The fluctuations in the cosmic microwave background (CMB) on large angular scales (> few degrees) are caused by perturbations in the gravitational field via the Sachs--Wolfe effect. On intermediate scales, 0.1^\circ\lsim\theta\lsim 2^\circ, the dominant contribution is due to coherent oscillations in the baryon radiation plasma before recombination. Unless the universe is reionized at some redshift z>50, these oscillations lead to the `Doppler peaks' in the angular power spectrum. In structure formation scenarios based on inflation the position of the first peak is typically at \ell\sim 200, with a height which is 4 -- 6 times that of the Sachs--Wolfe `plateau'. Here we present a corresponding study for perturbations induced by global textures. We find that the first Doppler peak is reduced to an amplitude comparable to that of the Sachs--Wolfe contribution, and that it is shifted to \ell\sim 350. We believe that our analysis can be easily extended to other types of global topological defects and general global scalar fields

Durrer, R

Gangui, A

Sakellariadou, M

CERN Document Server

ZH-TH13/95 SISSA{83/95/A
Doppler Peaks: A Fingerprint of Topological Defects.
Ruth Durrer
?
, Alejandro Gangui
y
and Mairi Sakellariadou
?
?
Institut fur Theoretische Physik
der Universitat Zurich, Winterthurerstr. 190,
CH-8057 Zurich, Switzerland
y
SISSA { International School for Advanced Studies
Strada Costiera 11, 34014 Trieste, Italy
Abstract
The uctuations in the cosmic microwave background (CMB) on large an-
gular scales (> few degrees) are caused by perturbations in the gravitational
eld via the Sachs{Wolfe eect. On intermediate scales, 0:1

<
 
<
 2

, the
dominant contribution is due to coherent oscillations in the baryon radiation
plasma before recombination. Unless the universe is reionized at some red-
shift z > 50, these oscillations lead to the `Doppler peaks' in the angular power
spectrum. In structure formation scenarios based on ination the position of
the rst peak is typically at `  200, with a height which is 4 { 6 times that
of the Sachs{Wolfe `plateau'. Here we present a corresponding study for per-
turbations induced by global textures. We nd that the rst Doppler peak is
reduced to an amplitude comparable to that of the Sachs{Wolfe contribution,
and that it is shifted to `  350. We believe that our analysis can be easily
extended to other types of global topological defects and general global scalar
elds.
PACS numbers: 98.80-k 98.80.Hw 98.80C
Presently there are two main classes of models to explain the origin of large scale
structure formation. Initial perturbations can either be due to quantum uctuations
of a scalar eld during an inationary era[1], or they may be seeded by topological
defects formed during a symmetry breaking phase transition in the early universe[2].
The CMB anisotropies are a powerful tool to discriminate among these models by
1
purely linear analysis. Usually CMB anisotropies are parameterized in terms of C
`
's,
dened as the coecients in the expansion of the angular correlation function
h
T
T
(n)
T
T
(n
0
)i



(nn
0
=cos#)
=
1
4
X
`
(2` + 1)C
`
P
`
(cos #):
For scale invariant spectra of perturbations `(` + 1)C
`
is constant on large angular
scales, say `
<
 50. Both ination and topological defect models lead to approxi-
mately scale invariant spectra on large scales.
Large scale CMB anisotropies are mainly caused by inhomogeneities in the space-
time geometry via the Sachs{Wolfe eect[3]. On smaller angular scales (0:1

<


<
 2

) the dominant contribution comes from coherent oscillations in the baryon{
radiation plasma prior to recombination. On even smaller scales the anisotropies are
damped due to the nite thickness of the recombination shell, as well as by photon
diusion during recombination (Silk damping).
Disregarding Silk damping, gauge invariant linear perturbation analysis leads
to[4]
T
T
=

