pnasresearcharticle A. Del Popolo 2 E-mail<EMAIL_ADDRESS>
# Improved Lemaitre-Tolman model and the mass and turn-around radius in group
of galaxies

Antonino Del Popolo  Dipartimento di Fisica e Astronomia, University Of
Catania, Viale Andrea Doria 6, 95125, Catania, Italy  Institute of Astronomy,
Russian Academy of Sciences, Pyatnitskaya str. 48, 119017 Moscow, Russia
Maksym Deliyergiyev  Département de Physique Nucléaire et Corpusculaire,
University of Geneva, CH-1211 Geneve 4, Switzerland  Man Ho Chan  Department
of Science and Environmental Studies, The Education University of Hong Kong,
Tai Po, New Territories, Hong Kong

###### Abstract

We extended the modified Lemaitre-Tolman model Ref.(1, 2) taking into account
the effect of angular momentum and dynamical friction. The inclusion of these
quantities in the equation of motion modifies the evolution of a perturbation,
initially moving with the Hubble flow. Solving the equation of motions we got
the relationships between mass, $M$, and the turn-around radius, $R_{0}$.
Knowing $R_{0}$, the quoted relation allows the determination of the mass of
the object studied. The relationships for the case in which also the angular
momentum is taken into account gives a mass $\simeq 90$ % larger than the
standard Lemaitre-Tolman model, and two times the value of the standard
Lemaitre-Tolman model, in the case also dynamical friction is taken into
account. As a second step, we found relationships between the velocity, $v$,
and radius, $R$, and fitted them to data of the Local Group, M81, NGC 253,
IC342, CenA/M83, and to the Virgo clusters obtained by Ref.(1, 2). This
allowed us to find optimized values of the mass and Hubble constant of the
objects studied. The fit gives values of the masses smaller with respect to
the $M-R_{0}$ relationship method, but in any case 30-40% larger than the
$v-R$ relationship obtained from the standard Lemaitre-Tolman model.
Differently from mass, the Hubble parameter becomes smaller with respect to
the standard Lemaitre-Tolman model, when angular momentum, and dynamical
friction are introduced. This is in agreement with Ref.(1, 2), who improved
the standard Lemaitre-Tolman model taking into account the cosmological
constant.

Finally, we used the mass, $M$, and $R_{0}$ of the studied objects to put
constraints to the dark energy equation of state parameter, $w$. Comparison
with previous studies show different constraints on $w$.

###### keywords:

dwarf galaxies $|$ galaxy clusters $|$ modified gravity $|$ mass-temperature
relation

While the mass-to-light ($M/L$) ratios of group of galaxies was in the past
estimated through the virial theorem to be typically of the order of $\simeq
170M_{\odot}/L_{B,M_{\odot}}$ Ref.(3), new measurements based on high quality
data, and estimating methods different from the Virial theorem Ref.(4) give
much smaller results in the range $10-30M_{\odot}/L_{B,M_{\odot}}$. This means
that the local matter density should be a fraction of the global one. It is
well known that the virial theorem gives reliable results if the system is in
dynamical equilibrium. This condition is often assumed if the crossing time is
less than the Hubble time. This assumption has been shown to be often not
correct by Ref.(5), whose analysis showed that there is no correlation between
the virial ratio $\frac{2T}{W}$, being $T$, and $W$ the kinetic and potential
energy, and the crossing time. By means of methods used by observers, Ref.(5)
showed that $\simeq 20\%$ of the studied groups were not gravitationally
bound. Ref.(6) and Ref.(7) proposed an alternative approach to the virial
theorem based on the Lemaitre-Tolman (LT) model Ref.(8, 9) giving a good
description of a central core gravitationally bound located inside an
homogeneous region whose density decreases till reaching the background value.
The model describes the evolution of the system in a similar way to that done
by the spherical collapse model. Considering a shell of given radius
containing a mass $M$, it initially expands following the Hubble flow. When
the density overcomes a critical value the shell reaches a maximum radius,
known as turn-around radius, $R_{0}$, characterized by zero velocity, and
collapses. Then in the LT model there is a central region in equilibrium,
surrounded by a region which reaches its maximum expansion and collapses, and
a zero totally energy region constituted by shells still bound to the
structure and unbound ones. Because of its characteristics, the LT model gives
a good description of a group of galaxies dominated by one or two central
galaxies embedded into a cloud of smaller ones. If using the velocity field
around the main bodies allows the determination of the turn-around radius
$R_{0}$, the mass can be obtained through the relation

$M=\frac{\pi^{2}R_{0}^{3}}{8GT_{0}^{2}}$ (1)

Refs.(7, 1, 2), where $T_{0}$ is the age of the universe. The quoted model was
applied to the local group Ref.(7) and to the Virgo cluster Refs.(10, 11, 12).
The model was modified taking into account the cosmological constant by
Ref.(1, 2) applying it to the Virgo cluster, the pair M31-MW, M81, the
Centaurus A-M83 group, the IC342/Maffei-I group, and the NGC 253 group. As
shown in Refs.(1, 2) the introduction of the cosmological constant modifies
the mass, $M$, turn-around radius, $R_{0}$, relation. As a consequence for a
given $R_{0}$, the value of the mass of the system is $\simeq 30\%$ larger
with respect to (1) Refs.(1, 2), while the Hubble constant of the modified
model is smaller than the standard LT.

