Tracking Field-Space Perturbations through Cosmic Inflation: Update 2

Since my last post, I have started the numerical analysis of my work, using the program Mathematica.  This is because my work has reached integrals and interpolations which are impossible to do analytically.  This is to say that there are no longer functions which are equal to the integral of the functions I have, in the way that x^2 = Integral [2x].  However, it is still possible to measure the area under the plot of the function I’m integrating, which is where the program comes in.

In a little refresher, I have a cosmic inflation model, given by the Lagrangian Screenshot 2016-07-05 16.16.55

where the V is defined as

Screenshot 2016-07-05 11.37.56

which can be visualized as

Spiral_Potential.

Using the relationship of

Screenshot 2016-07-07 10.50.21

and the relation

Screenshot 2016-07-07 10.52.39

which holds the inflaton to follow the bottom of the trench which spirals around the inside of the potential, I used Mathematica to find an equation θ(I).

Combining this with the equation for r, and the fact that that equation causes the equation for V to reduce to just V = λr^4, this gives V(λ,r(θ(I,a))), making V a function of λ, a, and I, where λ and a are variable coefficients and I is a single effective field, taking the place of the fields r and θ. Using this equation, I found equations for the CMB observables which tell me how well the model agrees with reality.  This is done using the equations

Screenshot 2016-07-03 16.04.41 Screenshot 2016-07-03 16.04.49 Screenshot 2016-07-03 16.04.57,

which can be combined to give the observables

Screenshot 2016-07-03 16.05.15

Screenshot 2016-07-03 16.06.13

Screenshot 2016-07-03 16.05.06

 

Screenshot 2016-07-03 16.06.04

Screenshot 2016-07-03 16.09.42

In relation to the picture on the left below, n_s is the relation between two points, sweeping an angle around the first point, which tells the probability that the second point will have the same density (here, color) as the first point.  n_r tells how that relationship changes as the distance between points increases.  r’ is related to the power in gravity waves that the model expects, as the presence of large powers in primordial gravitational waves is one of the predictions of cosmic inflation.  Evidence for these primordial gravitational waves is being searches for in terms of patterns in density perturbation plots such as the one by BICEP shown on the right.

Planck_CMBUnknown

Δ_R^2 represents the size of the first spike seen the angular plot of teh CMB shown below.

Unknown

N_e is the number of times the universe expanded by a factor of e between I_i, at which point the primordial temperature fluctuations, which led to the CMB and which are shown in the BICEP plot, formed, and I_f, at which point inflation ended.  The number is expected to be between 40 and 60.

After checking my potential’s output values against those of a previous paper to make sure it was working correctly, I ploted the values against a to see how changing the perturbation in field space affected values of the observables.n_s plot with a n_r plot with a r' plot with a DeltaR2

I am still working to get a graph of N_e, and I am currently working on finding a range of a where it has a more significant affect.