Effective Field Theory in the
Inflation, Axions and Baryogenesis
(Collaborations with A. Anisimov, T. Banks, M. Berkooz, M. Graesser,
Berkeley Oct. 2004
In very early universe cosmology, one is exploring
physics at scales well above those which have been
probed experimentally; the relevant laws of nature
are at best conjectural.
How to proceed? What can we hope to learn?
Might hope to guess the correct theory. More likely,
success will come from looking at classes of theories
(Supersymmetric? Large dimensions? Axions?)
Hopefully with connections to experiment.
Crucial tool: effective action, appropriate to the
relevant scale of energies, temperature, curvature
Problems to which we might apply these methods:
•Inflation (superluminal expansion, fluctuations)
In each of these cases, early analyses took some
renormalizable field theory, and analyzed as if space-time
were flat. More realistic analysis often yields a
qualitatively different picture. Different possible
phenomena; sometimes problems arise, sometimes
problems are resolved.
Today we will consider:
1. Axions as Dark Matter
2. Moduli in cosmology
We will do axions in the greatest
Axions and the Strong CP
Strong CP Problem:
QCD is a very successful theory. But aside from the coupling
constant, it has an additional parameter:
In QED, such a term would have no effect, but in QCD,
can show generates CP violating phenomena, e.g.
Why is q so small?
Axion solution (Peccei, Quinn, Wilczek, Weinberg):
Postulate field, a(x), symmetry:
QCD generates a potential for a(x). If q eff =0, QCD preserves
CP, so ignoring weak interactions, minimum of axion
potential will lie at q eff=0. Can calculate the axion potential
A light particle:
Constraints from particle searches, red giants:
Cosmology can potentially constrain as well.
The axion hypothesis, at first sight, is troubling:
Postulate a symmetry, which is not broken by anything
but small effects in QCD.
The tiniest effects due to unknown interactions at some
high energy scale would lead to too large a value of q.
In the language of effective actions, one needs to
suppress possible operators of very high dimension
which might break the symmetry.
String theory: produces exactly such axions. Typically
f a » Mp
but could be smaller. (This is one of the attractive
features of string theory).
Conventional Axion Cosmology
In FRW space-time
Since the mass is so small, at early times overdamped.
At late times, behaves like a coherent state of zero
momentum particles, diluted by the expansion:
Assuming that the initial angle, q = a/f a is of order one, the
axion initially has energy density
The relative proportion of axions and radiation
grows as 1/T; one requires that the axions not
dominate the energy density before recombination
time. This limits f_a.
The axion mass is actually a strong function of temperature.
It turns on quickly somewhat above the QCD phase transition
One obtains a limit:
fa < 3 £ 1011 GeV
Assumptions which go into the axion limit:
•Peccei-Quinn phase transition: the PQ
symmetry is a spontaneously broken
symmetry; axion is the corresponding
Goldstone boson. Usually assumed that there
is a transition between an unbroken and a
broken symmetry phase. Somewhat different
limits if before inflation (above); after
(production of cosmological defects)
•Universe in thermal equilibrium from, say,
period of inflation until decoupling.
•No other light particles which, like the
axion, might store energy.
These assumptions are all open to question. E.g.
consider supersymmetry. This is a hypothetical
symmetry between fermions and bosons. A broken
symmetry; scale might well be 100’s of GeV (I.e.
masses of scalar partners of electrons, quarks,
photon… of order 100’s of GeV.) Many physicists
believe that this explains why the weak scale is so
much smaller than the Planck scale (m w ¿ Mp). If
correct, will soon be discovered at accelerators.
If nature is supersymmetric, the axion is related by
the symmetry to other fields: a fermion and another
scalar field. This scalar field is often called the
If the saxion exists, it poses much more serious
cosmological problems than the axion.
The Moduli Problem
The saxion is an example of a modulus – a scalar field with
a nearly vanishing potential. Such fields are common in
string theory; their potentials are expected to vanish in the
limit that supersymmetry is unbroken.
m3/2 is gravitino mass; size of susy breaking
In the early universe, the equation for f is:
The system is overdamped for H>m3/2. At later times it
evolves like dust. Assuming that f starts a distance Mp
from its minimum, when it starts to oscillate, f stores
energy comparable to H2 Mp2, so quickly dominates.
