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The kinetic theory of the perturbations in the isotropic universe
Volume 64A, number 2
PHYSICS LETTERS
12 December 1977
THE KINETIC THEORY OF THE PERTURBATIONS IN THE ISOTROPIC UNIVERSE A.V. ZAKHAROV Department of Physics, State University, Kazan 420008, USSR Received 23 September 1977 The short wave and long wave gravitational perturbations in the isotropic universe are considered by using the general relativistic kinetic theory. It is shown that the short wave perturbations propagate in the ultrarelativistic collisionless gas with the velocity of light.
The behaviour of gravitational perturbations in a relativistic gas istheory. studiedThe by solution applyingdescribing the generaltherelativistic kinetic isotropic cosmological model [1] is chosen as an unperturbed solution of the Einstein—Vlasoff system. The gas is considered to be collisionless. It is reasonstagestoofdothe of the universe. At the earlier able so expansion at nonrelativistic and late ultrarelativistic stages the collisional frequency is high, so the behaviour of the gravitational perturbations at these stages is described hydrodynamically and is investigated in [2,31. The kinetical examination of the problem of the perturbations in the homogeneously expanding collisionless gas in the frame of Newtonian theory of gravitation is made in [4—61. Kinetic study of the perturbations in the relativistic cosmology has been restricted to gravitational waves in high frequency w ~> H (cf. [7]),Inwhere w the is the frequency, H limit: is Hubble’s constant. this paper we consider both the high frequency and the low frequency (w ~ H) limits. Moreover, we consider not only gravitational waves, but also scalar and vector perturbations. The explicit solution cosmological of the Einstein—Vlasoff system describing the isotropic model is [1]: ds2 = a2(77) [di~2 dr2 —
—
p2(r) (do2 + sin2O dp2)]
,
(1)
where p = r, sinr, shr for fiat, closed and open models respectively, and a(i~)obeys the equation a~2+~a232~2kfdqq2[m2c2a2+q2~h/2F 3 0 0(q2),
(2)
3c
where k is the gravitational constant, 6 = 0, —1, +1 for flat, closed and open models respectively, and the
single-particle distribution function F0 is an arbitrary 2 (ri) pcxpli (cf. [8]). Here is function q~of = a the space of part metric (l),Pcx the space components of the particle momentum (ct, j3, y, = 1, 2, 3). We may choose F 0 in the form [8], ...
2) = F0(q
4ir(mc)3K
ex ~
( [ c
2
1/2\
2(X) a2(s~0)i 2/T, n where A = mc 0 is the particle density at the (3) moment 77 77 T is the temperature of the gas at = ~ At this moment the distribution (3) coincides with Maxwell’s relativistic distribution. 5g~ Consider small perturbations 1of the metric tensor (i, I, k = 0, 1,2,3) and of the distribution function 8F(x, p) in a region which is much smaller than a. In this region the metric (1) may be written as 2 = a2(i~) (d~2 dx2 dy2 dz2). (4) ds Following [2], we can assume that 6g~ 4= 0. We’ll 2 ~h~ write h~ = —6g~~,h~ a 7,h~’ (a/3’q)h~. Linearized with respect to eq. (4) the Einstein— Vlasoff system is 2] 1/2 qc~ [222 +q — —
—
—
—
~L+
=
dF
~
0
2c2a2+q2]h/2h~~q~ql3
,
(Sa)
2t~dq [m —
—
6G~=87rk rd3q[m2c2a2+q2]1/2f
(5b)
6G~~J_~ifd3qq~f, c3a4 8irk jd3q = [~2 c2a2 + q2] 1/2 qPq~f,
(5c)
—
-~-~-
(Sd) 167
Volume 64A, number 2
PHYSICS LETTERS
h~(r~) = ~(~)G~
where fis defined according to the formula
f= 6F(x, F)— (1/2q)h~q~q~(dF0/dq), 2Pa
qa
=
q
a
ö
=
cS
q13
all
q2
=
‘
