Nuclear Physics B237 (1984) 226-236
© North-Holland Publishing Company
G A U G E - I N V A R I A N T C O S M O L O G I C A L F L U C T U A T I O N S OF
UNCOUPLED FLUIDS*
L.F. ABBOTT
Physics Department, Brandeis University, Waltham, MA 02254, USA
Mark B. WISE
California Institute of Technology, Pasadena, CA 91125, USA
Received 25 July 1983
(Revised 17 October 1983)
The gauge-invariant approach to cosmological perturbations introduced by Bardeen is extended to the case of multiple fluids which are uncoupled, except for gravitational interactions.
The most general scalar, vector, and tensor fluctuations are considered. The resulting equations are
applied to fluctuations involving axions and radiation and axions and baryons in the early
universe.
1. Introduction
There is considerable evidence [1] that the energy density of the universe contains
a large dark matter component along with the ordinary luminous component we see.
Massive neutrinos [2], photinos [3], gravitinos [4] and axions [5] have all been
proposed as candidates for the dark matter. The luminous and dark matter appear to
be uncoupled, except for gravitational interactions. The dark matter resides in
galactic halos and presumably in galactic clusters and superclusters and it is likely
that it played an important role in the formation of these structures [5-9]. The
problem of the evolution of structure in the universe has thus become a problem of
multiple uncoupled gravitating fluids.
The most elegant and powerful formalism for treating cosmological perturbations
is the gauge-invariant approach of Bardeen [10,11]. In its original formulation, this
approach applies to a single fluid. Here we extend the gauge-invariant formalism to
multiple uncoupled fluids. Scalar, vector and tensor perturbations are considered
respectively in sects. 2, 3 and 4. The resulting equations are ideally suited for
* Work supported in part by US Department of Energy contract no. DE-AC03-76-ER03230.A011 and
by contract no. DE-AC03-81-ER40050 and the Alfred P. Sloan Foundation.
226
L.F. Abbott, M.B. Wise / Uncoupledfluids
227
treatments of fluctuations involving both dark and luminous matter, especially when
the fluctuations are outside of the horizon. In sect. 5 we apply the equations for the
evolution of scalar perturbations to axions and radiation in a radiation-dominated
universe and to axions and baryons in an axion-dominated universe [5-9].
We consider fluctuations involving N fluids in a spatially flat Robertson-Walker
background spacetime. The letters a and b, running from 1 to N are used to denote
the N fluids. The letters i and j, running from 1 to 3 are spatial indices while/~ and p,
running from 0 to 3 are spacetime indices. Because the N fluids are uncoupled,
except for gravitational interactions, their energy-momentum tensors are individually covariantly conserved:
(1)
D ~'T~?~= O .
This, along with the Einstein gravitational equations
N
a.~ --= -8~rG Z T~,
(2)
a=l
determine the dynamics. The background Robertson-Walker metric is
ds 2 = - d r 2 + R ( t ) 2 d x
(3)
•dx,
and the background stress tensors take the perfect fluid form
To~° = - 0 . ( t ) ,
T)"i = p . ( t ) 8j,
Tff° = T0" i = 0.
(4)
It follows from eqs. (1) and (2) that R ( t ) satisfies the Friedman equations.
{1
\
.--.
N
ldZR
--R dt 2
dRm2=8~rG2..O.,
dt }
a=l
N
~rG ~-'~ (0. + 3p.),
(5)
a=l
and 0,(t) and p , ( t ) satisfy the conservation equations
do.
dt
3 dR
R dt ( 0 . + P . ) .
(6)
Following Bardeen [10], we find it convenient to introduce a conformal time ~"
satisfying
d~"
1
1
dt - R(t)
- S(r)
'
(7)
so that the background metric is
as 2 = S2(~-)[-dT 2 + d x. d x ] .
(3a)
L.F. Abbott, M.B. Wise / Uncoupledfluids
228
In conformal time eqs. (5) and (6) have the form
2
8"IrGS2 E Pa,
a=l
--4~rGS2 Z (Pa + 3pa)
a=l
(5a)
.
where a super-dot is used to denote differentiation with respect to conformal time.
2. Scalar fluctuations
The physical spacetime consists of the Robertson-Walker metric in eq. (3) and the
perfect fluid energy-momentum tensor in eq. (4), plus small fluctuations. Scalar
fluctuations are expanded in scalar harmonics (i.e. plane waves) Q(x) satisfying the
Helmholtz equation
wZQ + kZQ = O,
(8)
and in derivatives of Q,
1
Qi =- - -~ OiQ,
Qij ~ ~ OiOjQ + 18ijQ.
