The production of topological defects, especially cosmic strings, in extended inflation models was considered. In extended inflation, the Universe passes through a first-order phase transition via bubble percolation, which naturally allows defects to form at the end of inflation. The correlation length, which determines the number density of the defects, is related to the mean size of bubbles when they collide. This mechanism allows a natural combination of inflation and large scale structure via cosmic strings

Copeland, Edmund J.

Kolb, Edward W.

Liddle, Andrew R.

English

NASA Technical Reports Server

tFermi National Accelerator Laboratory
SUSSEX-AST 90/4-1
FNAL-PUB-90/56-A
(April,1990)
Topological Defects in Extended Inflation
Edmund J. Copeland, x Edward W. Kolb, 2 and Andrew R. Liddle s ._? _3_._-d_)
1Department of Physics / / '_"
University of Sussez
Brighton BNI gQtt U.K.
_NASA/Fermilab Astrophysics Center
Fermi National Accelerator Laboratory
Batavia, Illinois 60510-0500 U.S.A.
and
Department of Astronomy and Astrophysics
Enrico Fermi Institute
The University of Chicago
Chicago, Illinois 60637 U.S.A.
3 Astronomy Centre
Department of Physics
University of Sussez
Brighton BN1 9QIt U.K.
Abstract
We considerthe production oftopologicaldefects,especiallycosmic strings,
in extended inflation models. In extended inflation, the Universe passes
through a tlrst-order phase transition via bubble percolation, which natu-
rally allows defects to form at the end of inflation. The correlation length,
which determines the number density of the defects, is related to the mean
size of bubbles when they collide. This mechanism allows a natural com-
bination of inflation and large-scale structure via cosmic strings.
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There has been a recent renewed interest in the possibility of inflation via a first-
order phase transition in the early Universe, in the manner of Guth's "old" inflation
scenario. 1 In Guth's model, the Universe was trapped in a false-vacuum state and
the energy density of the Universe became dominated by the vacuum energy density.
This resulted in an exponential increase in the scale factor of the Universe, solving
several cosmological puzzles. The phase transition proceeded by quantum tunnelling
to the true vacuum state, 2 which lead to the nucleation of bubbles of true vacuum.
Having appeared with a characteristic size determined by microphysics (assuming that
gravitational effects upon bubble nucleation are not too large), s the bubbles then grew
at the speed of light. Unfortunately, because the regions of the Universe in the false-
vacuum state were expanding exponentially, the expansion was more effective than
the decay to the true vacuum state, bubbles of true vacuum never percolated, and
the energy density in the expanding walls of the true vacuum bubbles could not be
converted to radiation through bubble wall collisions. This problem became known
as the "graceful exit" problem of old inflation. 4
La and Steinhardt 5 have recently revived the idea of a first-order phase transition
in the context of theories in which the gravitational constant may vary, such as
Jordan-Brans-Dicke theories, and term their model "extended" inflation. In such
theories the expansion of the scale factor of the Universe is a power law rather than
exponential during inflation (for example, in a Brans-Dicke theory with parameter
w, the scale factor increasesas t_+t/2 at late times). In such cases the exponential
bubble nucleation rate willeventually overcome the power-law expansion rate, and
the Universe willexit from the inflationaryera. Such a model may relievemany of
the fine-tuningproblems associated with new and chaotic infiation.
As emphasised by Wcinberg s and by La, Steinhardt and Bertschinger,z the exit
from extended inflationbased on the Jordan-Brans-Dicke theory isnot without prob-
lems due to the existence of large vacuum bubbles at the end of inflation (the spectrum
for bubble formation is nearly scale independent). Large bubbles will not have time to
thermalize before recombination, and hence will lead to distortions in the microwave
background; arranging for these to be sufficiently small requires the Brans-Dicke pa-
rameter w certainly to be less than 50 and possibly much smaller, which is in conflict
with the lower bound from time-delay experiments of _o > 500.
To resolve this conflict, several more complicated (but perhaps more realistic)
models have been proposed, for instance allowing ¢0 to vary with time, s or by in-
corporating the scenario into either an induced-gravity model 9 or a Kaluza-Klein
model, 1° or by modifying the gravitational couplings of the inflaton, xl All these mod-
