We study a model where the domain walls are generated through a cosmological phase transition involving a scalar field. We assume the existence of a coupling between the scalar field and dark matter and show that the interaction between domain walls and dark matter leads to an energy dependent reflection mechanism. For a simple Yakawa coupling, we find that the vacuum expectation value of the scalar field is theta approx. equals 30GeV - 1TeV, in order for the model to be successful in the formation of large scale 'pancake' structures

Chakravorty, Alak

Massarotti, Alessandro

English

NASA Technical Reports Server

Fermi National Accelerator Laboratory
7f/y
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FERMILAB-Pub-91/306-A
November 1991
Could the electroweak scale be linked to the large scale
structure of the Universe?
Alak Chakravorty l and
Alessandro Massarotti li2
Department of Physics
The University of Chicago l
Chicago, IL 60637 and
NASA / Fermilab Astrophysics Center
Fermi National Accelerator Laboratory 2
Batavia, IL 60510
Abstract
We study a model where domain walls are generated through a cosmological phase
transition involving a scalar field. We assume the existence of a coupling between the
scalar field and dark matter and show that the interaction between domain walls
and dark matter leads to an energy dependent reflection mechanism. For a simple
Yukawa coupling, we find that the vacuum expectation value of the scalar field is
(o) ~ SOG'eV — iTeV, in order for the model to be successful in the formation of very
large scale "pancake" structures.
r
 (NASA-CR-190122) COULD THE ELECTROWEAK N92-20490
SCALE 8E LINKED TO THE LARGE SCALE STRUCTURE
OF THE UNIVERSE? (Fermi National
Accelerator Lab.) 7 p CSCL 038 Unclas
G3/90 0078142
V
Operated by Universities Research Association Inc. under contract with the United States Department of Energy
https://ntrs.nasa.gov/search.jsp?R=19920011248 2020-03-17T13:17:59+00:00Z
The existence of a very large scale structure of the Universe is one of the most
interesting problems of modern Cosmology. The recent data of the so called "pencil
beam" surveys.1 scanning small portions of the sky up to redshifts close to unity,
strongly suggests that clustering extends up to scales close to 100/i~1Mpc. These
data prompted new interest in scenarios for the formation of large scale structure
alternative to the standard cold dark matter perturbation model.
Domain walls, generated through a second order cosmological phase transition,
could be an avenue of research in this field, since they provide an alternative model
for structure formation. In the first domain wall models, these topological defects were
assumed to couple to the matter solely through gravitation.2"3 As a consequence of
the weakness of the interaction, domain walls would stretch undisturbed under their
surface tension and rapidly reach relativistic speeds, with the resulting network scale
close to tha.t of the horizon.'1"0 Recent work by one of the authors6"' has indicated that
the assumption of a non-gravitational coupling between the walls and a cosmologically
significant neutral component of dark matter can drastically change the features of
the model. Briefly summarizing, it was shown that if such a coupling leads to particle
reflection, then the domain walls can be slowed down very efficiently. At the same
time, the walls can sweep large quantities of matter in their motion and, at a late
stage of their evolution, give rise to wakes of 1 — !Qh~lMpc thickness. The model
also predicts comoving "interwake" distances of the order of 10 — !QQh~lMpc. It was
found that fermioris of mass m ~ 1 — lOeV would be an ideal candidate for the dark
matter interacting with the walls.
In this Letter, we will study the particle-kink interaction that derives from intro-
ducing a. simple model coupling between the field of the walls <f> and that of the dark
matter '</;:
(1)
. 4
The scalar potential is chosen so as to give rise to a symmetry breaking, generating
the domain walls and at least part of the fermion mass. Notice that any potential
generating domain walls (e.g. a sine-Gordon potential) is well approximated by this
"6'ln scalar Lagrangian. Our choice of the coupling is by no means exhaustive, but
it gives us an opportunity to introduce the formalism apprpriate to the study of the
more general wall-particle scattering problem. We will not attempt to investigate the
fundamental theory from which the £e// could originate.
r
1
We take the classical kink soliton solution for the scalar field.8 9 The field 4>
has vacuum expectation values (c6) = +d>0 or — d>0 and has the usual form <j>c(z) =
00 tanh(z/A), where A gives the thickness of the domain wall. In order to consider
the behavior of the fermions as they pass through domain wall boundary we look at
the stationary solutions to the Dirac equation, coming from the Lagrangian above,12
(5 • p -\- gj3f (&c(z)))i(? = -£/?/>, (2)
( I \a I, i^a , ibb being two component spinors. We can write the Dirac/
equation (in the Dirac representation) as 10
E
-a
-
(3)
Solving for •</'& in terms of •(/-•„ we get:
E
-a • V
- a - V
gf(oc(z))
(4)
which is equivalent to
•06
-V^a +
(5)
E-gffa)
As one might expect there is no spin dependence and we can write the expression
in one dimensional form:
2 df(oc) diba
and a. similar expression for '0(,.
For particular choices of /(CD) one can simplify eq.(6). In particular, we studied
the simple Yukawa case f(6) = <j>. The interaction term can be interpreted as a
Majorana neutrino mass term, where we can use weak singlets to define i/>. If one
th inks of it as a Dirac neutrino mass, this term is clearly anaesthetic, since it breaks
explicitly the S't/2weafc symmetry of the interaction Lagrangian. Its presence should