(9a)
(9b)
The homogeneity and isotropy of the Robertson-Walker background ensures that, in
linear perturbation theory, scalar fluctuations corresponding to different wavenumbers, k, evolve independently. The indices on Q~ and Q,j will be raised and lowered
with the ordinary Kronecker delta.
For scalar fluctuations of wave number k, the physical metric and energymomentum tensors are
g00 = - S 2 ( z ) [ 1 + 2 A ( ' c ) Q ( x ) ] ,
(10a)
go, = - S2( ~") B( ~")Q,( x ) ,
(10b)
gij= S2(r){ [1 + 2 H t ( , r ) Q ( x ) ] 8ij+ 2Hw(,r)Qij(x)} ,
(10c)
To~° = - p~(~')[1 + 8 , ( r ) Q ( x ) ] ,
(lla)
T~ i= - (O,,('r) + p ~ ( r ) ) V a ( r ) Q i ( x ) ,
(11b)
T.~° = (pa(r) +pa(r))(v~(r) - B ( r ) ) Q i ( x ) ,
(11c)
T y ' = p , ( ~ ) { [1 + ~r{(T)Q(x)] 8; +~r~-(z)QS(x)).
(lld)
LF. Abbott, M.B. Wise / Uncoupled fluids
229
The quantities A, B, H L, H T, 6,, v~, ~r~ and ¢r~ which parameterize the time
dependences of the perturbations are not invariant under scalar coordinate transformations of wavenumber, k, in the physical spacetime. Their use can lead to
confusion in treating fluctuations outside the horizon [12] and makes the ultimate
equations for the evolution of fluctuations needlessly complicated.
Under a scalar coordinate redefinition of wavenumber k,
-- • + T ( r ) Q ( x ) ,
(12a)
2, = x, + L ( ' r ) Q i ( x ) ,
(12b)
the quantities A, B, HL, HT, 6~, va, ~r~ and ~r~-become
g = A - J~- ( S / S ) T ,
(13a)
b = B + L + kT,
(13b)
[-IL = H L --
(13c)
(~k ) L - ( S / S ) T ,
(13d)
I-I T = H T -4- k L ,
~a = ~a 7t-(3(Pa
+Pa)/Pa)(S/S)
T,
(laa)
Oa= v. + L,
(14b)
•?r~ = ~r~_- ( p . / p . ) T,
(14c)
g'~- = 7r~..
(14d)
Bardeen noted [10] that linear combinations of the above quantities can be formed
which are invariant under coordinate transformations. These gauge-invariant quantities are
1.
1
• A=AA-~BA-~(-~)B---~(ffITA-(-~)f-tT),
(15a)
~H=HL+½HT+~(-~)B---~(~)f-tT,
(15b)
O.
1
k
( v ~ - B),
•
v~ = va - ~-H T,
(15d)
~. = 7 r ~ - c ] ( ~ ) ( ~ . ,
(15e)
Try-,
where
2 =_ d p a / d p a is the square of the speed of sound in the ath fluid.
c a
(15c)
(15f)
LF. Abbott, M.B. Wise / Uncoupledfluids
230
The program is to re-express the dynamics of fluctuations in terms of the gauge
invariant variables defined in eqs. (15). In terms of these variables the Einstein
equations (2) imply
2k 2
U
(16a)
S2 t~H = 87gG E eaPa,
a=l
k2
N
-~5( ~A + ~ H ) = -8¢rG E ~r~Pa,
(16b)
a=l
and the conservation eqs. (1) imply
07+ ( S~ ) va; - k ~ A
k
2
2
a
(Pa*la+CaPaea--XpaCrT}=O.
Pa + Pa
(18)
Combining these equations some straightforward, but tedious, algebra yields N
coupled equations for the variables ea. Defining
E a = PaeaR3 ,
H a =-paTr,~R3 ,
N
N
P - ~ P~,
P =-- E Pa'
a=l
a=l
N a -ParlaR 3 ,
(19a)
1 dR
q -- k / R ,
H = ~ d--T'
these equations are (in terms of the ordinary time variable t):
d2Ea +(2 + 3 c 2 ) H ~ _ ~
dt 2
×
12~rGH
qZ + 12~rG(o + p )
~dEa
b~= l [(1 + 3C2)(pb -'kpb)]--(Pa -'[-Pa) by'~
=l
N
2
-4~rG(pa+Pa) ~_, [Eb]+q 2CaE
a
b=l
{
+12~rG ( p + p ) c ~ E a - ( P a + P a )
N
~,, [C~Eb
b=l
1}
(1 +
3c2)---~]
(19b)
L.F. Abbott, M.B. Wise / Uncoupledfluids
= -q2Na+12~rG (Oa+P~) Y'~ [Nbl-(o+p)Na
231
+2q2H~
b=l
+ 8 ~ G / L ( ~ P + 2P - 2alP) - 2~/
24~GH2
+ q2 + 12~G(p + p )
× (O~+p,) ~_, [ ( 1 + 3e~)Hb]-H ~ Y~ [(1 + 3 c ~ ) ( p ~ + p , ) ]
b=l
.
b=l
(20)
Note that the terms in eq. (20) in brace brackets cancel for a single fluid so for
N = 1 our result agrees with the single fluid result of Bardeen [10] (expressed in
ordinary time). The quantity q is the physical wave number of the fluctuation while
k is the coordinate wavenumber. The fluctuation crosses the horizon when q = H.
