#  Turnaround radius in $\Lambda$CDM, and dark matter cosmologies II: the role
of dynamical friction

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, 119017, Pyatnitskaya str., 48 , Moscow  INFN
sezione di Catania, Via S. Sofia 64, I-95123 Catania, Italy
<EMAIL_ADDRESS>Man Ho Chan  Department of Science and Environmental
Studies, The Education University of Hong Kong, Tai Po, New Territories, Hong
Kong<EMAIL_ADDRESS>
###### Abstract

This paper is an extension of the paper by Del Popolo, Chan, and Mota (2020)
to take account the effect of dynamical friction. We show how dynamical
friction changes the threshold of collapse, $\delta_{c}$, and the turn-around
radius, $R_{t}$. We find numerically the relationship between the turnaround
radius, $R_{\rm t}$, and mass, $M_{\rm t}$, in $\Lambda$CDM, in dark energy
scenarios, and in a $f(R)$ modified gravity model. Dynamical friction gives
rise to a $R_{\rm t}-M_{\rm t}$ relation differing from that of the standard
spherical collapse. In particular, dynamical friction amplifies the effect of
shear, and vorticity already studied in Del Popolo, Chan, and Mota (2020). A
comparison of the $R_{\rm t}-M_{\rm t}$ relationship for the $\Lambda$CDM, and
those for the dark energy, and modified gravity models shows, that the $R_{\rm
t}-M_{\rm t}$ relationship of the $\Lambda$CDM is similar to that of the dark
energy models, and small differences are seen when comparing with the $f(R)$
models. The effect of shear, rotation, and dynamical friction is particularly
evident at galactic scales, giving rise to a difference between the $R_{\rm
t}-M_{\rm t}$ relation of the standard spherical collapse of the order of
$\simeq 60\%$. Finally, we show how the new values of the $R_{\rm t}-M_{\rm
t}$ influence the constraints to the $w$ parameter of the equation of state.

Dwarf galaxies; galaxy clusters; modified gravity; mass-temperature relation

###### pacs:

98.52.Wz, 98.65.Cw

##  I Introduction

In the past several decades, observations revealed that some missing mass
exist in our universe Sanders (2010). Many physicists believe that the
existence of some unknown particles called cold dark matter (CDM) can account
for the missing mass Profumo (2017). On the other hand, cosmological
observations suggest that the expansion of our universe is accelerating. Many
cosmologists propose that the existence of a new kind of energy called dark
energy can help explain the accelerating expansion Li  _et al._ (2011). In
standard cosmological model, the amount of dark energy can be represented by
the cosmological constant $\Lambda$. This standard cosmological model is now
known as the $\Lambda$CDM model. The $\Lambda$CDM model can give good
agreements for observations in large-scale structures Del Popolo (2013); Ade
(2016). However, there are some discrepancies between the predictions of the
$\Lambda$CDM model and the observations in small-scale structures.
Specifically, the core-cusp problem de Blok (2010), the missing satellites
problem Moore  _et al._ (1999) and the mass-discrepancy acceleration relation
problem McGaugh (2004); McGaugh  _et al._ (2016) are three classical problems
challenging the $\Lambda$CDM model. Moreover, currently no compelling particle
dark matter signal has been detected directly or indirectly. The current
direct-detection and indirect-detection constraints of dark matter have ruled
out a large parameter space of potential particle dark matter models Aprile
(2017); Abecrcrombie (2020); Ackermann (2015); Chan and Leung (2017); Chan
_et al._ (2019); Chan and Lee (2020). Also, the cosmological constant
$\Lambda$ suffers from the cosmological constant fine-tuning problem and the
cosmic coincidence problem Weinberg (1989); Velten  _et al._ (2014).
Therefore, despite some success in the cosmological scale, the $\Lambda$CDM
model is still being challenged by many recent studies.

Based on the above problems, some studies propose alternative models for the
universe accelerated expansion. Dark energy (DE) effects are generated by
additional matter fields (e.g., quintessence (Copeland  _et al._ , 2006)), or
that the dynamical effects of dark matter and/or dark energy might originate
from modified gravity (MG) models(Horndeski, 1974; Milgrom, 1983; Zwiebach,
1985; Moffat, 2006; Nojiri  _et al._ , 2005; Bekenstein, 2010; De Felice and
Tsujikawa, 2010; Linder, 2010; Milgrom, 2014; Lovelock, 1971; Hořava, 2009;
Rodríguez and Navarro, 2017; Horndeski, 1974; Deffayet  _et al._ , 2010, 2010)

Several popular modified gravity theories have been proposed to compete with
the standard $\Lambda$CDM model, including Emergent Gravity Verlinde (2017),
f(R) gravity Buchdahl (1970) and scalar-tensor-vector gravity Moffat (2006).
Therefore, it is very important to motivate some theoretical framework to
differentiate the dynamical effects of the $\Lambda$CDM model and the modified
gravity. Some recent studies proposed that using the turnaround radius (TAR)
can be a clue to test the standard $\Lambda$CDM model and modified gravity
models Bhattacharya  _et al._ (2017); Lopes  _et al._ (2018, 2019); Pavlidou
and Tomaras (2014); Pavlidou  _et al._ (2014); Faraoni  _et al._ (2015). The
TAR has been claimed to be a well-defined and unambiguous boundary of a
structure (e.g. galaxy clusters) in simulations Pavlidou and Tomaras (2014).
Different cosmological models and modified gravity models might have different
general relations of the TAR. Therefore, determining the TAR of different
structures precisely would be crucial to test and constrain different
cosmological models (Lopes  _et al._ , 2018), DE, and disentangle between
$\Lambda$CDM model, DE, and MG models (Pavlidou and Tomaras, 2014; Pavlidou
_et al._ , 2014; Faraoni  _et al._ , 2015; Bhattacharya  _et al._ , 2017;
Lopes  _et al._ , 2018, 2019).