els suppress large bubbles (those formed early in the inflationary epoch) and hence
allow sufficient thermalization of the bubbles before decoupling to evade microwave
background constraints. It has been suggested that the bubbly nature of space after
extended inflation may be reflected in observations of frothy large-scale structure, z
However there might very well be a different mechanism for the formation of
structure after extended inflation, namely the formation of topological defects in
the inflaton field formed as it passes through the phase transition. Calculations of
the false-vacuum decay rate made so far consider the evolution from a false-vacuum
state to a unique true-vacuum state. However, the inflaton is far more likely to
have degenerate minima, especially if it is part of a grand-unifled Higgs sector. For
example, if the inflaton is a real scalar field _b, then it would be common for it to have
two discrete minima at _b = +_0 and _ = -_b0, in which case there is the possibility
of forming domain walls. More interestingly, if the scalar field is complex, with a
potential of the form such that the manifold of possible ground states form a circle of
degenerate minima, e.g. V(_b) = _([_l 2 - _0_)2, there will be the possibility of cosmic
string solutions. Yet more complex models would allow monopole solutions, or hybrid
creatures such as monopoles connected by strings or walls bounded by strings. The
types of defects formed in the phase transition of any given model will depend on the
properties of the manifold of degenerate vacuum states.
Recall the standard picture of cosmic string formation in a smooth second-order
phase transition. 12 At early times the universe was very hot and the fields describing
interactions were in a highly symmetric phase. However as the universe expanded and
cooled, symmetry breaking processes spontaneously occurred, which may have left
behind remnants of the old symmetric phase, possibly in the form of strings, domain
walls or monopoles. Here, we concentrate on strings, which occur when the potential
has a circle of degenerate minima parametrized by a phase angle 8. As the system
cooled below the critical temperature, (To _" _0), the _ field began to fall to the
minima of its potential. Due to thermal fluctuations of the _ field, domains formed
of size _2 _" (v_0) -1 (the length over which the _ field is spatially correlated).
In separate domains _ pointed in arbitary directions in the vacuum manifold, but
matched smoothly at the boundary, with 8 varying so as to cause defects to form on
the edges common to certain domains. This is easily seen. Consider one such edge
where 8 varies by 27r in encircling the edge, i.e., all around the edge we continuously
encircle the minimum of the potential. Such regions line up to minimize the spatial
gradient energy, forming a defect line or cosmic string. This corresponds to a thin
tube of false-vacuum energy, and these strings are either in the form of closed loops
or are infinitely long.
Strings appropriate to galaxy formation are required to have a llne density of
C# _,, 10 -s, where # _ _02, corresponding to a breaking scale of 10 -s Planck masses.
Unfortunately, generic new and chaotic inflationary scenarios occur at or below this
energy scale, and hence the strings form before or early in the inflationary epoch and
are rapidly inflated away. It has been demonstrated that the universe cannot be made
to reheat after inflation to sumciently high temperatures as to restore the symmetry of
the string-forming field and allow a new phase of string formation after inflation. 13,14
This leads to the incompatibility of cosmic strings with new or chaotic inflation. These
arguments apply whether the inflaton and the cosmic string fields are the same field
or different ones (in chaotic inflation the inflaton field can never be identified with
the cosmic string field as the symmetry is broken even initially). In the case where
the cosmic string field is distinct from the inflaton field, models have been proposed
which resolve the conflict. The model of Vishniac, Olive, and Secke114 couples the
inflaton and the string field in a particular way, but the only motivation for doing this
is to solve the strings-inflation problem, so their solution appears unnatural. More
recently, Yokoyama is has suggested that a non-minlmal coupling to gravity of the
string field can hold it in its symmetric phase during inflation, and allow strings to
form at the end of inflation.
Now consider the picture of string formation in extended inflation, where the fact
that the transition is first order has crucial consequences. As the Universe cools
from high temperatures, a complex scalar field is trapped in a false-vacuum state and