In order to obtain the mass of the previously quoted objects, Refs.(1, 2),
differently from Ref.(7), did not use the standard LT (SLT) $M-R_{0}$ relation
((1)). They built up a velocity-distance relationship, $v-R$, describing the
kinematic status of the systems studied. Knowing the values of $v$, and $r$
for the members of the groups studied, the mass of the group, $M$, and the
Hubble parameter can be obtained by means of a non-linear fit of the $v-R$
relation to the data.

In the present paper, we will further extend the modified Lemaitre-Tolman (MLT
model) by taking into account the effect of angular momentum (JLT model) and
dynamical friction (J$\eta$LT model). The effect of these two quantities on
the spherical collapse model (SCM) and its effect on the clusters of galaxies
structure and evolution, the turn-around, the threshold of collapse, their
mass function, their mass-temperature relation, have been studied in Refs.(13,
14, 15, 16, 17, 18, 19, 20, 21).

Similarly to Ref.(1, 2), we will find the $v-R$ relation by solving the
equation of the SCM, and then fit it to the data of the Virgo cluster, the
pair M31-MW, M81, the Centaurus A-M83 group, the IC342/Maffei-I group, and the
NGC 253 group.

The paper is organized as follows. In Section Model, we introduce the model,
and solve it. In Section The velocity-radius relation, we find the velocity-
radius relation for the JLT, and J$\eta$LT models. In Section Application to
near groups and clusters of galaxies, we applied the $v-R$ relation to groups
and clusters of galaxies. In Section Effects of cosmological constant, angular
momentum, and dynamical friction, we studied the impact of the angular
momentum and dynamical friction on the $M-R_{0}$ relation. In Section
Constraints on the DM EoS parameter, we showed how the obtained values of $M$
and $R_{0}$ may constrain the dark energy equation of state parameter, $w$.
Section Conclusions is devoted to conclusions.

## Model

The simplest form of the SCM was introduced by Ref.(22). It is a simple and
popular method to study analytically the non-linear evolution of perturbations
of dark matter (DM) and dark energy (DE). As previously reported, the model
describes the evolution of a spherical symmetric over density which initially
expands with the Hubble flow, then detaches from it, when the density
overcomes a critical value, reaches a maximum radius, dubbed turn-around
radius, and finally collapse and virialize. SCM is a very simple model
assuming that matter moves in a radial fashion Ref.(22, 23, 24). Tidal angular
momentum Ref.(25, 26), random angular momentum Refs.(27, 28, 29), dynamical
friction (Refs.(30, 31)), etc., are not taken into account. Later the SCM was
improved in several papers Ref.(32, 33, 34, 27, 35, 36, 29), adding the
cosmological constant Ref.(37), and tidal and random angular momentum(27, 38,
39, 40, 41, 42, 43, 44, 36, 29, 45)111Particles angular momenta is randomly
distributed in random such that the mean angular momentum at any point in
space is zero Ref.(40, 42) then conserving spherical symmetry and angular
momentum.. Dynamical friction was studied in Refs.(30, 31), while Refs.(46,
47, 48) discussed the role of shear in the gravitational collapse.

The SCM with negligible DE perturbations was extensively investigated in
literature Refs.(see, e.g. 49, 50, 51, 52, 53, 54, 55), while DE fluid
perturbation were taken into account in Refs.(see 56, 57, 58, 59, 60, 61, 62,
63, 64). Using the non-linear differential equations for the evolution of the
matter density contrast derived from Newtonian hydrodynamics in Ref.(54),
Ref.(65) showed that the parameters of the SCM become mass dependent.

Refs.(66, 65) studied the effects of shear and rotation in smooth DE models.
The effects of shear and rotation were investigated in Refs.(66, 65) for
smooth DE models, Ref.(67) in clustering DE cosmologies, and Ref.(68) in
Chaplygin cosmologies.

In this paper, we are interested in describing a system constituted by a
dominant mass concentration, and satellites that are not contributing
significantly to the group mass, and that further mass accretion is neglected.

The equation of motion of the system may be obtained as follows. We consider
some gravitationally growing mass concentration collecting into a potential
well. Let us assume that the probability of a particle, located at $[r,r+{\rm
d}r]$, having angular momentum $L=rv_{\theta}$, defined in the range
$[L,L+{\rm d}L]$, with velocity $v_{r}={\dot{r}}$, defined in the range
$[v_{r},v_{r}+{\rm d}v_{r}]$, has the following form

${\rm d}P=f(L,r,v_{r},t){\rm d}L{\rm d}v_{r}{\rm d}r.$ (2)

The term $L$ takes into account ordered angular momentum generated by tidal
torques and random angular momentum (see Appendix C.2 of Ref.(31)). The radial
acceleration of the particle Refs.(69, 70, 37, 13, 14) is:

$\frac{{\rm d}v_{R}}{{\rm
d}t}=-\frac{GM}{R^{2}}+\frac{L^{2}(R)}{M^{2}R^{3}}+\frac{\Lambda}{3}R-\eta\frac{{\rm
d}R}{{\rm d}t}\,,$ (3)

with $\Lambda$ being the cosmological constant and $\eta$ the dynamical
friction coefficient. The previous equation can be obtained via Liouville’s
theorem Ref.(14). The last term, the dynamical friction force per unit mass,
$\eta$, is explicitly given in Ref.(31) (Appendix D, Eq. D5). A similar
equation (excluding the dynamical friction term) was obtained by several
authors Refs.(e.g., 71, 72, 65)) and generalized to smooth DE models in
Ref.(73).

