# Question on inflation as a phase transition

I have just finished watching the following video http://www.youtube.com/watch?v=beQ9fZ0jVdE where Laughlin, Gross and some students discuss e.g. about inflation. The following question is risen: Is inflation a phase transition (e.g. of the geometry of space time)?

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The inflating universe can for example be described by the FLRW metric

$$d\tau = dt^2-a(t)^2(dx^2 + dy^2 + dz^2)$$

The scale factor $a(t)$, which describes the expansion, is obtained from the appropriate Friedman equation which contains only the vacuum energy $\rho_0$ as source of gravity

$$\frac{\dot{a}}{a} = \sqrt\frac{8\pi G \rho_0}{3} = H$$

with the exponential solution

$$a = K e^{H t}$$

For inflation with this exponential expansion to occur, the the Hubble constant $H$ and therefore $\rho_0$ has to be constant.

During an inflationary phase it is assumed that the most important contrubution to the vacuum energy is due to the potential energy $V(\phi)$ of the scalar inflaton field $\phi$ which has the Lagrangian

$$\frac{\dot{\phi}^2}{2} + \frac{(\nabla\phi)^2}{2} - V(\phi)$$

So an inflational period itself, for example the inflation of the early universe, can not be described by a phase transition; the potential energy density of the inflaton field is assumed to be about constant, sitting in the left local minimum in the figure below.

However the end of this early inflation which can be described by a jump to the lower value of the potential energy density (and much slower inflation) we observe today, corresponds to a phase transition during which the difference in the potential energy density of the inflaton field between the left and the right local minimum is released as some kind of latent heat (reheating) and converted to kinetic energy and finally to particles and seeds of the galaxies we observe today. More details about how the end of the early inflation can be described as a phase transition can be found for example in this paper.

As said above the inflaton field is assumed to be a scaler field; plenty of scalar fields occur for example for different physical reasons in string theory, some of them are described here or here. Some people are trying to reconstruct the potential of the inflaton field from observations.

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Thank you for the reply. By the way. What is the physical origin of the inflaton field? Why has the potential this particular form? –  Hamurabi Jan 6 '13 at 14:24
Hi @Hamurabi I am not deep enough in inflationary cosmology unfortunately, but what I have heared there are many different physical mechanisms considered that could give rise to an appropriate inflaton field and I am not sure if there exist theoretical models which can exactly reproduce such a potential of the inflaton. An alternative explanation I have heared about why the early faster inflation (corresponding to a larger vacuum energy density) could have slowed down to the slower present inflation (corresponding to a very small vacuum energy density) is tunneling between two such vacua. –  Dilaton Jan 7 '13 at 23:46
Maybe you could ask the two interesting questions of your comment separately to get specific answers focused on each of them? –  Dilaton Jan 7 '13 at 23:47