Improved Lemaitre-Tolman model and the mass and turn-around radius in group of galaxies

We extended the modified Lemaitre-Tolman model 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.[New Astronomy 11(4):325, A&A 488(3):845]. 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.[New Astronomy 11(4):325, A&A 488(3):845], 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 $w$.

W hile 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 ≃ 170M⊙/LB,M ⊙ 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⊙/LB,M ⊙ . 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 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 ≃ 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 reach-ing 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, R0, 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 R0, the mass can be obtained through the relation Refs. (1,2,7), where T0 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, R0, relation. As a consequence for a given R0, the value of the mass of the system is ≃ 30% larger with respect to Eq. (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 − R0 relation (Eq. (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 (JηLT 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 , we introduce the model, and solve it. In Section , we find the velocityradius relation for the JLT, and JηLT models. In Section , we applied the v − R relation to groups and clusters of galaxies. In Section , we studied the impact of the angular momentum and dynamical friction on the M −R0 relation. In Section , we showed how the obtained values of M and R0 may constrain the dark energy equation of state parameter, w. Section is devoted to conclusions.
Refs. (65,66) studied the effects of shear and rotation in smooth DE models. The effects of shear and rotation were investigated in Refs. (65,66) 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 + dr], having angular momentum L = rv θ , defined in the range [L, L + dL], with velocity vr =ṙ, defined in the range [vr, vr + dvr], has the following form dP = f (L, r, vr, t)dLdvrdr.
[2] * Particles 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. 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. (13,14,37,69,70) is: with Λ being the cosmological constant and η 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, η, 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., 65,71,72)) and generalized to smooth DE models in Ref. (73).
In terms of the specific angular momentum J = L M , and Ω Λ = ρ Λ ρc , where ρc is the critical density, Eq. (3) can be written as [4] where w is the DE equation of state (EoS) parameter. DE is modeled by a fluid with an EoS P = wρ, where ρ is the energy density. a is the expansion parameter. Eq. (4) satisfies equation . [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 α , with α = 1, in agreement with Ref.(74) † , and k constant. In terms of the variables y = R/R0, t = x/H0, Eq. (4), and Eq. (5) can be written as where Eq. (6) has a first integral, given by 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 dy dx = 0, and y = 1 are satisfied. A second way to get A, is to use the equation for the velocity (Eq. (8)).
Let's show this second method in the case cosmological constant, and angular momentum are present (JLT case) having the first integral At the turn-around point Eq. (10) gives: At high redshifts (z = 1000), or y ≪ 1, as was described the gravitational term dominates, and by a Taylor expansion one gets the relation y ≃ ( 9A 4 ) 1/3 x 2/3 . Assuming an initial value of y, yi = 0.001, corresponding approximately to 1 kpc, the initial time xi can be obtained. The initial value of the velocity ui can be obtained, when A is known, through Eq. (10), recalling that yi = 0.001. The value of A is obtained as follows. Eq. (10) can be written as Eq. (7), recalling that a 0 a = 1 + z, can be written as 3 = 0.964. [12] For Ω Λ = 0.7, Ωm = 0.3, Kj = 0.78 ‡ , x = 0.964, Eq. (11) can be solved to get A = 5.037. In the case, Kj = 0, A = 3.6575, and if Kj = 0, Ω Λ = 0, the SLT gives A = 2.655. In the case JηLT (Eqs.(6)- (8)), A can be obtained similarly to the previous case (JLT) solving numerically Eq. (8) with the initial condition on yi, and varying A until the condition dy dx = 0, and y = 1 are satisfied. Similarly, we can solve Eq. (6) with the initial condition yi, and varying A until the condition 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. Eq. (9) can be solved with the conditions yi = 0.001, and [13] In Fig.1, we plot the result of the solution. The red line corresponds to the case K = −A − Ω Λ = −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 ≃ 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.

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 = − 1.4054 y n + 1.4054y [14] where n = 0.6293. This can be written in terms of the physical units as [16] or 4054H0R [17] ‡ The value of K j was obtained recalling that term related to angular momentum, L, in Eq. (4), is 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) § .
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 where n = 0.7549, and in the complete case (cosmological constant, angular momentum, and dynamical friction) 3436H0R [19] where n = 0.9107. In Fig.2 we plot, from left to right, the velocity profile of the MLT, JLT, and JηLT cases, using adimensional variables. All the previous equations satisfy the condition v(R0) = 0. In the following, we will apply Eq. (19), related to the JηLT 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 JηLT 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 R0 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)(18)(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 § This was confirmed via a private discussion with one of the authors of Ref. (2), namely de Freitas Pacheco.
heliocentric to the Local Group rest frame. The distance can be written as where the angle θ is the angle between the center of mass and the galaxy, D the distance from the galaxy to the center of mass, and Dg is the distance to the galaxy. Indicating with V , and Vg 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) = Vg cos α − V cos β [21] being α = D sin θ Dg −D cos θ , and β = α + θ. 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)

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 Eq. (1), namely For the MLT, the value of A can be obtained combining Eq. (11), and Eq. (12), and one gets A = 3.6575. By the definition of A = 2GM (1 + z) Ω Λ + Ωm(1 + z) 3 [23] we obtain, for Ω Λ = 0.7 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, R0, may be obtained also from Eq. (17) and M = 3.065 × 10 12 h 2 R 3 0 M⊙ [28] In Fig.4, we plot the v − y(R) relations for the MLT, the JLT, and the JηLT cases. For distances smaller than R0, the plot shows that the JηLT 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 JηLT 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 −R0 relation in the case SLT (Eq. (27)) is smaller than that of the MLT case (Eq. (17)). The last is smaller than the mass obtained with the JLT (Eq. (18)), and this is smaller that that of JηLT case (Eq. (19)). For example, fitting the data by means of Eq. (27) (case SLT), the mass is ≃ 10% smaller than that obtained with Eq. (17)  The differences between the two methods can be explained as follows. In the method based on the fitting, the turnaround is obtained through R0 = ( 2GM AH 2 0 ) 1/3 , and depends from M , and H, obtained through the fit.
In the method based on the M −R0 relation, R0 is obtained by any method allowing the determination of this quantity, and then the M − R0 relationship gives the mass.
Another interesting point, is the decrease of h from the SLT model, to the JηLT model. The maximum differences for the groups and clusters studied is ≃ 30%.   Table 2. 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. (66,73,(95)(96)(97), to show how R0 is modified by the presence of vorticity, and shear in the equation of motion. We also showed how the M − R0 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 R0 of galaxies, groups, and clusters. Some data where taken from Ref. (90), and others from Ref. (1,2).

Constraints on the DM EoS parameter
With the revised value of mass, M , and R0 presented in this paper, we recalculate the constraints showed in Ref. (21).  Table 2 (case JηLT).
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 R0 by means of the virial theorem or the LT model.

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 (JηLT model). After solving the equation of motion, we got the relationships between mass, M , and the turn-around radius R0, 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 R0, 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 (JηLT 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 Eq. (19) (JηLT model) are larger than those obtained by means of Eq. (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 R0 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 R0 by means of the virial theorem or the LT model.

ACKNOWLEDGMENTS.
The authors wish to express their grat-itude to S. Peirani and A. De Freitas Pacheco for a fruitful discussion.