Figure 1:  Evolution of shell radius for different values of $K$. The red,
cyan, and green lines correspond to $K=-5.737$, $K=-6.2$, and $K=-5.1$,
respectively.

In terms of the specific angular momentum $J=\frac{L}{M}$, and
$\Omega_{\Lambda}=\frac{\rho_{\Lambda}}{\rho_{c}}$, where $\rho_{c}$ is the
critical density, (3) can be written as

$\frac{{\rm d}v_{R}}{{\rm
d}t}=-\frac{GM}{R^{2}}+\frac{J^{2}}{R^{3}}-\frac{1+3w}{2}\Omega_{\Lambda}H_{0}^{2}\left(\frac{a_{0}}{a}\right)^{3(1+w)}R-\eta\frac{{\rm
d}R}{{\rm d}t}\,,$ (4)

where $w$ is the DE equation of state (EoS) parameter. DE is modeled by a
fluid with an EoS $P=w\rho$, where $\rho$ is the energy density. $a$ is the
expansion parameter. (4) satisfies equation

$H=\frac{\dot{a}}{a}=H_{0}\sqrt{\Omega_{\rm
m}\left(\frac{a_{0}}{a}\right)^{3}+\Omega_{\Lambda}\left(\frac{a_{0}}{a}\right)^{3(1+w)}}.$
(5)

In the following, we will treat the case $w=-1$, in other words we assume that
DE is the cosmological constant. With this assumption, and assuming that
$J=kR^{\alpha}$, with $\alpha=1$, in agreement with Ref.(74)222In that paper
$\alpha=1.1\pm 0.3$, and $k$ constant. In terms of the variables $y=R/R_{0}$,
$t=x/H_{0}$, (4), and (5) can be written as

$\frac{d^{2}y}{dx^{2}}=-\frac{A}{2y^{2}}+\Omega_{\Lambda}y+\frac{K_{j}}{y}-\frac{\eta}{H_{0}}\frac{dy}{dx},$
(6)

where $K_{j}=k\frac{1}{(H_{0}R_{0})^{2}}$, $A=\frac{2GM}{H_{0}^{2}R_{0}^{3}}$,
and

$H=H_{0}\sqrt{\Omega_{m}\left(\frac{a_{0}}{a}\right)^{3}+\Omega_{\Lambda}}.$
(7)

(6) has a first integral, given by

$\displaystyle u^{2}$ $\displaystyle=\left(\frac{dy}{dx}\right)^{2}$
$\displaystyle=\frac{A}{y}+\Omega_{\Lambda}y^{2}+2K_{j}\log{y}-2\frac{\eta}{H_{0}}\int\left(\frac{dy}{dx}\right)^{2}dx+K$
(8)

where $K=\frac{2E}{(H_{0}R_{0})^{2}}$, and $E$ is the energy per unit mass of
a shell.

Eqs.(5), and (6) where solved as described in Ref.(1, 2). There are a couple
of ways of doing that. A first way, is to obtain the value of the scale
parameter and the corresponding time for a given redshift. At high redshift,
the gravitational term dominates and through a Taylor expansion one can get
the initial conditions. In order to get the parameter $A$, it is varied until
the condition $\frac{dy}{dx}=0$, and $y=1$ are satisfied. A second way to get
$A$, is to use the equation for the velocity ((Model)).

model | $\eta/H_{0}$ | $K_{J}$ | $b$ | $n$ | $A$
---|---|---|---|---|---
MLT | – | $0.0$ | $1.4054$ | $0.6293$ | $3.6575$
JLT | – | $0.78$ | $1.3759$ | $0.7549$ | $5.0370$
J$\eta$LT | $0.5$ | $0.78$ | $1.3436$ | $0.9107$ | $6.0500$
Table 1:  The constant $A$, and the fitting parameters $b$, and $n$ of the
velocity-distance ($v-R$) relations, for the MLT, the JLT, and the J$\eta$LT
model.

Let’s show this second method in the case cosmological constant, and angular
momentum are present (JLT case)

$\frac{d^{2}y}{dx^{2}}=-\frac{A}{2y^{2}}+\Omega_{\Lambda}y+\frac{K_{j}}{y}$
(9)

having the first integral

$\displaystyle
u^{2}=\left(\frac{dy}{dx}\right)^{2}=\frac{A}{y}+\Omega_{\Lambda}y^{2}+2K_{j}\log{y}+K$
(10)

At the turn-around point (10) gives: $K=-A-\Omega_{\Lambda}$.

At high redshifts ($z=1000$), or $y\ll 1$, as was described the gravitational
term dominates, and by a Taylor expansion one gets the relation
$y\simeq(\frac{9A}{4})^{1/3}x^{2/3}$. Assuming an initial value of $y$,
$y_{i}=0.001$, corresponding approximately to 1 kpc, the initial time $x_{i}$
can be obtained. The initial value of the velocity $u_{i}$ can be obtained,
when $A$ is known, through (10), recalling that $y_{i}=0.001$. The value of
$A$ is obtained as follows. (10) can be written as

$x=\int_{y_{i}}^{1}\frac{dy}{\sqrt{\frac{A}{y}+\Omega_{\Lambda}y^{2}+\frac{K_{j}}{y}-A-\Omega_{\Lambda}}}$
(11)

(7), recalling that $\frac{a_{0}}{a}=1+z$, can be written as

$x(y=1)=\int_{0}^{\infty}\frac{dz}{(1+z)\sqrt{\Omega_{\Lambda}+\Omega_{\rm
m}(1+z)^{3}}}=0.964.$ (12)

For $\Omega_{\Lambda}=0.7$, $\Omega_{\rm m}=0.3$, $K_{j}=0.78$333The value of
$K_{j}$ was obtained recalling that term related to angular momentum, L, in
(4), is given by $\frac{L^{2}}{M^{2}R^{3}}$., $x=0.964$, (11) can be solved to
get $A=5.037$. In the case, $K_{j}=0$, $A=3.6575$, and if $K_{j}=0$,
$\Omega_{\Lambda}=0$, the SLT gives $A=2.655$.

