A&A 387, 759-777 (2002)
DOI: 10.1051/0004-6361:20020399
A. Del Popolo^{1,2,3}
1 - Dipartimento di Matematica, Università Statale di Bergamo,
via dei Caniana 2, 24129 Bergamo, Italy
2 -
Feza Gürsey Institute, PO box 6 Çengelköy, Istanbul, Turkey
3 -
Boaziçi University, Physics Department,
80815 Bebek, Istanbul, Turkey
Received 30 October 2001 / Accepted 18 February 2002
Abstract
I study the role of shear fields
on the evolution of density perturbations
by using an analytical approximate solution for the equations of
motion of homogeneous ellipsoids embedded in a homogeneous
background.
The equations of motion of a homogeneous ellipsoid (Icke 1973;
White & Silk 1979, hereafter WS) are modified in order to take
account of the tidal field, as done in Watanabe (1993) and then are
integrated analytically, similar to what was done in WS. The
comparison of the analytical solution with numerical simulations
shows that it is a good approximation of the numerical one. This
solution is used to study the evolution of the configuration of
the ellipsoids, to calculate the evolution of the density contrast
and that of the axial peculiar velocity of the ellipsoids for
several values of the amplitude of the external tidal field,
and is compared again with numerical simulations. In order to
calculate the evolution of the density contrast at turn-around and
collapse velocity at the epoch of collapse, as a function of the
ratio of the initial value of the semi-axes, I use the previously-obtained
approximate solution to modify the analytical model
proposed by Barrow & Silk (1981) for the ellipsoid evolution in
the non-linear regime. The density contrast at turn-around and the
collapse velocity are found to be reduced with respect to that
found by means of the spherical model. The reduction increases
with increasing strength of the external tidal field and with
increasing initial asymmetry of the ellipsoids. These last
calculations are also compared with numerical solutions and they
are again in good agreement with the numerical ones.
Key words: cosmology: theory - large-scale structure of Universe - galaxies: formation
On the accuracy of the Bertschinger & Jain collapse theorem there is not fully agreement in the literature. For example, according to the previrialization conjecture (Peebles & Groth 1976; Davis & Peebles 1977; Peebles 1990), initial asphericities and tidal interactions between neighboring density fluctuations induce significant non-radial motions which oppose the collapse. This kind of conclusion was supported by BS, Szalay & Silk (1983), Villumsen & Davis (1986), Bond & Myers (1993a,b) and Lokas et al. (1996).
In a more recent paper, Audit et al. (1997) have proposed some analytic prescriptions to compute the collapse time along the second and the third principal axes of an ellipsoid, by means of the "fuzzy'' threshold approach. They pointed out that the formation of virialized clumps must correspond to the third axis collapse and that the collapse along this axis is slowed down by the effect of the shear rather than being accelerated by it, in contrast to its effect on the first axis collapse. They concluded that spherical collapse is the fastest, in disagreement with Bertschinger & Jain's theorem.
Van de Weygaert & Babul (1994) studied the influence of shear fields on the evolution of galactic scale perturbations. They found that the tidal forces induced by large-scale inhomogeneities can affect the mass in and around primordial density peaks and that in some peculiar conditions the shear can break up a primordial peak into two distinct halos. Shear is even able to produce the collapse of a void (van de Weygaert 1996).
Internal shear has been studied by means of collapsing homogeneous ellipsoids (Icke 1973; WS; BS). These models show that the evolution of an isolated homogeneous ellipsoid (namely an ellipsoid not exposed to external tidal force) proceeds through a series of uniform ellipsoids, or in other terms the shape of a perturbation does not change much until it acquires a significant overdensity with respect to the background. The collapse time of perturbations of a given initial overdensity, decreases with increasing eccentricity and the collapse is faster for near-oblate configurations than for near prolate ones. BS showed that the density contrast at turn-around and the collapse velocity at pancake formation are reduced relative to the spherical case. Summarizing, the main conclusions are that internal shear can alter the collapse history of structures (Icke 1973; WS). Note that the ellipsoid model has also been used to calculate the cosmological density parameter (WS; BS; Watanabe 1993).
It is important to notice that while the homogeneous ellipsoid model has been widely used in the cosmological context (Palmer 1983; WS; Peebles 1980; BS; Hoffman 1986; Monaco 1995; Eisenstein & Loeb 1995; Bond & Myers 1996a,b; van de Weygaert 1996; Audit & Alimi 1996), only a few papers deal with the study of the effect of shear on the ellipsoid evolution and the analytical studies are even fewer (WS; BS). Moreover, the effect of internal and external shear has been studied separately: for example WS, BS and Watanabe & Inagaki (1991, 1992) neglect the role of tidal forces; Watanabe (1993) studied only the effect of external shear.
