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On the influence of degeneracy on gravitational instability
Asrroph\s. Vol. 22, No. 4. pp. 380-385. 1998 of Acts Astruphys. .fin. Vol. 18. No. 3, pp. 229-234, 1998 0 1998 Elsevier Science B.V. All rights reserved Printed in Great Britain 0275-1062/98 $19.00 + 0.00
Chin. Asrrurt.
Pergamon
A translation
PII: SO2751062(98)00049-6
On the influence of degeneracy gravitational GAO Jian-gong’ ‘Xinjiang
The influence system
The collapse
time and the Landau
tions
on gravitational
with a semi-degenerate
on Jeans
in the theory
in a collision-
function
is discussed.
time are calculated.
and on stability
formation
instability
distribution
damping
wave number
of galaxy
Urumqi, 830008
University, Italy
of degeneracy
less particle of degeneracy
R. Ruffini2
of Technology,
2 Roma
Abstract
instabilityt
S. Filippi2
Institute
on
in a universe
The influence
have important dominated
implica-
by fermionic
dark matter. Key words:
cosmology-dark
matter-gravitational
instability
1. INTRODUCTION In the field of astrophysics, formation density
there has been a long-standing
and the large scale structure perturbation
background
by thermal
radiation
in recent
of the universe,
motion.
Observations
years have placed
the difficulty.
Now, a widely accepted
of the universe
a large component
is dark matter.
of dark matter
“inos”, such as cosmic
neutrinos,
cannot
rest-mass
The existence
matterl’l,
of initial
of the microwave
on the perturbations
and so further
years is that
of the constraint
be baryonic
of non-zero
over from the early phase of the universe.
constraints
transparent,
idea of recent Because
in the study of galaxy
the smoothing-out
on the isotropy
strong
that could exist at the time when the universe became matter
difficulty
namely,
underlined
the greater
part of the
from He abundance, but is most
and extremely of a background
such
likely to be
weak interaction of dark matter
left has
a direct influence on the evolution of inhomogeneities in the universe. One of the possibilities is that, before the de-coupling of matter and radiation, inhomogeneities in the distribution of the dark matter
could have survived and undergone
to a new way of overcoming
the difficulty
evolution.
in the formation
These considerations
of the galaxies
structure. t Supported Received
by National 1997-11-13;
Natural revised
Science version
Foundation 1997-12-15
380
and Xinjiang
Educational
point
and large-scale
Commission
Degeneracy on Instability
381
A system of weakly interacting inos, like any stellar system of any scale (star clusters, galaxies, galaxy clusters), can be treated as a collisionless system of particles. To study the evolution of perturbations in such systems, then, we can use the dynamical methods that have been established in the stability studies of collisionless plasmas. Although similar to certain extent, there are some fundamental differences between collisionless dynamical systems and collisionless plasmas. A most important difference is that gravitational force is always attractive whereas plasmas on large scales are neutral. Hence, a plasma can form a homogeneous equilibrium system, whereas a gravitating system cannot. This circumstance greatly complicates the study of the stability of gravitating systems. Nonetheless, when we study cases of short wavelength, then within a small region inside an inhomogeneous system we can suppose the gravitational potential to be approximately a constant and use the methods for homogeneous plasmas for that region. There have already been many papers on the behavior of perturbations in an infinite gravitating system121. In these works, the distribution function of the unperturbed particles is mostly taken to be the Maxwellian distribution. Ref. [3] have studied gravitational instability for completely degenerate particle systems. When dealing with ino systems left over from the early universe, we must consider the degree of degeneracy of the distribution function. In the standard cosmological model dark matter particles like the cosmic neutrinos were decoupled from thermal equilibrium at an early time and kept their Fermi distribution before decoupling:
n(p)+ = 4ffhm3p2 [ exp
(“‘zg
“)
+ I] -rdp
,
where E(p) is the energy of the particle with momentum p, /.Lis the chemical potential, T, the absolute temperature and K, the Boltzmann constant. Degeneracy of neutrinos was considered by Weinberg141. The Maxwellian and the completely degenerate distributions are two extreme cases. The more general case is where there is a certain degree of degeneracy, or the case of semi-degeneracy. In computation the more general case is much more complicated and difficult than the two extreme cases. In this paper we study systems with a certain degree of degeneracy and the Landau damping, with the aim of finding out whether the introduction of a degree of degeneracy will Section 2 gives the necessary formulae, influence the growth or damping of perturbations. Section
We start
3. the results
of our calculation.
