Tracking Field-Space Perturbations through Cosmic Inflation: Final Update

Since my last post, I have found regions of significant movement for the CMB observable values for varying values of a.  Graphs of these are posted below.

deltaR2 graph nr graph ns graph r graph

These graphs are only shown in pieces due to the time required to run the interpolation function of V over large ranges of a.  Instead of wasting actual days waiting for the functions to run for all values, I decided to choose a few shorter ranges and allow them to speak to the shape of the general plots, which I believe they succeed in doing.  At this point, I examining my function for n_r to see why is grows at such a large rate.  As you can see from these graphs, it grows through three orders of magnitude between 0 and 300, which is at the very least surprising.

nr a(0,.1) nr a(0,1) nr a(0,40) nr a(160,200)

I am also investigating the holes that can be seen in the plots at a = 20 and a = 180.  The one at a = 180 is of particular interest because this is also the point at which the plot of n_s asymptotically approaches -infinity.

ns graph

I will investigate this by graphing the curvature, K, which is a function of the field r and a.  This was calculated using the Riemann curvature tensor, which is a function of Christoffel symbols, which are functions of the components of the field-space metric.  Below is the equation for a Christoffel symbol, where g with lower indices are various components of the field-space metric, while g with upper indices is the inverse of the its lower-indices twin.

Screenshot 2016-07-25 12.36.44 (taken from Wikipedia)

For my work, the indices can be r or θ, and the Christoffel symbol is calculated for every combination of i, k, and l being either of those indices.  The right hand side of the equation is also a summation over each version of the statement with m taking on a different value, r or θ, and x^(l,k,m) are fields with respect to which the derivatives of the metrics can be taken. An example of this is shown below.

IMG_2088  (I apologize for the image quality.)

The Riemann curvature tensor, shown below, works similarly, where  λ is the summed-over index.

Screenshot 2016-07-25 12.51.29

The Riemann curvature tensor is then used to calculate the scalar curvature, using the Ricci curvature tensor, which is the Riemann curvature tensor where ρ and μ are the same, and so drop out. The scalar curvature is given below, where the right side is the sum of all possible combination of i and j, where each can be r or θ, and g with upper indices are again the inverses of the field-space metric components.

Screenshot 2016-07-25 13.03.26

I also calculated the Gaussian curvature, K, which should be equal to 1/2 S, using the following equations, where ζ is a summed index.  Here, R_{abcd} is the same as R_{ρ σμν}.


Screenshot 2016-07-25 13.24.10

Screenshot 2016-07-25 13.23.50

Using these processes, I calculated K to be

Screenshot 2016-07-25 15.20.55.

I will be plotting this equation and seeing its behavior, especially at a = 20 and a=180.  After that is complete, I will be writing up my final report on my work.


  1. balenhart says:

    Whoa, sick data man. I can tell you’ve worked pretty hard on this.
    I had a difficult time understanding what your findings represented. Granted, I have no background in Astrophysics. But your findings look promising, maybe in the future you could clarify what some of the graphs are refering too, such as the nature of “a” and “cmb?”