 
1
4
D
(r)
g
  V
j
n
j
 	+ 

f
i
+
Z
f
i
(	
0
  
0
)d ; (1)
where  and 	 are quantities describing the perturbations in the geometry and
V denotes the peculiar velocity of the radiation uid with respect to the overall
Friedman expansion. D
(r)
g
species the intrinsic density uctuation in the radiation
uid. There are several gauge invariant variables which describe density uctuations;
they all dier on super{horizon scales but coincide inside the horizon. Below we use
another variable, D
r
, for the radiation density uctuation[5]. Since the coherent
oscillations giving rise to the Doppler peaks act only on sub{horizon scales, the
choice of this variable is irrelevant for our calculation.
 , 	 and D
(r)
g
in Eq. (1) determine the anisotropies on large angular scales
1
, and
have been calculated for both ination and defect models [6, 7, 8, 9]. Generically,
1
One might think that D
(r)
g
leads just to coherent oscillations of the baryon radiation uid, but
this is not the case. Note that, e.g., for adiabatic CDM models without source term one can derive
(1=4)D
(r)
g
=  (5=3)	 on super{horizon scales. Since for CDM perturbations  =  	 and 	
0
' 0,
2
a scale invariant spectrum is predicted and thus the Sachs{Wolfe calculations yield
mainly a normalization for the dierent models. On the other hand the amplitude of
the Doppler peak, which most probably will be measured in the near future, might
be an important discriminating tool between them. In this Letter we present a com-
putation for the Doppler contribution from global topological defects; in particular
we perform our analysis for 
3
{defects, textures [10], in a universe dominated by
cold dark matter (CDM). We believe that our main conclusion remains valid for all
global defects.
The Doppler contribution to the CMB anisotropies is given by
"
T
T
(x;n)
#
Doppler
=
1
4
D
r
(x
rec
; t
rec
) +V(x
rec
; t
rec
)  n; (2)
where x
rec
= x   nt
0
. In the previous formula n denotes a direction in the sky
and t is the conformal time, with t
0
and t
rec
the present and recombination times,
respectively. To evaluate Eq. (2) we calculate D
r
and V at t
rec
. We consider a
two{uid system: baryons plus radiation, which prior to recombination are tightly
coupled, and CDM. The evolution of the perturbation variables in a at background,

 = 1, is described by[5]
V
0
r
+
a
0
a
V
r
= k	+ k
c
2
s
1+w
D
r
V
0
c
+
a
0
a
V
c
= k	
D
0
r
  3w
a
0
a
D
r
= (1 + w)[3
a
0
a
	  3
0
  kV
r
 
9
2

a
0
a

2
k
 1
(1 +
w
r

)V
r
]
D
0
c
= 3
a
0
a
	  3
0
  kV
c
 
9
2

a
0
a

2
k
 1
(1 +
w
r

)V
c
;
(3)
where subscripts
r
and
c
denote the baryon{radiation plasma and CDM, respectively;
D; V are density and velocity perturbations; w = p
r
=
r
, c
2
s
= p
0
r
=
0
r
and  = 
r
+
c
.
The only place where the seeds enter this system is through the potentials 	 and .
These potentials can be split into a part coming from standard matter and radiation,
and a part due to the seeds, 	 = 	
(c;r)
+	
s
and  = 
(c;r)
+ 
s
, where 	
s
and 
s
the usual Sachs{Wolfe result T=T = (1=3)	(x
rec
; t
rec
) is recovered. Neglecting D
(r)
g
, the result
would be 2	 and therefore wrong by a factor of 6!
3
are determined by the energy momentum tensor of the seeds. In this way, nally
the seed source terms below arise[4].
From Eqs. (3) we derive two second order equations for D
r
and D
c
, namely
D
00
r
+
a
0
a
[1 + 3c
2
s
  6w + F
 1

c
]D
0
r
 
a
0
a

c
F
 1
(1 + w)D
0
c
+4Ga
2
[
r
(3w
2
  8w + 6c
2
s
  1)  2F
 1
w
c
(
r
+ 
c
)
+
c
(9c
2
s
  7w) +
k
2
4Ga
2
c
2
s
]D
r
  4Ga
2