then be explained in the context of a larger theory in order to be realistic. Let us now
consider the relative magnitude of the two terms on the l.h.s. of eqn.(6). Far away
from the kink d<j)c/dz —> 0, so this is just the Klein-Gordon equation for fermions of
mass 771, as we would expect. In the vicinity of the wall we can define a "momentum"
pefj = J E2 — m l f A z ) , where meff = y<l>c(z}. The second term in eq.(6) is,
where max[gS0/(E + g<f>c)} = 0(1). The first term in eq.(6) is, on the other end,
of the order of P l r r - The problem simplifies it, during the interaction with the kink,
we ca,n ignore the second term a,s compared to the first, i.e. if p2,^ 3> p ef j / \~ l , or
Peff ^ ^-1- We <^an achieve this by imposing the condition A ^> m"1, since within
the kink pejj ~ mL, — gcj)0 >> A"1. This turns out to be the condition we would
like for the reflection of the fermions to occur without pair creation. In fact, the
probability of this happening is approximately given by P ~ (m/A)2 exp(— rrmA),
which is clearly suppressed by our condition on the barrier thickness. u Ignoring then
the gradient term, we can write eq.(6) as:
or. with m = g<j)0, we have
>
B = 0 ( 8 )2m2 dz2 V 2
This is a Schroedinger-like equation for a pa,rticle in a V(z) — 1/2 tanh2(.r/A)
"potential", with "mass" m = m2 and energy E = £2/2m2. This is a potential
barrier with E > V0. The reflection coefficient can be calculated exactly for this
process and is given by14
sinh (7T.EA) +
Examining this reflection coefficient for different values of the fermion velocity, we
find that clearly R(v) —> 1 as v —> 0 and R(v) —> 0 as v —> 1. In the intermediate
range, i.e. for m^A ^> v~2 but v <C 1, we get
3
exp(-7?v6-2A) + 1
This exponential behavior of the reflection coefficient in this regime was actu-
ally well approximated by a step function in ref.(7), in which we just supposed the
existence of a threshold energy t0. Apart from the detailed knowledge of the reflec-
tion coefficient, eq.(10) tells us that the threshold energy is simply correlated to the
thickness of the walls, since e0 ~ A"1.
A similar result could be obtained in an axion-like model, like the one explored in
ref.(12). There, the fermion couplings with the axion field were written as m/e'75*^0,
and the resulting -Dirac equation contains a spin dependent fermion mass term m* =
nif + Szdz<j)c. The domain walls are a barrier or a well, of potential 6m* ~ A"1, for
opposite helicity states.
For all the cases in which the threshold energy e0 is correlated to the wall thickness
a.s 60 ~ A"1, we can use the results of ref.(7) to determine the constants in the domain
wall potential of eq.(l). In the model elaborated in ref.(7), "light" domain walls form
some time well before recombination. The dynamics of the domain wall network is
strongly influenced by the interaction of walls with the neutrino background. By
being partly reflected from the walls, the fermions gives rise to a friction pressure
PJ. which is a function of the wall speed v and of the temperature of the fermion
gas. In the first stages of the wall evolution the friction turns out to be irrelevant.
At a later stage, when the friction becomes dominant, the domain walls slow down
and the comoving scale of the network at that epoch (which is roughly the scale
of the horizon) remains frozen in. The comoving network scale at the freeze-in is
typically f ~ 10 — WQh~ lA4pc and the matter wakes are 1 — lOh~1Mpc thick, if
e0 ~ 10~4 - 10-5e\/. The freeze in takes place at a redshift z ~ 103 - 104.
From eq.(10) we can infer that A ~ 104 — lQ°eV~ l. Since other dynamical consid-
erations pin down the range for the wall surface density to a ~ 10"1 — IMeV3, using
the values of a and A we can determine all of the parameters in the Lagrangian.
From the relations a ~ A©3, and A"1 ~ X60 we conclude that A ~ 10~15 — 10"1',
rh0 ~ 10Ko — 103G'eV/. This statement would be correct in all models of interaction
where one obtains e0 ~ A"1.
If one does not dismiss this concordance as a coincidence, the result may suggest
a possible connection between the field 0. originating the topological defects, and the
scalar (Higgs) of the Weinberg-Salam model, which gives rise to the electroweak sym-
metry breaking (stable domain walls cannot arise from the doublet itself). Because of
the small value of the A coupling constant, the phase transition that gives rise to the
domain walls actually takes place at much lower temperature than the electroweak
scale. We can simply evaluate what is called the Ginzburg temperature by recalling
that Tg ~ X2(f>o'. we obtain Tg ~ \03'5eV, which is the temperature at which stable
domain walls actually form and begin stretching under their surface tension.
The smallness of the A constant is typical of "light" domain wall models. Its
value constitutes one of the outstanding problems to solve, since it is not easy to
a,ccomodate the cosmological constraints using the simplest "realistic" particle physics
models available. More research is needed in this direction.
Special thanks to E. Kolb. D. Schramm, J. Harvey, J. Rosner, A. Trivedi, L.
Amendola and A. Capozziello for useful comments. Work supported in part by DOE
and NSF grants and by NASA grant NAGW2381at NASA/Fermilab.
References
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2. Y.B. Zel'dovich et al., JETP 40, 1, 1 (1974).
3. C. Hill et al., Com. Nuc. Par.Phys. 19, 25 (1989).
4. W.H. Press et al., Ap.J. 347, 590 (1989).
5. L. Kawano, Phys.Rev. D, 41, 1013 (1990).
6. A. Massarotti, Phys. Rev. D 43, 346, Jan. 1991.
7. A. Massarotti, Fermi lab preprint 91/104-A, submitted.
8. A. Vilenkin, Phys. Rep. 121, 5, 263 (1985).
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Could the electroweak scale be linked to the large scale structure of the Universe?

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