It is thus interesting to study the evolution of the homogeneous ellipsoid taking account of both internal and external shear and to look for analytical solutions of the equations describing the evolution of an ellipsoidal perturbation.
In this paper, I shall study the effect of internal and external shear, finding an approximate analytical solution to the equations of motion given in Watanabe (1993). These equations are the equations of Icke (1973) and WS, modified to take account of the effect of the tidal distortion. Similarly to WS, I find an approximate solution to Watanabe's equations which is then compared with numerical simulations.
The paper has two aims:
1) to find an analytical solution for the equations describing the
evolution of an ellipsoidal perturbation and to study the changes
in configuration, axial velocity fields, and density contrast in
terms of the internal shear (intrinsic initial asphericity) and
external shear;
2) To study how internal and external shear affect the ellipsoid
overdensity at turn-around and axial velocity at the collapse
epoch. The final aim is to have some insights in the
previrialization conjecture.
I develop this last item by improving the BS model, namely I use the analytical solution of the equations of motion, previously found, to calculate the density contrast at turn-around and the axial velocity at the collapse epoch, similar to what is done by BS.
The plan of the paper is the following: in Sect. 2, I introduce the equations of motion of an unisolated ellipsoid. In Sect. 3, I find an approximate analytical solution to the previous equations, similar to what is done by WS for the isolated ellipsoid model. In Sect. 4, I use the previous solution to find the evolution of the density contrast and the axial peculiar velocity. In Sect. 5, the density contrast at turn-around and the velocity at collapse are calculated in order to have some insights on the effects of shear on the collapse of the structure. Section 6 is devoted to conclusions.
The equations of motion of an irrotational homogeneous ellipsoid
with semiaxis lengths a_{1}, a_{2} and a_{3} embedded in a
Friedmann-Robertson-Walker background was given by Icke (1973), and WS as:
Before going further, let us consider the effects of the
over-simplifications of the model:
1) The ellipsoidal approximation, described by the
Eqs. (1)-(4), applies mainly in the
immediate vicinity of the density
extrema (maxima and minima), where the leading terms in the
gravitational potential are the quadratic ones. Previous works (see Eisenstein & Loeb 1995) has
shown that the inner region of a perturbation is well approximated by the
ellipsoid approximation (Eisenstein & Loeb 1995 apply this model to a
variety of mass scales in the range
and WS to superclusters.)
2) The homogeneous isolated ellipsoid model does not take
account of the role of the tidal forces due to other nearby objects. This problem can
be solved as shown in the rest of this paper or as shown by Eisenstein & Loeb (1995).
3) The homogeneous ellipsoid model does not take account of
the inhomogeneity and of the substructure internal to the system. This
last item is a natural limitation of the model: by definition, a
homogeneous ellipsoid cannot represent the substructure or
inhomogeneity of the object. This limitation, on the one hand,
prevents us from treating the distribution of matter and angular
momentum within the collapsing object and, on the other hand, has the
effect of underestimating the effect of previrialization, and in
particular the value of the overdensity at virialization,
(Peebles 1990). This means that the
effect of the shear on the evolution of the density contrast at
turnaround and the velocity at collapse, estimated with the ellipsoid model,
is even smaller than
what one expects in the collapse of a real protostructure.
4) Another limitation of the model is the effect of the matter in
the ellipsoid on the smooth background. One expects that the
ellipsoid matter causes the density of its immediate surroundings
to deviate from the cosmic mean. Then, as a back reaction, tidal
fields due to the perturbed external material should induce
departure from homogeneity in the ellipsoid. However, when the
density inside the ellipsoid,
,
is very close to the
background density,
,
this effect is negligibly small
(Watanabe & Inagaki 1991; WS), while if
,
the evolution will be determined by its self-gravity (WS).
Moreover, numerical simulations have shown that it is a good
approximation to ignore departures from homogeneity when one
calculates the evolution of the axis ratio (Hui & Bertschinger 1995).
In spite of the uncertainties listed above, the ellipsoid model gives a good approximation for the evolution of any object that collapses in a fairly homogeneous manner (WS; Eisenstein & Loeb 1995).
While it is not possible to take account of inhomogeneity,
the homogeneous isolated ellipsoid model can be modified in
order to take account of external shear as done, for example, by
Watanabe (1993). In order to get the quoted goal, one has to get a
multipole expansion of the gravitational potential, .
The
gravitational potential at time t and comoving coordinates
(), due to the field outside a comoving radius X, is given
by:
= | |||
= | (5) |
(7) |
(8) |
The equations of motion are obtained by adding the force due to
the potential given by Eq. (6) into
Eq. (1). Assuming
that the principal axes of the external tidal tensor are always
oriented along the principal axes of the mass tensor,
the evolution equations reduces to three equations for the three
semiaxes of the ellipsoid and are given by (Watanabe 1993;
van de Weygaert 1996):
In order to have an estimate of the value of Q_{0},
for a cluster interacting with a neighboring one, I use the simple
model in Watanabe (1993). I consider a cluster which has a
neighboring cluster with a mean density contrast
,
a comoving separation (0,0,x_{3}), and a comoving size
.
