# Neutrino Halo profiles: HR-DEMNUni simulation analysis

Beatriz Hernández-Molinero  Carmelita Carbone  Raul Jimenez  Carlos Peña Garay

###### Abstract

Using the high-resolution HR-DEMNUni simulations, we computed neutrino
profiles within virialized dark matter haloes. These new high-resolution
simulations allowed us to revisit fitting formulas proposed in the literature
and provided updated fitting parameters that extend to less massive haloes and
lower neutrino masses than previously in the literature, in accordance with
new cosmological limits. The trend we observe for low neutrino masses is that,
for dark matter halo masses below $\sim 4\times 10^{14}$$h^{-1}M_{\odot}$, the
presence of the core becomes weaker and the profile over the whole radius is
closer to a simple power law. We also characterized the neutrino density
profile dependence on the solid angle within clustered structures: a forward-
backward asymmetry larger than 10% was found when comparing the density
profiles from neutrinos along the direction of motion of cold dark matter
particles within the same halo. In addition, we looked for neutrino wakes
around halo centres produced by the peculiar motion of the halo itself. Our
results suggest that the wakes effect is observable in haloes with masses
greater than $3\times 10^{14}$$h^{-1}M_{\odot}$ where a mean displacement of
$0.06$$h^{-1}\mathrm{Mpc}$ was found.

##  1 Introduction

Cosmology remains the most promising route to prove the absolute mass of
neutrinos. Current cosmological surveys are already providing very stringent
upper limits that are very close to the total mass inferred from underground
experiments that measure the mass splits, i,e, $0.059$ eV [1], which in turn
confidently suggests, within the framework of Bayesian evidence, that the
hierarchy is normal [2, 3]. However, in order to confirm that we have actually
discovered neutrinos and not something else, it is important to have
additional tests about the nature of this ‘hot’ component of the Universe
density budget. Some test can also be obtained by measuring the sound speed
and viscosity of this extra component [4]. Another important probe is to study
the clustering of neutrino haloes around their corresponding cold dark matter
ones. This is the probe we explore in this work.

If massive neutrinos exist in the Lambda Cold Dark Matter (LCDM) model, then
there should be clear signatures of how they cluster around Cold Dark Matter
(CDM) haloes. It is these signatures, which could be measured by e.g. weak
lensing surveys, that could provide evidence that we have indeed detected
massive neutrinos. In that sense, in the present work we aim to provide high
fidelity signatures about neutrino haloes by cold dark matter and neutrino
density profiles inside and around virialized haloes, making emphasis in their
angular dependence within the neutrino halo.

In particular, we revise the fitting formulas proposed in [5] but using
improved simulations which have a significantly larger mass resolution for
both Cold Dark Matter (CDM) and neutrino particles, in agreement with current
cosmological simulations, which allows us to get more reliable results.
Firstly, we wanted to see how previous formulas stand for new simulations
where the neutrino mass is twice smaller. We will see how these new
simulations push higher the applicability limit of fitting function. Moreover,
taking advantage of the high resolution of these new simulation, we also
explore the angular dependence of neutrino overdensity distributions around
haloes. Neutrino density profiles have been obtained always assuming isotropy,
but, in the surroundings of super-clusters, some preferred direction of motion
could exist depending on the dark matter particles distribution. If neutrinos
follow a particular direction, some asymmetry could be created. We investigate
and quantify this effect for different kinds of clusters through the neutrino
density profiles. In addition, we also compute neutrino density maps in the
centre of haloes to confirm the neutrino wakes signatures proposed in [6, 7].

This paper is organised as follows. In § 2 we present the new HR-DEMNUni
simulations that have been used to carry out all analysis presented in this
work as well as the method used to compute the density profiles. § 3 gathers
our main results, several density profiles for cold dark matter and neutrino
particles in haloes with different masses. Some of these profiles take into
account the angular distribution of the neutrinos around dark matter haloes.
Finally, the discussion and conclusions are addressed in § 4.

##  2 Methodology

In this section we describe the simulations employed and the method applied to
compute density profiles, starting with the details of the high-resolution
simulations in § 2.1 and the expressions for the profiles in § 2.2.

