Effective Field Theory in the Early Universe Inflation, Axions and Baryogenesis Michael Dine Berkeley Oct (Collaborations with A. Anisimov, T. Banks, M. Berkooz, M. Graesser, T. Volansky)

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: Dark matter Inflation (superluminal expansion, fluctuations) Baryogenesis 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 3.Inflation We will do axions in the greatest detail.

Axions and the Strong CP Problem 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  so small? Axion solution (Peccei, Quinn, Wilczek, Weinberg): Postulate field, a(x), symmetry: QCD generates a potential for a(x). If  eff =0, QCD preserves CP, so ignoring weak interactions, minimum of axion potential will lie at  eff =0. Can calculate the axion potential (current algebra):

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 . 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 » M p 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,  = a/f a is of order one, the axion initially has energy density vs 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 temperature. One obtains a limit: f a < 3 £ 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 ¿ M p ). 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 ``saxion.” 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. m 3/2 is gravitino mass; size of susy breaking

In the early universe, the equation for  is: The system is overdamped for H>m 3/2. At later times it evolves like dust. Assuming that  starts a distance M p from its minimum, when it starts to oscillate,  stores energy comparable to H 2 M p 2, so quickly dominates. Lifetime:

Catsrophic for Nucleosynthesis! What sorts of solutions are proposed for this puzzle: No supersymmetry, no moduli Moduli heavier than m 3/2 (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.

Axion Dilution Axions start to oscillate when H » m a. At this time,  a = m a 2 f a 2, 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 < 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 scale. 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 = m 3/2. 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 relaxed.

Not only are the conventional limits relaxed, but the basic 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 topological defects.

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” (  ) and the inflaton (  ).

(Berkooz, Volansky, M.D.):  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 fluctuation spectrum. V  

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. Terms like: arise automatically in these models; they break the would- be symmetry and also alter the  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.