The expressions for the perturbed components of Einstein’s tensor can be found [2, 3]. It seemed possible to analyze eqs. (5) in two extreme cases: for the ultrarelativistic gas and the nonrelativistic gas. The gas with distribution (3) is nonrelativistic 2 -~a2(77)/a2(77 if the condition T/mc 0)is satisfied. In this limit we can assume that mca ~ q in eqs. (2), (5). Neglecting the thermal velocity straggling, we get 0. In this case the eqs. (Sd) are identical to Lifshits’s equations [2], obtained in the cluster case using the hydrodynamical approach. The distribution2(3) becomes ultrarelativistic under ~‘a(77)/a(77 the condition T/mc 0),which is satisfied at sufficiently early stages of expansion. Assume that the gas is collisionless in this case too (for example, aneutrinogas considered from r 0.2 sec after the expansion began [9]). In the ultrarelativistic limit we can assume q mca in eqs. (2) and (5). Then on the condition 77 ~ 1 we obtain from eq. (2)a ~ Thus the eqs. (5) are considerably simplified. The coefficients of eqs. (5) are independent of xa, so the solution of eqs. (5) may be written as h0(77) exp (in~xa).As in [21, all the perturbations divide in three types: scalar, vector and tensor perturbations. Consider the solutions obtained for the ultrarelativistic case: ~‘
(~ö~
=
p
=
2’P
—
1, n
nan’3/n2) + ~-p(77)b~
11a~’
2 = ,
(6a)
+ C2
sin ni)
;
(6b)
(2) vector perturbations: U(77)(SaflP+ Sllna)/Sn
,
4(C
u(~)= ii
3 cos ~77 + C4 sin fl77),
(7a) (7b)
where Sa = constant is the perturbation’s polarization vector, Sll = 6allSa, ~2 = SaS°, San°~ = 0; (3) tensor perturbations: 168
(8a)
= ~ (C5 COSfl77 + C6 Sin n77); (8b) where G~= constant is the perturbation polarization tensor,G~=0,G~nll = 0,G2=G~G~=2. Here C 1,C2, C3, C4, C5, C6 are arbitrary constants. With the help of these expressions it is clear that all the perturbations propagate in the ultrarelativistic, collisionless gas with the speed of light (under the condition ni~~‘ 1). The solution of eqs. (5) in the limit nri ‘~ 1 has a rather complex form. We shall fix only the main characteristic features of the obtained solutions. Scalar and vector perturbations appearing at the moment ~ 1/n undergo a significant growth up to i7~ 2.7 ~ü After the moment 77 = ~ the significant growth stops. Scalar and vector perturbations increase if = i~, the perturbation of the macroscopic momenturn density of the gas, is nonzero. At 77 = ~1 the perturbations of the macroscopic momentum density of the gas (6T~)vanish. The amplitude of the tensor perturbations under 1177 <~1 decreases as ~~_uI2. A detailed variant of this article is expected to appear later [10].
I am grateful to L.G. Ignaty ev G.G. Ivanov and V.R. Kaygorodov for discussions. Rf e erences [21 Lifshits, Zh. Eksp. Fiz.83.16(1946)587. [1] E.M. L.Bel,AstrophysJ. 155Teor. (1969)
[3] E.M. Lifshits and LM. Khalatnikov, Usp. Fis. Nauk 80 (1963) 391. [4] G.S. Bisnovaty-Kogan and Ya.B. Zeldovich, Astron. Zh. 47 (1970) 949.
ir’(C
=
1 cos 1177
=
,
i(ri)
qaq
n° = (1)6allnll): scalar perturbations(1/n ~77 ~
12 December 1977
[5] V.B. Magalinsky, Astron. Zh. 49 (1972) 1017. [61V.V. Seliverstov, Astron. Zh. 51(1947) 293. [7] E. Asseo, D. Gerbal, I. Heyvaerts and M. Signore, Phys. Rev. D13 (1976) 2724. [8] I. Ehlers, P. Gear and R.K. Sachs, J. Math. Phys. 9 (1968) 1344. [9] Ya.B. Zeldovich, and ID. Novikov, Structure and evolution of the universe (Science, Moscow, 1975). [10] A.V. Zakharov, Izvestia Vuzov, USSR (Fizika), to appear.