c
(1 + w)D
c
= (1 + w)S ; (4)
D
00
c
+
a
0
a
[1 + (1 + w)F
 1

r
(1 + 3c
2
s
)]D
0
c
 
a
0
a
(1 + 3c
2
s
)F
 1

r
D
0
r
 4Ga
2

c
D
c
  4Ga
2

r
(1 + 3c
2
s
)[1  2(
r
+ 
c
)F
 1
w]D
r
= S ; (5)
where F  k
2
(12Ga
2
)
 1
+ 
r
(1 + w) + 
c
and S denotes a source term, which in
general is given by S = 4Ga
2
( + 3p)
seed
. In our case, where the seed is described
by a global scalar eld , we have S = 8Ga
2
(
0
)
2
. From numerical simulations one
nds that the integral of a
2
j
0
j
2
over a shell of radius k, can be modeled by[9]
a
2
hj
0
j
2
i(k; t) =
1
2
A
2
p
t[1 + (kt) + (kt)
2
]
; (6)
with  denoting the symmetry breaking scale of the phase transition leading to
texture formation. The parameters in (6) are A  3:3,    0:7=(2) and  
0:7=(2)
2
. On super{horizon scales, where the source term is important, this t is
accurate to about 10%. As we argue later, analytical estimates support this nding.
On small scales the accuracy reduces to a factor of 2. By using this t in the
calculation of D
r
and D
c
from Eqs. (4), (5) we eectively neglect the time evolution
of phases of (
0
)
2
; the incoherent evolution of these phases may smear out subsequent
Doppler peaks[11], but will not aect substantially the height of the rst peak.
From D
r
and D
0
r
we calculate the Doppler contribution to the C
`
's according to
C
`
=
2

Z
dk
"
k
2
16
jD
r
(k; t
rec
)j
2
j
2
`
(kt
0
) + (1 + w)
 2
jD
0
r
(k; t
rec
)j
2
(j
0
`
(kt
0
))
2
#
; (7)
where j
`
denotes the spherical Bessel function of order ` and j
0
`
stands for its deriva-
tive with respect to the argument. As we will see below, the angular power spectrum
4
`(` + 1)C
`
yields the Doppler peaks.
In order to solve Eqs. (4), (5) we need to specify initial conditions. For a given
scale k we choose the initial time t
in
such that the perturbation is super{horizon and
the universe is radiation dominated. In this limit the evolution equations reduce to
D
00
r
 
2
t
2
D
r
=
4
3
A
p
t
; (8)
D
00
c
+
3
t
D
0
c
 
3
2t
D
0
r
 
3
2t
2
D
r
=
A
p
t
; (9)
with particular solutions
D
r
=  
16
15
At
3=2
; D
c
=  
4
7
At
3=2
: (10)
In the above equations we have introduced   4G
2
, the only free parameter in
the model. We consider perturbations seeded by the texture eld, and therefore
it is incorrect to add a homogeneous growing mode to the above solutions. With
these initial conditions, Eqs. (4), (5) are easily integrated numerically, leading to
the spectra for D
r
(k; t
rec
) and D
0
r
(k; t
rec
) [see, Fig. 1].
Integrating Eq. (7), we obtain the Doppler contribution to the CMB anisotropies
[see, Fig. 2]. For ` < 1000, we nd three peaks located at ` = 365, ` = 720 and
` = 950. Silk damping, which we have not taken into account here, will decrease the
relative amplitude of the third peak with respect to the second one; however it will
not aect substantially the height of the rst peak.
Our most important results regard the amplitude and position of the rst Doppler
peak, for which we nd
`(` + 1)C
`



`360
= 5
2
: (11)
It is interesting to notice that the position of the rst peak is displaced by `  150
towards smaller angular scales than in inationary models [6]. This might be due
to the dierence in the growth of super{horizon perturbations, which is D
r
/ t
3=2
in our case, and D
r
/ t
2
for inationary models.
5
Figure 1: The dimensionless power spectra, k
3
jD
r
j
2
(solid line) and kjD
0
r
j
2
(dashed line)
in units of (A)
2
, are shown as functions of k. These are exactly the quantities which enter
in the expression for the C
`
's. We set h = 0:5 ; 

B
= 0:05 and z
rec
= 1100.
One may understand the height of the rst peak from the following analytic
estimate: matching the sub{horizon with the super{horizon solutions of Eq. (5), in
the matter dominated era, one nds
D
c
  
2
5
A(k=2)
1=2
t
2
:
From Eq. (4) we then obtain in this limit D
r
 Ak
 3=2
. Plugging this latter value
into Eq. (7) we get roughly
`(` + 1)C
`
 (A)
2
;
for the height of the rst peak.
Let us now compare our value for the Doppler peak with the level of the Sachs{
Wolfe plateau [7, 8, 9],
`(` + 1)C
`



SW
 (6  14)
2
: (12)
It is apparent from Eqs. (11) and (12) that the Doppler contribution from textures
is substantially smaller than for inationary models.
6
Figure 2: The angular power spectrum for the Doppler contribution to the CMB
anisotropies is shown in units of 
2
. We set cosmological parameters h = 0:5 ; 

