Stress Tensor in De Sitter Space

Abstract

The Hadamard regularization calculation is carried out to obtain the renormalized stress tensor of scalar fields with respect to the de Sitter invariant vacuum. We discuss how the condition of de Sitter invariance fixes the form of the Hadamard series expansion of the Hadamard elementary function. We obtain the value of the mass scale appearing in the logarithmic term in the Hadamard series expansion. We also compare the renormalized stress tensor of conformally invariant field in the de Sitter invariant and conformal vacua. For the purpose of investigating the role of quantized fields in cosmology, quantum field theory in de Sitter space is an interesting problem because the high symmetry of de Sitter space enables us to obtain the simple exact solution of the field equation. It is also expected to give fruitful information on the inflationary model of the early universe. It is one of the most important subjects to obtain the renormal­ ized expectation value of stress tensor of quantized fields with respect to certain vacuum states,. because the particle concept is very obscure in curved space and the renormalized stress tensor acts as the source term in the Einstein equation by con­ sidering the semi-classical back-reaction of quantized fields on the background geome­ try. The problem of how to select a physically meaningful vacuum state, however, remains unsolved. Vacuum states are sometimes characterized by symmetric prop­ erties or coordinatization of the background manifold. The most symmetric vacuum in de Sitter space is the de Sitter invariant one, which was first described by Chernikov and Tagirov 1 ) and have been studied by Schomblond, Spindel,2) Mottola 3 ) and Allen. 4 ) Several authors have evaluated the renormalized stress tensor with respect to the de Sitter invariant vacuum. Bunch and Davies investigated the behaviour of quantized scalar field and evaluated the renormalized stress tensor in de Sitter space with point-splitting regularization. 5 ) Their regularization method was formulated in the Friedmann universe not in the general curved spacetime. 6 ) Thus the subtracted divergent terms can be ambiguous. Dowker and Critchley evaluated the renormal­ ized stress tensor with the zeta function regularization,7) and Birrell and Davies did with the dimensional regularization. 8 ) Since they evaluated only the trace of the stress tensor with the effective Lagrangian, they could not obtain the state-dependent contribution. Allen, Bernard and Folacci applied their Hadamard regularization scheme to the de Sitter invariant vacuum. 9 ),IO) They used the Lagrangian with an additional term to give the limiting value in the flat space limit. There have been many works on this problem, but a few works with straightforward calculation. And the results for massive scalar fields are obscure. It is also still not clear how the