspherical
manifold such that $Com({\bf R}^{n},M)_{0}$ is an algebraic group and the
dimension of its orbits is $n-1$. Then $M$ is the total space of a fibre space
whose fibre is a nilmanifold of dimension superior or equal to $1$.

The corollary 3.5 implies:

Corollary 4.1. Let $M$ be an $n$-dimensional compact polynomial manifold such
that $dim(Com({\bf R}^{n},M)_{0})=n-1$. Then $M$ is the total space of a fibre
space whose fibre is a nilmanifold of dimension superior or equal to $1$.

In dimension $3$, we can precise these results: The solvable $2$-dimensional
Lie groups are either isomorphic to ${\bf R}^{2}$, or to $GA$, the group of
affine transformations of the real line. The action of $GA$ has been studied
by Ghys . We can deduce:

Theorem 4.2. Let $M$ be a differentiable ${\bf R}^{3}$-aspherical manifold.
Suppose that the dimension of $Aff(M)$ is $2$, and $Aff(M)_{0}$ is not
commutative and preserves a volume form on $M$. Then $M$ is a solvmanifold, or
the quotient of $\widetilde{SL(2,{\bf R})}$ by a lattice.

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