The Strong Cosmological Principle

The first presentation of Layzer's Strong Cosmological Principle apparently was at a conference on The Nature of Time at Cornell in 1963, organized by

Thomas Gold. The attendees included Hermann Bondi, Subramanyan Chandrasekhar,

Richard Feynman,

Gold, Martin Harwit,

Roger Penrose, Philip Morrison, and John Wheeler, among others.

Five years later, Layzer presented a paper, entitled "Cosmogonic Processes," at the Brandeis Summer Institute of Theoretical Physics in 1968, published as *Astrophysics and General Relativity*, two volumes, edited by Max Chrétien, Stanley Deser, and Jack Goldstein, Gordon and Breach, NY, 1971.

Layzer begins with a hypothesis of Ludwig Boltzmann that depends on the universe being infinite. Since Boltzmann was attempting to explain how we could have departed from equilibrium enough to live in a universe in which entropy was low and obviously increasing, it is clearly the intellectual origin of the problem of growth of order. It also appears to have inspired Layzer's Strong Cosmological Principle.

*Boltzmann's hypothesis*

Boltzmann pointed out that in an infinite universe in a state of thermodynamic equilibrium every finite configuration whose structure does not violate the laws of physics has a finite probability of occurring. Hence the observable universe of Boltzmann's day could represent a statistical fluctuation in a universe that, as a whole, was in a stationary
state.
(*Astrophysics and General Relativity*, vol.2, p.156)

Layzer's theory of the

growth of order in the universe shows that "the initial state of local thermodynamic equilibrium is uniquely defined, and it is a state of zero specific information."

Subsequently, as the universe expands, the specific information must increase, and this increase defines the arrow of time — at least on the cosmological level. But can we attribute objective significance to the concept of information? Information is, after all, defined only in the context of a statistical description. The quantity *n* that figures in definition (2.27) is a probability density in phase space. If we had chosen to specify the state of our ideal gas through the actual occupation numbers of cells in phase space instead of through a distribution function, the specific entropy would have been precisely zero and would have remained zero for ail time. A complete microscopic description of a closed system always contains the same quantity of information, and is hence completely time-symmetric.
This argument — that the asymmetry between past and future disappears when we pass from the macroscopic to the microscopic level of description — is undoubtedly valid for closed macroscopic systems.

It has been raised against statistical theories of irreversibility in macroscopic systems since the time of Boltzmann, and even today it stands in the way of a completely satisfactory theory of macroscopic irreversibility. To avoid it, some physicists have adopted the view that irreversible processes occur only in systems that are not quite closed [allowing the environment to disturb the system and introduce disorder]. Interactions between a system and its environment give rise to genuine indeterminacy in the microstate of the system and thus afford an objective basis for a statistical description.
An alternative and equally objective basis for a statistical description not only of macroscopic systems but of the universe as a whole is provided by an assumption that I shall call the *strong cosmological principle*. This asserts that the universe is characterized completely by random functions and that no statistical property of the description serves to define a preferred position or direction in space. A Poisson distribution of particles filling all space is the simplest model of a universe that satisfies this postulate. For the sake of definiteness, we imagine space to be partitioned into cells of volume *V*. The distribution of particles among the cells is completely determined by the parameter *nV*, where *n* is the mean number density of particles. This infinite and unbounded Poisson distribution has two properties that are not shared by any finite or bounded distribution.

(a) From a single realization we can evaluate the defining parameter *nV* — or indeed any average quantity pertaining to the distribution with arbitrary precision.

This property depends on the law of large numbers and on the fact that the distribution occupies an infinite volume.

(b) There do not exist two distinguishable realizations of a given Poisson distribution. For the set of occupation numbers that characterizes any finite part of a given realization has a finite and calculable probability of occurrence and must therefore have an exact counterpart — indeed infinitely many exact counterparts — in a second realization of the same distribution. Thus any two realizations can be made to coincide over any finite volume. It follows that "different" realizations of the same Poisson distribution are operationally indistinguishable.

We are therefore forced to the rather startling conclusion that, for any finite value of the cell size *V*, a Poisson distribution of particles in an infinite Euclidian space contains ail the information needed to define it (namely, the value of *n*) but does not contain any "microscopic" information. The proviso that the cell size *V* be finite is essential. If the positions of particles were specified precisely through their coordinates, the distance between any two particles would suffice to characterize the distribution uniquely. In reality we are of course concerned with distributions of particles in phase space, whose cellular structure is guaranteed by the laws of quantum mechanics. Thus the indeterminacy resulting from the absence of microscopic information in a cosmic distribution satisfying the strong cosmological principle is closely related to the quantal indeterminacy expressed by Heisenberg's uncertainty relations. Nevertheless the two kinds of indeterminacy are distinct.

We conclude that objective significance can be attached to the growth of information in a Friedmann universe expanding from an initial state of local thermodynamic equilibrium.

(*Astrophysics and General Relativity*, vol.2, pp.165-6)

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