the Universe enters a phase of rapid power-law expansion. Bubbles of true vacuum
then begin to nucleate and grow at the speed of light. Due to the presence of event
horizons in the inflating Universe they grow to a constant comoving volume which
depends on their time of formation. The important ingredient to our scenario is that
each bubble forms independently of the rest, and so there is no correlation between
the choice of true vacuum made in each bubble from the selection of degenerate true
vacua. Eventually the bubbles grow and collide, finally percolating the Universe and
bringing the inflationary era to an end.
At the end of inflation, the collision of bubble walls (in which all the energy is
held) produces particles and causes thermalization of the energy. However, because
4
the scalar field is only correlated on the scale of a bubble, we can expect topological
defects to be present. The usual arguments state that there is typically of order one
cosmic string per correlation volume of the scalar field, and hence we expect roughly
one string per mean bubble size at the end of inflation.
All attempts to solve the big-bubble problem of the original extended inflation
model have the common feature that large bubbles (bubbles formed early in inflation)
are suppressed. Therefore we expect that in any successful extended inflation model
large bubbles should be very rare, and there should be a rapid "turn on" of bubble
nucleation at the end of inflation. Therefore to a first approximation we will assume
that at the end of inflation nearly all of the volume of the Universe is taken up by
bubbles nucleated near the end of inflation, with only very few rare large bubbles. The
distribution of large bubbles clearly depends upon the model for extended inflation,
while the fact that most of the volume of the Universe must be in bubbles nucleated
near the end of inflation should be true in any successful extended inflation model.
It is possible to estimate the effective correlation length for the string network
formed at the end of extended inflation. In the thin-wall picture a bubble is nucle-
ated with physical size Re = 3(AV) -1 f d_ _-_, where the integral is evaluated
between the true and false vacuum values of _, and AV is the potential difference
between the false and true vacua driving inflation? For typical polynomial potentials,
AV ,,_ eA¢_, where A is an overall coupling constant for the potential, _0 sets the scale
for the vacuum expectation value, and e describes the difference in potential energy
between the false and true vacua. To get a lower limit on the correlation length, we
assume that there has been little growth in the coordinate radius of the mean bubble
from the time it was nucleated, _ until the end of inflation, _cnd (bubbles nucleated
toward the end of inflation have little growth in their coordinate radius). Then the
physical size of the mean bubble at the end of inflation (i.e., the effective correlation
Tlength) is _,_r "" Re[aCt,,.d)laCf)].
This leads to a correlation length greater than that typical of the Kibble mech-
anism where cosmic strings form via a second-order thermal phase transition, and
hence a more dilute network will be formed in our model. For example, consider
a potential of the form V(¢) = A(¢ 2 -¢o2) 2 - _Aes_b0. In the limit of small e,
Re _ v/'_¢g/AV ,., (v_e¢0) -1. Therefore _,e _ (v/Xe¢o)-l[a(t,_d)/a(t)]. As
previously mentioned, a second-order phase transition in such a model leads to a
correlation length of _2 "" (x/_¢0)-1, so the comparison of the effective correlation
length in our model to the usual correlation length in a second-order transition is
_,fr/_2 = e-X[a(t,_i)/a(E)]. Note that e << 1 is required for sufficient inflation. On the
other hand, we expect that in successful extended inflation scenarios [a(t,,,a)/a(E)] is
not too much larger than unity.
This model for the formation of strings allows for the existence of large voids,
which would be a consequence of the rare large bubbles. Although the typical string
separation at the end of inflation is _e_, extended inflation allows for the possibility
of rare large bubbles, formed by quantum tunnelling early in inflation. The true
vacuum formed inside bubbles contains no matter (any matter originally in that
volume is assumed to be inflated away while the scalar field dominates the energy
density). All the energy of the Universe after inflation is contained in the walls of
the expanding bubbles which collide to form matter and to cause thermalization of
the energy density. After collisions, matter will flow back into the void, though as
it cannot travel faster than light, we can calculate the minimum time the bubble
