Causality constraints on corrections to Einstein gravity (2201.06602v1)

Abstract: We study constraints from causality and unitarity on $2\to2$ graviton scattering in four-dimensional weakly-coupled effective field theories. Together, causality and unitarity imply dispersion relations that connect low-energy observables to high-energy data. Using such dispersion relations, we derive two-sided bounds on gravitational Wilson coefficients in terms of the mass $M$ of new higher-spin states. Our bounds imply that gravitational interactions must shut off uniformly in the limit $G \to 0$, and prove the scaling with $M$ expected from dimensional analysis (up to an infrared logarithm). We speculate that causality, together with the non-observation of gravitationally-coupled higher spin states at colliders, severely restricts modifications to Einstein gravity that could be probed by experiments in the near future.

• The paper demonstrates how causality and unitarity constraints on 2-to-2 graviton scattering yield strict bounds on higher-derivative terms in Einstein gravity.
• It derives dispersion relations linking low-energy observables with high-energy physics, thereby constraining the Wilson coefficients in gravitational effective field theories.
• The results imply that significant modifications to general relativity require new higher-spin states, suggesting observable consequences at energies below the Planck scale.

Overview of Causality Constraints on Corrections to Einstein Gravity

The paper "Causality constraints on corrections to Einstein gravity," authored by Simon Caron-Huot, Yue-Zhou Li, Julio Parra-Martinez, and David Simmons-Duffin, presents a detailed exploration of the constraints imposed by causality and unitarity on graviton scattering processes within the framework of effective field theories (EFTs). The authors address how these constraints can inform the permissible modifications to Einstein's general relativity, particularly through higher-derivative contributions to the gravitational action.

General relativity has been remarkably successful but is frequently scrutinized in relation to cosmic phenomena like dark energy and dark matter. In addressing potential modifications to general relativity, the paper centers on higher-derivative terms in gravitational theories, which naturally emerge in string theories and other UV-complete quantum gravity settings. These terms lead to significant effects at short distances, potentially contributing to the resolution of aforementioned cosmic puzzles.

Key Concepts and Methodology

The paper investigates $2 \to 2$ graviton scattering within four-dimensional, weakly-coupled EFTs. The authors utilize causality and unitarity to derive dispersion relations that link low-energy observables to high-energy data. They particularly focus on constraints for the Wilson coefficients of gravitational interactions, which scale according to the mass $M$ of new higher-spin states.

The authors structure their analysis around several foundational assumptions:

1. Spectrum Assumptions: At low energies, the existence of a massless graviton field is complemented by a finite number of fields with spins less than or equal to two. These fields are described by an effective low-energy action, often modified by higher-derivative terms including contractions of Riemann tensors.
2. Causality Constraints: The principle that signals cannot travel faster than light governs the constraints applied to the corrections of Einstein gravity. Such constraints are formulated through dispersion relations and positivity bounds.
3. Unitarity Constraints: The partial wave expansion of scattering amplitudes must abide by conditions that ensure positive semi-definite matrices for probabilities, thereby respecting unitarity.
4. Analyticity and Crossing Symmetry: These play crucial roles in relating amplitudes across different energy regions, allowing the derivation of bounds from known principles of scattering theory.

Results and Implications

The paper provides precise numerical bounds for cubic and quartic gravitational couplings $g_3$ and $g_4$. Notably, expressions such as $|g_3|^2M^8 \leq 24.9 \log(M/m_{\text{IR}}) - 27.6$ hint at the requirement of new higher-spin states if significant corrections to general relativity are to be observed. A salient finding is that dimensionless ratios $g_k M^{2(k-4)}/g_4$ are constrained by order-unity constants, suggesting tight bounds on the possible corrections to gravity at low energies.

The implications of this research are multifaceted:

• Theoretical: The bounds provide insight into the structure of gravitational theories that can be reconciled with causality and unitarity. They suggest that modifications to Einstein gravity are severely constrained unless accompanied by new, potentially observable states at energies below the Planck scale.
• Practical: Collider experiments probing energies above scales predicted by the bounds should be able to detect higher-spin states if substantial deviations from general relativity exist at astrophysical scales.
• Future Developments: The research indicates directions for future inquiries into the nature of gravity with considerations for higher-dimensional theories and the role of string theory-like structures in formulating consistent gravitational interactions.

Conclusion

Overall, the paper extends the understanding of how causality constraints shape modifications to Einstein gravity, emphasizing that significant departures from general relativity require new physics at accessible scales. The rigorous mathematical treatment, coupled with potential implications for experiments, underscores the relevance of dispersion relations in governing the structure of gravitational theories. The authors articulate a vision where gravity's interaction dynamics are delicately traced to the presence or absence of new high-energy states, paving the way for both theoretical advancements and experimental exploration.