Well inside the horizon e, = 8~ = 80~/0~ so eq. (20) becomes an equation for 8OJO~.
For fluctuations taking the form of a perfect fluid the entire right-hand side of eq.
(20) vanishes. Eq. (20) is our main result and in sect. 5 we apply it to the evolution of
scalar perturbations for axions and radiation and axions and baryons in the early
universe.
3. Vector fluctuations
Vector fluctuations are expanded in vector harmonics
v2Q}') + k2Q} 1)= O,
Q}l)(x) satisfying
OjQ}I)= o,
(21)
OjQ(1))
(22)
and in derivatives of Q}I),
Q(1,~
__ 2_~(0kp(1)_~
kj
Indices on Q}n and c~(1)
~ j k will be raised and lowered with the ordinary Kronecker
delta. The divergenceless condition, OjQ} 1)= O, ensures that vector fluctuations
evolve independently of scalar fluctuations in linear perturbation theory. Again the
homogeneity and isotropy of the Robertson-Walker background implies that, in
linear perturbation theory, vector fluctuations corresponding to different wavenumbers, k, evolve independently.
232
L.F. Abbott, M.B. Wise / Uncoupledfluids
For vector fluctuations of wavenumber k, the physical metric and energy-momenturn tensor are
goo = - S 2 ( z ) ,
(23a)
goj = - S 2 ( r ) B(')( r )Q}I)( x ) ,
(23b)
gij = S 2 ( r ) [ ~ij + 2 H(Tx)(r) Q~I)(x)],
(23c)
Toa° = - P a ( r ) ,
(24a)
Tja° = (p~ ( r ) +p~ (r))(va(l)(r) - B ( 1 ) ( r ) ) Q J l ) ( x ) ,
(248)
T(~i = - ( Pa("r ) + p , ( r ) )v(~l)( z )Q(1)i( x ) ,
(24c)
TY i= Pa( r )[ ¢$j + qr~O) ( r ) Q}I)i( x ) ] .
(24d)
Under a vector coordinate redefinition of wavenumber, k,
= r,
(25a)
2 j = x j + L°)(r)Q°)J(x),
(258)
the quantities B (1), H{,1), v~(1) and ~r-~o) transform like
k (1)
=
B (1) + L (1) ,
/I(T1) = H(T1) + k L 0) ,
~(1) = V(1)+ L(1),
~ 0 ) = ~r~,O).
(26)
(268)
(27a)
(27b)
Gauge invariant linear combinations of B (1), H~ 1), v~(l), and ~r~(1) are [10]
~p = B (1) - 1//(1)
(28a)
Ua = Ua(1) -- B (1) ,
(28b)
rr~ (1) .
(28c)
k
T ,
In terms of these variables Einstein equations (2) imply
N
½k2~ = 89rGS2 Y'~ (Pa + Pa) vc,
a=l
(29)
L.F. Abbott, M.B. Wise / Uncoupled fluids
233
and the conservation equations (1) imply
0~ +
1
3C] )Oa
(30)
Note that (unlike scalar fluctuations) vector fluctuations in the N stress tensors Tfl ~'
evolve independently.
4. Tensor fluctuations
Tensor fluctuations are expanded in the tensor harmonics Q}]}(x) which satisfy
/,. 2t~ (2) : 0,
vZQ} 2) + '~
~jk
(31a)
Q(2}
jk = Q~k~,
O/Q}~ ) = 0,
(31b)
0(2)
~..jj = 0.
(31c)
Eq. (31c) implies that tensor fluctuations evolve independently of scalar and vector
fluctuations and the isotropy and homogeneity of the Robertson-Walker background
ensures that tensor fluctuations with different wave numbers k evolve independently.
Indices on t3(2)
"x~k j are raised and lowered with the ordinary Kronecker delta.
For tensor fluctuations of wave number, k, the physical metric and energy
momentum tensors are
g00 = - $ 2 ( ~ ' ) ,
(32a)
gOj = 0 ,
(32b)
gij = $2( r)[ 3ij + 2 H{2) ( r ) Q}2)( x ) ] ,
(32c)
-
(33a)
Tja° = TO"J= 0,
(33b)
Tj:+ = po(~') [ ~j + 27r+t2)(T)Q}2)'(x)].