B
= 0:05
and z
rec
= 1100.
Normalizing with COBE{DMR experiment[12]
`(` + 1)C
`



COBE
 8 10
 10
;
one nds the symmetry breaking energy scale, and from this   10
 5
. This value
for  depends of course on the numerical simulations[7, 8, 9].
We believe that our result, stating that the rst Doppler peak has a height
comparable to the Sachs{Wolfe plateau, is valid for all global defects. This depends
crucially on the 1=
p
t behavior of (a
0
)
2
=
_

2
on large scales (cf. Eq. (6)), which is
a generic feature of global defects: on super{horizon scales,
_

2
(k) represents white
noise superimposed on the average given by
_

2
(k = 0) /
p
V =t
2
. Since there are
N = (L=t)
3
independent patches in a simulation of linear size L, the amplitude of
_

2
(k) is proportional to L
3=2
=(t
2
p
N) / 1=
p
t. (Notice that this argument does not
apply for local cosmic strings.)
Based on our analysis we conclude that if the existence of Doppler peaks is indeed
conrmed and the height of the rst peak is larger than about twice the level of the
7
Sachs{Wolfe plateau, namely
`(` + 1)C
`



peak
> 2 10
 9
;
and if the rst peak is positioned at ` < 300, then global topological defects are
ruled out. On the other hand, if the rst Doppler peak is positioned at `  350 and
its height is below the above value, global defects are strongly favored if compared
to inationary models. To our knowledge this is the rst clear ngerprint within
present observational capabilities, to distinguish among these two competing models
of structure formation.
As we were completing our work, a preprint[15] on the same issue, but follow-
ing a dierent approach, came to our attention. The authors calculate the Doppler
peaks from cosmic textures in the synchronous gauge. A main assumption of that
analysis, which is not shared by us, is that spatial gradients in the scalar eld are
frozen outside the horizon, and therefore time derivatives are negligible. Even though
we basically agree with the shape and position of their Doppler peaks, we draw a
stronger conclusion regarding the dierence between texture and inationary power
spectra, since we also calculate the height of the Doppler peaks, for which these
authors do not present any estimates.
Acknowledgement
We thank Leandros Perivolaropoulos, who participated in the beginning of this
project and Mark Hindmarsh, for helpful discussions and in particular for his skills
with MATLAB. One of us (R.D.) acknowledges stimulating discussions with Neil
Turok. A.G. thanks Nuno Antunes and Dennis Sciama for encouragement, the In-
stitut fur Theoretische Physik, Zurich for hospitality and The British Council for
partial nancial support. This work was partially supported by the Swiss NSF.
8
References
[1] P.J. Steinhard, Class. Quantum Grav. 10, S33, (1993).
[2] T.W.B. Kibble, Phys. Rep. 67, 183 (1980).
[3] R.K. Sachs and A.M. Wolfe, Astrophys. J. 147, 73 (1967).
[4] R. Durrer, Fund. of Cosmic Physics 15, 209 (1994).
[5] H. Kodama and M. Sasaki, Prog. Theor. Phys. Suppl. 78, 1 (1984).
[6] P.J. Steinhard, Cosmology at the Crossroads, preprint (1995).
[7] D. Bennett and S.H. Rhie, Astrophys. J. 406, L7 (1993).
[8] U.{L. Pen, D.N. Spergel and N. Turok, Phys. Rev. D49, 692 (1994).
[9] R. Durrer and Z.H. Zhou, in preparation.
[10] N. Turok, Phys. Rev. Lett. 63, 2625 (1989).
[11] A. Albrecht, D. Coulson, P. Ferreira and J. Magueijo, Imperial Preprint
/TP/94-95/30, astro-ph/9505030.
[12] G.F. Smoot et al., Ap. J. 396, L1 (1992); K.M. Gorski et al., Astrophys. J.
430, L85 (1994).
[13] D.P. Bennett, A. Stebbins and F. Bouchet, Astrophys. J. 399, L5 (1992).
[14] D. Scott, J. Silk and M. White, Science 268, 829 (1995).
[15] R.G. Crittenden and N. Turok, Princeton University Preprint, PUPT-94-1545,
astro-ph/9505120.
9


Doppler peaks: a fingerprint of topological defects

Abstract

Similar works

Full text

Available Versions

CERN Document Server