The Q_{33} quadrupole component is given by:
(12) |
Another way of estimating Q_{0} is by using the anisotropy of the velocity field in the LSC from data of Lilje et al. (1986). If we indicate with the component of the largest absolute value of the anisotropic velocity, one gets: (Watanabe 1993). Since Lilje et al. (1986) deduced a value of 0.1-0.2 at the distance of the Local Group from Virgo, we have that .
Before going on, it is important to discuss a basic difference between the present paper and that of Watanabe (1993). Unlike from Watanabe (1993), in this paper I assume that protostructures have an initial asphericity, while the paper of Watanabe (1993) (similarly to those of van de Weigaert 1996 and Palmer 1983) assume that the initial configuration is spherical, so that the principal axes of the external tidal tensor will be oriented along the principal axes of the mass tensor and the equations of motions reduce to three equations involving the diagonal components of the traceless tidal tensor. Our assumption of initial asphericity of protostructures is motivated by the fact that previous analyses of the topology of the constant-density profiles in the neighborhood of the peaks of the Gaussian field showed that the isodensity surfaces are simply connected and approximately ellipsoidal (Doroshkevich 1970; Bardeen 1986). We also know that the initial asphericity has a certain role in shaping the final configuration of the structure (Icke 1973; WS; BS). By means of this assumption, we have the noteworthy advantage of studying the joint effect of "internal and external shear" (see the final part of Sect. 3 for a discussion). The approach of this paper, assuming that the principal axes of the external tidal tensor are always oriented along the principal axes of the mass tensor, is chosen for mathematical simplicity. Note that, at the same time, the assumption is not strange or without motivation: for example van de Weygaert & Babul (1994), in order to study the effect of shear fields on the evolution of galactic scale density peaks, make a similar assumption, namely that the shear tensor, at the location of the peak representing the structure, is oriented so that it is diagonal. Moreover, in a recent paper Porciani et al. (2002) find a strong alignement between the principal axes of the inertia and the shear tensor, in contradiction to usual assumption that the two tensors are largely uncorrelated (Hoffman 1986b; Heavens & Peacock 1988; Catelan & Theuns 1996).
Finally, I want to stress that the approximate solution found in the present paper give a more general representation of structure formation than those described, as examples, in the following sections. For example, the assumptions and results of Watanabe (1993) are re-obtained assuming that the three axes of the ellipsoid are equal, or in other words Watanabe's result is a "particular case" of those of this paper when a_{1}(t_{i})=a_{2}(t_{i})=a_{3}(t_{i}). Moreover, in the present paper, I derive analytical solutions for the equations of motion of the ellipsoid, while Watanabe (1993) solve the same equations numerically.
An analytical approximation of the solution describing the
evolution of the ith axis of the isolated ellipsoid was
found by WS. The equations of motion can be integrated
analytically if one assumes that:
1) The evolution of the configuration is self-similar, which means
that
.
2) The time dependences of
and
are the same as in the spherical model, namely:
Coming back to the goodness of the approximation of Eq. (15)
we have, for example, in the first case
(a_{1}:a_{2}:a_{3}=1:1.25:1.5), that the error in
is
while that in
is ,
where
is the collapse time of the first axis. For a
configuration a_{1}:a_{2}:a_{3}=1:1.5:3, Fig. 1b shows that Eq. (15) gives a worse approximation going from less
asymmetric to more asymmetric configurations, especially in the
case of the shortest axis. This trend is confirmed by Fig. 1c,
representing the initial configuration a_{1}:a_{2}:a_{3}=1:1.5:5. The
same problem was encountered by Watanabe & Inagaki (1991) in the
calculation of the axial peculiar velocity using the WS approximate
solution.
Figure 1: a)-c) Evolution of isolated homogeneous ellipsoidal perturbations in an EdS universe with , and axial ratio 1:1.25:1.5 a), 1:1.5:3 b), 1:1.5:5 c). The solid lines represent numerical solutions obtained using Bulirsh-Stoer algorithm while the dotted ones the approximate analytical solution of WS. | |
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c_{1} | = | ||
c_{3} | = | 1.002 a_{10}^{0.1}a_{20}^{-0.035}a_{30}^{-0.065} | (19) |
Figures 1d-f shows that Eqs. (16)-(18)
give a better representation of numerical results, with respect to
Eq. (15), for all of the three axes (the initial axial ratio
is the same as Figs. 1a-c).
