Generation of perturbations on Inflation

This course is offered for the graduate students of the Astronomy Department. If you would like to attend, either for credit or not, e-mail me at

There will be 7 lectures on Mondays and Fridays at 10(05)am starting Friday Nov.22nd. The classes will take place on 15th floor.

The aim of the course is to provide you with basic (but operational) understanding of the mechanism of generation of inhomogeneities during inflation. This is the cornerstone mechanism of the generation of initial perturbations in modern cosmological paradigm. I hope you will get a feeling what you should/can expect for a spectrum of "initial" fluctuations it gives. Later Llyod Knox will present a course on how further "Evolution of Fluctuations" gives rise to cosmological structures/CMB anisotropy we observe. To get the full list of minicourses offered, click here .

If to specify a single reference this course is close to it is:
A.Linde: 1990, Particle Physics and Inflationary Cosmology, Chapter 7.

Actually, even closer (but difficult) reference is :
V.Mukhanov,H.Feldman,R.Brandenberger, 1992, Theory of Cosomological Perturbations, Physics Reports, 215(5-6)

Here is the current list of problems suggested. The due date is mid January.

Dmitry Pogosyan

The Course Outline

(I'll try not to overlap too much with your standard course on cosmology). All references are available either in Astronomy or in Physics library - although I may hold some of them at the moment !
  1. Basic information needed
    GR - equations, FRW metrics, horizon, what else ?
    ( Landau, Lifshits: Classical Theory of Fields )
    scalar fields - potential of self-interaction, Lagrangian, energy-momentum tensor
    (Ya. Zel'dovich: 1990, My Universe, pp.61-96 )
  2. How one gets inflationary regime if the Universe is dominated by (suitable) scalar field. Phase-space diagram of the background (homogeneous and isotropic) solutions. Slow-rolling regime and its conditions. See Problem #1.
    (V.A.Belinskii et al, 1985, Sov.Phys.JETP, 62(2), p.195 )
    (for phase space technique see , C.Bender, S.Orszag: Advanced mathematical methods for scientists and engineers, p.171 )
  3. Perturbation theory - scalar/vector/tensor modes, gauge-invariant description (hopefully useful for L.Knox course as well).
    (J.Bardeen, 1980, Phys.Rev.D, 22, p.1882 )
  4. Quantum fluctuations of scalar field. "Flat" spectrum of fluctuations after "freezing out" in long wave regime in the simplest inflationary models. I'll try to be somewhat more rigorous describing the time evolution of the amplitudes of Fourier modes, but not quantum field questions. See Problem #2.
    (A.Linde: 1990, Particle Physics and Inflationary Cosmology, Chapter 7.)
    (V.Mukhanov, 1988, Sov.Phys.JETP, 67(7), p.1297)
    This last paper is contained in full in:
    (V.Mukhanov,H.Feldman,R.Brandenberger, 1992, Theory of Cosomological Perturbations, Physics Reports, 215(5-6),p.203)
  5. How "flatness" of initial spectrum can be altered if one considers more general cases of inflationary models (for example when slow-rolling conditions are broken or there are several scalar fields in a play). This is one of the recent arguments for a position that "inflation lost its predictive power".
  6. Generation of the gravitational waves (tensor mode) and their spectrum. See Problem #3.

Suggested Problems

Problem #1

Present a (as complete as possible) discussion (at least qualitative, best with some quantitative estimates on the level of what we had during the class) of the phase space in the model with scalar field potential of W-type

V(\phi)= - (m^2 \phi^2) /2 + (lambda \phi ^4)/4 + A

m, and \lambda are two parameters, notice a "-" sign in front of "m^2" term.

A=const. Discuss what should be value of A so that t->inf behaviour of the model is Friedman dust-like (as it was during the class). Introduce the notion of "effective mass" to analyse t->inf asymptotics.

I'd like you to find out what inflationary regimes are present (there are more than one) and approximate them.

Problem #2

Derive the formula for the spectrum of perturbations (for grav potential) in the theory with V(\phi)=lambda/4 \phi^4. Obtain the correct amplitude and logarithmic correction to the flat spectrum.

Problem #3

Derive the spectrum of gravitational waves in the inflationary model with V(\phi)=m^2 \phi^2 /2. Obtain the amplitude from quantum considerations, the shape of spectrum and compare both the amplitude and the with the results for scalar mode.

Hint1: gravitational waves obey the equation h''+ 2(a'/a) h' + k^2 = 0 here ( )' is the derivative with respect of conformal time - so it is exactly as in the case of scalar field.

Hint2: There are 2 tensor degrees of freedom (2 polarizations of grav.waves), each separately obeys the same equation.