###  2.1 Numerical Simulations

In this analysis, we have used the “Dark Energy and Massive Neutrino Universe”
(DEMNUni) suite of large N-body simulations [8]. The DEMNUni simulations have
been produced with the aim of investigating the large-scale structure of the
Universe in the presence of massive neutrinos and dynamical DE, and they were
conceived for the nonlinear analysis and modelling of different probes,
including dark matter, halo, and galaxy clustering [9, 10, 11, 12, 13, 14, 15,
16, 17, 18, 19], weak lensing, CMB lensing, Sunyaev-Zel’dovich and integrated
Sachs-Wolfe effects [8, 20, 21, 22], cosmic void statistics [23, 24, 25, 26,
27, 28], as well as cross-correlations among these probes [29, 30].

In particular, we have used the new high-resolution (HR) simulations with 64
times better mass resolution than previous standard runs: the HR-DEMNUni
simulations are characterized by a comoving volume of
$(500\>h^{-1}\mathrm{Gpc})^{3}$ filled with $2048^{3}$ dark matter particles
and, when present, $2048^{3}$ neutrino particles. The simulations are
initialized at $z_{\rm in}=99$ with Zel’dovich initial conditions. The initial
power spectrum is rescaled to the initial redshift via the rescaling method
developed in [31]. Initial conditions are then generated with a modified
version of the N-GenIC software, assuming Rayleigh random amplitudes and
uniform random phases. The HR-DEMNUni set consists of two simulations with
total neutrino masses of $\sum m_{\nu}=0,\,0.16\,{\rm eV}$ considered in the
degenerate mass scenario with three active neutrinos. The other cosmological
parameters of the simulations are based on a Planck 2013 [32] LCDM reference
cosmology (with massless neutrinos), in particular: $n_{\rm s}=0.96$, $A_{\rm
s}=2.1265\times 10^{-9}$, $h=H_{0}/[100\,{\rm km}\,s^{-1}{\rm
Mpc}^{-1}]=0.67$, $\Omega_{\rm b}=0.05$, and $\Omega_{\rm m}=\Omega_{\rm
CDM}+\Omega_{\rm b}+\Omega_{\nu}=0.32$; $H_{0}$ is the Hubble constant at the
present time, $n_{\rm s}$ is the spectral index of the initial scalar
perturbations, $A_{\rm s}$ is the scalar amplitude, $\Omega_{\rm b}$ the
baryon density parameter, $\Omega_{\rm m}$ is the total matter density
parameter, $\Omega_{\rm CDM}$ the cold dark matter density parameter, and
$\Omega_{\nu}$ the neutrino density parameter. In the presence of massive
neutrinos, $\Omega_{\rm b}$ and $\Omega_{\rm m}$ are kept fixed to the above
values, while $\Omega_{\rm CDM}$ is changed accordingly. Tab. 1 summarizes the
masses of the CDM and neutrino particles together with the neutrino fraction
$f_{\nu}\equiv\Omega_{\nu}/\Omega_{\rm m}$.

HR-DEMNUni
---
$\sum m_{\nu}$ [eV] | $f_{\nu}$ |  $m_{\rm p}^{\rm CDM}$ [$h^{-1}M_{\odot}$ ] |  $m_{\rm p}^{\nu}$ [$h^{-1}M_{\odot}$ ]
0 | 0 | $1.2921\times 10^{9}$ | $0$
0.16 | 0.012 | $1.2767\times 10^{9}$ | $1.5441\times 10^{7}$
Table 1:  Summary of particle masses and neutrino fractions implemented in the
HR-DEMNUni simulations. The first column shows the total neutrino mass, the
second the fraction of neutrinos and matter density parameters, and the last
two columns show the corresponding mass of CDM and neutrino particles
implemented in the simulations.

Dark matter haloes are identified using a friends-of-friends (FoF) algorithm
[33], with a linking length of 0.2 times the mean particle separation, applied
to both CDM and neutrino particles in the simulation, with a minimum number of
particles per species fixed to 32, corresponding to a mass of $\sim 4\times
10^{10}h^{-1}M_{\odot}$. FoF haloes are further processed with the subfind
algorithm [34, 35] to produce subhalo catalogues. With this procedure, some of
the initial FoF parent haloes are split into multiple substructures. In
particular, in this work we use the spherical overdensity halo catalogues, so-
called $M_{\rm 200b}$, identified by the subfind algorithm, and in the
following we will refer to them with the term “halo”.

###  2.2 Computing Density Profiles

In order to compute smoothed density profiles we use the kernel-based
algorithm proposed in [36]. This density estimator is based on a variable
window width.

$\hat{\nu}(r)=\sum_{i=1}^{N}\frac{1}{h_{i}^{3}}\tilde{K}(r,r_{i},h_{i}),$
(2.1)

where $h_{i}$ is the window width associated with the $i$th particle and
$\tilde{K}$, a Gaussian kernel.