(33c)
7"o" 0 =
There are no tensor coordinate redefinitions so H.~2) and ~r~<2}are gauge invariant.
Einsteins eqs. (2) imply that
N
H~Z) + 2 ( S / S ) [I~ 2) + k 2H~2} = 81rGS 2 Y'~ p~ (z) ~r~-t2) .
The conservation eqs. (1) are automatically satisfied by tensor fluctuations.
(34)
L.F. Abbott, M.B. Wise / Uncoupledfluids
234
5. A x i o n s
Models which break a U(1) Peccei-Quinn symmetry at an energy scale of about
1011-12 GeV produce a large density of axions when the universe is about 1 0 - 6
seconds old [5]. These axions are non-relativistic, although they are very light, and
they begin to dominate the energy density of the universe at about 1000 years.
There are two periods during which the multi-fluid eqs. (20) are useful. Before
1000 years, when the universe is radiation dominated, fluctuations in ev grow
linearly with time and then begin to oscillate. The axion fluctuations eA follow this
growth and then grow logarithmically. After 1 000 years, when the universe is axion
dominated, the axion fluctuations eA grow like t 2/3 and after recombination, baryon
fluctuations e B follow the axion perturbations leading to the structures we now see.
Thus, we will first consider radiation and axion fluctuations in a radiation dominated
universe and then axion and baryon fluctuations in an axion dominated universe.
Some of the results we will obtain have previously been derived by other methods
[5-9], however, we present them here as an example of the gauge invariant approach
to the evolution of fluctuations.
We treat the axions, radiation and baryons as perfect fluids. In a radiation
dominated universe the radiation fluctuations obey the equation
d2
dt 2
dEr
(35)
+ (2 + 3c )
where c v2 _ ~_ and py = ]Ov. Defining the variable
X =
qcv
---~ - ,
(36)
which is the ratio of the sound horizon length for 2~r times the fluctuation
wavelength, the solution to eq. (35) is
[ sinx xXCO +
where a v and fly are constants. Note that eq. (36) implies that for small x the terms
in the brackets of eq. (37) behave like t and 1 / ~ - , since x goes like t 1/2. For a
galactic-size fluctuation x = 2 × 103 t~(in years) so that x = 1 0 - 5 when the non-relativistic axions are produced at t = 10 -6 sec. The value x ~ 1 corresponds to
t ~ 10 4 years and axions begin to dominate the universe when x---3 000. When
x ~ 1 the radiation fluctuations given by eq. (37) oscillate. Between the time the
non-relativistic axions are produced and the time x ~ 1, ev grows by about a factor
of 109 .
L.F. Abbott, M.B. Wise / Uncoupledfluids
235
In a radiation dominated universe eq. (20) implies' that axion fluctuations obey
d2
[
d~eA+2H
1
1+
]d.
l + q2/12~rG(pv + pv )
dt
1 + q~/12~-~(o., +p~,) ]~, dt -/-h.,
(38)
,
or in terms <)f the variable x of eq. (36)
d2eA + 1 1 1 +
dx 2
X
1
+
,
2
] deA
½x 2 dx
311
x = e Y + ~-x
l+½x 2
dx
x
"
For small x we find that
R.
eA = ~e, + ~A + ~
t
(40)
'
where o/A and flA are constants. Since % is growing linearly with time, EA will rapidly
reach the value 9 e v regardless of its initial value and then grow linearly in time
following ev. When x gets of order one, and e v begins to oscillate, eA starts to level
off and logarithmic growth starts. The behavior of e v and eA as a function of x is
shown in fig. 1.
After axions dominate the energy density of the universe at t = 1000 years axion
perturbations obey
d2eA
de A
dt 2 + 2 H - ~ - - 47rGPAe A = 0,
~A
(41)
~~
/
/
/
/
/
,/
/
~× or ~A
0
2
4~
8
I0
Fig. 1. e v and eA as a function of x during the radiation dominated era,
236
LF. Abbott, M.B. Wise / Uncoupled fluids
which has the solution
eA = O~At2/3 + flat
2
(42)
The axion perturbations grow like t 2 / 3 once axions dominate the energy density of
the universe. After recombination, baryons form a non-relativistic perfect fluid.
Fluctuations in eB larger than the baryon Jeans mass (i.e., about 105 solar masses)
obey
d2eB + 2 H ~ t B =
dt 2
4rrGPAe A .
(43)
Eq. (43) has the solution
eB = EA q- aB -~- fib t
-1/3,
(44)
where a B and fib are constants. Regardless of its initial value, eB, will rapidly
approach eA and follow it growing like t 2/3 [6, 7].
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