Contrarily to the previous claim, we already showed in (Del Popolo  _et al._ ,
2013a, b; Pace  _et al._ , 2014a; Mehrabi  _et al._ , 2017; Pace  _et al._ ,
2019; Del Popolo  _et al._ , 2020) that shear, and vorticity modifies the non-
linear evolution of structures. In this paper, we will also show that
dynamical friction further modifies the structure formation, and consequently
modifies TAR, and that TAR generally depends on baryons physics Del Popolo
_et al._ (2020). By using an extended spherical collapse model, the TAR
depends on the effects of shear and vorticity. Taking into account of the
effects of shear and vorticity, the relation between TAR and total mass can
differ by 30% from that omit these effects, especially in galaxies Del Popolo
_et al._ (2020). In the present paper, we show that the effect of dynamical
friction is also significant.

TAR was calculated by (Pavlidou and Tomaras, 2014; Pavlidou  _et al._ , 2014)
for the $\Lambda$CDM model smooth DE model, while (Faraoni  _et al._ , 2015)
obtained TAR in generic gravitational theories.

In this paper, we extend the results of (Del Popolo  _et al._ , 2020), based
on an extended spherical collapse model (ESCM) introduced, and adopted in (Del
Popolo, 2013; Del Popolo  _et al._ , 2013a; Pace  _et al._ , 2014a; Mehrabi
_et al._ , 2017; Pace  _et al._ , 2019). The ESCM takes into account the
effect of shear, vorticity and dynamical friction on the collapse, to show how
the TAR is changed. Apart the typical parameters of the spherical collapse,
shear, vorticity, and dynamical friction change the two-point correlation
function (Del Popolo and Gambera, 1999), the weak lensing peaks (Pace  _et
al._ , 2019), and the mass function (Del Popolo, 2013; Del Popolo  _et al._ ,
2013a; Pace  _et al._ , 2014a; Mehrabi  _et al._ , 2017). Similarly, to (Del
Popolo  _et al._ , 2020), the aim of the paper is to show how the parameters
of the spherical collapse are changed, together with the $R_{\rm t}$-$M_{\rm
t}$ relation for MG models, DE models, and to compare to $\Lambda$CDM model
predictions.

The paper is organized as follows. Section II describes the model used to
derive the $R_{\rm t}$-$M_{\rm t}$ relation. Section III is devoted to the
discussion of our results. Section IV is devoted to conclusions.

##  II The Model

In the following, we will use an improved version (Fillmore and Goldreich,
1984; Bertschinger, 1985; Hoffman and Shaham, 1985; Ryden and Gunn, 1987;
Subramanian  _et al._ , 2000; Ascasibar  _et al._ , 2004; Williams  _et al._ ,
2004) of the spherical collapse model introduced by Gunn and Gott (1972). The
model describes the evolution of perturbation from the linear to the non-
linear phase, when them decouple from Hubble flow, reach a maximum radius, the
TAR, collapse, and viriliaze forming a structure. As reported the initial
model of Gunn and Gott (1972) was extended to take account of angular momentum
(Ryden and Gunn, 1987; Gurevich and Zybin, 1988a, b; White and Zaritsky, 1992;
Sikivie  _et al._ , 1997; Nusser, 2001; Hiotelis, 2002; Le Delliou and
Henriksen, 2003; Ascasibar  _et al._ , 2004; Williams  _et al._ , 2004; Zukin
and Bertschinger, 2010), of dynamical friction (Antonuccio-Delogu and
Colafrancesco, 1994; Del Popolo, 2009), shear (Hoffman, 1986, 1989; Zaroubi
and Hoffman, 1993), and the effects of the DE fluid perturbation (see Mota and
van de Bruck, 2004; Nunes and Mota, 2006; Abramo  _et al._ , 2007, 2008,
2009a, 2009b; Creminelli  _et al._ , 2010; Basse  _et al._ , 2011; Batista and
Pace, 2013). Del Popolo  _et al._ (2013a, b) studied the effects of shear and
rotation in smooth DE models, Pace  _et al._ (2014b) in clustering DE
cosmologies, and Del Popolo  _et al._ (2013c) in Chaplygin cosmologies.

###  II.1 The ESCM

Here, we show how the evolution equations of $\delta$ in the non-linear regime
can be obtained.

