Holographic renormalization in no-boundary quantum cosmology

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Recent results of Hartle-Hawking wave functions on asymptotic dS boundaries show non-normalizability, while the bulk origin is not clear. This paper attempts to addresse this problem by studying (Kerr-)dS₃ cosmology in Einstein gravity deformed by a minimally coupled scalar field. Various saddle-point contributions exp[i(saddle-point action)] to the Hartle-Hawking wave functions are computed with mini-superspace formalism. The saddle-point actions are first obtained in the spacetime bulks by direct computation, then on the asymptotic dS boundary by holographic renormalization. It is found that the imaginary part of the saddle-point actions, as functions of scalar field deformation, are generally bounded in the bulk, but can diverge to -infty on the boundary. This can probably be a source of the scalar field-related non-normalizability of the boundary Hartle-Hawking wave functions. For Kerr-dS₃ cosmology, some saddle-point actions have imaginary parts diverging to -infty both in the bulk and on the boundary, when the boundary T² is stretched to infinitely long. This can be an origin of non-normalizability of the boundary Hartle-Hawking wave functions related to temperature divergence. Finally the holographic renormalization is extended to scalar-deformed dSd₊₁ (d>= 3) cosmologies for the imaginary part of the saddle-point actions. The result is tested on a 5d example, and saddle points leading to divergent contribution to the boundary Hartle-Hawking wave function are shown to exist.