Section
2. FORMULAE
FOR
with the linearized,
collisionless
4 is a discussion.
COMPUTATION
Boltzmann
equation,
(1) and the Poisson
Assuming
equation,
the unperturbed
initial
J
.
fid3v
V2&
= 4aG
state
to be uniform
and time-independent,
fo(r, v, t> = fo(v) 1
(2)
382
GAO Jian-gong et al.
and following
Jeans,
we set 40 = 0, and assume solutions fi(z,
v,t)
= fa(v) exp[i(k = & exp[i(k
&(z,t) We then obtain
the dispersion
4aGm
J
system,
exp
dispersion
Taking
.
(4)
of the particle,
k-2
(2’) -q
function
can be expressed
as
(61
7
$1
g is its spin state
and 71= p/KT is the degeneracy
k to be along the z-axis,
(5)
the distribution
fo= g
velocity
. z -w-k)]
k+v_wd3v=0.
A?
gravitating
where v is the velocity
(3)
. v - ut)] ,
relation,
1+For a semi-degenerate
of the form
number,
c = (KT/m)‘I’
is the
parameter.
we have
+cx,
p =
8n2Gm4g
J
h3
and the Jeans
wave number
t71
-ccl
is
‘cj” = P(w
= 0) = 8fiz;m4go
. F_,,2(71)
(8)
,
where
Fk(77) =
ykdy
Jexp(y- rl) + 1 . 0
In terms of the Jeans
wave number
we can express
the dispersion
relation
as
(9) where
00
JV=
J
udu
1
(10)
--oo
co D=
The quantity
(11)
w = w7 +iw; being complex valued, we shall discuss three cases separately.
Degeneracy on Instability
time.
383
Case i. wi > 0 This is an unstable mode, and the perturbation grows exponentially Let (Y = w/H. The imaginary part of the integral n/(a) is then
in
05
I,(N)
= ffi
udu J (es-” _-oo
the dispersion relation requires this part to be zero, and this implies y = u2/2c2, the two integrals in the dispersion relation (12) become 00
JV = Re(N)
= 2&
J
(12)
+ l)[(u - CX,)~+ cyf] ’
eY_“ny, 1 -$ [&j - --j$-
0
tan-l CT
w, = 0. In terms
(_
of
(13)
00
D=&r Case ii. wi = 0 This corresponds
-1/2dy
y
(14)
J ev-q + 1 ’ -ccl
to undamped
oscillation.
In this case, the integral
hf
where p denotes the Cauchy principal value of the integral. The principal value is real, hence the imaginary part of n/ is zero, and this requires w, = 0. Hence, undamped oscillating solution does not exist. Case iii. wi < 0 This is damped solution. To simply the calculation we considered case w; < O,w, = 0. Integrating along the Landau circuit and writing w = -ir,r positive, we have
JV=
the real
2fiu_l ey-~+l~[~&tan-l (+a)] (16) IZnIexp[-
It is easy to see that damped solution.
(&j2-y]
$1
*
in this case, we have M//2) > 1. Hence, for lc > lci, what we obtain
3. RESULTS
OF NUMERICAL
is
CALCULATION
In the numerical calculation, for the sake of generality we take (Gp)-lj2 as the unit of time, as the unit of wave number, G, p, m, n being the gravitational and Ice = (G~~~ni/~/h~)l/~ constant, mass density, particle mass and number density, respectively. For different values of the degeneracy parameter we integrated numerically equation (8) and obtained the relation between the Jeans wave number and degeneracy parameter shown in Fig. 1.
GAO Jian-gong et al.