The equations of evolution of $\delta$ in the non-linear regime were obtained
by Bernardeau (1994); Padmanabhan (1996); Ohta  _et al._ (2003, 2004); Abramo
_et al._ (2007); Pace  _et al._ (2010). In order to obtain the equation, we
used the Neo-Newtonian expressions for the relativistic Poisson equation, the
Euler, and continuity equations (Lima  _et al._ , 1997)

$\displaystyle\frac{\partial\rho}{\partial
t}+\nabla_{\vec{r}}\cdot(\rho\vec{v})+\frac{P}{c^{2}}\nabla_{\vec{r}}\cdot\vec{v}$
$\displaystyle=$ $\displaystyle 0\;,$ (1)
$\displaystyle\frac{\partial\vec{v}}{\partial
t}+(\vec{v}\cdot\nabla_{\vec{r}})\vec{v}+\nabla_{\vec{r}}\Phi+{\bf\frac{c^{2}}{c^{2}\rho+P}\nabla
P}$ $\displaystyle=$ $\displaystyle 0\;,$ (2)
$\displaystyle\nabla^{2}\Phi-4\pi G\left(\rho+\frac{3P}{c^{2}}\right)$
$\displaystyle=$ $\displaystyle 0\;,$ (3)

where the equation of state (EoS) is given by $P=w\rho c^{2}$, $\vec{r}$
indicates the physical coordinate, $\Phi$ the Newtonian gravitational
potential, and $\vec{v}$ the velocity in three-space. Writing and combining
the perturbation equation as in (Del Popolo  _et al._ , 2020), we obtain the
non-linear evolution equation in a dust ($w=0$) universe

$\begin{split}\ddot{\delta}+2H\dot{\delta}-\frac{4}{3}\frac{\dot{\delta}^{2}}{1+\delta}-4\pi
G\bar{\rho}\delta(1+\delta)-\\\
(1+\delta)(\sigma^{2}-\omega^{2})&=0\;\end{split}$ (4)

Eq. (4) is Eq. 41 of Ohta  _et al._ (2003), and a generalization of Eq. 7 of
Abramo  _et al._ (2007) to the case of a non-spherical configuration of a
rotating fluid.

In Eq. (4), $H$ is the Hubble function, $\overline{\rho}=\rho-\delta\rho$ the
background density, $\sigma^{2}=\sigma_{ij}\sigma^{ij}$, and
$\omega^{2}=\omega_{ij}\omega^{ij}$ are the shear, and rotation term,
respectively. The shear term is related to a symmetric traceless tensor,
dubbed shear tensor, while rotation term is related to an antisymmetric
tensor.

In terms of the scale factor, $a$, the nonlinear equation driving the
evolution of the overdensity contrast can be rewritten as:

$\begin{split}\delta^{\prime\prime}+\left(\frac{3}{a}+\frac{E^{\prime}}{E}\right)\delta^{\prime}-\frac{4}{3}\frac{\delta^{\prime
2}}{1+\delta}-\frac{3}{2}\frac{\Omega_{\mathrm{m},0}}{a^{5}E^{2}(a)}\delta(1+\delta)-&\\\
\frac{1}{a^{2}H^{2}(a)}(1+\delta)(\sigma^{2}-\omega^{2})&=0\;,\end{split}$ (5)

where $\Omega_{m,0}$ is DM density parameter at $t=0$ ($a=1$), and $E(a)$ is
given in Eq. 11 of (Del Popolo  _et al._ , 2020).

Since $\delta=\frac{2GM_{\rm m}}{\Omega_{m,0}H^{2}_{0}}(a/R)^{3}-1$, where $R$
is the effective perturbation radius, inserting into Eq. (4), it is easy to
check that the evolution equation for $\delta$ reduces to the spherical
collapse model (SCM) (Fosalba and Gaztan̈aga, 1998; Engineer  _et al._ , 2000;
Ohta  _et al._ , 2003)

$\displaystyle\ddot{R}$ $\displaystyle=$ $\displaystyle-\frac{GM_{\rm
m}}{R^{2}}-\frac{GM_{\rm de}}{R^{2}}(1+3w_{\rm
de})-\frac{\sigma^{2}-\omega^{2}}{3}R=$ (6) $\displaystyle-\frac{GM_{\rm
m}}{R^{2}}-\frac{4\pi G\bar{\rho_{\rm de}}R}{3}(1+3w_{\rm
de})-\frac{\sigma^{2}-\omega^{2}}{3}R\,,$

where $M_{\rm de}$ is the mass of the dark-energy component enclosed in the
volume, $M_{\rm m}=\frac{4\pi R^{3}}{3}(\bar{\rho}+\delta\rho)$, $w_{\rm de}$,
and $\bar{\rho}_{\rm de}$, are the DE equation-of-state parameter and its
background density, respectively (Fosalba and Gaztan̈aga, 1998; Engineer  _et
al._ , 2000; Ohta  _et al._ , 2003; Pace  _et al._ , 2019).

In the case $w=-1$, namely the cosmological constant, $\Lambda$, case, Eq.
(6), for $w=-1$, can be written as

$\ddot{R}=-\frac{GM_{\rm
m}}{R^{2}}-\frac{\sigma^{2}-\omega^{2}}{3}R+\frac{\Lambda}{3}R\ $ (7)

The previous equation is clearly similar to the usual expression for the SCM
with cosmological constant, and angular momentum (e.g. Peebles, 1993; Nusser,
2001; Zukin and Bertschinger, 2010):

$\frac{d^{2}R}{dt^{2}}=-\frac{GM}{R^{2}}+\frac{L^{2}}{M^{2}R^{3}}+\frac{\Lambda}{3}R=-\frac{GM}{R^{2}}+\frac{4}{25}\Omega^{2}R+\frac{\Lambda}{3}R,$
(8)

The last right term in Eq. 8 is obtained recalling that $L=I\Omega$, and the
momentum of inertia of a sphere, $I=2/5MR^{2}$.

