Early Universe scalar field, inflaton and analogies in electric field

[say_cheese]

I have been trying to get my head around this topic for a while. As I go through the description of scalar fields, the inflation and the potential inflaton, (in description as in ned.ipac.caltech.edu), I constantly miss a concept. There must be a fundamental difference between the type of (scalar) potential electric field has and the scalar field of Higgs, the Dilaton and Massive fields.

In electric field, the potential is the scalar quantity ∅(x,t) and its gradient is the field. When we describe the energy state, we describe the energy of an interacting particle, such as an electron- in principle, two conceptual entities, field and the particle. So when we state that in the inflation related scalar field, the inflaton is the ∅ and V(∅) is the potential energy of the scalar field, ∅ seems to have the particle analogy and V has the potentiai analogy (but it is written as V(∅) ? For electric field this is similar to writing the potential as ∅(electron)). 1. Is this related to some kind of self organizing system description? In other words, ∅ is the system (universe) itself and the energy density of the system is all its potentialities? So when we say (for example) ∅ rolls down or in the false minimum, what is rolling or is in a minimum?

Confused!

 

[bapowell]

Don't think "particle" during inflation. As a quantum field, while the inflaton field can in general be associated with particles, during inflation the field is not in its vacuum and the very notion of "particle" is ill-defined.

I don't think your analogy is useful, and is likely causing you trouble. We are talking about the potential energy (scalar quantity) of a scalar field (scalar quantity), which is different than the electric potential of a particle. The potential energy of the inflaton is the energy associated with the scalar field that is distributed throughout space. At each spacetime point, there is a field energy. It happens to be parameterized by the value of the field itself, although this field value is not itself physically relevant. What really matters is that the field value, in turn, is a function of coordinate time, t. So, really you have $V(\phi(t))$. If you prefer, you can think of the statement "phi rolls down to a minimum" as instead saying "at a given point in space, the potential energy associated with the scalar field phi is changing as a function of t according to $V(\phi(t))$. So while it's true that it's easy to think of phi rolling down the potential energy hill (to think of V as a function of \phi), what's really happening is that phi is rolling down in time and the end measurable result is that the potential energy in space is changing in time.

 

[say_cheese]

Thanks for the kind explanation. I do realize that the analogy is poor because the electric field is a vector, while phi (the field quantity) is a scalar. I guess that makes all the difference. As you point out, the terminology has been the same as that of a particle. Yes, that is very misguiding. Let me attempt a description, please let me know if I am right:

The energy density V of a scalar field is a function of phi and there is a system/configuration of the scalar field phi and its derivatives with time. The system variationally approaches a state of minimum energy (and energy density allowing homogeneity). The slow rolls and deep drops describe the gradients in V as a function of phi and show the direction in which the system will move towards. But very importantly these do not reflect the speed (rate) of things (changes in phi) that happened or are to happen in the Universe.

 

[bapowell]

Yes, that's all quite close. A couple points: you can imagine that the scalar field takes on a different value at each point in space (say, on some suitably defined hypersurface at some initial time). These initial condition could be "chaotic" in the sense that they might vary widely across the universe, or they might be well-correlated across Hubble volumes (although this is only likely if the universe is in equilibrium prior to inflation). If the energy of the scalar field dominates the energy density of the universe, those regions will inflate. As the universe inflates, the energy density drops: at each point, the field rolls down to the minimum of the potential energy function, and the time rate of change of phi is determined by the gradient of the potential (if the potential is not very steep but instead rather flat, the field evolves slowly -- it is said to slow roll). But that's the classical motion of the field. Things get much more interesting when you consider the fact that the inflaton field is in fact a quantum field. Now, while the classical tendency is for the field to roll down (for the energy density to drop at each point), quantum fluctuations add some fuzziness to this process -- the field might hop back up the potential, or down a bit more. As the inflaton field approaches the state of minimum energy, it doesn't do so in a fully uniform matter -- the quantum nature of the field ensures that in some parts of the universe the field hits the minimum before it does in some other parts. What you end up with is a mostly uniform universe -- uniform except for small fluctuations in the final density.

 

[say_cheese]

Excellent! Thanks Some give the analogy of a line of soldiers marching and then falling. An initial undulation in the line amplifies the end non-uniformity.