In the case J$\eta$LT (Eqs.(6)-(Model)), $A$ can be obtained similarly to the
previous case (JLT) solving numerically (Model) with the initial condition on
$y_{i}$, and varying $A$ until the condition $\frac{dy}{dx}=0$, and $y=1$ are
satisfied. Similarly, we can solve (6) with the initial condition $y_{i}$, and
varying $A$ until the condition $\frac{dy}{dx}=0$, and $y=1$ are satisfied. In
this way, one gets $A=6.05$.

Now, we show the solution for the case JLT. (9) can be solved with the
conditions $y_{i}=0.001$, and

$\begin{split}u(0)&=\sqrt{\frac{5.037}{y_{i}}+0.7y_{i}^{2}+2K_{J}\log{y_{i}}-0.7-5.037}\\\
&=70.833943.\end{split}$ (13)

In Fig.1, we plot the result of the solution. The red line corresponds to the
case $K=-A-\Omega_{\Lambda}=-5.737$, being $A=5.037$. This solution is the one
that has just reached the maximum expansion, or turn-around, and the collapse
happens in $\simeq 13.8$ Gyr. The cyan line is characterized by $K=-6.2$. It
reached the turn-around in the past. Turn-around will happen only for
$K<-5.56812$, for larger values the collapse will never occur, as the case of
the green line characterized by $K=-5.1$.

Figure 2:  The velocity profile in the three cases studied. The left panel
shows the MLT model. The central panel, the JLT. The right panel, the
J$\eta$LT case. Data points are obtained as follows. Fig. 1, shows some
solutions of Eq. 6 for different values of K: K = -6.2, -5.737, and -5.1. We
obtained several solution relative to different values of K: $K_{1}$, $K_{2}$,
$K_{3}$….. The intersection of each solution with the vertical axis $x=0.964$,
gives a corresponding value of y: $y_{1}$, $y_{2}$, $y_{3}$,…. The solution of
Eq. 6 for different values of K at $x=0.964$ gives also $u_{1}$, $u_{2}$,
$u_{3}$…. All this gives us a pair of values $(y_{i},u_{i})$ for each
intersection of the vertical line with the curves. The solid black lines are
the fit to the points in the examined models.

## The velocity-radius relation

In order to get the mass, and turn-around radius of some groups of galaxies,
we will find a relation between the velocity, and radius, $v-R$, that will be
fitted to the data. The $v-R$ relation is obtained as follows. Let’s consider
Fig.1. The vertical line corresponds to $x=0.964$. Its intersection with the
curves, solution of the equations described in the previous section, gives the
value $y(x)=y(0.964)$. The solution of the equations of the previous section,
also gives the velocity, allowing us to find $u(x)=u(0.964)$. We will get a
couple of value $(y,u)$ for each intersection of the vertical line with the
curves (see Fig.2 caption for an extended description). This allows us to find
a series of points that can be fitted with a relation of the form
$u=-b/y^{n}+by$. For example in the case of the MLT, we get

$v=-\frac{1.4054}{y^{n}}+1.4054y$ (14)

where $n=0.6293$. This can be written in terms of the physical units as

$v(R)=-bH_{0}R_{0}\left(\frac{R_{0}}{R}\right)^{n}+bH_{0}R$ (15)

where $b=1.4054$. Substituting in this equation,
$R_{0}=(\frac{2GM}{H_{0}^{2}})^{\frac{1}{3}}$, we get

$v(R)=-b\frac{H_{0}}{R^{n}}\left(\frac{2GM}{AH_{0}^{2}}\right)^{\frac{n+1}{3}}+bH_{0}R$
(16)

or

$v(R)=-\frac{-1.013H_{0}}{R^{n}}\left(\frac{GM}{H_{0}^{2}}\right)^{\frac{n+1}{3}}+1.4054H_{0}R$
(17)

This relation is slightly different from that obtained by Ref.(2), probably
due to the noteworthy sensitivity of the solution of the equation to initial
conditions, and to the fact we used more digits in the initial condition for
$u(0)$ 444This was confirmed via a private discussion with one of the authors
of Ref.(2), namely de Freitas Pacheco..

For this reason, in the rest of the paper, we also consider the MLT case,
already studied by Ref.(2). In a similar way, we can obtain the $v-R$ relation
in the case the of the JLT model

$v(R)=-\frac{-0.80155H_{0}}{R^{n}}\left(\frac{GM}{H_{0}^{2}}\right)^{\frac{n+1}{3}}+1.3759H_{0}R$
(18)

where $n=0.7549$, and in the complete case (cosmological constant, angular
momentum, and dynamical friction)

$v(R)=-\frac{-0.66385H_{0}}{R^{n}}\left(\frac{GM}{H_{0}^{2}}\right)^{\frac{n+1}{3}}+1.3436H_{0}R$
(19)

where $n=0.9107$. In Fig.2 we plot, from left to right, the velocity profile
of the MLT, JLT, and J$\eta$LT cases, using adimensional variables.