Angular momentum is related to vorticity by $\Omega=\omega/2$ (see also
Chernin (1993)), in the case of a uniform rotation with angular velocity
$\Omega=\Omega_{\rm z}{\bf e}_{\rm z}$. As in (Del Popolo  _et al._ , 2013a,
b, 2020), we define the dimensionless, but mass dependent, quantity $\alpha$
as the ratio between the rotational and the gravitational term in Eq. (8):

$\alpha(M)=\frac{L^{2}}{M^{3}RG}$ (9)

In order to solve Eq. (4), the relation between the term
$\sigma^{2}-\omega^{2}$, and the density contrast, $\delta$ is needed. This
connection can be obtained recalling the relation between angular momentum and
shear, and recalling that Eq. (8) from which $\alpha$ was obtained is
equivalent to Eq. (6) which is also equivalent to Eq. (4).

Calculating the same ratio between the gravitational and the extra term
appearing in Eq. (4) we obtain

$\frac{\sigma^{2}-\omega^{2}}{H^{2}_{0}}=-\frac{3}{2}\frac{\alpha\Omega_{\rm
m,0}}{a^{3}}\delta.$ (10)

This reasonable assumption (see (Del Popolo  _et al._ , 2013b)) was also used
in (Del Popolo  _et al._ , 2013a, b; Pace  _et al._ , 2014a; Mehrabi  _et al._
, 2017).

The nonlinear equation to solve is obtained substituting Eq. (10) into Eq. (4)

$\begin{split}\ddot{\delta}+2H\dot{\delta}-\frac{4}{3}\frac{\dot{\delta}^{2}}{1+\delta}-4\pi
G\bar{\rho}\delta(1+\delta)-\\\
-\frac{3}{2}H_{0}^{2}(1+\delta)\frac{\alpha\Omega_{\rm
m,0}}{a^{3}}\delta&=0\;\end{split}$ (11)

Solving Eq. (11) following the method described in (Pace  _et al._ , 2010), or
solving Eq. (6), the threshold of collapse, and the turnaround, can be
obtained.

At this point, we want to take into account also dynamical friction in our
analysis. Then we notice that Eq. (6), can be written in a more general form
taking into account dynamical friction (Kashlinsky, 1986, 1987; Lahav  _et
al._ , 1991; Bartlett and Silk, 1993; Antonuccio-Delogu and Colafrancesco,
1994; Peebles, 1993; Del Popolo and Gambera, 1998; Del Popolo  _et al._ ,
1998; Del Popolo, 2006, 2009; Del Popolo  _et al._ , 2019)

$\begin{split}\ddot{R}=-\frac{GM}{R^{2}}+\frac{L^{2}(R)}{M^{2}R^{3}}+\frac{\Lambda}{3}R-\eta\frac{{\rm
d}R}{{\rm d}t}=\\\ -\frac{GM_{\rm m}}{R^{2}}-\frac{GM_{\rm
de}}{R^{2}}(1+3w_{\rm
de})-\frac{\sigma^{2}-\omega^{2}}{3}R-\eta\frac{dR}{dt}.\end{split}$ (12)

being $\eta$ the dynamical friction coefficient. Eq. (12) can be obtained via
Liouville’s theorem (Del Popolo and Gambera, 1999), and the dynamical friction
force per unit mass, $\eta\frac{{\rm d}R}{{\rm d}t}$, is given in (Del Popolo,
2009)(Appendix D, Eq. D5), and (Del Popolo, 2006), Eq. 5).

A similar equation (excluding the dynamical friction term) was obtained by
several authors (e.g., Fosalba and Gaztan̈aga, 1998; Engineer  _et al._ ,
2000; Del Popolo  _et al._ , 2013b)) and generalized to smooth DE models in
Pace  _et al._ (2019).

Eq. (12), and Eq. (6) differs for the presence of the dynamical friction term.
Dynamical friction similarly to rotation, and cosmological constant delays the
collapse of a structure (perturbation). (Antonuccio-Delogu and Colafrancesco,
1994; Del Popolo, 2009, 2006; Del Popolo  _et al._ , 2017, 2019). The
magnitude of the effect of cosmological constant, rotation, and dynamical
friction, are of the same order with differences of a few percent (see Fig. 1
of (Del Popolo  _et al._ , 2017), and Fig. 11 of (Del Popolo, 2009)).

Notice that, by means of the relation $\delta=\frac{2GM_{\rm
m}}{\Omega_{m,0}H^{2}_{0}}(a/R)^{3}-1$, Eq. (12) can be written in terms of
$\delta$, similarly to Eq. (4).

Our SCM model depends not only from shear, vorticity, as in (Del Popolo  _et
al._ , 2020), but also on dynamical friction. Since shear, rotation, and
dynamical friction depends from the mass, the SCM results depend from the
baryon physics, differently from what claimed by (Pavlidou and Tomaras, 2014;
Lopes  _et al._ , 2018; Bhattacharya  _et al._ , 2017).

##  III Results

Figure 1: The threshold of collapse $\delta_{c}$, as function of redshift, and
mass. In the left panel, $\delta_{c}$ vs redshift is plotted. The prediction
for $\delta_{c}(z)$ for the $\Lambda$CDM model are represented by the red
lines. The values of $\alpha$ varies from 0.05 (solid line) (galactic mass
scale), to 0.03 (long dashed line), to 0.01 (dotted line) (cluster mass
scale), and to 0 (dashed line). The right panel plots $\delta_{c}$ vs mass.
The solid red line represents the result of the ESCM model, the red dashed
line that of the elliptictal collapse model of (Sheth  _et al._ , 2001), and
the brown line the $\Lambda$CDM expectation when no shear and rotation are
taken into account.