PHYSICS LETTERS
12 December 1977
THE KINETIC THEORY OF THE PERTURBATIONS IN THE ISOTROPIC UNIVERSE A.V. ZAKHAROV Department of Physics, State University, Kazan 420008, USSR Received 23 September 1977 The short wave and long wave gravitational perturbations in the isotropic universe are considered by using the general relativistic kinetic theory. It is shown that the short wave perturbations propagate in the ultrarelativistic collisionless gas with the velocity of light.
The behaviour of gravitational perturbations in a relativistic gas istheory. studiedThe by solution applyingdescribing the generaltherelativistic kinetic isotropic cosmological model [1] is chosen as an unperturbed solution of the Einstein—Vlasoff system. The gas is considered to be collisionless. It is reasonstagestoofdothe of the universe. At the earlier able so expansion at nonrelativistic and late ultrarelativistic stages the collisional frequency is high, so the behaviour of the gravitational perturbations at these stages is described hydrodynamically and is investigated in [2,31. The kinetical examination of the problem of the perturbations in the homogeneously expanding collisionless gas in the frame of Newtonian theory of gravitation is made in [4—61. Kinetic study of the perturbations in the relativistic cosmology has been restricted to gravitational waves in high frequency w ~> H (cf. [7]),Inwhere w the is the frequency, H limit: is Hubble’s constant. this paper we consider both the high frequency and the low frequency (w ~ H) limits. Moreover, we consider not only gravitational waves, but also scalar and vector perturbations. The explicit solution cosmological of the Einstein—Vlasoff system describing the isotropic model is [1]: ds2 = a2(77) [di~2 dr2 —
—
p2(r) (do2 + sin2O dp2)]
,
(1)
where p = r, sinr, shr for fiat, closed and open models respectively, and a(i~)obeys the equation a~2+~a232~2kfdqq2[m2c2a2+q2~h/2F 3 0 0(q2),
(2)
3c
where k is the gravitational constant, 6 = 0, —1, +1 for flat, closed and open models respectively, and the
single-particle distribution function F0 is an arbitrary 2 (ri) pcxpli (cf. [8]). Here is function q~of = a the space of part metric (l),Pcx the space components of the particle momentum (ct, j3, y, = 1, 2, 3). We may choose F 0 in the form [8], ...
2) = F0(q
4ir(mc)3K
ex ~
( [ c
2
1/2\
2(X) a2(s~0)i 2/T, n where A = mc 0 is the particle density at the (3) moment 77 77 T is the temperature of the gas at = ~ At this moment the distribution (3) coincides with Maxwell’s relativistic distribution. 5g~ Consider small perturbations 1of the metric tensor (i, I, k = 0, 1,2,3) and of the distribution function 8F(x, p) in a region which is much smaller than a. In this region the metric (1) may be written as 2 = a2(i~) (d~2 dx2 dy2 dz2). (4) ds Following [2], we can assume that 6g~ 4= 0. We’ll 2 ~h~ write h~ = —6g~~,h~ a 7,h~’ (a/3’q)h~. Linearized with respect to eq. (4) the Einstein— Vlasoff system is 2] 1/2 qc~ [222 +q — —
—
—
—
~L+
=
dF
~
0
2c2a2+q2]h/2h~~q~ql3
,
(Sa)
2t~dq [m —
—
6G~=87rk rd3q[m2c2a2+q2]1/2f
(5b)
6G~~J_~ifd3qq~f, c3a4 8irk jd3q = [~2 c2a2 + q2] 1/2 qPq~f,
(5c)
—
-~-~-
(Sd) 167
Volume 64A, number 2
PHYSICS LETTERS
h~(r~) = ~(~)G~
where fis defined according to the formula
f= 6F(x, F)— (1/2q)h~q~q~(dF0/dq), 2Pa
qa
=
q
a
ö
=
cS
q13
all
q2
=
‘