$\tilde{K}(r,r_{i},h_{i})=\frac{1}{2(2\pi)^{3/2}}\left(\frac{rr_{i}}{h_{i}^{2}}\right)^{-1}\left[e^{-(r_{i}-r)^{2}/2h_{i}^{2}}-e^{-(r_{i}+r)^{2}/2h_{i}^{2}}\right]$
(2.2)

The window width varies with position in such a way that the bias-to-variance
ratio of the estimate is relatively constant. We chose the window width in the
same way as [36], $h_{i}\propto r^{1/2}$ is set at $0.1r_{vir}$ as
$h_{0.1}=0.05r_{vir}$. We use this kernel to compute all density profiles in
this paper.

##  3 Density profiles

In this Section we present our main results. Firstly, we will focus on density
profiles integrated in angles both for cold dark matter and neutrino
distributions within haloes of different masses, studying if new density
profiles measured from the HR-DEMNUni simulations fit the formulas proposed by
[5] with the aim to reproduce them. Secondly, we will consider how density
profiles could change depending on the neutrino direction and compute them in
solid angles ahead of and behind the halo centres along the direction of
motion of cold dark matter particles. Lastly, we will look at the centre of
the selected haloes and look for backward neutrino wakes as those proposed in
[6, 7]. All analyses have been carried out over a redshift zero realization of
the HR-DEMNUni simulations.

###  3.1 Profiles as a function of radius

As a first check, we have calculated the CDM density profiles for four
different halo masses at redshift zero (Figure 1) and fit them to the standard
NFW profile (dashed lines in Figure 1). For these and the following fits, the
python package lmfit111https://lmfit.github.io/lmfit-py/ based on non-linear
least-squares minimisation has been used.

Figure 1: CDM overdensity profiles in haloes with masses $M_{\rm h}$ in
different mass bins. Dashed lines correspond to a NFW density profile fit.
Vertical dotted lines set the halo radii $\rm{R}_{200}$ defined by an
overdensity threshold $\Delta=200$ with respect to the mean background total
matter density $\rho_{b}$.

The outcome is as expected, we see the plateau in the innermost regions, the
higher the halo mass is the larger is this plateau, then the profiles drop
quickly to the mean density around the virial radius and inside and outside
over-densities differ by several orders of magnitude. In addition, and as it
is already known, NFW profiles just return a good fit around the virial
radius. These profiles are well known and have been calculated several times
through literature, we use them here just to show that the algorithm used to
compute the density profiles works correctly, so it can be applied in the
calculation of neutrino profiles.

Now, we repeat the same procedure for the neutrino component, also at redshift
zero, but in smaller halo mass bins. We plot the overdensity for neutrinos and
use the formulas (3.1) below to fit the profiles. We present the results for
haloes with masses greater than $10^{14}\,h^{-1}M_{\odot}$ in the left panel
of Figure 2, along with the results of the fit to Equation (3.1a) which is
plotted as dashed lines. Only some of them are displayed in the left panel of
Figure 2 for ease of visualisation, but all fit results are included in Table
2. Likewise, in the right panel of Figure 2, the density profiles obtained for
haloes with masses between $10^{11}\,h^{-1}M_{\odot}\leq M_{\rm h}\leq
10^{14}h^{-1}M_{\odot}$ are shown combined with the fitting result to Equation
(3.1b), plotted as dashed lines too. As well, only a few are displayed in the
right panel of Figure 2 but all results are included in Table 3. Due to
computational limitations, the angle-averaged profiles, corresponding to each
of the halo masses listed in Tables 2 and 3, has been calculated by averaging
over neutrino profiles measured from 200 haloes. All available haloes has been
used for masses close to $10^{15}$$h^{-1}M_{\odot}$. In Figure 2 the error
bars represent the dispersion around the mean density profile, rather than the
dispersion in the mean density profile as instead assumed in [5]. So our
errors are more conservative and mildly depend on the number of analysed
haloes. We performed our fits using the equations below:

$\displaystyle\delta_{\nu}(r)=$
$\displaystyle\frac{\rho_{\nu}(r)-\bar{\rho}_{\nu}}{\bar{\rho}_{\nu}}=\frac{\rho_{c}}{1+(r/r_{c})^{\alpha}}\hskip
28.45274pt$ $\displaystyle\text{for halo masses M}>10^{14}\,h^{-1}M_{\odot}$
(3.1a) $\displaystyle\delta_{\nu}(r)=$ $\displaystyle\kappa/r^{\alpha}$
$\displaystyle\text{for halo masses M}<10^{14}\,h^{-1}M_{\odot}$ (3.1b)

|
---|---
Figure 2: Mean neutrino overdensity profiles for different halo masses. For each mass we select a maximum of $200$ haloes randomly selected within a mass bin of width $\pm 5\%$ of the central value displayed in the legend. The error bars correspond to $1\sigma$ dispersion around the calculated mean value of the overdensity at each radius. LEFT: Neutrino overdensity profiles for $M_{\rm h}>10^{14}\,h^{-1}M_{\odot}$. In dashed lines the fits to formula (3.1a). Vertical dotted lines set the core radius ($r_{c}$ parameter) obtained from the fit. RIGHT: Mean neutrino overdensity profiles for $M_{\rm h}<10^{14}\,h^{-1}M_{\odot}$. In dashed lines the fit to formula (3.1b). $M_{\rm h}\ [h^{-1}M_{\odot}]$ | $\rho_{c}$ | $r_{c}\ [h^{-1}\rm{Mpc}]$ | $\alpha$
---|---|---|---
$1\times 10^{14}$ | $0.58\pm 0.12$ | $1.31\pm 0.57$ | $1.04\pm 0.16$
$2\times 10^{14}$ | $1.23\pm 0.21$ | $0.68\pm 0.22$ | $1.05\pm 0.09$
$4\times 10^{14}$ | $2.55\pm 0.30$ | $0.53\pm 0.11$ | $1.03\pm 0.05$
$6\times 10^{14}$ | $2.96\pm 0.33$ | $0.85\pm 0.16$ | $1.25\pm 0.08$
$8\times 10^{14}$ | $3.58\pm 0.44$ | $0.92\pm 0.18$ | $1.30\pm 0.10$
$9\times 10^{14}$ | $3.98\pm 0.33$ | $0.94\pm 0.12$ | $1.38\pm 0.07$
Table 2: Coefficients of Equation (3.1a) fitted to the mean neutrino overdensity profiles. Results for neutrinos within haloes with $M_{\rm h}>10^{14}$$h^{-1}M_{\odot}$ in HR-DEMNUni simulations. For each mass we analyze a maximum of $200$ haloes randomly selected within a mass bin of width $\pm 5\%$ of the central value displayed in the first column. $M_{\rm h}\ [h^{-1}M_{\odot}]$ | $\kappa$ | $\alpha$
---|---|---
$4\times 10^{11}$ | $0.140\pm 0.020$ | $0.62\pm 0.12$
$7\times 10^{12}$ | $0.140\pm 0.020$ | $0.61\pm 0.13$
$1\times 10^{13}$ | $0.136\pm 0.018$ | $0.53\pm 0.11$
$4\times 10^{13}$ | $0.215\pm 0.020$ | $0.56\pm 0.08$
$7\times 10^{13}$ | $0.250\pm 0.022$ | $0.64\pm 0.09$
Table 3: Coefficients of Equation (3.1b) fitted to the mean neutrinos
overdensity profiles. Results for neutrinos within haloes with $M_{\rm
h}<10^{14}$$h^{-1}M_{\odot}$ in the HR-DEMNUni simulations. For each mass we
analyze a maximum of $200$ haloes randomly selected within a mass bin of width
$\pm 5\%$ of the central value displayed in the first column.

In [5], the authors found that Equations (3.1) described very well the average
neutrino overdensity profiles over a wide range of radii. The physical meaning
of the parameters in the profile (3.1a) is very simple: $r_{c}$ and $\rho_{c}$
represent the length and the overdensity of the core in the overdensity
profile of the neutrino halo, while $\alpha$ is a parameter that controls how
fast the overdensity profile falls on large radii. In [5] it was also claimed
that for haloes with mass below $\sim 10^{13.5}\,h^{-1}M_{\odot}$ the
resolution in their N-body simulations was not large enough to properly
resolve the core in the neutrino density profiles, so Equation (3.1b) was
proposed to reproduce the outskirts of the neutrino density profiles. In our
case, even though our simulations have much larger mass resolution, the
neutrino mass is smaller than in [5] and we can not resolve the core either,
if any core exists for such small neutrino masses. So we use the same
expression (3.1b) to fit the neutrino profiles in haloes with masses below
$10^{14}\,h^{-1}M_{\odot}$. It is important to emphasize that in our case the
small neutrino mass, therefore the high thermal neutrino velocity, is
preventing the neutrino clustering in haloes with masses less than
$10^{14}\,h^{-1}M_{\odot}$, which results in no neutrino core formation.