Figure 1: d)-f) Same as the previous figures Figs. 1a-c but now the WS analytical approximate solution is improved by using some free parameters fitted to the numerical solution by means of the least-square method (see text for a description). | |
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Similarly to the case of an isolated ellipsoid, it is
possible to obtain an analytical solution of Eq.
(9), describing the evolution of an unisolated
ellipsoid. In this case the solution can be written in the form:
(23) |
(24) |
(25) |
In the
case of prolate spheroids, with axial ratio 1:1:a_{3} and
,
a better approximation to the
is:
(26) |
(27) |
(28) |
Figure 2: a)-c) Evolution of unisolated homogeneous ellipsoidal perturbations in an EdS universe with , , axial ratio is 1:1.25:1.5, b_{i}=(-0.5,-0.5,1), and Q_{0}=0.1 a), Q_{0}=0.2 b), Q_{0}=0.4 c). | |
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In Figs. 2d-e, I plot the evolution of the ellipsoids for a fixed value
of Q_{0} and b_{i} and for different values of the initial axial
ratio, in order to study the effect of the internal shear. In
Figs. 2d-e, the axial ratio is 1:1.25:2,
1:1.25:5, respectively, Q_{0}=0.1 and
b_{i}=(-0.5,-0.5,1).
Figure 2: d)-e) Same as the previous Fig. 2a but now the axial ratio is 1:1.25:2 d), 1:1.25:5 e), while Q_{0}=0.1 and b_{i}=(-0.5,-0.5,1). | |
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As discussed in Sect. 2, in this paper, unlike Watanabe (1993), I assume
that protostructures have an initial asphericity.
By means of this assumption,
I have the noteworthy advantage of being able to study the joint effect of "internal and external shear".
So it is interesting to analyse the contribution to the asphericity of real clusters of galaxies (e.g. LSC)
coming from inner shear and external shear.
In the case of an isolated ellipsoid the length of the uncollapsed axes at collapse can be
obtained similarly to WS, by means of Eqs. (17), (18):
(29) |
In the case of an unisolated ellipsoid the length of the uncollapsed axes at collapse can be
obtained by means of Eqs. (21)-(22):
(30) |
The previous are only some examples (summarized in Table 1) of how the model can be used to obtain information on the effect of external and internal shear on structure formation.
The approximate solution obtained in the previous section
can be used to calculate the
evolution of the axial peculiar velocity. I use
Eqs. (20)-(22) to calculate the
peculiar velocity along the axis of the ellipsoid in units of the
Hubble expansion velocity, H:
Figure 3: a) Evolution of the axial velocity. The solid lines represent, from up to bottom, the numerical result for the peculiar velocity along the shortest, medium and longest axis. The dotted line represents the velocity obtained from Eq. (31) using the approximation for the semiaxes (Eqs. (22)-(24)). The value of the axial ratio is 1:1.25:1.5, while Q_{0}=0.1, and b_{i}=(-0.5,-0.5,1). | |
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Figure 3: b)-c) Same as Fig. 3a but now the value of the axial ratio is 1:1.25:5 b), 1:2:3 c), while Q_{0}=0.1, and b_{i}=(-0.5,-0.5,1). | |
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Figure 3: d)-e) Evolution of the axial velocity. The solid line represents the peculiar velocity for an ellipsoid of axial ratio 1:1.25:1.5 d) and 1:2:3 e) when no external field is present, while the dotted and dashed lines represent the case Q_{0}=0.1, b_{i}=(-0.5,-0.5,1), and Q_{0}=0.2, b_{i}=(-0.5,-0.5,1), respectively. | |
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The effect of the external field is shown in Figs. 3d-e. The solid line in Fig. 3d represents the peculiar velocity for an ellipsoid of axial ratio 1:1.25:1.5 when no external field is present, while the dotted and dashed lines represent the case Q_{0}=0.1, b_{i}=(-0.5,-0.5,1), and Q_{0}=0.2, b_{i}=(-0.5,-0.5,1), respectively. The situation is similar to that seen when I described the evolution of the semiaxes for different values of the external field. External shear produces different effects on the axes: the evolution of the longest axis, which is characterized by the smallest velocity, tends to be slowed down with increasing strength of the external field (dotted and dashed lines). The effect of external shear on the shortest and medium axes is opposite to that on the longest one, namely shear produces an acceleration in their evolution. Figure 3e is the same as the previous one, but now the axial ratio is 1:2:3. Similar to the previous plot, increasing the strength of the external field produces an acceleration of evolution in the shortest and medium axes and the opposite effect on the longest one. Another feature shown by a comparison of Figs. 3d and 3e is that the evolution of the shortest axis is accelerated with increasing asymmetry of the structure while the opposite is true for the longest axis: as noted before, external and internal shear have a qualitatively similar effect on the evolution of the shortest and longest axes. The difference between the velocities along the longest and shortest axes is larger for unisolated ellipsoids than for the isolated ones and this difference increases with increasing strength of the external field.