The effect of shear (Hoffman, 1986, 1989; Zaroubi and Hoffman, 1993), and
rotation (Del Popolo and Gambera, 1998, 1999, 2000; Del Popolo  _et al._ ,
2001; Del Popolo, 2002a; Del Popolo  _et al._ , 2013b; Pace  _et al._ , 2019)
on the collapse are manifold.

One general feature is that of slowing down the collapse (Peebles, 1990; Audit
_et al._ , 1997; Del Popolo  _et al._ , 2001; Del Popolo, 2002a). Mass
function (Del Popolo and Gambera, 1999, 2000; Del Popolo  _et al._ , 2013b;
Pace  _et al._ , 2014a; Mehrabi  _et al._ , 2017; Del Popolo  _et al._ , 2017;
Pace  _et al._ , 2019), two-point correlation function, (Del Popolo  _et al._
, 2005), scaling relations like the mass-temperature, and luminosity-
temperature relation (Del Popolo  _et al._ , 2005) (Del Popolo, 2002b; Del
Popolo  _et al._ , 2019), are modified. This is connected to the change of the
typical parameters of the SCM.

###  III.1 Threshold of collapse with shear, rotation, and dynamical friction

Figure 2: Turnaround radius $R_{\rm t}$ vs mass, $M_{\rm t}$. The solid red
line represents the TAR predicted by the standard SCM, and the dashed red line
ESCM result for the $\Lambda$CDM model. The black dotted line the result of
the ESCM for the AS model.

Fig. 1 shows how shear, rotation, and dynamical friction (shortly SRD) change
the collapse threshold, $\delta_{c}$. We show the dependence from redshift in
the left panel, and mass in the right panel. The red line represents the
predictions of the ESCM for $\delta_{c}(z)$ for the $\Lambda$CDM model, while
the black line the ESCM in the case of one DE model, the Albrecht-Skordis
(Albrecht and Skordis, 2000) (AS) (black line).

In the left panel, from top to bottom, the value of $\alpha$ varies from 0.05
(solid line) corresponding to a mass $\simeq 10^{11}M_{\odot}$, to 0.03 (long
dashed line), corresponding to a mass $\simeq 10^{13}M_{\odot}$, to 0.01
(dotted line), corresponding to a mass $\simeq 10^{15}M_{\odot}$, and to 0
(dashed line). The value of $\delta_{c}(z)$ for $\alpha=0.05$, is $\simeq
30\%$ larger than in the $\alpha=0$ case.

In the case, of no shear, rotation ($\alpha=0$) and dynamical friction,
$\delta_{c}$, has a weak dependence from redshift in the $z$ range $[0,2]$,
and then assumes the value predicted by the Einstein de Sitter model.

In other terms, SRD gives rise to a non-flat threshold, $\delta_{c}$, which is
monotonically decreasing with redshift. Moreover, the larger is $\alpha$, the
larger is the difference between the values of $\delta_{c}(z)$, as shown by
the different curves.

Of the DE quintessence models in literature, we plotted the AS model because
the other models considered in previous papers (Pace  _et al._ , 2010; Del
Popolo  _et al._ , 2013a) (INV1 ($w_{0}=-0.4$), INV2 ($w_{0}=-0.79$), 2EXP
($w_{0}=-1$), CNR ($w_{0}=-1$), CPL ($w_{0}=-1$), SUGRA ($w_{0}=-0.82$) (see
(Pace  _et al._ , 2010; Del Popolo  _et al._ , 2013b))111$w_{0}$ is the value
of $w$ nowadays) are contained in the envelope between the region included in
the $\Lambda$CDM and the AS model (see Fig. 4 of (Pace  _et al._ , 2010)).

The right panel of Fig. 1 plots $\delta_{c}(M)$ versus the mass. In absence of
SRD the value of $\delta_{c}$ is constant (brown line). in presence of SRD,
$\delta_{c}$ becomes mass dependent, and monotonically decreases with mass.
This means that in order a less massive perturbations (e.g., galaxies) form
structures must cross a higher threshold than more massive ones. This
behavior, is related to the anticorrelation of the angular momentum acquired
by the proto-structure and its height222The peak height is defined as
$\nu=\delta_{c}/\sigma(M)$, where $\sigma(M)$ is the mass variance. We have
that the specific angular momentum, $j$ is given by $j\propto\nu^{-3/2}$
(Hoffman, 1986; Del Popolo, 2009; Polisensky and Ricotti, 2015). Since low
peaks acquire larger angular momentum than high peaks, they need a higher
density contrast to collapse and form structures (Del Popolo and Gambera,
1998; Del Popolo  _et al._ , 2001; Del Popolo, 2002a, 2009; Ryden, 1988;
Peebles, 1990; Audit  _et al._ , 1997).

As shown in the left and right panel of Fig. 1, the effect of dynamical
friction is that of increasing the values of $\delta_{c}(z)$, and
$\delta_{c}(M)$ with respect to the case it is not present as shown in (Del
Popolo  _et al._ , 2020).

Figure 3: Stable structures mass-radius relation 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 results of the ESCM are represented by the dashed lines. The dots
with error bars, are data as in (Pavlidou  _et al._ , 2014).

