Fractal Geometry in Quantum Cosmology

Fractal geometry introduces a paradigm in which spacetime exhibits scale-dependent structure characterised by non-integer dimensions. In quantum cosmology, such structures arise naturally when gravitational and quantum effects intertwine at the Planck scale, leading to a dynamical dimensional flow that departs from the smooth manifold of classical general relativity. Fractional calculus provides the mathematical toolkit to describe these phenomena through derivatives and integrals of arbitrary order, allowing models to interpolate between classical and fractal regimes. Central to this field is the study of modified field equations—most notably the Wheeler–DeWitt equation in minisuperspace—where fractional operators encode memory effects and non-locality, yielding new insights into singularity resolution, the emergence of inflationary dynamics and the evolution of black-hole interiors without classical divergences. Multifractional spacetimes further enrich this picture by assigning distinct fractal measures to different directions or scales, offering potential observational signatures in the cosmic microwave background and gravitational-wave propagation. Collectively, these approaches aim to bridge the gap between quantum gravity and cosmological observations, proposing novel routes to address the flatness problem, the nature of dark energy and the initial conditions of the Universe.

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Technical terms

Fractal geometry: A mathematical framework describing structures that exhibit self-similarity and non-integer dimensionality across scales.

Fractional derivative: A generalisation of integer-order differentiation allowing operations of arbitrary real or complex order, used to capture memory effects and scale-dependent dynamics.

Wheeler–DeWitt equation: A fundamental equation in canonical quantum gravity that governs the quantum state of the universe’s geometry within the minisuperspace approximation.

Dimensional flow: A phenomenon in quantum gravity whereby the effective dimensionality of spacetime changes with observational scale, often reducing from four to lower dimensions at high energies.

Minisuperspace: A reduced configuration space in quantum cosmology that restricts the universe’s geometry to a finite number of degrees of freedom by assuming homogeneity and isotropy.

References

1. A vacuum solution of modified Einstein equations based on fractional calculus. European Physical Journal C (2023).
2. Multifractional theories: an unconventional review. Journal of High Energy Physics (2017).
3. From Fractional Quantum Mechanics to Quantum Cosmology: An Overture. Mathematics (2020).
4. Cosmic microwave background and inflation in multi-fractional spacetimes. Journal of Cosmology and Astroparticle Physics (2016).
5. Does fractal universe describe a complete cosmic scenario ?. European Physical Journal C (2018).
6. Inflation and Fractional Quantum Cosmology. Fractal and Fractional (2022).
7. Dimension and Dimensional Reduction in Quantum Gravity. Universe (2019).