All the previous equations satisfy the condition $v(R_{0})=0$. In the
following, we will apply (19), related to the J$\eta$LT model to some groups
of galaxies and clusters. In Table 1, we summarize the parameters of the
different models that were described in this paper. The first line corresponds
to the MLT model. The second line to the JLT model, and, the last line to the
J$\eta$LT case,

Table 1, as well as Fig.2 shows that including the angular momentum, and
dynamical friction steepens the velocity profile, and increases the parameter
$A$. This means that for a given $R_{0}$ the mass of the structure increases,
while the radius of the zero-gravity surface decreases.

## Application to near groups and clusters of galaxies

Now, we will apply Eqs.(17-19) to near groups and a cluster of galaxies. To
this aim, we need for each galaxy its velocity and distance with respect to
center of mass. We will use data obtained by Refs.(1, 2). Velocities were
transformed from the heliocentric to the Local Group rest frame. The distance
can be written as

$R=\sqrt{D^{2}+D_{g}^{2}-2DD_{g}\cos{\theta}}$ (20)

where the angle $\theta$ is the angle between the center of mass and the
galaxy, $D$ the distance from the galaxy to the center of mass, and $D_{g}$ is
the distance to the galaxy. Indicating with $V$, and $V_{g}$ the center of
mass velocities, and that of the galaxy with respect the Local Group rest
frame, the velocity difference along the radial direction between both object
is

$V(R)=V_{g}\cos{\alpha}-V\cos{\beta}$ (21)

being $\alpha=\frac{D\sin_{\theta}}{D_{g}-D\cos{\theta}}$, and
$\beta=\alpha+\theta$.

Since in the list given by Ref.(2) unbound objects, and uncertain distances
and velocities were excluded, an error of 10% was considered for velocities
and distances by Ref.(2). This value of uncertainty is a weighted mean of data
including measurement errors and data reported without errors Ref.(76, 81).

In the case of the group M31-MW, the data were obtained by Ref.(1) from
Ref.(75) data. We used the data of Ref.(1) also for the case of the Virgo
cluster.

Fig.3 plots the $v-R$ relationships for the groups studied: the M31-MW group
(top left panel), the M81 group (top right panel), the NGC 253 group (central
left panel), the IC 342 group (central right panel), the CenA/M83 group
(bottom left panel), the Virgo cluster (bottom right panel). The red squared
are the data from Refs.(1, 2).

Table 2:  Characteristic parameters of the examined groups. The rows 1-3 represent the value of the Hubble parameters for the MLT, JLT, and J$\eta$LT models. The rows 4-6 the masses of the groups in units of $10^{12}M_{\odot}$ for the same cases, and the rows -9 the values of the turn-around radius, $R_{0}$, in Mpc, for the same cases. The last three rows give the velocity dispersion resulting from the fit of data to the $v-R$ relation for the same cases.  | M31/MW | M81 | NGC 253 | IC 342 | CenA/M83 | Virgo
---|---|---|---|---|---|---
h($\Omega_{\Lambda}=0.7$) | $0.73\pm 0.04$ | $0.68\pm 0.04$ | $0.63\pm 0.06$ | $0.58\pm 0.10$ | $0.57\pm 0.04$ | $0.71\pm 0.08$
h(j) | $0.70\pm 0.04$ | $0.65\pm 0.04$ | $0.63\pm 0.06$ | $0.56\pm 0.10$ | $0.55\pm 0.04$ | $0.65\pm 0.09$
h(j,$\eta$) | $0.69\pm 0.04$ | $0.65\pm 0.04$ | $0.63\pm 0.05$ | $0.55\pm 0.10$ | $0.55\pm 0.04$ | $0.59\pm 0.09$
[2.5pt/5pt] M($\Omega_{\Lambda}=0.7$) [$10^{12}M_{\odot}$] | $2.49\pm 0.50$ | $1.14\pm 0.10$ | $0.14\pm 0.15$ | $0.22\pm 0.12$ | $2.16\pm 0.50$ | $1493\pm 200$
M(j) [$10^{12}M_{\odot}$] | $3.090\pm 0.50$ | $1.320\pm 0.10$ | $0.195\pm 0.10$ | $0.263\pm 0.10$ | $2.655\pm 0.50$ | $1585\pm 200$
M(j,$\eta$) [$10^{12}M_{\odot}$] | $3.570\pm 0.40$ | $1.398\pm 0.10$ | $0.244\pm 0.10$ | $0.292\pm 0.10$ | $3.015\pm 0.40$ | $1525.55\pm 200$
[2.5pt/5pt] $R_{0}(\Omega_{\Lambda}=0.7)$ [Mpc] | $1.038\pm 0.10$ | $0.840\pm 0.05$ | $0.440\pm 0.10$ | $0.540\pm 0.09$ | $1.160\pm 0.08$ | $8.850\pm 0.80$
$R_{0}(j)$ [Mpc] | $1.04\pm 0.10$ | $0.81\pm 0.05$ | $0.44\pm 0.10$ | $0.53\pm 0.09$ | $1.14\pm 0.08$ | $8.67\pm 0.80$
$R_{0}(j,\eta)$ [Mpc] | $1.02\pm 0.10$ | $0.78\pm 0.05$ | $0.44\pm 0.10$ | $0.52\pm 0.09$ | $1.13\pm 0.08$ | $8.56\pm 0.80$
[2.5pt/5pt] $\sigma(\Omega_{\Lambda}=0.7)$ [km/s] | $37.3$ | $51.16$ | $45.58$ | $33.49$ | $44.72$ | $352.1$
$\sigma(j)$ [km/s] | $38.3$ | $53.24$ | $45.8$ | $33.93$ | $44.75$ | $352.88$
$\sigma(j,\eta)$ [km/s] | $38.8$ | $54.77$ | $45.9$ | $34.48$ | $44.81$ | $355.1$