The expressions for the perturbed components of Einstein’s tensor can be found [2, 3]. It seemed possible to analyze eqs. (5) in two extreme cases: for the ultrarelativistic gas and the nonrelativistic gas. The gas with distribution (3) is nonrelativistic 2 -~a2(77)/a2(77 if the condition T/mc 0)is satisfied. In this limit we can assume that mca ~ q in eqs. (2), (5). Neglecting the thermal velocity straggling, we get 0. In this case the eqs. (Sd) are identical to Lifshits’s equations [2], obtained in the cluster case using the hydrodynamical approach. The distribution2(3) becomes ultrarelativistic under ~‘a(77)/a(77 the condition T/mc 0),which is satisfied at sufficiently early stages of expansion. Assume that the gas is collisionless in this case too (for example, aneutrinogas considered from r 0.2 sec after the expansion began [9]). In the ultrarelativistic limit we can assume q mca in eqs. (2) and (5). Then on the condition 77 ~ 1 we obtain from eq. (2)a ~ Thus the eqs. (5) are considerably simplified. The coefficients of eqs. (5) are independent of xa, so the solution of eqs. (5) may be written as h0(77) exp (in~xa).As in [21, all the perturbations divide in three types: scalar, vector and tensor perturbations. Consider the solutions obtained for the ultrarelativistic case: ~‘
(~ö~
=
p
=
2’P
—
1, n
nan’3/n2) + ~-p(77)b~
11a~’
2 = ,
(6a)
+ C2
sin ni)
;
(6b)
(2) vector perturbations: U(77)(SaflP+ Sllna)/Sn
,
4(C
u(~)= ii
3 cos ~77 + C4 sin fl77),
(7a) (7b)
where Sa = constant is the perturbation’s polarization vector, Sll = 6allSa, ~2 = SaS°, San°~ = 0; (3) tensor perturbations: 168
(8a)
= ~ (C5 COSfl77 + C6 Sin n77); (8b) where G~= constant is the perturbation polarization tensor,G~=0,G~nll = 0,G2=G~G~=2. Here C 1,C2, C3, C4, C5, C6 are arbitrary constants. With the help of these expressions it is clear that all the perturbations propagate in the ultrarelativistic, collisionless gas with the speed of light (under the condition ni~~‘ 1). The solution of eqs. (5) in the limit nri ‘~ 1 has a rather complex form. We shall fix only the main characteristic features of the obtained solutions. Scalar and vector perturbations appearing at the moment ~ 1/n undergo a significant growth up to i7~ 2.7 ~ü After the moment 77 = ~ the significant growth stops. Scalar and vector perturbations increase if = i~, the perturbation of the macroscopic momenturn density of the gas, is nonzero. At 77 = ~1 the perturbations of the macroscopic momentum density of the gas (6T~)vanish. The amplitude of the tensor perturbations under 1177 <~1 decreases as ~~_uI2. A detailed variant of this article is expected to appear later [10].
I am grateful to L.G. Ignaty ev G.G. Ivanov and V.R. Kaygorodov for discussions. Rf e erences [21 Lifshits, Zh. Eksp. Fiz.83.16(1946)587. [1] E.M. L.Bel,AstrophysJ. 155Teor. (1969)
[3] E.M. Lifshits and LM. Khalatnikov, Usp. Fis. Nauk 80 (1963) 391. [4] G.S. Bisnovaty-Kogan and Ya.B. Zeldovich, Astron. Zh. 47 (1970) 949.
ir’(C
=
1 cos 1177
=
,
i(ri)
qaq
n° = (1)6allnll): scalar perturbations(1/n ~77 ~
12 December 1977
[5] V.B. Magalinsky, Astron. Zh. 49 (1972) 1017. [61V.V. Seliverstov, Astron. Zh. 51(1947) 293. [7] E. Asseo, D. Gerbal, I. Heyvaerts and M. Signore, Phys. Rev. D13 (1976) 2724. [8] I. Ehlers, P. Gear and R.K. Sachs, J. Math. Phys. 9 (1968) 1344. [9] Ya.B. Zeldovich, and ID. Novikov, Structure and evolution of the universe (Science, Moscow, 1975). [10] A.V. Zakharov, Izvestia Vuzov, USSR (Fizika), to appear.