363. Vector inflation

Alexey Golovnev is a research associate in the Arnold Sommerfeld Center for Theoretical Physics , U. of Munich. Dmitry.

I would like to blog about the recently proposed model of vector inflation, Golovnev, Mukhanov, Vanchurin, arXiv:0802.2068. (See also Golovnev, Mukhanov, Vanchurin, arXiv:0810.4304 and Golovnev, Vanchurin, arXiv:0903.2977 for further developments.) Inflation is one of the basic concepts in cosmology, it solves the well-known cosmological problems (for which I refer the reader to any modern textbook on cosmology, see below) by a period of very rapid (nearly exponential) increase of the size of the Universe. By a straightforward inspection of the Friedman equations, this period of accelerated expansion can be achieved with a matter content of negative pressure. As calculating the pressure basically amounts to taking the difference of kinetic and potential energies, the most natural idea to realize this type of equation of state is to “freeze” an inflaton field at a high value of its potential energy.

Successful models of inflation usually invoke a scalar inflaton field which rolls slowly down the hill of potential energy. The Universe expansion provides a viscous friction term making the slow roll regime possible. Indeed, if we assume chaotic inflation on a mass term, the equation of motion reads and the D’Alambert operator can be calculated as the divergence of the gradient: . In the physical time with the metric element we get where stands for the Hubble parameter. The coefficient refers to the Universe expansion. More physically it can be understood if we consider taking divergence of some conserved vector current . In this case the term shows that the charge density exponentially decays because the physical volume grows at the relative rate of . For cosmology it means that the highly excited scalar field does not relax fast, but rather rolls down slowly providing a large enough period of inflation.

Vector fields have not received that much attention in the cosmological context due to several reasons. First of all, vector fields induce anisotropy. We resolve this problem either exactly by taking a triad of vector fields or approximately by introducing a large random collection of them. (The former solution was known before, stemming from some specific Einstein-Yang-Mills configurations, see the reference list in our article.) After that one can deduce the equations of motion in an inflationary background. Everything proceeds exactly along the lines indicated above, and the final result for a massive vector field shows that in the homogeneous test field approximation we have and for the spatial components the slow-roll regime is possible. Clearly, it is not what we want to have. The potential energy rapidly decays as . These vector fields can not source the inflation. There is no surprise in that. Neglecting the potential term, the vector fields are conformal invariant and shouldn’t really care about the relevant length scale.

A better choice of variables is ; it is these quantities which have to change slowly during inflation. One can easily find the following equation of motion for them: . During inflation it implies a very large effective mass of order which renders the friction term inefficient and the slow roll impossible. But note that where is the scalar curvature; it shows that this unfortunate contribution can be compensated exactly by a suitable non-minimal coupling of the vector fields to gravity.

It brings us to original Lagrangian of the vector inflaton fields from 0802.2068 arXiv paper: with and . We also take the standard Einstein-Hilbert action for gravity and sum over a large number of independent randomly oriented vector fields. At the background level the solution is basically the same as for scalar inflaton fields but we can have some anisotropy for free. And one would like to consider also a more general proposal before proceeding with the perturbation analysis.

Being that simple for inferring the background evolution, the vector fields turn into a real disaster already at the linear level in perturbations. The reason is very simple. Different types of perturbations (scalar, vector and tensor) do not decouple even for perfectly isotropic backgrounds because, for example, terms like couple tensor (transverse and traceless) metric perturbations to the scalar part of the linear perturbation equations. Due to this problem, we don’t have at the moment a full rigorous analysis of linear perturbations in vector inflation. Nevertheless, we were able to provide some approximate consideration in our paper 0903.2977 which strongly suggests that, most probably, there should be some considerable correlations between scalar and tensor modes in CMB. And, of course, another very important consequence is that we expect some anisotropy. It is very important in view of the current observational progress in cosmology. See D. Wands, Nature Physics, Volume 5, Issue 2, pp. 89-90 (2009) for an interesting expert opinion.

A better analysis is needed to have more detailed predictions and to choose a proper form of inflationary potential . As I’ve stated above, it is not available yet. However, a simple consideration of gravitational waves only allowed us to exclude a large class of models in arXiv:0810.4304. A catastrophic growth of gravitational waves in most of the models with large values of vector fields can be seen directly in the equations of motion, but an easier way of doing this is to expand the action of the theory up to the second order in the tensor metric perturbation neglecting variations of all the other variables. It shows that the gravitons acquire an effective mass because we have quadratic in terms in the action which depend on background values of the vector fields and come about due to raising the indices in and . (Of course, it is of crucial importance here that the fields are in the slow-roll regime which makes the mass approximately constant.) It is clear that for models with large vector fields the mass should also be large. And in fact, it is usually tachyonic and makes the large fields vector inflationary models badly unstable. (There are some weird exceptions like ). On the other hand, the small fields models are generically stable in this respect. For example, one can use the Coleman-Weinberg type of potential for vector inflaton fields.

I have to mention here that in D. Lyth et al., arXiv:0809.1055 it was claimed that gravitational waves are stable in vector inflation. The authors have moved to the Einstein frame without finding it explicitly and then they have made some arguments of slightly philosophical nature for rather tame behaviour of the gravity waves. This statement can’t be trusted because the raising of indices occurs in any frame and, after all, there is no reason for such a striking difference between results obtained in different frames (unless some wild instability of the conformal factor takes place).

There is also one more controversy about vector inflation in the literature. It is the problem of stability raised in B. Himmetoglu, C. Contaldi, M. Peloso, Physical Review D79, 063517 (2009). It was noticed that the longitudinal component of a tachyonic vector field looks like a ghost making the quantum theory problematic. (I remind to the reader that a large tachyonic contribution to the effective mass is absolutely crucial to insure the slow roll.) At the classical level we don’t see this instability. The tricky point here is that can be allowed to grow (even exponentially), but what should be stable is . Any theory in terms of the components should be in a sense unstable. The tachyonic mass (at least for the large fields models) translates exactly to this purely coordinate effect. The more physical fields behave like if there were no tachyonic mass. If there is no way to construct a stable quantum theory in this case, then this translation of the purely coordinate growth to a fundamental catastrophic instability sounds very unsatisfactory for me. Actually, some arguments in favour of quantum stability are given in D. Lyth et al., arXiv:0809.1055 although they are not very conclusive. Ghost instabilities present a very serious issue for quantum field theory and deserve somewhat more serious attitude than just counting the total energy of a state with a very small occupation number.

So, I have to admit that I have no answers to many perfectly legal questions. But I think, at the very least, it is important to understand the possibility of playing with vector fields too instead of restricting oneself to a purely scalar world of inflation. Moreover, higher spin fields are also possible, see T. Kobayashi, S. Yokoyama, arXiv:0903.2769 and T.Koivisto, D. Mota, C. Pitrou, arXiv:0903.4158.

Some literature

1. V. Mukhanov, Physical foundations of cosmology.
2. A. Liddle, D. Lyth, Cosmological inflation and large-scale structure.
3. A. Linde, Particle physics and inflationary cosmology.