Catsrophic for Nucleosynthesis!
What sorts of solutions are proposed for this puzzle:
•No supersymmetry, no moduli
•Moduli heavier than m3/2 (1 TeV?); say 100 TeV
In this case, universe reheats when the moduli decay to
about 10 MeV. Nucleosynthesis restarts. It is
necessary to create the baryons at this time.
Implications for axions:
1. If moduli decays reheat universe to 10 MeV, they dilute
the axions. This relaxes substantially the constraint on
f a. In these circumstances, saxions decay early and are
not a problem.
2. Might not have Planck-scale moduli, but in an axion
model, must at least have saxion. Saxion is not a
problem if decays early enough (small enough f a); then
no axion problem.
Axions start to oscillate when H » ma. At this time,
ra = ma2 f a2 , so
Since both moduli and axions dilute like dust, this
ratio is preserved until moduli decay. T=10 MeV,
radiation domination. Need radiation domination to
persist until recombination. This gives
f a < 1015.5 GeV
This is a much weaker limit than the conventional one. It
relies on speculative but plausible physics. Even the
conventional axion implicitly assumes some new
dynamics at a scale well below the unification of Planck
Suppose that there are no moduli, other than the saxion.
In this case, the limits may be relaxed as well. Consider,
first, a model for axions with a range of f a:
Saxions: perhaps lower decay constants. One can construct
field theory models for this.
where q carries color and perhaps weak isospin and
hypercharge; c is a constant with dimensions of mass cubed, m is
some large integer.
The S vev is of order:
Now we need to consider the saxion cosmology. If we take
this potential as the potential relevant to the early universe,
then again the saxion starts to oscillate once H = m3/2. The
saxion lifetime might be expected to be:
There are a number of possibilities. The saxion may
decay before it dominates the energy density. If not, it
will dilute the axions at least to some extent. Again,
the constraints on the axion decay constant are
Not only are the conventional limits relaxed, but the
picture of the PQ phase transition may also be altered.
There need be no phase transition at all!
In our model, the PQ symmetry is an accident, resulting
from discrete symmetries. The symmetry is explicitly
broken by very high dimension operators. But if the
inflaton transforms under the discrete symmetries, the
symmetry can be explicitly – and badly – broken during
inflation. As a result, there need be no PQ phase
transition. This leads to relaxation of constraints from
isocurvature fluctuations. It can eliminate production of
General lesson: considerations of the form of the effective
field theory reveal a range of phenomena beyond those seen
in the lowest dimension, renormalizable terms.
Another application: inflation.
There are presently many models. At the moment, models
involving colliding branes are attracting great attention.
But it was long ago noted by Guth and Randall that
supersymmetric models provide a particularly natural
framework for ``hybrid inflation.”
Their proposal, however, was harshly criticized. It was
argued to lead to excessive production of defects (cosmic
strings, domain walls); the end of inflation was said to be
complicated, with a problematic fluctuation spectrum.
Both criticisms result from an overly restrictive view of the
effective action. (In fairness to the critics, these were
features of the original analysis).
Basic idea: inflation at the weak (supersymmetry
breaking) scale. Two fields, the ``waterfall field” (c) and
the inflaton (f).
(Berkooz, Volansky, M.D.): c naturally a modulus. To solve
moduli problem, need to be heavy – 100 TeV. This leads to
sufficient inflation and a suitable fluctuation spectrum!
The possible problem (Linde et al): discrete symmetry at the
end of inflation, leads to production of defects, problematic
But (BDV): this problem arises from oversimplifying the potential.
Realistic potentials (realistic forms of susy breaking) have no such
symmetry and a smooth end of inflation.
arise automatically in these models; they break the would-
be symmetry and also alter the c dynamics, bringing a
rapid end to inflation.
In this ``moduli context”, there is also a natural way
to produce baryons. Moduli: flat directions of the
MSSM. AD baryogenesis.
Low scale inflation raises a number of issues of initial
conditions, but this is a relatively compact, simple
model. Like virtually all inflation models, there is a
fine tuning, but this solves problems at once. Again,
thinking carefully about the effective action –
considering all of the expected terms – opens up a
broader and more realistic set of possibilities.