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Quantum estimation of cosmological parameters

Michał Piotrak, Thomas Colas, Ana Alonso-Serrano, Alessio Serafini

Abstract

Understanding how well future cosmological experiments can reconstruct the mechanism that generated primordial inhomogeneities is key to assessing the extent to which cosmology can inform fundamental physics. In this work, we apply a quantum metrology tool - the quantum Fisher information - to the squeezed quantum state describing cosmological perturbations at the end of inflation. This quantifies the ultimate precision achievable in parameter estimation, assuming ideal access to early-universe information. By comparing the quantum Fisher information to its classical counterpart - derived from measurements of the curvature perturbation power spectrum alone (homodyne measurement) - we evaluate how close current observations come to this quantum limit. Focusing on the tensor-to-scalar ratio as a case study, we find that the gap between classical and quantum Fisher information grows exponentially with the number of e-folds a mode spends outside the horizon. This suggests the existence of a highly efficient (but presently inaccessible) optimal measurement. Conversely, we show that accessing the decaying mode of inflationary perturbations is a necessary (but not sufficient) condition for exponentially improving the inference of the tensor-to-scalar ratio.

Quantum estimation of cosmological parameters

Abstract

Understanding how well future cosmological experiments can reconstruct the mechanism that generated primordial inhomogeneities is key to assessing the extent to which cosmology can inform fundamental physics. In this work, we apply a quantum metrology tool - the quantum Fisher information - to the squeezed quantum state describing cosmological perturbations at the end of inflation. This quantifies the ultimate precision achievable in parameter estimation, assuming ideal access to early-universe information. By comparing the quantum Fisher information to its classical counterpart - derived from measurements of the curvature perturbation power spectrum alone (homodyne measurement) - we evaluate how close current observations come to this quantum limit. Focusing on the tensor-to-scalar ratio as a case study, we find that the gap between classical and quantum Fisher information grows exponentially with the number of e-folds a mode spends outside the horizon. This suggests the existence of a highly efficient (but presently inaccessible) optimal measurement. Conversely, we show that accessing the decaying mode of inflationary perturbations is a necessary (but not sufficient) condition for exponentially improving the inference of the tensor-to-scalar ratio.

Paper Structure

This paper contains 27 sections, 141 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notations and units.
  3. Symplectic geometry in field theory
  4. Symplectic geometry
  5. The symplectic group
  6. Early universe cosmology
  7. Background evolution
  8. Cosmological perturbations
  9. Quantization.
  10. Evolution.
  11. Covariance
  12. Squeezing.
  13. Quantum Fisher information
  14. Gaussian parameter estimation
  15. Fisher information from the characteristic function
  16. ...and 12 more sections

Figures (3)

  • Figure 1: Entries of the covariance matrix $\sigma_ {\cal R}$ as a function of the number of e-folds $N_{\mathrm{ef}} \equiv - \log (- k \eta)$
  • Figure 2: Squeezing parameters as a function of the number of e-folds $N_{\mathrm{ef}} \equiv - \log (- k \eta)$. Left: Squeezing parameter $r$. Right: Squeezing angle $\theta$.
  • Figure 3: Left: Fisher information (in units of $1/\epsilon^2_1$), as a function of the number of e-folds. In the far past, the Fisher information is approximately constant and at positive e-folds it begins to grow exponentially. Right: the function $F(\eta)$, which determines the Cramér-Rao bound. One can notice the variance is lower bounded by a constant value at negative e-folds, and unbounded for positive.