As the halo mass becomes smaller, the neutrino overdensity profile becomes
flatter, meaning a larger neutrino clustering in more massive haloes.
Comparing the neutrino profiles in the left and right panels of Figure 2 with
the CDM overdensity profiles in Figure 1, it is possible to observe that
neutrino profiles are more extended than CDM ones, due to the larger CDM
particles clustering as compared to hot neutrinos. If we compare neutrino
profiles for large and low halo masses, we see a clear overdensity decrease of
around a factor of two in the latter case, which is a consequence of the
decrease of the halo mass and the associated gravitational potential well.
Paying attention to the tails of the neutrino profiles, we can verify that
they are well reproduced by the proposed fitting functions. At this point, it
is worth noting that formulas (3.1), which were proposed studying simulations
with larger neutrino masses, work reasonably well also for these new
simulations where the total neutrino mass is twice smaller.

Figure 3: Derived parameters of the fitting functions 3.1a (right) and 3.1b
(left) used to characterize the neutrino overdensity profiles. In both columns
we show the dependence of the respective parameters as a function of the
masses of their host halo.

In Figure 3, we show the parameters derived from the fitting functions against
the mass of the halo hosting the neutrino halo for the two cases of neutrino
profiles (3.1a)-(3.1b), with the aim to check if the trend is the same as in
[5]. We find that our results for the $\alpha$ parameters are quite comparable
and, in both cases, the tendency is alike to that obtained in [5]. The slopes
of the fitting functions are close to zero, suggesting that there is not much
mass dependence for the two $\alpha$ parameters. Regarding the core density
parameter, $\rho_{c}$, the trend we find is comparable as well: the density of
the neutrino core increases with the mass of the host halo, nevertheless the
results of the fit are slightly different than in [5]. We get a fit for the
$\kappa$ parameter: its trend is a some constant regime from $10^{11}$ up to
$1\times 10^{13}\,h^{-1}M_{\odot}$ and then it starts to grow. On distances
much larger than the core radius, $r\gg r_{c}$, the profile (3.1a) reduces to
(3.1b), with $\kappa=\rho_{c}r_{c}^{\alpha}$, so we expected a decreasing of
$\kappa$ as the halo mass decrease and so it is. Particularly relevant is the
trend observed for $r_{c}$ for which we do not get a fit, as shown in the
$r_{c}$ panel of Figure 3. In [5] $r_{c}$ grows with the halo mass, but what
we have found from our analysis is that it decreases with the halo mass for
$M_{\rm h}\lesssim 4\times 10^{14}$$h^{-1}M_{\odot}$, while it starts to grow
for $M_{\rm h}>4\times 10^{14}$$h^{-1}M_{\odot}$, recovering a similar trend
as in [5] (see Table 2). As the expected tendency, also considering what
happens for the CDM core, is that the radius of the neutrino core grows with
the mass of the host halo, this result suggest that Equation (3.1a) may not be
applicable for haloes with masses lower than $4\times
10^{14}$$h^{-1}M_{\odot}$. A higher resolution simulation and, especially, a
lower total neutrino mass seem to have pushed the limit: for haloes with
$M_{\rm h}<4\times 10^{14}$$h^{-1}M_{\odot}$ the obtained profiles are too
flat for formula (3.1a) to resolve the core, so the core information is washed
out.

This could mean that, for neutrino total masses less than $\sim 0.16$ eV and
CDM halo masses less than a few $10^{14}$$h^{-1}\mathrm{Mpc}$, neutrinos
behave more as light-like particles and this prevents neutrino core formation.
In [5], their results suggest that the fitting formula does not work for
haloes with masses less than $10^{13.5}$$h^{-1}M_{\odot}$, now, for less
massive neutrinos we put the limit higher suggesting $4\times
10^{14}$$h^{-1}M_{\odot}$. It is possible that the limit can be push higher,
maybe around $10^{15}$$h^{-1}M_{\odot}$ with N-body simulations implementing
total neutrino masses lower than $0.16$ eV.

Following the previous argument, such a low neutrino mass produces a very flat
profiles with very large error bars in the innermost halo regions, so a limit,
where we are dominated by the dispersion of the profiles, is reached meaning
that no reliable information can be obtained in these regimes.

Our simulations thus predict what averaged neutrino profiles should look like
in the presence of CDM, although some new limits have been found. In order to
obtain more conclusive results, more high-resolution simulations like the one
used in this work with different and smaller neutrino masses are needed.
Nevertheless, we can conclude that, if the signal is measured by future weak
lensing surveys, this should provide a clear signature that massive neutrinos
do exist.