### M31-MW

We applied Eqs. (17-18), and (19) to the Ref.(1) data. Both the mass and the
Hubble parameter were allowed to vary. The results are shown in Table 2.
Ref.(75) estimated a turn-around radius of $0.94\pm 0.1$ Mpc and using the SLT
model obtained a mass of $1.5\times 10^{12}M_{\odot}$, which is much smaller
than the estimate of Ref.(1) $(2.5\pm 0.7)\times 10^{12}M_{\odot}$,
$R_{0}=1.0\pm 0.1$ Mpc, and $h=0.74\pm 0.04$. The value of the mass is larger
than that of Ref.(75), that used the SLT model. A tendency of the LT models is
that of giving higher masses, and smaller $h$ if the effect of the
cosmological constant, angular momentum, and other effects which contribute
with positive terms in the equation of motion are taken into account. In fact,
Ref.(1) found a values of $h=0.87\pm 0.05$ when using the SLT model, and
$h=0.73\pm 0.04$ in the MLT case.

The values of $R_{0}$, in all three cases (MLT, JLT, and J$\eta$LT) are in
agreement, within the estimated uncertainties, with estimate reported in
Ref.(1). Our values of $h$, and $M$ are in agreement to that of Ref.(1) in the
MLT, JLT, cases, while in the J$\eta$LT the value is slightly larger. The
average value of $h$ is smaller in the JLT, and J$\eta$LT models, while the
reverse happens to the mass. We recall that the errors, come from the fitting
procedure.

### The M81 group

The M81 group was studied by Refs.(75, 76, 77). The authors found
$R_{0}=0.89\pm 0.05$ Mpc, smaller than our estimates, and $M=(1.03\pm
0.17)\times 10^{12}M_{\odot}$, in agreement only with our MLT case. Ref.(2)
found $M=(0.92\pm 0.24)\times 10^{12}M_{\odot}$, smaller than our cases JLT,
and J$\eta$LT and $h=0.67\pm 0.04$, in agreement with all our cases. Our JLT,
and J$\eta$LT model estimates, as in the previous, and in all cases, gives
average values of the mass, $M$ larger than the average of the estimates of
Refs.(77, 2).

### The NGC253 group

Concerning this group, Ref.(78) obtained $R_{0}=0.7\pm 0.1$ Mpc, smaller than
our estimates, and $M=(5.5\pm 2.2)\times 10^{11}M_{\odot}$, larger than our
estimates. Ref.(2) found $M=(1.3\pm 1.8)\times 10^{11}M_{\odot}$ whose larger
uncertainties is probably due to incompleteness in the data. They also found
$h=0.63\pm 0.06$. Both their estimates for $h$, and $M$, are in agreement with
all our cases.

### The IC342 group

According to Ref.(79), the group has $R_{0}=0.9\pm 0.1$, and $M=(1.07\pm
0.33)\times 10^{12}M_{\odot}$, both larger than our estimates. Ref.(2) found a
smaller value of the mass, $M=(2.0\pm 1.3)\times 10^{11}M_{\odot}$, and also
$R_{0}$ ($\simeq 0.53$ Mpc), while $h=0.57\pm 0.10$. Our values of mass, turn-
around radius, and $h$ agree with Ref.(2) estimates.

### The CenA/M83 group

This group was studied by Refs.(80, 81). From the distances, and velocities of
the group member, taking into account the cosmological constant they found
$R_{0}=1.55\pm 0.13$ Mpc, and $M=(6.4\pm 1.8)\times 10^{12}M_{\odot}$, larger
than our estimates. Ref.(82), using different mass indicators found a larger
mass ($M=(9.2\pm 3)\times 10^{12}M_{\odot}$). Ref.(2), found values 3-4 times
smaller ($M=(2.1\pm 0.5)\times 10^{12}M_{\odot}$), and $h=0.57\pm 0.04$. In
our analysis, both $M$, and $h$ are in agreement with Ref.(2).

### The Virgo cluster

Concerning Virgo, several estimates for the mass were done by means of the SLT
model Ref.(10, 83), by means of the Virial theorem Ref.(11) finding masses
smaller than $10^{15}M_{\odot}$, except Ref.(83) who found a value of
$1.3\times 10^{15}M_{\odot}$. Using the MLT model Ref.(1) found $M=(1.10\pm
0.12)\times 10^{15}M_{\odot}$, $h=0.65\pm 0.09$, and $R_{0}=8.6\pm 0.8$ Mpc.
Our estimates are in agreement with the value of $h$, $R_{0}$ of Ref.(1),
while the masses in the cases JLT, and J$\eta$LT are larger than in Ref.(1).

In summary, our estimates usually agree with the estimates of Ref.(1, 2),
especially in the case MLT, and JLT. In some cases there are discrepancies
with the predictions of our J$\eta$LT model. Moving from the SLT model to the
MLT, JLT, and J$\eta$LT, the values of the cosmological constant decreases,
and the opposite happens to the mass, $M$.