The evolution of the density contrast can be calculated using the
usual definition:
Figure 4: a)-b) The evolution of the density contrast. The axial ratio of the ellipsoid is 1:1.25:1.5 a) and 1:2:3 b), the lines from bottom to top represent the case of an isolated ellipsoid ( b_{i}=(0,0,0)), Q_{0}=0.1, b_{i}=(-0.5,-0.5,1), and Q_{0}=0.2, b_{i}=(-0.5,-0.5,1), respectively. | |
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Q_{0} | a_{1}:a_{2}:a_{3} | |
0 | 1:1.25:1.5 | 1.9 |
0.1 | 1:1.25:1.5 | 2.5 |
0.2 | 1:1.25:1.5 | 3.3 |
0.4 | 1:1.25:1.5 | 5.9 |
Q_{0} | a_{1}:a_{2}:a_{3} | |
0 | 1:1.25:1.7 | 2.8 |
0.1 | 1:1.25:1.7 | 3.7 |
0 | 1:1.25:2 | 4.6 |
0.1 | 1:1.25:2 | 6 |
As reviewed in the introduction, in the literature there is not full agreement on the effect of shear on the collapse of density perturbations: while according to Bertschinger & Jain's (1994) collapse theorem the spherical perturbations are the slowest in collapsing, several other studies conclude opposite (Peebles & Groth 1976; Davis & Peebles 1977; BS; Szalay & Silk 1983; Villumsen & Davis 1986; Peebles 1990; Bond & Myers 1993a,b; Lokas et al. 1996; Audit et al. 1997).
In other words,
the results concerning the effect of shear on collapse,
are of two opposite kind:
1) Shear slows down the collapse (Peebles & Groth 1976; Davis &
Peebles 1977; BS; Szalay & Silk 1983; Villumsen & Davis 1986;
Peebles 1990; Bond & Myers 1993a,b; Lokas et al. 1996; Audit
et al. 1997). This result is obtained if, for example, one uses
homogeneous ellipsoids to model an extended mass
distribution and not vanishing mass elements and collapse is
followed even after the collapse of the first axis: the collapse
of one axis is "frozen'' when it becomes smaller than a certain
value, and the collapse of the other axes is followed until the
collapse of the third axis, which defines the collapse time. This
is done to simulate the virialization process. In fact, as
remarked by Eisenstein & Loeb (1995), after the short axis
collapses, it makes a small contribution to the quadrupole moment
of the ellipsoid (due to the fact that the
quadrupoles are proportional to the difference between the squares
of the lengths of the axes) and then, in order to take
into account the acquisition of angular momentum it is necessary
to follow the collapse after the first axis collapse.
2) Shear accelerates the collapse (e.g., Hoffmann 1986a, 1989;
Evrard & Crone 1992; Bertschinger & Jain 1994). This result
is obtained in papers dealing with the evolution of individual
mass elements for which the collapse corresponds to the collapse
of the first axis.
Figure 5: The density contrast at turnaround for a prolate spheroid for several values of the longest axis, a_{3}, (the other two axes have fixed value a_{1}:a_{2}=1:1). The solid lines, from top to bottom, represent numerical results for the density contrast for an isolated spheroid ( b_{i}=(0,0,0)), and for unisolated spheroids with Q_{0}=0.1, b_{i}=(-0.5,-0.5,1) and Q_{0}=0.2, (-0.5,-0.5,1), respectively. The dashed lines represent the approximate solution (Eq. (37)). The upper dotted line represents the value of the density contrast at turnaround for a spherical perturbation. | |
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Figure 6: Comparison between density contrast predicted by spherical and ellipsoidal model. The solid line represents the density contrast predicted by the spherical model, while the dotted, short-dashed, long-dashed and long-dashed-short-dashed lines represents the prediction of the ellipsoid model with Q_{0}=0.1, b_{i}=(-0.5,-0.5,1) and initial axial ratio 1:1.25:1.5, 1:1.25:3, 1:1.25:5, and 1:1.25:8respectively. | |
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In the remainder of this section, I shall show that, even if the effects of substructure are neglected, by approximating the structure formation by means of a homogeneous ellipsoid, and even if one assumes that the collapse is stopped when the first axis collapses, the shear slows down the rate of growth of the density contrast by lowering the peculiar velocity (WS; BS; Szalay & Silk 1983).
To this aim, in the following, I re-derive the fundamental equations in BS for an homogeneous ellipsoid model also taking into account the external field (in BS, only the collapse of an isolated ellipsoid model was studied). These equations will be used to study the velocity at collapse and the overdensity at turnaround.
