Inflationary Theory of Primordial Cosmological

Perturbation

Project for General Relativity(Instructor: Prof.Whiting)

Sohyun Park

Brief introduction about the title

It is firmly established by observation that our universe• is homogeneous and isotropic on large scales (~

100Mpc).• has well developed inhomogeneous structure on

small scales - p.4

Mukhanov

There should be primordial perturbations to be the seeds of the structure formation! -> Inflation gives a very plausible mechanism for that!Inflationary Theory of

Primordial Cosmological Perturbation

Two systems

homogeneous, isotropic and spatially flat (zero spatial curvature) background

1st order perturbations around the inflationary background

+

Spatially flat, FRW geometry

Two functions in terms of a

Definition of Inflation : accelerated expansion

Brief History of Universe

q ~ -1 q = 1 q = 1/2 q(t_0) = -0.6

Today (t_0): inflating too, so the early one called Primordial Inflation!

Einstein eqns for FRW cosmology

Symmetry of FRW -> only two nonzero parts for stress tensor:t only

Energy density, pressure

Q) What causes inflation? See

A) ANYTHING with

How do we find something with ?

Many!, some are very complicated, but we have two simple ones:

• Positive cosmological constant

• Minimally coupled scalar with a flat potential

OK

Scalar eq for FRW

Hubble Friction

Basic inflation mechanism: particle production

Take the massless, minimally coupled scalarV=0 good approxfor flat potential

With FRW metric

Fourier transf, no coupling between different modes, consider oneAll free QFT: infinite collection of HO’s

Bunch-Davis vacuum

Basic inflation mechanism: particle production

Eq of motion

Hubble Friction

Solve it for the de Sitter case:

Solution: where

canonically normalized:

The energy of state

Now QM

?

Basic inflation mechanism: particle production

1st order Perturbations around the inflationary background

Perturbation equations

Observables: power spectra(which we can compare with data eg.CMB)

For scalar perturbations the power spectrum in the post-inflationary, radiation-dominated epoch

For metric perturbations the power spectrum

Conclusion

Inflation “washes out’’ all pre-existing classical inhomogeneities by stretching them to very large scales. It sometimes called thatInflation removes the “classical hairs.’’ However, it cannot remove Quantum fluctuations (“quantum hairs”). In place of the stretched quantum fluctuations, new ones “are generated” via the Heisenberg uncertainty relation.

-p.328 Mukhanov

The formation of our galaxy began with a quantum fluctuation…