###  III.2 Comparison of TAR, in the $\Lambda$CDM, ESCM, and DE models

Shear, rotation, and dynamical friction modify the TAR. In order to show this,
we may compare the predictions of the $\Lambda$CDM, that of the ESCM, and DE
models. (Pavlidou and Tomaras, 2014) and (Pavlidou  _et al._ , 2014)
calculated the maximum TAR, MTAR, that is, the radius of the surface where
$\ddot{R}=0$. (Pavlidou  _et al._ , 2014) found

$R_{\rm ta}=\left[\frac{-3M}{4\pi\rho_{\rm de}(1+3w)}\right]^{1/3}$ (13)

which in the case of the $\Lambda$CDM model ($w=-1$) reduces to

$R_{\rm ta}=\left[\frac{3GM}{\Lambda}\right]^{1/3}$ (14)

(Pavlidou and Tomaras, 2014).

In their estimation, they assumed that shear and rotation were not present.
Their expression can be generalized to the case shear, and rotation are non
zero. This can be done using Eq. (6), obtaining

$R_{\rm ta}=\left[\frac{-3M}{4\pi\rho_{\rm
de}(1+3w)+(\sigma^{2}-\omega^{2})}\right]^{1/3}$ (15)

In the paper, we will get and plot the TAR, not MTAR, since we compare with
(Lopes  _et al._ , 2018), which calculated the TAR.

Fig. 2, shows the TAR predicted by the $\Lambda$CDM model (solid red line),
that of the ESCM model (taking into account shear rotation, and dynamical
friction) (red dashed line) for the $\Lambda$CDM model, and the AS model
(black dashed line). When shear, rotation and dynamical friction are taken
into account the collapse is slowed down, and the the TAR is smaller. The
difference between the $\Lambda$CDM model, and the ESCM preditions increases
going towards smaller masses. This is mainly related to the larger rotation of
smaller objects, and reach a maximum difference of $\simeq 60\%$. The black
dashed line, as already reported, is the AS model, which has a slightly larger
TAR with

###  III.3 Constraints on DE EoS parameter

Table 1: The allowed ranges of $w$ based on the ESCM model. 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 | $w\geq-2$
Figure 4: Turnaround radius, $R_{\rm t}$ vs mass, $M_{\rm t}$. The plot
represents the result obtained by (Lopes  _et al._ , 2018) for $R_{\rm ta}$ vs
mass for the $\Lambda$CDM model, the model with $\epsilon=1/3$, and $f(R)$
with $f_{\rm R0}=10^{-6}$, $10^{-5}$, and $10^{-4}$. The gray band represents
our prediction for the $\Lambda$CDM model, obtained with the ESCM, with the
68% confidence level region. Figure 5: The turnaround radius, $R_{\rm t}$ vs
mass, $M_{\rm t}$ obtained with the ESCM for the DE models. Left panel: from
top to bottom, the cyan, blue, brown, magenta, black, red, and green lines
represent the INV1, INV2, SUGRA, $w_{09}$333Namely the model having $w=-0.9$,
AS, $\Lambda$CDM without shear, and rotation, and $\Lambda$CDM with shear, and
rotation, respectively. Right panel: the solid lines are the $\Lambda$CDM, and
$f(R)$ models of (Lopes  _et al._ , 2018), while the blue dashed line, and the
green dashed lines, are the INV2 model, and the $\Lambda$CDM with shear,
rotation, and dynamical friction, plotted in the left panel.

Knowing the relation between TAR, the structure mass, and the EoS parameter,
it is possible to obtain some constraints on the DE EoS ($w$). This was tried
by (Pavlidou  _et al._ , 2014) by means of Eq. (13). In general, a great
number of DE models are described through the equation of state parameter,
$w(a)$. The last depends on its present value $w_{0}$, on that at matter-
radiation equality epoch, and some other parameters at the same epoch (see Eq.
23 of (Pace  _et al._ , 2010)). If one wants to constrain the EoS parameter,
$w$ evolution, high redshift structures are needed. If by converse, one wants
to constrain the values of $w_{0}$ small $z$ cosmic structures may be used.
(Pavlidou  _et al._ , 2014) compared the predicted TAR at different $w$ in the
plane $M_{t}$-$R_{t}$, finding constraints on $w_{0}$. To this aim data from
several structures (e.g., Milky Way (MW), M81/M82 group, Local Group, Virgo
cluster, Fornax-Eridanus group) were used. It is of fundamental importance to
note, that the constrain depends from the model used. As shown by (Peirani and
de Freitas Pacheco, 2008), the mass of the structure, and its TAR changes when
to the standard SCM (considering only the gravitational potential) is also
added the cosmological constant (Eq. 1 of (Peirani and de Freitas Pacheco,
2008)). This was also pointed out comparing the values predicted by several
Karachentsev’s paper (e.g., (Karachentsev  _et al._ , 2002; Karachentsev,
2005), for M81, the local group, and neighboring groups) with those of
(Peirani and de Freitas Pacheco, 2008).

