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Anisotropic quantum universe in Hořava-Lifshitz gravity

Vikramaditya Mondal1,* and Shinji Mukohyama2,3,4,†

  • *Contact author: vikramaditya.academics@gmail.com
  • †Contact author: shinji.mukohyama@yukawa.kyoto-u.ac.jp
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Phys. Rev. D 112, 083542 – Published 22 October, 2025

DOI: https://doi.org/10.1103/6wp2-b17k

Abstract

We quantize a Bianchi IX universe in Hořava-Lifshitz theory. For analytical tractability, we consider the small anisotropy limit of the Bianchi IX, that is, a perturbative anisotropic deformation of a closed, homogeneous, and isotropic universe. In the case of the projectable theory we further set the “dark matter as integration constant” to zero by assuming that the space consists of only one connected piece. In that limit and under the assumption, we first study the semiclassical Wentzel-Kramers-Brillouin (WKB) solutions to the Wheeler-DeWitt equation. We find the wave function of the universe, up to an overall normalization, and estimate the semiclassical tunneling probability for the emergence of an expanding universe. We establish a dictionary of correspondence between the WKB wave functions in general relativity and Hořava-Lifshitz theory in the large-scale factor (or IR) limit. For a small universe (UV limit), on the other hand, due to contributions from higher-dimensional operators, the anisotropies decouple from the scale factor, a behavior significantly different from general relativity, and analytic solutions to the Wheeler-DeWitt equation beyond the WKB approximation can be found. The wave function of the scale factor satisfies the DeWitt criterion, whereas the wave functions of anisotropies resemble those of quantum harmonic oscillators. The quantum prediction for the initial condition of anisotropies is obtained in terms of the coupling parameters of Hořava-Lifshitz theory. We find a bound on the coupling parameters from the normalizability of the wave functions of anisotropies. Further, we calculate the expectation values for squared anisotropic shear and squared anisotropies in both the large universe and the small universe limits.

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Phys. Rev. D 112, 083542– Published 22 October, 2025
  • Received 29 May 2025
  • Accepted 26 September 2025
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