384
Degeneracyparameter Fig. 1 Dependence of the Jeans wave number on the degeneracy parameter LI
’
’
.
’
I
’
I
10
5
k Fig. 2 Collapse time as function of the wave number, for the indicated values of the degeneracy parameter As in the hydrodynamic separates
stable
grow. We numerically relation
integrated
(9) we obtained
values of the degeneracy damping
treatment
and unstable
solutions:
perturbations
the equations
the collapse parameter.
time, shown in Fig. 3.
of gravitation
instability, with larger
the Jeans
wave number
wave numbers
cannot
(13) and (14), and then from the dispersion
time as a function
of the wave number,
See Fig. 2. Substituting
(16) for (14),
for different
we obtained
the
Degeneracy
on Instability
385
loo
150
k Fig. 3 Damping
time as function
of the wave number,
the degeneracy
4.
RESULTS
AND
for the indicated
values of
parameter
DISCUSSION
1) When the degeneracy parameter is large (10 in the Fig. 3), its influence on the damping time is not marked, but the gravitational instability of the system is sensitive to the degeneracy parameter, when the latter is around 0. 2) The larger the wave number, the closer is the damping time to the case of large degeneracy parameter. This is to say, if we confine ourselves to discussing short wave perturbations, then we can use the results of the extreme case of complete degeneracy, which would greatly simplify the discussion. 3) For k > lcj, because of the decay of the perturbation, the linear theory remains reliable throughout and we can derive from it such useful results as the damping time, etc.. 4) Since the effect of self-gravity of an equilibrium system is small on small scales, it follows that the assumption of homogeneity and the Jeans neglect of gravitational potential (setting do = 0) w h en d iscussing damped solution were justifiable.
References [l]
FM&xi FL, Song D. J., Tareglio
[ 2]
Binney J., Tremaine S., Galactic Dynamics,
[ 31
Shen Tian-zeng
[ 41
Weinberg 1980
S., A&A,
& Gao Jian-gong,
S., Gravitation
1988,
190, 1
Princeton
Kexue Tongbao,
and Cosmology
Univ.
Press, 1987
1996, 41(g),
(Chinese translation),
780 Beijing, Kexue Chubanshe,
Chin. Asrrurt.
Pergamon
A translation
PII: SO2751062(98)00049-6
On the influence of degeneracy gravitational GAO Jian-gong’ ‘Xinjiang
The influence system
The collapse
time and the Landau
tions
on gravitational
with a semi-degenerate
on Jeans
in the theory
in a collision-
function
is discussed.
time are calculated.
and on stability
formation
instability
distribution
damping
wave number
of galaxy
Urumqi, 830008
University, Italy
of degeneracy
less particle of degeneracy
R. Ruffini2
of Technology,
2 Roma
Abstract
instabilityt
S. Filippi2
Institute
on
in a universe
The influence
have important dominated
implica-
by fermionic
dark matter. Key words:
cosmology-dark
matter-gravitational
instability
1. INTRODUCTION In the field of astrophysics, formation density
there has been a long-standing
and the large scale structure perturbation
background
by thermal
radiation
in recent
of the universe,
motion.
Observations
years have placed
the difficulty.
Now, a widely accepted
of the universe
a large component
is dark matter.
of dark matter
“inos”, such as cosmic
neutrinos,
cannot
rest-mass
The existence
matterl’l,
of initial
of the microwave
on the perturbations
and so further
years is that
of the constraint
be baryonic
of non-zero
over from the early phase of the universe.
constraints
transparent,
idea of recent Because
in the study of galaxy
the smoothing-out
on the isotropy
strong
that could exist at the time when the universe became matter
difficulty
namely,
underlined
the greater
part of the
from He abundance, but is most
and extremely of a background
such
likely to be
weak interaction of dark matter
left has
a direct influence on the evolution of inhomogeneities in the universe. One of the possibilities is that, before the de-coupling of matter and radiation, inhomogeneities in the distribution of the dark matter
could have survived and undergone
to a new way of overcoming
the difficulty
evolution.