The evolution of the ellipsoid can be obtained using Eqs. (20)-(22):
Following BS, it is easy to find that the density contrast
is given by:
(39) |
(40) |
Figure 7: Comparison between density contrast predicted by spherical and oblate spheroidal models. The solid line represents the density contrast predicted by the spherical model, while the dotted, short-dashed, and long-dashed lines represents the prediction of the model with Q_{0}=0.1, b_{i}=(-0.5,-0.5,1) and initial axial ratio 2:2:1, 4:4:1, 8:8:1, respectively. | |
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Figure 8: Density contrast at virialization. The solid line refers to an prolate spheroid. | |
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Figure 9: Turnaround epoch for a prolate spheroid. The solid, short-dashed and long-dashed lines, represent respectively the time of turnaround for an isolated spheroid and for spheroids having Q_{0}=0.1, b_{i}=(-0.5,-0.5,1) and Q_{0}=0.2, (-0.5,-0.5,1). The upper dotted line represents the value of the density contrast at turnaround for a spherical perturbation. | |
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Figure 10: Axial velocity at collapse as function of the ratio of the initial value of the axes, a_{3}/a_{1}. The solid lines represent the numerical results of the collapse velocity for a prolate spheroid, the dotted lines the result obtained from the approximate solution. The top curve represents the velocity for an isolated spheroid, the medium one the same quantity for a spheroid having Q_{0}=0.1, b_{i}=(-0.5,-0.5,1), and the last bottom curve the velocity for a spheroid having Q_{0}=0.2, and b_{i}=(-0.5,-0.5,1). | |
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The density contrast at virialization is a bit more difficult to
calculate. To begin with, it is important to recall the difference
between virialization and collapse: this last term describe a
state of the system in which the density approaches infinity,
while virialization is characterized by |U|=2 K, where U and Kare, respectively, the potential and kinetic energy. Only in the
case of perfectly spherical infall, are collapse and virialization
synonymous (although in the case of a bound system one rapidly follows
the other). In this case, the infall cannot be halted and
it proceeds towards a singularity, with all mass of the system
collapsing to a single point which means that the density
contrast becomes infinite. This result is physically unacceptable
and to prevent the system from reaching this state it is necessary to
introduce, by hand, the assumption that the collapse halts when
spherical shells reach a particular radius
,
(where ,
and
are the final radius, the virialization radius and the turnaround
radius, respectively)^{}. Then, in
the spherical infall model, the density contrast of the virializing
structure is calculated assuming that, after the shell collapses,
the final radius is
:
this leads
to the conclusion that
.
The density contrast
of a non-spherical virialized system can be calculated using the
same definition given for the spherical system, namely:
(41) |
The time at collapse of the shortest axis can be obtained by means
of the second part of Eq. (36) once the parameter
at the collapse time, characterized by
,
is known. After some calculations it is possible
to show that
can be obtained by solving the
following equation:
(43) |
The results in Figs. 5-8 could seem in disagreement with Figs. 4a-b, since in those figures the value of increases with increasing initial asymmetry of the ellipsoid. The reason why decreases with increasing initial asymmetry is due to the fact that the turn-around , and also the collapse epoch, moves towards lower values of time, t, for larger values of initial asymmetry of a given perturbation and larger strength of the external field. This last effect is shown in Fig. 9: the solid, short-dashed and long-dashed lines, represent respectively the time of turn-around for an isolated spheroid and for spheroids having Q_{0}=0.1, b_{i}=(-0.5,-0.5,1) and Q_{0}=0.2, (-0.5,-0.5,1). More asymmetric structures are characterized by a smaller value of turn-around time, and the external field contributes to this reduction of . In the case of oblate spheroids, the collapse time can be reduced to values for initial axial ratio 8:8:1.
Another interesting quantity that can be obtained is the collapse
velocity at the time of collapse. I shall calculate the collapse
velocity using the same steps followed by BS, with the difference
that the spheroids considered in the following are prolate. Their
evolution is obtained as before, putting Eqs. (33)-(35) in the equation for the
collapse velocity down the shortest axis:
(46) |
(47) |
(48) |
(50) |
(51) |
In Fig. 10, I plot
,
as a function of the ratio of the initial value of the axes,
a_{3}/a_{1}. The solid lines represent numerical results of the
collapse velocity for a prolate spheroid (
a_{1}=a_{2}<a_{3}), the
dotted lines represent the result obtained from the approximate solution.
The top curve represents the velocity for an isolated
spheroid, the medium one the same quantity for a spheroid having
Q_{0}=0.1,
b_{i}=(-0.5,-0.5,1), and the bottom curve the
velocity for a
spheroid having Q_{0}=0.2,
b_{i}=(-0.5,-0.5,1).
