How deep can a cosmic void be? Voids-informed theoretical bounds in Galileon gravity

Voids-informed theoretical bounds in Galileon gravity establish a new consistency test for scalar-tensor theories by linking non-linear void dynamics to cosmic expansion history, which yields a redshift-dependent upper bound on void depth and promotes this requirement to a new viability condition complementary to standard stability criteria.

The gist

This work introduces a void-based consistency test for Galileon scalar-tensor theories, showing that the previously reported unphysical breakdown of the predicted Newtonian force in certain Galileon models is controlled by a single condition linking non-linear void dynamics to the cosmic expansion history. This connection yields a redshift-dependent upper bound on the allowed depth of voids and promotes this requirement to a new viability condition, complementary to standard stability criteria.

Modified Newtonian Force and Pathological Regime

The study begins by introducing modifications to the Newtonian potential in Galileon scalar-tensor theories. At the non-linear level, the Newtonian gravitational potential, Ψ(t, x), is related to the non-linear density contrast, δ(t, x), through a modified Poisson equation: ∇2 a2Ψ = 4πG µNL ρ¯m δ (1). This modification is encapsulated in an effective coupling µNL(t, x) that encodes non-linear effects associated with screening. The analysis utilizes Galileon scalar-tensor theories characterized by the time-dependent α-basis, focusing on parameters αB and αM. Under the quasi-static (QS) approximation, the nonlinear solution for gravitational potentials depends on the matter source only through the enclosed density contrast ∆. The nonlinear modification to the effective gravitational coupling takes a specific form: µNL(a, R) = M2pl / M2 (1 + 2 M2/pl µL − 1)! R/RV3 × s1 + RV/R3 − 1 (2).

Void-Informed Consistency Criterion

The core contribution is the introduction of a simple, physically motivated criterion to identify the region of parameter space where the modified gravity force remains well defined in QS, spherically symmetric void configurations. This requires ensuring that the square-root argument in Eq. (2) to be non-negative for all void configurations (δ ≥ −1) (7), thereby guaranteeing a real non-linear coupling, µNL, and a well-defined fifth force. The criterion translates into the following condition: max 0 ≤ z ≤ zin fMG(z) > 1 ⇒ model excluded (8). This background function fMG is defined as the quantity in Eq. (6), which relates the ratio of screening scale to void radius cubed: RV/R3 = 4(αB + αM)(2αM + αB) Ωm / M2pl (α c2s)2 δ.

Implications for Parameter Space and Void Depth

Applying this criterion allows for the delineation of the viable parameter space. The condition max fMG > 1 excludes models where underdense configurations can reach δ = −1, meaning "any model for which fMG exceeds unity at any redshift during the structure-formation era necessarily admits configurations with fMG δ < −1, for which µNL becomes imaginary and the theory loses viability (8). The criterion also provides a practical prescription to bound void depth: the minimum allowed density contrast at a given redshift is δmin(z) = max [-1, -1/fMG(z)] (9). This yields a model-dependent upper limit on the depth of voids: in theories that never develop a pathology, one can reach δ = −1, while whenever fMG(z) > 1 voids with δ ∈ [−1, −1/fMG(z)] are forbidden at that redshift" (9).

Results and Conclusion

The application of this background-level diagnostic across the parameter space reveals that about 60% of the scanned models are ruled out by Eq. (8), and for almost all of them, the maximum of fMG lies at zmax ≲ 10 (5). This demonstrates that a single background function is sufficient to compute the minimum allowed under-density without solving non-linear dynamics. The findings position cosmic voids as sharp, theory-informed filters for viable MG, enabling more informed priors and parameter space choices in future cosmological inference. The method can be extended to any Modified Gravity (MG) theory exhibiting the same square-root structure in its non-linear force law.

The gist

This work introduces a void-based consistency test for Galileon scalar-tensor theories, showing that the previously reported unphysical breakdown of the predicted Newtonian force in certain Galileon models is controlled by a single condition linking non-linear void dynamics to the cosmic expansion history. This connection yields a redshift-dependent upper bound on the allowed depth of voids and promotes this requirement to a new viability condition, complementary to standard stability criteria.

How it works

Here are specific improvements to AI systems derived from the insights in this scientific paper, along with what those improved systems could do:

1. Enhanced Cosmological Model Constraint Engine:

The core contribution is a method for using background functions (like the function of Modified Gravity, fMG) to impose constraints on void depth before running computationally expensive non-linear simulations or N-body codes.

Improvement: Develop an AI module capable of rapidly calculating and mapping the viability region in parameter space based solely on the background evolution of a modified gravity theory (like Galileon). This module would directly implement the criterion from Eq. (8): "Exclude models where max fMG(z) > 1."

Improved AI System Capability: This system could be used in large-scale cosmological parameter estimation pipelines (e.g., those aiming to constrain Dark Energy or modified gravity). Instead of testing every model configuration through full simulations, the AI would instantly prune the parameter space, focusing computational resources only on models that satisfy this fundamental theoretical consistency check, drastically reducing inference time and computational cost while ensuring theoretical viability is maintained from the outset.

2. Automated Phenomenological Prior Generation for Structure Formation:

The paper shows how a single background function can yield a model-dependent upper limit on void depth, even without specific non-linear dynamics (Eq. 9).

Improvement: Create an AI agent that takes an observed cosmological dataset (e.g., galaxy surveys providing constraints on density fields) and automatically generates theory-informed priors for future structure formation studies. This agent would use the derived bounds, such as the minimum allowed density contrast, δmin(z), to define physically meaningful limits on void sizes or depths that are consistent with a specific MG theory (e.g., Galileon).

Improved AI System Capability: When analyzing N-body simulation results or observational data concerning cosmic voids (which are sensitive probes of MG), this AI could automatically filter out unphysical results—those that imply voids deeper than the model allows without triggering an imaginary force. This provides researchers with a theory-informed lens to interpret complex, noisy observational data, distinguishing between genuine physical signals and mathematical artifacts stemming from unphysical theory configurations.

3. Model Comparison and Diagnostic Selection Tool:

The paper establishes a systematic procedure: (1) Define the background function fMG(a), (2) Apply the threshold check max fMG > 1 to define viable parameter space, and (3) Calculate δmin(z).

Improvement: Develop an AI system that acts as a Diagnostic Selector. Given a set of competing Modified Gravity theories, this system could automatically generate the necessary background functions (fMG), compute the exclusion boundaries in the parameter space, and rank theories based on their robustness against this specific pathology.

Improved AI System Capability: This tool would allow theorists to efficiently benchmark new MG extensions against established constraints. It moves beyond simple stability checks by introducing a powerful, structure-sensitive diagnostic that directly addresses unphysical breakdown of the predicted Newtonian force in certain Galileon models. It could rapidly identify which theoretical extensions are most likely to be ruled out by realistic cosmological observations related to large-scale structure.

4. Real-Time Parameter Space Exploration (Bayesian Inference Enhancement):

The method provides a clear, background-dependent filter that can be applied before detailed non-linear evolution is considered.

Improvement: Integrate the void consistency criterion (Eq. 8) directly into Bayesian inference frameworks used for cosmological parameter estimation, particularly when exploring complex parameter spaces like those of scalar-tensor theories.

Improved AI System Capability: During Markov Chain Monte Carlo (MCMC) runs or Nested Sampling algorithms, the AI would use this criterion as a strong prior or an immediate rejection filter. This prevents the sampler from wasting computational cycles exploring regions of parameter space that are fundamentally non-viable due to unphysical forces, leading to faster convergence to reliable posterior distributions and more accurate constraints on modified gravity parameters.