###  3.2 Profiles as a function of angle

It has always been assumed that neutrino haloes, produced in areas with a high
dark matter density, are spherical; but if neutrinos fall in those areas
because of dark matter, the CDM distribution could produce a preferential
direction of neutrino in-falling which would break the isotropy, resulting in
some kind of ovoid neutrino haloes. For some dark matter haloes, we expect a
deformed associated neutrino halo that would present a higher neutrino density
in the front part in the direction of motion of the CDM and a lower neutrino
density behind the halo centre along the direction of motion as well.

In order to search for this effect, and as we want to see an effect on the
neutrino density profiles as a function of the angle, a reference direction,
with respect to which the angles are measured, is needed. We set the reference
direction for each analysed halo as the direction resulting from averaging the
velocity vectors of all CDM particles within the halo, assuming that the
haloes extend up to 20 $h^{-1}$Mpc, since neutrino haloes have much larger
radii than the corresponding CDM haloes. Then, we compute the neutrino
overdensity profiles ahead of and behind the halo centre along this reference
direction.

In practice, the procedure followed is: first, find the average CDM particle
velocity in the halo, then rotate all the system, CDM and neutrino particles
(up to a distance of 20 $h^{-1}$Mpc radius from the center of the halo) so
that the average halo CDM velocity is aligned to the x-axis, i.e.
$\theta=90^{\circ}$ and $\phi=0^{\circ}$ in spherical coordinates; then, the
neutrino position distribution, after the rotation, is computed.

Another anisotropy effect is the one proposed in [6, 7]. This produces
neutrino wakes in the inner regions of the neutrino halo due to the peculiar
motion of the halo itself. In order to investigate such an effect we look at
the centre of the halo in a box of 6 $h^{-1}$Mpc side.

####  3.2.1 Front loading neutrinos

Figure 4: Neutrino overdensity in spherical coordinates for some selected
haloes of different masses. Triangle markers represent mean CDM and neutrino
velocity vector in spherical coordinates. Red solid line marks the neutrinos
ahead of the halo centre in the direction of the CDM motion and red dashed
line marks neutrinos behind.

As noted above, neutrinos falling in a particular halo will see a preferential
direction, i.e. the average CDM direction of motion, and will fall following
it. Following that direction, the closer the neutrinos travel to the halo
centres, the more gravitational force they will suffer and the more they will
deviate from their trajectory towards the halo central axis. As a result, a
region of overdensity will form ahead of the halo centres in the direction of
CDM motion.

In practice, if neutrinos are to cluster further ahead of the centre of the
halo, the expected mean angle values are $\theta=90^{\circ}$ and
$\phi=0^{\circ}$ for construction, because we have rotated each halo system
such that the CDM mean velocity vector is aligned to the x-axis
($\theta=90^{\circ}$ and $\phi=0^{\circ}$, in spherical coordinates). To
analyze this effect, we have randomly selected a total of 120 typical haloes
of different masses. In all of these haloes we observe that the largest
clustering occurs ahead of the halo centre in the direction of motion. Some
examples are shown in Figure 4.

Figure 5: Neutrino profiles for haloes with masses between $1\times 10^{14}$
and $9\times 10^{14}\ h^{-1}M_{\odot}$. Solid line represent profiles
integrated in angles. Dashed line show profiles obtained from neutrinos
located ahead of the halo centre in the direction of the CDM motion. Dotted
line show profiles obtained from neutrinos located behind the halo centre in
the direction of CDM motion.

From the simulations we see that the neutrino mean velocity is more or less
aligned to the CDM mean velocity for each analyzed halo, as shown in Figure 4,
where black triangles represent the CDM mean velocity and green triangles the
neutrino mean velocity. This was the first lineal effect studied in the cosmic
neutrino background framework. It can also be observed how neutrinos cluster
along the direction of CDM motion ($45^{\circ}<\theta<135^{\circ}$) and how
they cluster more in the areas located ahead of the halo centres
($\theta=90^{\circ}$ and $\phi=0^{\circ}$) marked by a red solid line square,
and less in the areas located behind the halo centres ($\theta=90^{\circ}$ and
$\phi=180^{\circ}$) marked by a red dashed line square. The effect is present
in all halo masses and its amount is not so different among the analysed
haloes (see Figure 6). This effect depends on the CDM distribution of the halo
and on the haloes surrounding it; actually, we made an extended analysis for
about hundreds of haloes and selected those where the effect is larger, as the
ones shown in Figure 4. The results we present in this work correspond to an
analysis of a total number of 120 haloes of different masses.