In Fig. 3, the solid lines obtained from the equation of TAR (Eq. 13),
correspond to $w=-2.5$, -2, -1.5, -1, -0.5, from bottom to top when shear,
rotation, and dynamical friction are absent. The dashed lines are the
corrections obtained when shear, rotation, and dynamical friction are taken
into account. The range of $w$ for which no stable structures exist is given
by the parameter space above each line. (Pavlidou  _et al._ , 2014) discussed
some constraints to $w$, based on the highest mass objects. As shown by the
dashed lines, one has to expect that at masses smaller than $10^{13}M_{\odot}$
the TAR is modified by the presence of shear, rotation, and dynamical
friction. As a consequence, structures at smaller masses can give different
constraints to $w$. At the same time, following (Peirani and de Freitas
Pacheco, 2008), taking also the effect of dynamical friction, we see that the
constraints on the cosmic structure studied, and plotted in Fig. 3, is
noteworthy different from that of (Del Popolo  _et al._ , 2020). The values of
TAR and mass for each of the objects in Fig. 3 were obtained using the SCM
with dynamical friction, and using a method described by (Peirani and de
Freitas Pacheco, 2006, 2008) (see Appendix). In Table 1, we report the
constraints we obtained.

###  III.4 Comparison with TAR in $f(R)$ theories

In this section, we compare the evolution of the TAR in the ESCM, and the
$f(R)$ theories. The evolution of TAR in General Relativity (GR), and in the
$f(R)$ theories was investigated by (Lopes  _et al._ , 2018). Modified Gravity
(MG) effects were introduced in the equation of the evolution of overdensity
(their Eq. 3.3) by means of the parameter $\epsilon(a,k)$, where $k$ is the
(angular) wavenumber. GR is recovered when $\epsilon(a,k)$ is zero, and our
Eq. (5), is recovered in the case $\epsilon(a,k)$ is zero, and shear, and
rotation are put equal to zero. In other terms, our Eq. (5) is a
generalization of Eq. 3.3 of (Lopes  _et al._ , 2018), for the case
$\epsilon(a,k)$ is zero. Consequantly, their Figs. 1, 3, similarly to the left
panel of our Fig. 1, and to the $\Lambda$CDM, and DE models without shear, and
rotation, as shown in (Pace  _et al._ , 2010), shows a monotonic increase of
$\delta_{c}(z)$. Their Fig. 4 shows an almost flat behavior of
$\delta_{c}(M)$, with variations from constancy of the order of 1%. The
behavior of $\delta_{c}(z)$, and $\delta_{c}(M)$ in (Lopes  _et al._ , 2018)
disagrees with the prediction of several papers (e.g., (Sheth  _et al._ ,
2001; Del Popolo  _et al._ , 2013a, b; Pace  _et al._ , 2014a; Mehrabi  _et
al._ , 2017; Del Popolo  _et al._ , 2017)). The previous papers showed that in
order to have a mass function reproducing simulations, the threshold must be a
monotonic decreasing function of mass. Since in (Lopes  _et al._ , 2018)
$\delta_{c}(M)$ is practically constant, and $\delta_{c}(z)$ is a monotonic
increasing function of redshift, this implies that (Lopes  _et al._ , 2018)
results cannot reproduce the mass function obtained in simulations, and
observations. This is due to the the fact (Lopes  _et al._ , 2018) discarded
the effect of shear, and rotation, and in general of aspericities. A more
detailed discussion of this aspect can be found in (Del Popolo  _et al._ ,
2020).

Going back to our main goal, namely the use of the TAR to disentangle between
GR and MG, and GR and DE models, discussed by several authors (e.g., (Lopes
_et al._ , 2018)), we will compare (Lopes  _et al._ , 2018) prediction on the
$M_{t}$-$R_{t}$ relation with that of our ESCM. In Fig. 4, the result obtained
by (Lopes  _et al._ , 2018) for $R_{\rm t}$ vs mass for the $\Lambda$CDM
model, the model with $\epsilon=1/3$, and $f(R)$ with $f_{\rm R0}=10^{-6}$,
$10^{-5}$, and $10^{-4}$, are plotted. Our prediction for the $\Lambda$CDM
model, obtained with the ESCM, with the 68% confidence level region, are
represented by the grey band. The 68% confidence level was obtained, as in
(Del Popolo  _et al._ , 2020), by means of a Monte Carlo simulation. The
result of the plot is slightly different from Fig. 4 of (Del Popolo  _et al._
, 2020). Because of the presence of dynamical friction, which further
contribute to slow down the collapse, the TAR has smaller values.
Consequently, the TAR in the ESCM model does not completely overlap with that
of (Lopes  _et al._ , 2018), as happened in (Del Popolo  _et al._ , 2020).
This means that the study of the top values of the $M_{t}$-$R_{t}$ relation
could disentangle the GR predictions from that of the $f(R)$ theories.
Choosing peculiar values of the TAR would be possible to disentangle between
GR, and $f(R)$ theories.

The predictions of some of the quintessence DE models previously cited are
compared in Fig. 5 to the same $f(R)$ models of (Lopes  _et al._ , 2018)
plotted in Fig. 4. All the curves are obtained by means of the ESCM applied to
the DE models of the TAR. From top to bottom, the cyan, blue, brown, magenta,
black, red, and green lines represent the INV1, INV2, SUGRA, $w_{09}$444Namely
the model having $w=-0.9$, AS, $\Lambda$CDM without shear, and rotation, and
$\Lambda$CDM with shear, and rotation, respectively.

The comparison of the (Lopes  _et al._ , 2018) predictions for TAR with that
of the DE models are plotted in the right panel. In this plot, we show only
INV2 (blue dashed line), and the $\Lambda$CDM with shear, rotation, and
dynamical friction (green dashed line). With exception of INV1, the two quoted
curves contain all other DE models.