The figure shows
two trends:
a) the collapse velocity is reduced with increasing initial
asymmetry. For example for
a_{1}/a_{3}=0.3 the collapse velocity is
reduced to the Hubble velocity in the plane of collapse (plane of
the pancake for oblate spheroids ) (
), while
in the case of more extreme "flattening''
a_{1}/a_{3}=0.125, the
collapse velocity is reduced by a factor of 2.5 with
respect to the previous value. In the case of oblate spheroids, for
this initial asymmetry, this value
is 6.
b) The collapse velocity is reduced with increasing strength of
the external field. In the case of the bottom curve, for
a_{1}/a_{3}=0.125, the collapse velocity is reduced by a factor of
3. Similar to item (a), in the case of oblate
spheroids, for this last initial asymmetry, this value can be
larger than 6.
In other words, the slowing down of the rate of growth of density contrast produces a lowering of the peculiar velocity in qualitative and quantitative agreement with BS and Szalay & Silk (1983).
The results obtained help to clarify the controversy relative to the previrialization conjecture. According to this paper and with WS and BS and in agreement with Hoffman (1986a) and Bertschinger & Jain's collapse theorem, it is surely true that the effect of the shear is to reduce the collapse time of perturbations. As remarked in item "b" of Sect. 3 and in agreement with WS and BS: The collapse time of perturbations of fixed initial overdensity, for a fixed background, decreases with increasing initial asymmetry.
The decrease of the collapse time compensates the effect of the increase of density contrast and collapse velocity produced by the shear. To be more clear, it is useful to concentrate in Figs. 3, 4. If the collapse of the ellipsoids occurred at a fixed value of t, just like for the spherical model ( ), the larger increase in density contrast or velocity produced by the initial asymmetry, in comparison with the spherical model, should have as a result that at collapse both the density contrast and velocity should be enhanced (with respect to the spherical model). But in the ellipsoidal collapse, the collapse time decreases with increasing initial anisotropy, and as we have previously seen, in case of initial axial ratio 8:8:1 it is given by . As a consequence, the values of density contrast and collapse velocity at collapse time are always reduced with respect to the spherical collapse, in agreement with WS, BS, Szalay & Silk (1983).
I also add that
in the real collapse other effects have an important role, (e.g.,
the effects of small scale substructure). Both the Hoffman (1986a) and
Bertschinger & Jain (1994) results are valid for a fluid element,
that has no substructure by definition, while a small scale
substructure produces a slowing down of the collapse at least
in two ways:
1) encounters between infalling clumps and substructure internal
to the perturbation (Antonuccio-Delogu & Colafrancesco 1994;
Del Popolo & Gambera 1997; Del Popolo & Gambera 1999);
2) tidal interaction of the main proto-structure with substructure
external to the perturbation (Peebles 1990; Del Popolo
& Gambera 1998).
Moreover, it should be pointed out that, as more small-scale power
is present, the collapse of a perturbation may be slowed down in a
way that could inhibit the effect of shear.
Similar to Bertschinger & Jain (1994), the model presented in
this paper does not take account of the substructure internal to
the system.
I, however, recall that the same shortcoming was present in
Peebles (1990): in that paper the substructure was
suppressed, since it adopted a homogeneous Poisson distribution
of particles within the protocluster (Peebles 1990). This limit
has the effect of underestimating the effect of previrialization,
(Peebles 1990). In other words, the slowing down of the collapse
obtained in this paper (similarly to that of Peebles 1990)
is smaller than that we would find if we had used a system
having internal substructure, as in the above point 1.
Before concluding, I want to discuss the impact of the results of this paper on our view of structure formation.
The reduction of the rate of growth of overdensity and collapse velocity has several consequences for structure formation. A first consequence is a change of the mass function, the two-point correlation function, and the mass that accretes on density peaks. These last consequences are connected to the effects of the shear (Audit et al. 1997, 1998). According to Audit et al. (1997, 1998), the mass function depends on two parameters, a density threshold and a shear threshold . According to the previous authors, structures results from a collapse along their third principal axis, which is slowed down by the effect of the shear (in agreement with our results). Therefore on small scales, where the shear is statistically greater, structures need on average a higher density contrast to collapse and as a consequence the number of objects with ^{} decreases as compared to the collapse of the second or first axis, and so the mass function is much below the standard Press-Shechter prediction. Even the two-point correlation function of galaxies and clusters of galaxies are strongly modified since the two-point correlation function of the collapsed halos is directly connected to the number of objects of a given mass (see Peebles 1993; Sheth & Jain 1997; Del Popolo & Gambera 1999; Del Popolo et al. 1999). Another important consequence of the results described is connected to the value of the density parameter, . Since the initial anisotropy (internal shear) and the tidal interaction with external objects (external shear), slows down the collapse infalling velocity, when using the spherical infall model we underestimate the value of the density parameter (Szalay & Silk 1983; Lee et al. 1986; Taruya & Soda 2000). The previously described effects are even larger if the structure acquires angular momentum during evolution (Del Popolo & Gambera 1999; Del Popolo & Gambera 2000). In our model, we assumed that the principal axes of the external tidal tensor are always oriented along the principal axes of the mass tensor and this implies that the linear angular momentum should be zero (at least from the linear phase to shell-crossing), and so while it is legitimate to speak about the effects of shear on structure formation, we do not take account of the effects of angular momentum acquisition on structure formation, at least before gravitational collapse^{}. As a result, since tidal forces produces effects similar to that due to shear, the quoted limit has the consequence of underestimating the global effect on structure formation of the interaction of a protostructure with neighboring ones (see also Del Popolo & Gambera 1999; Del Popolo & Gambera 2000).
