Study Bounds Field Excursions Along Null Geodesics, Revealing Implications for Cosmology and Naked Singularities

Scalar fields, fundamental to many theoretical physics models, frequently possess a range of possible values, and changes in these values can produce detectable gravitational effects, from the earliest moments of the universe to the regions surrounding black holes. Aidan Herderschee of the Institute for Advanced Study and Aron C. Wall of the University of Cambridge, along with their colleagues, now demonstrate a quantitative limit on how much these scalar fields can vary along the paths light travels, a result they term a field excursion bound. The team derives this bound using established principles governing gravity, specifically the Raychaudhuri equation, and assumes a common condition regarding energy distribution in spacetime. This achievement significantly constrains theoretical models of inflation and offers new insights into scenarios where the universe might select its properties based on the conditions favourable to life, because it establishes a clear relationship between the extent of field variations and the expansion of the universe.

Early Universe Cosmology and Inflationary Theory

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Scalar Field Excursions Constrained by Raychaudhuri Equation

Scientists have developed a quantitative limit on how much scalar fields can vary along light paths, building upon the Raychaudhuri equation, a fundamental result in general relativity. This field excursion bound (FEB) constrains these variations, relevant in cosmology and near black holes where scalar fields can influence gravitational effects. The team rigorously derived this bound, demonstrating its saturation in spacetimes containing timelike naked singularities. To quantify these constraints, researchers calculated that, in a standard slow-roll inflationary scenario, a minimum of approximately 4√π 3 10150/ns e-folds is required to expect at least one viable vacuum.

This calculation considers the number of scalar fields and its relationship to observational constraints. They further explored the geometry of hyperbolic space, demonstrating that the ratio of volumes scales exponentially with the number of e-folds, and established that an excursion of at least (ns −1)∆φ L ≳150 log(10) ≈345 is necessary. Researchers connected this geometrical analysis to the required number of e-folds, deriving the bound ∆N ≳ (4√π) ns −1150 log(10) L ≈(2. 5 × 103) L ns −1. This bound suggests more e-folds may be required than current observations allow, unless a substantial number of scalar fields are present. To extend the analysis to quantum regimes, the study considered potential violations of the null energy condition due to quantum effects like Hawking radiation, and proposed a generalization of the FEB that accounts for entropy variation, explicitly retaining Newton’s constant to account for renormalization effects. This innovative approach allows for a more comprehensive understanding of the constraints on scalar fields in both classical and quantum cosmological models.

Field Excursions Constrained by Geodesic Expansion

Scientists have derived a quantitative field excursion bound (FEB) that constrains variations in scalar fields along light paths, expressing this bound in terms of the expansion parameter, θ, which quantifies how light rays are focused or defocused by gravity. The work builds upon previous research demonstrating access to regions deep within the vacuum manifold using extremal black holes, where entropy grows exponentially with field distance. The core principle underlying the FEB is that coherent variations in scalar field expectation values induce gravitational focusing of light rays. The team found that the FEB establishes a relationship between the change in θ and the geodesic distance between two spacetime points, governed by the equation log(θ2/θ1) ≥ 4r / (2π(D-2d(X2, X1))), where ‘r’ represents a constant, ‘D’ is the spacetime dimension, and d(X2, X1) is the geodesic distance calculated using the kinematic metric.

This bound holds whenever the signs of θ1 and θ2 are the same, indicating a consistent focusing or defocusing of light rays. Researchers verified the FEB in spacetimes containing a timelike naked singularity, demonstrating that the bound is saturated under these extreme conditions. In cosmological settings, the FEB implies that the extent of large field excursions is linearly upper-bounded by the number of e-folds, independent of the underlying cosmological model. This has significant implications for anthropic models of inflation, suggesting that the inflationary epoch should have lasted many e-folds, a prediction that stands in tension with the Hartle-Hawking wavefunction, which favors initial conditions with minimal inflation.

Researchers also proposed a potential generalization of the FEB to semiclassical spacetimes that violate the null energy condition, relying on a strengthened Quantum Focusing Condition (QFC). Although a formal proof of the QFC remains an open question, the team notes it leads to desirable properties of the spacetime beyond a quantum generalization of the FEB. While these quantum effects currently lack immediately evident phenomenological applications, they represent a significant formal interest and may prove relevant in future contexts.