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I will give a broad overview of what has become the standard paradigm in cosmology. I will briefly explain the evidence for this paradigm and then review some of the most significant conceptual challenges it contains. These challenges can be split into two classes: (1) Classical issues to do with, for example, the infinite spatial volume of the universe and (2) Quantum issues, which revolve around trying to apply quantum theory to cosmology.

Einstein’s equations are local four-dimensional equations, in that they are valid at every point in space-time. Following the ‘cosmological principle’ – an extension of the ‘Copernicus principle’ – that physics at every point in our universe should look the same, we are lead to the modern view of cosmology. This procedure is reasonably well understood for an exactly homogeneous universe, however the inclusions of small perturbations over this homogeneity leads to many interpretational/ conceptual difficulties. For example, in an (spatially) infinite universe perturbations can be arbitrarily close to homogeneous. To any observer, such a perturbation would appear to be a simple rescaling of the homogenous background and hence, physically, would not be considered an inhomogeneous perturbation at all. However, any attempt to choose the physically relevant scale at which perturbations should be considered homogeneous will break the cosmological principle i.e. it will make the resulting physics observer dependent. A more practical issue is that the universe is not static and hence such a scale will be time dependent. Thus what appears ‘physically homogeneous’ to an observer at one time will not appear so at another.

This issue is brought to the forefront by considering the canonical (space and time rather space- time) version of the theory. The phase space formulation of General Relativity, just as for any other theory, contains the shadow of the underlying quantum theory. This means that, although the formulation is still classical, many of the subtleties that are present in the quantum theory are already apparent. In the cosmological context the infinite spatial volume renders almost all expressions formal or ill-defined. In order to proceed, we must restrict our attention to some finite spatial ‘cell’, which we consider to be only a mathematically convenient construct. To extract physics we need to take the volume of this cell to infinity, mathematically this is clear, but the conceptual underpinnings are less so – is our universe (spatially) infinite or merely arbitrarily large?

These difficulties are IR or large (spatial) scale issues in cosmology, however in addition there are UV or short (spatial) scale problems that need to be tackled. There are the usual problems of renormalization, which are further complicated by the fact that the universe is not static, and also the issue of singularities. The latter should only be thought of a signaling the fact that General Relativity is being applied outside its domain of validity. It should not be considered the ‘beginning or end of time’, but rather the boundary of the classical era of the universe. There is a well-developed model of quantum cosmology (Loop Quantum Cosmology) that shows how this classical singularity is resolved, being replaces by a ‘quantum Big-Bounce’. ‘Time’ in this context is a relational time (a la Leibnitz) and it is only in the classical (low-density) phase of the universe that we can translate this to a useful notion of ‘proper’ or ‘cosmic’ time. Loop Quantum Cosmology then gives us a universe with an infinite temporal extent and it shows an important relationship between semi-classicality and evolution. Namely, for semi-classical states the evolution with respect to relational time is largely insensitive to the choices one makes in defining the quantum theory (the self-adjoint extension). This suggests that our intuitive notion of cosmological time is tied to our semi-classicality.