Finally, we want to recall, as previously reported, that in the case of an initial spherical configuration our model reduces to Watanabe's. In this case, the misalignment condition is verified and the sphere can acquire angular momentum.
I examined the effect of internal and external shear on the
evolution of non-spherical inhomogeneities in a EdS universe. The
study was based upon an approximate analytical solution of the
equation of motions of the axes of the ellipsoid. In the first
part of the paper, I found the analytical solution to the quoted
equations and I compared the result with the numerical solution of
Icke's (1973) equations, and in the case of isolated
ellipsoids with the WS analytical solution. The analytical
approximation is in good agreement with the numerical results both
for isolated and unisolated ellipsoids, and it gives a
better approximation to the numerical results with respect to the WS
analytical solution. The quoted solution was used to study the
effect of shear on the density contrast and peculiar velocity. The results show that:
a) The collapse time of perturbations of fixed initial
overdensity, for a fixed background, decreases with increasing
initial asymmetry and strength of the external field. To be more
precise, the evolution of the shortest axis is accelerated with
increasing asymmetry of the structure while the opposite is true
for the longest axis. The effect of a positive tidal force, along
a given axis, is that of slowing down the expansion of the
ellipsoid along this axis and the effect of a negative tidal field
is that of accelerating the expansion of the ellipsoid along that
axis: external and internal shear have a qualitatively similar
effect on the evolution of the shortest and longest axes.
b) The
difference between the velocities along the longest and shortest
axes is larger for unisolated ellipsoids than for the isolated ones and this difference increases with increasing
strength of the external field.
c) While for isolated ellipsoids or for unisolated
ones with small values of ,
the axial ratio does not
appreciably change until the perturbation enters a strongly
nonlinear regime, (the self-gravity is dominant); for large values
of ,
the collapse is anisotropic even for not large
values of .
This means that the collapsing region geometry
is strongly influenced by the external shear: if the external
field is strong enough,
then external shear is dominant (with respect
to internal shear) in shaping the region, in agreement with
Eisenstein & Loeb (1995) and Watanabe (1993).
d) Increasing the strength of the external field, the value of the
density contrast increases.
e) More asymmetrical initial configurations tend to have, at a
given time, larger values of .
Then internal and external
shear produce a more rapid evolution of
the density contrast.
In order to study the effect of shear on the density contrast at
turnaround and velocity at collapse, I re-derived the equations of
the density contrast at turn-around and the velocity at collapse
time of the BS model, taking account of both internal and
external shear. The results have shown that:
(aa) The values of density contrast and collapse velocity at
collapse time are always reduced with respect to the spherical
collapse, in agreement with WS, BS, Szalay & Silk (1983).
(bb) The effects of the slowing down of the collapse obtained in
this paper (similar to that of Peebles 1990) are
smaller than that we would find if we had used a system having
internal substructure.
(cc) The shear has a big impact on our view of structure
formation:
(cc1) its first consequences are a change in the mass function, the
two-point correlation function, and the mass that accretes on
density peaks (see also Del Popolo & Gambera 2000; Audit et al.
1997; Del Popolo & Gambera 1999; Del Popolo et al. 1999; Peebles
1993).
(cc2) Another important consequence of the results described is
connected to the value of the density parameter, .
When
using the spherical infall model we underestimate the value of the
density parameter (Szalay & Silk 1983; Lee et al. 1986; Taruya &
Soda 2000), since
shear slows down the collapse infalling velocity.
Almost all the results obtained with the analytical model were
tested against numerical solutions, always finding good agreement
with them.
Acknowledgements
I would like to thank Prof. E. Recami, E. Nihal Ercan, A. Diaferio, J. D. Barrow and Y. Eksi for some useful comments.
Finally, I would like to thank The Bo aziçi University Research Foundation for the financial support through the project code 01B304.