Cosmological perturbation theory

Cosmology is the study of the structure of the universe on the largest scales we can observe. More informally, it can be described as the study of the structure of the universe as a whole. Since it is impossible to model everything at the same time it is necessary to neglect the influence of structures on scales which are smaller than those which we are concentrating on. The simplest class of models are those which are homogeneous (all spatial points are equivalent) and isotropic (all directions at a given point are equivalent). This is formulated mathematically as the existence of a large group of symmetries, in fact a six-dimensional group (three translations plus three rotations). It has been found that this kind of highly symmetric model gives a reasonable description of observed quantities after averaging on a suitable scale. This fact and the advantage of simplicity lead to models which are homogeneous and isotropic playing a dominant role in theoretical cosmology. In the astrophysical literature there is little emphasis on rigorous mathematical analysis and so there is a lot of work to be done to bring ideas introduced in astrophysical cosmology into the domain of theorems and proofs. I find the question of the benefits which may arise from doing so very interesting but I will not comment on it further here.

It is clear that the actual universe we live in is not homogeneous or isotropic. If I look at the sky I see a star in some directions and I see none in others. How is this taken into account in the theory? The related question of the formation of structure is of great interest. We are immersed in a bath of microwaves which cannot be associated to any localized source (cosmic microwave background, CMB). This radiation has been propagating freely since a very early epoch in the development of the universe (about 300000 years after the big bang). It is highly isotropic. The variation in its temperature over directions in the sky is about one part in 100000. It can be concluded that the radiation was uniformly distributed at the time (decoupling) when it stopped interacting with other matter. Since before that there was a tight coupling between radiation and ordinary matter it follows that ordinary matter was also uniformly distributed at that time. The question is then how the inhomogeneous structures such as galaxies we observe today arose. In fact this picture is complicated by dark matter which interacts weakly with normal matter and radiation even before decoupling and may therefore consistently be highly inhomogeneous at decoupling. The task of explaining the origin of the inhomogeneity of the dark matter distribution remains.

It is evident that in order to describe the formation of localized structures it is necessary to go beyond the homogeneous and isotropic framework. If the structures develop from small deviations from homogeneity and isotropy we are dealing with quantities which are initially small. Then it is tempting to linearize the equations, discarding expressions which are quadratic or higher order in the magnitude of the deviation from homogeneity in the equations. In fact in the literature on cosmology the usual next step after considering homogeneous and isotropic models is to linearize about these. Then it is investigated whether solutions of the linear equations grow in time. In principle this a very similar procedure to what is done in the paper of Keller and Segel described in a previous post for a quite different application. In detail things are more difficult because in cosmology the equations to be solved are more complicated and the interpretation of the results presents special difficulties. In any case, this is what is meant by cosmological perturbation theory. The most useful source I have found for this subject is the book ‘Physical Foundations of Cosmology’ by V. Mukhanov. Unfortunately it is difficult to establish contact between the calculations and discussions arising in this context and rigorous mathematical statements. There is a serious problem of language at the very least. In the past I have published several papers addressing these issues in the homogeneous case. Now, in collaboration with Paul Allen, we are tackling mathematical issues raised by cosmological perturbation theory. We already have some preliminary results but I will leave a discussion of these to a later post. I will give a talk on this subject at a conference in Stockholm next week.

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