in the formation
These considerations
of the galaxies
structure. t Supported Received
by National 1997-11-13;
Natural revised
Science version
Foundation 1997-12-15
380
and Xinjiang
Educational
point
and large-scale
Commission
Degeneracy on Instability
381
A system of weakly interacting inos, like any stellar system of any scale (star clusters, galaxies, galaxy clusters), can be treated as a collisionless system of particles. To study the evolution of perturbations in such systems, then, we can use the dynamical methods that have been established in the stability studies of collisionless plasmas. Although similar to certain extent, there are some fundamental differences between collisionless dynamical systems and collisionless plasmas. A most important difference is that gravitational force is always attractive whereas plasmas on large scales are neutral. Hence, a plasma can form a homogeneous equilibrium system, whereas a gravitating system cannot. This circumstance greatly complicates the study of the stability of gravitating systems. Nonetheless, when we study cases of short wavelength, then within a small region inside an inhomogeneous system we can suppose the gravitational potential to be approximately a constant and use the methods for homogeneous plasmas for that region. There have already been many papers on the behavior of perturbations in an infinite gravitating system121. In these works, the distribution function of the unperturbed particles is mostly taken to be the Maxwellian distribution. Ref. [3] have studied gravitational instability for completely degenerate particle systems. When dealing with ino systems left over from the early universe, we must consider the degree of degeneracy of the distribution function. In the standard cosmological model dark matter particles like the cosmic neutrinos were decoupled from thermal equilibrium at an early time and kept their Fermi distribution before decoupling:
n(p)+ = 4ffhm3p2 [ exp
(“‘zg
“)
+ I] -rdp
,
where E(p) is the energy of the particle with momentum p, /.Lis the chemical potential, T, the absolute temperature and K, the Boltzmann constant. Degeneracy of neutrinos was considered by Weinberg141. The Maxwellian and the completely degenerate distributions are two extreme cases. The more general case is where there is a certain degree of degeneracy, or the case of semi-degeneracy. In computation the more general case is much more complicated and difficult than the two extreme cases. In this paper we study systems with a certain degree of degeneracy and the Landau damping, with the aim of finding out whether the introduction of a degree of degeneracy will Section 2 gives the necessary formulae, influence the growth or damping of perturbations. Section
We start
3. the results
of our calculation.
Section
2. FORMULAE
FOR
with the linearized,
collisionless
4 is a discussion.
COMPUTATION
Boltzmann
equation,
(1) and the Poisson
Assuming
equation,
the unperturbed
initial
J
.
fid3v
V2&
= 4aG
state
to be uniform
and time-independent,
fo(r, v, t> = fo(v) 1
(2)
382
GAO Jian-gong et al.
and following
Jeans,
we set 40 = 0, and assume solutions fi(z,
v,t)
= fa(v) exp[i(k = & exp[i(k
&(z,t) We then obtain
the dispersion
4aGm
J
system,
exp
dispersion
Taking
.
(4)
of the particle,
k-2
(2’) -q
function
can be expressed
as
(61
7
$1
g is its spin state
and 71= p/KT is the degeneracy
k to be along the z-axis,
(5)
the distribution
fo= g
velocity
. z -w-k)]
k+v_wd3v=0.
A?
gravitating
where v is the velocity
(3)
. v - ut)] ,
relation,
1+For a semi-degenerate
of the form
number,
c = (KT/m)‘I’
is the
parameter.
we have
+cx,
p =
8n2Gm4g
J
h3
and the Jeans
wave number
t71
-ccl
is
‘cj” = P(w
= 0) = 8fiz;m4go
. F_,,2(71)
(8)
,
where
Fk(77) =
ykdy
Jexp(y- rl) + 1 . 0
In terms of the Jeans
wave number
we can express
the dispersion
relation
as
(9) where
00
JV=
J
udu
1
(10)
--oo
co D=
The quantity
(11)
w = w7 +iw; being complex valued, we shall discuss three cases separately.
Degeneracy on Instability
time.