Using the selected haloes, we compute the neutrino overdensity profiles,
similarly to the previous ones in the left and right panels of Figure 2, but
now without averaging on the $4\pi$ solid angle. We select a region defined by
$\Delta\theta=60^{\circ}$ and $\Delta\phi=120^{\circ}$ with respect to the
centre placed at $\theta=90^{\circ}$ and $\phi=0^{\circ}$ in order to catch
neutrinos ahead the halo centre, and with centre placed at $\theta=90^{\circ}$
and $\phi=180^{\circ}$ in order to catch neutrinos behind the halo centre in
the direction of the CDM flow (see the squares drawn in Figure 4). Then, we
calculate the profiles of these neutrino subsets. The results for haloes with
masses between $1\times 10^{14}$ and $9\times 10^{14}\,h^{-1}M_{\odot}$,
together with the profile integrated in angles, are shown in Figure 5. The
effect becomes noticeable beyond a radius of 5 $h^{-1}$Mpc where the three
lines diverge: the dashed one, which corresponds to neutrinos ahead, increases
and the dotted one, which corresponds to neutrinos behind, decreases, while
the solid one, which corresponds to all neutrinos within the halo, stays in
between.

To quantify the asymmetry effect we have integrated the profiles from radius
equal to 6 up to 20 $h^{-1}$Mpc, for both ahead and behind profiles. The
result is an overdensity asymmetry above $0.1$ (Figure 6) which remains
constant over the dark matter halo mass and slightly increases for masses
close to $10^{15}\,h^{-1}M_{\odot}$.

Figure 6: Neutrino overdensity for neutrinos located ahead of and behind the
halo centre in the direction of the CDM motion as a function of the dark
matter halo masses. The difference between ahead overdensity and behind
overdensity means is plotted in the bottom subplot. The error bars represent
$1\sigma$ error of the mean values.

####  3.2.2 Wakes in neutrino haloes

Figure 7: Neutrino overdensity distribution in the centre of one halo of mass
$3\times 10^{14}\,h^{-1}M_{\odot}$. The black star represents the position
where the maximum value of the overdensity distribution is reached. The grey
arrow shows the average direction of CDM motion. Figure 8: Mean x and y
position of the maximum value of the 2D neutrino overdensity distribution in a
box of 6 $h^{-1}$Mpc side centred on the centre of the halo as a function of
the mass of the host halo. In the third subplot, the maximum value of the
neutrino overdensity is plotted. All error bars correspond to $1\sigma$ error
of the respective value.

In [6, 7], it was shown that the peculiar motion of haloes causes neutrino
particles to accumulate behind the moving halo, generating wakes that slow the
halo down due to dynamical friction, and reducing the cross-correlation
between neutrinos and CDM. Here, using the HR-DEMNUni simulations we look for
these wakes at the centre of the haloes. To this aim, we use the same
procedure as before: find the halo, determine the CDM mean velocity vector and
rotate all the system such that the CDM mean velocity vector matches the
x-axis at the centre of the halo. For each rotated halo, we select a box of 6
$h^{-1}$Mpc side centered at the halo centre and calculate the neutrino
density distribution over xy-axis. This time, neutrino overdensity
distributions are computed using a standard Gaussian kernel. One example of
the final result is shown in Figure 7, where the black star marks the position
of the maximum in the overdensity distribution. This plot is a clear example
of the effect, some wakes can be observed and the maximum of the overdensity
is displaced behind the centre along the CDM velocity direction which is
illustrated by the grey arrow.

We repeated this analysis for hundreds of haloes in the simulation and saved
the neutrino overdensity maximum value together with its location. These
values are shown in Figure 8 as a function of the halo mass. The mean value of
each parameter and the error bars corresponding to a $1\sigma$ error are
represented. One could say that the effect is observable for halo masses
greater than $3\times 10^{14}\,h^{-1}M_{\odot}$. For those haloes, the mean
position of the neutrino overdensity maximum is at $(-0.06,-0.06)\ h^{-1}$Mpc.