Another important issue that is shown by Table 2, is that the values of $h$
are in some cases smaller than the known estimates. In the past decade or so,
has been performed dozens of measurements of the Hubble constant, to try to
overcome the Hubble constant tension. As clear shown from Ref.(84), from the
year 2000 the constraints have changed from $72^{+8}_{-8}$ km/Mpc s, to the
range $67-75$ km/Mpc s. Recent constraints from the gravitational wave signal
of GW170817 gives $70.3^{+5.3}_{-5.0}$ km/Mpc s Ref.(85), $67.4^{+1.1}_{-1.2}$
km/Mpc s ($\rm DES+BAO+BBN$), and $67.5\pm 1.1$ km/Mpc s Ref.(86). The
previous constraints are in agreement with our results, except for CenA/M83
having $H=59$ km/Mpc s. The last discrepancy with observations may be due to
non completeness of the data used in 2008 by Ref.(2). Based on a large-scale
survey of the Centaurus group done in 2014-2015, a significant amount of faint
dwarf galaxy candidates were discovered Ref.(87). Therefore, the old data used
in Ref.(2) may contain some selection bias so that the resulting $H$ obtained
is systematically smaller.

Figure 3:  Velocity-distance plots for the M31-MW group (top left panel), the
M81 group (top right panel), the NGC 253 group (central left panel), the IC
342 group (central right panel), the CenA/M83 group (bottom left panel), the
Virgo cluster (bottom right panel). The black dots are the data from Ref.(1,
2). Red solid lines correspond to the fit with the MLT model, (17), cyan short
dashed lines – JLT model, (18), blue dashed lines – J$\eta$LT model, (19). The
$\chi^{2}/N_{dof}$ values were added into each panel only in illustrative
purpose, since the exact uncertainties of the data unknown.

## Effects of cosmological constant, angular momentum, and dynamical friction

As we wrote in the Introduction, the mass predicted by the LT model is given
by (1), namely

$M=\frac{\pi^{2}R_{0}^{3}}{8GT_{0}^{2}}=3.06\times
10^{12}h^{2}R_{0}^{3}M_{\odot}$ (22)

For the MLT, the value of $A$ can be obtained combining (11), and (12), and
one gets $A=3.6575$. By the definition of $A=\frac{2GM}{H_{o}^{2}R_{o}^{3}}$,
and recalling that in the $\Lambda$CDM model $H_{0}=f(\Omega)/T_{0}$, where

$f(\Omega)=\int_{0}^{\infty}\frac{dz}{(1+z)\sqrt{\Omega_{\Lambda}+\Omega_{\rm
m}(1+z)^{3}}}$ (23)

we obtain, for $\Omega_{\Lambda}=0.7$

$M=\frac{1.82875H_{0}^{2}R_{0}^{3}}{G}=\frac{1.69945R_{0}^{3}}{GT_{0}^{2}}=4.22\times
10^{12}h^{2}R_{0}^{3}M_{\odot}$ (24)

Comparing (1) (or (22)), and (24), we get a difference of 38%.

In the case of the JLT model, with $K_{j}=0.78$ the value of $A$ is 5.037,
then

$M=\frac{2.5185H_{0}^{2}R_{0}^{3}}{G}=\frac{2.3404R_{0}^{3}}{GT_{0}^{2}}=5.8148\times
10^{12}h^{2}R_{0}^{3}M_{\odot}$ (25)

and then the difference with the case LT is 90%. Finally, in the case of the
J$\eta$LT model, $A=6.05$

$M=\frac{3.025H_{0}^{2}R_{0}^{3}}{G}=\frac{2.8111R_{0}^{3}}{GT_{0}^{2}}=6.9843\times
10^{12}h^{2}R_{0}^{3}M_{\odot}$ (26)

which means that the mass in this case is more than double of the case LT. The
difference in mass between the previous cases is due to the modification of
the perturbation evolution due to the effect of angular momentum, and
dynamical friction as also shown in several papers Refs.(13, 14, 15, 16, 17,
18, 19).

The relation between mass, $M$, and turn-around radius, $R_{0}$, may be
obtained also from (17), (18), and (19), solving the equation $v(R_{0})=0$
with respect to $M$. In the case, LT, $A=2.655$, and the $v-R$ relation is
given by

$v(R)=-1.038\frac{GM}{R}+1.196H_{0}R$ (27)

and

$M=3.065\times 10^{12}h^{2}R_{0}^{3}M_{\odot}$ (28)

In Fig.4, we plot the $v-y(R)$ relations for the MLT, the JLT, and the
J$\eta$LT cases. For distances smaller than $R_{0}$, the plot shows that the
J$\eta$LT cases gives larger negative velocities than the JLT model, and this
larger negative velocities than the MLT model. This imply that the turn-around
happens earlier in J$\eta$LT with respect to the JLT model, and similarly the
turn-around happens earlier in JLT with respect to the MLT model. One
interesting point is that the mass obtained from the $M-R_{0}$ relation in the
case SLT ((27)) is smaller than that of the MLT case ((17)). The last is
smaller than the mass obtained with the JLT ((18)), and this is smaller that
that of J$\eta$LT case ((19)).

For example, fitting the data by means of (27) (case SLT), the mass is $\simeq
10\%$ smaller than that obtained with (17) (case MLT)555The differences
between the SLT, and MLT given by the $M-R_{0}$ relations (Eqs.(22), (24)) is
38%. . Fitting the data by means of (27), and (19) the mass differences become
larger (10% in the case of M81, 100% in the case of NGC 253, and around 40% in
the other cases, excluding Virgo).