will require to thermalize. A large bubble will have a coherent scalar field vacuum
and hence no strings will be formed within it--we can thus expect the interior of the
bubble to evolve into a large region void of strings. If cosmic strings are to provide
the seeds for galaxy formation, then we can expect to see few or no galaxies within
6
the void. The presence of voids is an additional property of this model which may
help explain observed large-scale structure.
In fact, at the time of percolation the bubbles may have a range of sizes, which can
lead to the formation of an initial string network differing from the usual one. As the
correlation length is essentially.just the bubble size, and because there would appear to
be no a priori reason why bubbles everywhere should be ezactly the same size (at small
sizes the assumption of a scale-invariant bubble size distribution would seem more
reasonable), the strings will be formed with a randomly spatially varying correlation
length. This will presumably lead to higher densities of strings in some regions than
others, which again may have implications for structure formation, depending on how
much the effects of the initial string distribution might be wiped out by the future
evolution and decay of strings. One desirable effect of a more dilute string network
would be to avoid the uncomfortable bounds from gravitational wave production, from
small string loops. TM The fact that the correlation length will generically be greater
(and in some models perhaps much greater) than that of the Kibble mechanism may
also have important implications, though perhaps not as great as one might naively
suppose if the small strings rapidly disappear from the network once string evolution
commences.
These formation arguments can be equally well applied to the cases of domain
walls and monopoles, again giving rise to an estimate of order one defect per bubble
at the time of bubble collision. In the case of domain walls this will give rise to
an excessive number, and will be disallowed on cosmological grounds. Hence, any
extended inflation model featuring a potential with domain wall solutions (i.e., a
disconnected vacuum manifold) can be ruled out. The situation is less clear for
monopoles, because the correlation length may well be substantially greater than
that of the Kibble mechanism and hence proportionally fewer monopoles are expected.
However, standard estimates of the cosmological monopole abundance 17 give values of
perhaps twenty orders of magnitude in excess of the Parker limit, is so the correlation
length would have to be increased by seven or eight orders of magnitude before being
within experimental limits--such an increase seems very unlikely.
If we consider the unification to be part of a grand-unified theory, the problem of
monopole overproduction must be addressed, s as any breaking to the symmetry of
the standard model must produce monopoles at some stage. The simplest method is
to arrange for monopoles to be formed in a partial symmetry breaking and then later
inflated away in a second transition. La, Steinhardt and Bertschinger 7 considered
this solution though their emphasis was on monopoles produced in a field other then
the inflaton.
We also point out that if the string network produced in extended inflation is
relevant for formation of large-scale structure, the llne density must be about 10 -e
in Planck units, corresponding to a symmetry breaking scale of 1016GeV. Thus ob-
servational limits on the string line density translate into bounds on the mass of the
inflaton.
In conclusion, we have shown that extended inflation and cosmic strings are quite
compatible. The cosmic string network formed in an extended inflation phase transi-
tion will in many ways be identical to the network formed in non-inflationary phase
transitions. Possible exceptions to this statement are !) a larger effective correlation
length, which will lower the density of small string loops, lessening the constraints
from gravitational wave production, and 2) perhaps rare large volumes void of strings,
which may be of importance in formation of large-scale structure. Extended inflation
does not solve the monopole or domain wall problem, at least in regard to monopoles
and domain walls produced by the inflaton.
7
Acknowledgements
EJC and ARL were supported by the SERC. EWK was an SERC-supported Visiting
Research Fellow at the University of Sussex. This work was supported by the DoE and
NASA (grant #NAGW-1340) at Fermilab. EWK would like to thank John Barrow
for his hospitality at the University of Sussex where this work was done.
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Topological defects in extended inflation

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