##  4 Discussion and conclusions

High-resolution cosmological simulations open up the possibility to analyze in
detail specific high precision features of the cosmological model. In this
work we have focussed on the properties of neutrino haloes, looking for
asymmetries in regions away from the centre of the haloes but also in regions
close to it. To this end, we have used the HR-DEMNUni simulations. These kinds
of studies are motivated by the current context of new experiments aimed at
improving our understanding of the Universe. If work continues in this
direction, it is possible that, in the future, neutrinos from the cosmic
background will be detected. Some evidences of the effects studied in the
present work will help to make sure that neutrinos are being detected. The
simulations used in this work allow us to obtain more reliable results as they
contain neutrinos with a total mass closer to the actual limits established by
cosmology.

We have obtained angle-averaged neutrino profiles for several halo masses,
using a kernel to obtain a smoother shape, similar to previous works, but this
time for a lower neutrino mass. We have shown that, due to the higher thermal
velocity of lower mass neutrinos, the clustering in haloes of even $M\sim
10^{14}$$h^{-1}M_{\odot}$ is more suppressed. The lower the neutrino mass the
more diffuse the neutrino halo core is, then problems arise when fitting to
the formula proposed. We put the limit of the applicability of this formula in
halos of $4\times 10^{14}$$h^{-1}M_{\odot}$. This result suggest that low mass
neutrinos can be understood mostly as light-like particles in the surrounding
of dark matter haloes with masses less than $4\times
10^{14}$$h^{-1}M_{\odot}$. With lower neutrino mass simulations that fit the
current cosmological limits close to the mass split from underground
laboratories of $0.06$ eV, this limit could be pushed further.

We have also investigated asymmetries that can arise when the neutrino
profiles are not averaged over the solid angle. As we know, cosmic neutrinos
will cluster around virialized dark matter haloes as a consequence of the huge
gravitational field produced by clusters of $10^{11}-10^{15}\,h^{-1}M_{\odot}$
of mass. As neutrinos will fall-in following the gravitational field, if this
field is not isotropic enough we could find some anisotropies in neutrino
field as well. In that sense, in the region located ahead of the dark matter
halo centre we expected a neutrino density higher than in the region located
behind, with respect to the direction of CDM motion, i.e., along the
anisotropy. In this work, we have shown that the predicted front loading of
neutrinos effect is indeed present in the simulations and has been quantified.
It is visible for $\text{R}>6\,h^{-1}\rm{Mpc}$ for a solid angle
$\Delta\theta=60^{\circ}$, $\Delta\phi=120^{\circ}$ centred at the centre of
the halo. We have also quantified its mass dependence, the result obtained is
an average overdensity around 0.1 and independent of the halo mass. Only for
haloes with masses close to $10^{15}\,h^{-1}M_{\odot}$ some enhancement of the
effect is observed.

We found that the HR-DEMNuni simulations have enough resolution to search for
the wakes in neutrino profiles recently proposed in [6, 7]. These wakes were
expected in regions close to the centre of the haloes and due to the relative
motions of neutrinos and CDM. We searched for them in the innermost regions
within the haloes, in a box of $6\ h^{-1}$Mpc side placed in the centre of the
halo, and we found that these wakes only take place in the most massive haloes
($M_{\rm h}>3\times 10^{14}\,h^{-1}M_{\odot}$). An average displacement of the
neutrino overdensity distribution maximum of $0.06\,$$h^{-1}\mathrm{Mpc}$ was
obtained.

An experimental detection of all these effects will cement the case for
neutrinos being observed in the sky and in addition, detailed study of these
effects will in turn unveil the nature of neutrinos and the Universe when it
was only one second old. Finally, we conclude that these very fascinating
features will be seen once cosmic neutrino background experiments manage to
distinguish direction and angle.

## Acknowledgments

The DEMNUni simulations were carried out in the framework of “The Dark Energy
and Massive Neutrino Universe" project, using the Tier-0 IBM BG/Q Fermi
machine, the Tier-0 Intel OmniPath Cluster Marconi-A1 and Marconi-100 of the
Centro Interuniversitario del Nord-Est per il Calcolo Elettronico (CINECA). CC
acknowledges a generous CPU and storage allocation by the Italian Super-
Computing Resource Allocation (ISCRA) as well as from the coordination of the
“Accordo Quadro MoU per lo svolgimento di attività congiunta di ricerca Nuove
frontiere in Astrofisica: HPC e Data Exploration di nuova generazione”,
together with storage from INFN-CNAF and INAF-IA2. This work was supported by
the “Center of Excellence Maria de Maeztu 2020-2023” award to the ICCUB
(CEX2019-000918-M) funded by MCIN/AEI/10.13039/501100011033. Funding for the
work of RJ was partially provided by project PID2022-141125NB-I00.

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