Tracking Perturbations in Space-Time Curvature through Cosmic Inflation

So far in my research, I’ve spent the majority of my time learning the requisite information and terminology to do my work.  I am working with equations of motion for field vectors involved in cosmic inflation.  These field vectors include time and multiple dimensions of space, such as x and y.  I find these equations of motion using the Lagrangian, L=T-V, where T is the sum of the kinetic energies of the different fields of the system and V is the potential energy of the system.  Using differentials, this equation can be manipulated to give the equations of motion for the system.  Before these equations can be calculated, a metric (which is a type of matrix), is multiplied to the kinetic terms of the Lagrangian.  In flat space-time, this metric does not affect the values of the kinetic terms.

Once an equation of motion is found, the fields are parameterized to an inflation vector, I, which follows a particular path based on the potential field described by V.  Once this is complete, the newly parameterized V function and derivatives of this function are used to calculate the values of certain observables in the cosmic microwave background, which has existed since the end of inflation and carries information about inflation’s operations in it.  I am dealing with two particular observables known as ε and η, where

Screenshot 2016-06-07 23.16.19Screenshot 2016-06-07 23.16.30.

 

I am starting with a spiral inflation field, such as the one below,

Spiral_Potential-1

Here, the inflation vector I, commonly referred to as the inflaton, rolls down the trench visible in the side of the bowl.  Inflation continues until the inflaton reaches the bottom  and stops moving.  Due to the circular nature, I will be using field vectors of r and θ, and applying small perturbations H to the flat space-time metric in order to see how such changes with show up in the equations for ε and η.

Thus far, the most difficult part of the work has been keeping track of terminology and symbolic meaning within the equations.   This confusion can be illustrated using the equation below, which is a general equation of motion for inflation dynamics in a potential V.

Screenshot 2016-06-07 23.39.00

For now, the work and derivations are confusing, but as I get a better grasp on the ‘mathspeak’, which mixes differential equations, linear algebra, and multivariable calculus, I hope to find it go smoother.

Comments

  1. Robert O'Gara says:

    Dear Seamus,

    I’ll be honest when I say that I do not understand a lot of this considering I am not a physicist. However, I would like to get a better understanding of the purpose/applicability of your project. In other words, what will tracking perturbations in space-time curvature through cosmic inflation teach us about physics? Does it help us better understand the concepts of space and time? And where would we expect to see such science used? Is it more of a theoretical concept or is it commonly used?

    Sincerely,
    Rob

  2. sherriman says:

    This research will build the store of information known about a class of inflation model. This class is one which has large powers in gravitational waves, and if these were detected in the Cosmic Microwave Background, it would be strong evidence towards the solidity of the theory of Cosmic Inflation as an explanation for early universe dynamics.