The differences between the two methods can be explained as follows. In the
method based on the fitting, the turn-around is obtained through
$R_{0}=(\frac{2GM}{AH_{0}^{2}})^{1/3}$, and depends from $M$, and $H$,
obtained through the fit.

In the method based on the $M-R_{0}$ relation, $R_{0}$ is obtained by any
method allowing the determination of this quantity, and then the $M-R_{0}$
relationship gives the mass.

Another interesting point, is the decrease of $h$ from the SLT model, to the
J$\eta$LT model. The maximum differences for the groups and clusters studied
is $\simeq 30\%$.

Figure 4:  $v-R$ relationship for the MLT model (red curve), JLT (green curve), and J$\eta$LT (brown curve).  Stable structure | range of $w$
---|---
M81 | $w\geq-1.5$
IC342 | $w\geq-1$
NGC253 | $w\geq-1$
CenA/M83 | $w\geq-1.5$
Local Group | $w\geq-2$
Virgo | $w\geq-1.5$
Table 3: The allowed ranges of $w$.

## Constraints on the DM EoS parameter

Recently, the turn-around radius, $R_{0}$ has been proposed as a promising way
to test cosmological models Ref.(88), DE, and disentangle between $\Lambda$CDM
model, DE, and MG models Refs.(89, 90, 91, 92, 88, 93).

Ref.(89) calculated $R_{0}$ for $\Lambda$CDM, and Ref.(90) did the same for
smooth DE. According to Ref.(93) $R_{0}$ is affected by modified gravity (MG)
theories. In MG theories Ref.(94) found a general relation for $R_{0}$, and
Ref.(91) found a method to get the same quantities in generic gravitational
theories. In Ref.(21), we used an extended spherical collapse model (ESCM)
introduced, and adopted in Refs.(95, 66, 96, 97, 73), to show how $R_{0}$ is
modified by the presence of vorticity, and shear in the equation of motion. We
also showed how the $M-R_{0}$ plane can be used to put some constraints on the
DE EoS parameter $w$, similarly to Refs.(89, 90). The constraints on $w$
depends on the estimated values of the mass and $R_{0}$ of galaxies, groups,
and clusters. Some data where taken from Ref.(90), and others from Ref.(1, 2).

With the revised value of mass, $M$, and $R_{0}$ presented in this paper, we
recalculate the constraints showed in Ref.(21).

Fig.5 plots the mass-radius relation of stable structures for different $w$.
The solid lines from top to bottom represent $w=-0.5$ (solid green line), -1
(black solid line), -1.5 (blue solid line) ,-2 (pink solid line), -2.5 (red
solid line). The dashed lines are the same of the previous lines, but they are
obtained using the model from Ref.(21). The dots with error bars, are data
obtained in the previous sections, and reported in Table 2 (case J$\eta$LT).

The constraints to $w$ are reproduced in Table 3. They are different from
previous ones obtained by Ref.(89, 90) based on the calculation of the mass,
$M$, and $R_{0}$ by means of the virial theorem or the LT model.

Figure 5:  Mass-radius relation of stable structures for different $w$. The
solid lines from top to bottom represent $w=-0.5$ (solid green line), -1
(black solid line), -1.5 (blue solid line) ,-2 (pink solid line), -2.5 (red
solid line). The dashed lines are the same of the previous lines, but they are
obtained in Ref.(21). The dots with error bars taken from Table 2.

## Conclusions

In this paper, we extended the modified LT (MLT) model Refs.(1, 2) to take
account the effect of angular momentum and dynamical friction. The inclusion
of these two quantities in the equation of motion modifies the evolution of
perturbations as described by the MLT model. The collapse of shells inside the
zero-velocity surface collapse earlier when adding the angular momentum (JLT
model), and dynamical friction term (J$\eta$LT model). After solving the
equation of motion, we got the relationships between mass, $M$, and the turn-
around radius $R_{0}$, similar to those obtained for the SLT model by Ref.(7),
and for the MLT model by Refs.(1, 2). The relationships show, for a given
$R_{0}$, a larger mass of the perturbation when angular momentum, and
dynamical friction are taken into account. If one can obtain by some method
the value of the turn-around, these relations show that the perturbation mass
is 90% (JLT model), and two times larger (J$\eta$LT model) with respect to the
SLT model. In the paper, we also found velocity, $v$, radius, $R$,
relationships for the cases considered depending on mass and the Hubble
constant. These were fitted to the data of the local group, M81, NGC 253,
IC342, CenA/M83, and Virgo. The values of the masses obtained fitting the data
by means of (19) (J$\eta$LT model) are larger than those obtained by means of
(27) (SLT model). The mass difference is 10% in the case of M81, 100% in the
case of NGC 253, and around 40% in the other cases.

The Hubble parameter becomes smaller when introducing angular momentum, and
dynamical friction with respect to the SLT model. The same happens when adding
the cosmological constant to the SLT model, as noticed by Refs.(1, 2).

Finally, we used the mass, $M$, and $R_{0}$ for the studied objects to put
constraints to $w$. The constraints obtained differ from those obtained in
previous papers Refs.(89, 90) based on the calculation of the mass, $M$, and
$R_{0}$ by means of the virial theorem or the LT model.

The authors wish to express their gratitude to S. Peirani and A. De Freitas
Pacheco for a fruitful discussion.

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