383
Case i. wi > 0 This is an unstable mode, and the perturbation grows exponentially Let (Y = w/H. The imaginary part of the integral n/(a) is then
in
05
I,(N)
= ffi
udu J (es-” _-oo
the dispersion relation requires this part to be zero, and this implies y = u2/2c2, the two integrals in the dispersion relation (12) become 00
JV = Re(N)
= 2&
J
(12)
+ l)[(u - CX,)~+ cyf] ’
eY_“ny, 1 -$ [&j - --j$-
0
tan-l CT
w, = 0. In terms
(_
of
(13)
00
D=&r Case ii. wi = 0 This corresponds
-1/2dy
y
(14)
J ev-q + 1 ’ -ccl
to undamped
oscillation.
In this case, the integral
hf
where p denotes the Cauchy principal value of the integral. The principal value is real, hence the imaginary part of n/ is zero, and this requires w, = 0. Hence, undamped oscillating solution does not exist. Case iii. wi < 0 This is damped solution. To simply the calculation we considered case w; < O,w, = 0. Integrating along the Landau circuit and writing w = -ir,r positive, we have
JV=
the real
2fiu_l ey-~+l~[~&tan-l (+a)] (16) IZnIexp[-
It is easy to see that damped solution.
(&j2-y]
$1
*
in this case, we have M//2) > 1. Hence, for lc > lci, what we obtain
3. RESULTS
OF NUMERICAL
is
CALCULATION
In the numerical calculation, for the sake of generality we take (Gp)-lj2 as the unit of time, as the unit of wave number, G, p, m, n being the gravitational and Ice = (G~~~ni/~/h~)l/~ constant, mass density, particle mass and number density, respectively. For different values of the degeneracy parameter we integrated numerically equation (8) and obtained the relation between the Jeans wave number and degeneracy parameter shown in Fig. 1.
GAO Jian-gong et al.
384
Degeneracyparameter Fig. 1 Dependence of the Jeans wave number on the degeneracy parameter LI
’
’
.
’
I
’
I
10
5
k Fig. 2 Collapse time as function of the wave number, for the indicated values of the degeneracy parameter As in the hydrodynamic separates
stable
grow. We numerically relation
integrated
(9) we obtained
values of the degeneracy damping
treatment
and unstable
solutions:
perturbations
the equations
the collapse parameter.
time, shown in Fig. 3.
of gravitation
instability, with larger
the Jeans
wave number
wave numbers
cannot
(13) and (14), and then from the dispersion
time as a function
of the wave number,
See Fig. 2. Substituting
(16) for (14),
for different
we obtained
the
Degeneracy
on Instability
385
loo
150
k Fig. 3 Damping
time as function
of the wave number,
the degeneracy
4.
RESULTS
AND
for the indicated
values of
parameter
DISCUSSION
1) When the degeneracy parameter is large (10 in the Fig. 3), its influence on the damping time is not marked, but the gravitational instability of the system is sensitive to the degeneracy parameter, when the latter is around 0. 2) The larger the wave number, the closer is the damping time to the case of large degeneracy parameter. This is to say, if we confine ourselves to discussing short wave perturbations, then we can use the results of the extreme case of complete degeneracy, which would greatly simplify the discussion. 3) For k > lcj, because of the decay of the perturbation, the linear theory remains reliable throughout and we can derive from it such useful results as the damping time, etc.. 4) Since the effect of self-gravity of an equilibrium system is small on small scales, it follows that the assumption of homogeneity and the Jeans neglect of gravitational potential (setting do = 0) w h en d iscussing damped solution were justifiable.
References [l]
FM&xi FL, Song D. J., Tareglio
[ 2]
Binney J., Tremaine S., Galactic Dynamics,
[ 31
Shen Tian-zeng
[ 41
Weinberg 1980
S., A&A,
& Gao Jian-gong,
S., Gravitation
1988,
190, 1
Princeton
Kexue Tongbao,
and Cosmology
Univ.
Press, 1987
1996, 41(g),
(Chinese translation),
780 Beijing, Kexue Chubanshe,