[SOLVED] AST3220 -Project 3a

Description

For this project, you can choose between two alternatives. The first option is a numerical study of inflation. The second is strictly outside the scope of this course, but it will give you insight into gravitational waves from binary black holes.

Alternative 1: Inflation without approximation

In the lectures and in the problems we have studied inflation analytically with the slow-roll approximation. If we forego nice, closed expressions we can however, solve the full equations numerically, and this is what you will do in this project.

I remind you that the Planck energy, Planck mass, and Planck length are defined by, respectively

.                                (1)

Assuming spatial flatness and that the scalar field dominates the energy density, the equations governing the evolution of the scalar field and the scale factor are

)    =    0                                                       (2)

.                  (3)

Before solving these equations numerically it is useful to rewrite them in terms of dimensionless quantities. First, define

,                                              (4)

where φ_i is the initial value of the field, and then introduce the variables

(5)

(6)

(7)

E_P

(8)

1. Check that these variables are dimensionless.
2. Show that equations (2) and (3) can be rewritten as

=    0                                                   (9)

(10)

We need to think about the initial conditions. It is convenient to shift the origin of the time coordinate so we can start at τ = 0. The condition for h is trivial: h(0) = H(0)/H_i = 1. For ψ = φ/E_P we should choose a value that makes sure that we get inflation, and that means that the slowroll conditions should be fulfilled. But since the equation for the scalar field is a second-order equation, we also need an initial value for dψ/dτ.

1. Give an argument for why

,                              (11)

where ψ_i is the inital value of ψ.

We are now ready to look at specific models. Let’s try

,                                           (12)

from the example starting on page 115 in the lecture notes.              Let mc² =

0.01E_P.

1. Use the slow-roll conditions to choose an appropriate initial value forthe field.
2. Solve equations (9) and (10) numerically and plot the results. Based onthe lectures, how would you expect ψ to behave? Does the numerical solution conform with your expectation?
3. In the same plot, plot the slow-roll solution from the lecture notes.When does it start to deviate significantly from the exact, numerical solution?
4. Using the result for h, plot ln[a(τ)/a_i], where a_i = a(τ = 0). (Hint: Start with the definition H = a/a˙ and integrate.) Estimate how many e-foldings we get and compare with the slow-roll result.
5. Show that in terms of the dimensionless variables

(13) i) What would you expect the ratio in equation (13) to be in the slow-roll regime? And in the oscillating phase? Plot the numerical result and compare.

1. Assume a(t) ∝ t^p at late times, after the slow-roll conditions break down, and use your numerical results to find the value of p by trial and error (or in a smarter way, if you can think of one). Which value would you expect? Why?
2. Repeat d)-g) with i) V (φ) = λφ⁴/(¯hc)³ and ii) V (φ) = V₀e^−λφ. Compare with results from the lectures and the weekly problems.

Alternative 2: Gravitational waves

If you choose this option, you will work your way through the article ”Gravitational waves from orbiting binaries without general relativity: a tutorial” by R. C. Hilborn (link on the course webpage). The article includes a number of exercises, of which you should do the following:

• Both on page 8
• All on page 10
• The one on page 14
• Both on page 17
• The exercises on page 19, 20, 21, 22
• The first exercise on page 23
• All exercises on page 24, 25, 26, 28, and 29
• The first and the third on page 32
• All on page 33, 34, 37, 38, 39, 40, 42, 43, 48, 50, and 51
• Extra exercise: Reproduce figure 6 in the paper
• Another extra exercise: Explain at least two important differences between the gravitational waves from inflation and the gravitational waves considered in this article