The other lines are the $\Lambda$CDM, and $f(R)$ models of (Lopes  _et al._ ,
2018). Differently from Fig. 4, the DE models contain the (Lopes  _et al._ ,
2018) prediction for TAR. As a consequence the TAR cannot be used to
disentangle DE, and $f(R)$ theories predictions.

##  IV Conclusions

In this paper, we discussed how shear, rotation, and dynamical friction change
the TAR, and some of the parameters of the SCM. The results were obtained
using an ESCM taking into account the effects of shear, vorticity, and
dynamical friction, to determine the $R_{\rm t}-M_{\rm t}$, in $\Lambda$CDM,
and in DE scenarios. We extended numerically the formula for maximum TAR
obtained in (Del Popolo  _et al._ , 2020), to take into account dynamical
friction. The value of TAR is reduced by shear, rotation, and dynamical
friction, especially at galactic scales. Using the $R_{\rm t}-M_{\rm t}$
relationship, and data from stable structures, one can obtain constraints to
$w$. Its values are smaller for structures with masses approximately smaller
that $10^{13}M_{\odot}$. In this paper, we recalculated the mass, and TAR of
M81/M82 group, Local Group, Virgo cluster, NGC253, IC342, CenA/M83 group
following (Peirani and de Freitas Pacheco, 2006, 2008).

A comparison of the $R_{\rm t}-M_{\rm t}$ relationship obtained for
$\Lambda$CDM, and DE scenarios with (Lopes  _et al._ , 2018) prediction of the
$f(R)$ theories, shows that $R_{\rm t}-M_{\rm t}$ relationship in the $f(R)$
models are practically identical to that of of DE scenarios. This implies that
the $R_{\rm t}-M_{\rm t}$ relationship is not a good probe to disentangle
between GR, and DE models predictions. The situation is different in the case
of the $\Lambda$CDM model. In this case, the 68% confidence level region does
not overlap with that of the $f(R)$ models. The higher values of the TAR could
be used to disentangle between $f(R)$ theories, and GR.

##  Appendix A Mass and TAR of structures

The most general equation taking account of shear, rotation, and dynamical
friction is Eq. (12). We will rewrite it in adimensional form. Assuming that
$J=kR^{\alpha}$, with $\alpha=1$, in agreement with (Bullock  _et al._ ,
2001)555In that paper $\alpha=1.1\pm 0.3$, and $k$ constant. In terms of the
variables $y=R/R_{0}$, $t=x/H_{0}$, Eq. (12) can be written as

$\frac{d^{2}y}{dx^{2}}=-\frac{A}{2y^{2}}+\Omega_{\Lambda}y+\frac{K_{j}}{y}-\frac{\eta}{H_{0}}\frac{dy}{dx},$
(16)

where $K_{j}=k\frac{1}{(H_{0}R_{0})^{2}}$666$K_{j}=0.78$, and
$\frac{\eta}{H_{0}}=0.5$, $A=\frac{2GM}{H_{0}^{2}R_{0}^{3}}$, and

$H=H_{0}\sqrt{\left(\frac{a_{0}}{a}\right)^{3}\Omega_{m}+\Omega_{\Lambda}}.$
(17)

Eq.(16) has a first integral, given by

$\displaystyle u^{2}=\left(\frac{dy}{dx}\right)^{2}$
$\displaystyle=\frac{A}{y}+\Omega_{\Lambda}y^{2}+2K_{j}\log{y}-2\frac{\eta}{H_{0}}\int\left(\frac{dy}{dx}\right)^{2}dx+K$
(18)

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

The previous equation can solved as described in (Peirani and de Freitas
Pacheco, 2006, 2008).

Figure 6:  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.

The mass, and turn-around radius of some groups of galaxies, is obtained
finding a relation between the velocity, and radius, $v-R$. The last will be
fitted to the data. The $v-R$ relation is obtained as follows. Let’s consider
Fig.6. This is a solution of Eq. (12) for different values of $K$. The
vertical line corresponds to $x=0.964$. Its intersection with the curves,
solution of Eq. (12) (cyan, red, green) gives, for each one a value
$y(x)=y(0.964)$. The solution of Eq. (12), 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. This allows us to find a
series of points that can be fitted with a relation of the form
$u=-b/y^{n}+by$ obtaining $u=-1.3436/y^{0.9107}+1.3436y$. This last relation
can be written in physical units as follows.

$v(R)=-bH_{0}R_{0}\left(\frac{R_{0}}{R}\right)^{n}+bH_{0}R$ (19)

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

$v(R)=-b\frac{H_{0}}{R^{n}}\left(\frac{2GM}{AH_{0}^{2}}\right)^{\frac{n+1}{3}}+bH_{0}R$
(20)

or

$v(R)=-\frac{-0.66385H_{0}}{R^{n}}\left(\frac{GM}{H_{0}^{2}}\right)^{\frac{n+1}{3}}+1.3436H_{0}R$
(21)

where $n=0.9107$. Eq. (21) satisfy the condition $v(R_{0})=0$. Fitting the
equation to the data of (Peirani and de Freitas Pacheco, 2006) (Fig. 2), and
(Peirani and de Freitas Pacheco, 2008) (Fig. 2) one obtains the value of the
Hubble parameter , $H_{0}$, and TAR. Solving Eq. (12) one can obtain the value
of $A$, and solving the equation $A=\frac{2GM}{H_{0}^{2}R_{0}^{3}}$, being
$H_{0}$, and $R_{0}$, one gets the mass, $M$.

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