This paper proposes a method using two independent non-Riemannian volume forms (integrand of a volume integral) to derive an effective potential for the scalar matter field that is capable of describing both the early universe expansion and dark energy in our universe today. In other words, they are able to derive an effective potential that produces accurate energy densities during the inflationary phase of the universe and our current universe containing dark energy.

This paper isn't very difficult, but does require some prerequisites to understand. You should be somewhat familiar with tensor calculus. If you understand what Lorena has been doing in her Numerical Relativity posts then you should be fine with the level of tensors here. Some familiarity with scale transformations would be nice so that you can understand their procedure. An understanding of manifolds would be great, but is not necessary for understanding the primary point of this paper.

Repeating the authors' calculations here would be redundant (especially due to the small size of the paper), so I will just mention specific things of interest and some information useful when reading the paper. However, I note that it is impossible for me to teach you all the material required to fully understand this paper. This would require a course in General Relativity.

The approach the authors use is to variate an action integral which is pretty standard. The peculiar thing here is that they use two independent non-Riemannian forms instead of one. Following standard procedure for the Lagrangians and measure densities, they obtain the following action

The approach the authors use is to variate an action integral which is pretty standard. The peculiar thing here is that they use two independent non-Riemannian forms instead of one. Following standard procedure for the Lagrangians and measure densities, they obtain the following action

$$ S = \int{d^{4}x \Phi_{1}(A)[R + L^{(1)}]} + \int{d^{4}x \Phi_{2}(B)[L^{(2)} + \frac{\Phi(H)}{\sqrt{-g}}]} $$

Let's analyze this equation and gain some understanding of what it means. Typically in Classical Mechanics, one is dealing with an action of the form $ S = \int{dt L(\Phi,\partial_{\mu}\Phi)} $, where $L$ is the well known Lagrangian. In Special Relativity the Lagrangian may be rewritten as $ L = \int{d^{3}x \mathcal{L}(\Phi,\partial_{\mu}\Phi)} $, where $\mathcal{L}$ is the Lagrangian density. This allows us to generalize the action to $ S = \int{d^{4}x \mathcal{L}(\Phi,\partial_{\mu}\Phi)} $ where we are now integrating over space-time.

Now we can make some comparisons between our standard special relativistic action integral and the one given in the paper/above. The terms $R$ and $g$ come from General Relativity. $R$ is the Ricci scalar which gives a measure of the curvature of the geometry in consideration. As a clarification to those who haven't encountered relativity or differential geometry of any sort; $g$ is not Earth's acceleration of gravity, rather it is $\|g_{\mu \nu}\|$ where $g_{\mu \nu}$ is the metric tensor for whatever geometry you would like to use. The measures $\Phi$ are the field strengths of typical gauge fields. This will tell you which field is the primary contributor for specific phenomena.

The most difficult part of this paper is where they pull out this gauge transformation that leaves the action invariant and therefore informs us of a symmetry. Gauge transformations can be a pain and aren't very intuitive. Moving along, they variate the action integral and discover two terms that are not invariant under the scale transformation and lead to spontaneous symmetry breaking.

In the end, what is obtained is an interesting effective Lagrangian and scalar field potential (see paper). The potential smoothly moves from one flat region at large negative values of the parameter to another flat region at large positive values, as can easily be seen in the figure at the end of the paper.

The key to the success of this calculation was the use of two volume forms instead of the typical one used by most physicists. Similar results can be obtained with one volume form, but this leads to nonlinear terms which are obviously ugly. It suffices to say that using two volume forms gets rid of this issue and presents a nice way of calculating the vacuum energy density for both the early universe inflationary phase and our present day dark energy phase.

That's about all I have to say about this paper. It's a great read that performs a quick calculation of some pretty cool stuff. I wouldn't get bogged down in the details of the paper if you aren't strong in General Relativity. Rather you should notice how awesome this result is! Hope you enjoyed the post as much as I did.