Conformal and pure scale-invariant gravities in d dimensions (2506.18775v1)

Abstract: We consider conformal and scale-invariant gravities in d dimensions, with a special focus on pure $R^2$ gravity in the scale-invariant case. In four dimensions, the structure of these theories is well known. However, in dimensions larger than four, the behavior of the modes is so far unclear. In this work, we explore this question, studying the theories in conformally flat spacetimes as well as anisotropic backgrounds. First, we consider the pure theory in d-dimensions. We show that this theory propagates no degrees of freedom for flat space-time. Otherwise, we find the theory in the corresponding Einstein frame and show that it propagates a scalar field and two tensor modes, that arise from Einstein's gravity. We then consider conformal gravity in d dimensions. We argue on the number of degrees of freedom for conformally flat space-times and show that for $d>4$, there exists a frame in which this theory can be written as the Weyl-squared gravity with a cosmological constant, and also generalize this formulation to the $f\left(W^2\right)$ theories. Then, we consider the specific model of conformal gravity in five dimensions. We find the analytical and numerical solutions for the anisotropic Universe for this case, which admits super-Hubble and exponential expansions. Finally, we consider the perturbations around these solutions and study the number of the degrees of freedom.

• The paper demonstrates that pure scale-invariant gravity in higher dimensions exhibits no propagating degrees in flat spacetime while unveiling additional modes in curved backgrounds after an Einstein frame transformation.
• It shows that conformal gravity, extended beyond four dimensions with transitional metrics and auxiliary scalar fields, potentially supports a richer spectrum including scalar and vector fields.
• The study presents analytical and numerical insights—such as Friedmann-like solutions and de Sitter behaviors—that offer promising implications for cosmology and unified field theories.

Conformal and Pure Scale-Invariant Gravities in Higher Dimensions

The paper "Conformal and Pure Scale-Invariant Gravities in d Dimensions" by Anamaria Hell and Dieter Lüst explores the complex domain of higher-derivative gravitational theories, specifically focusing on conformal and pure scale-invariant gravity across dimensions greater than four. Higher-derivative theories of gravity, which include terms of higher powers of curvature tensors in the gravitational action, have historically been considered as candidates for a renormalizable theory of gravity, albeit often at the price of introducing ghost modes—unphysical solutions with negative energy leading to potential instabilities.

Theoretical Framework and Approach

1. Pure Scale-Invariant Gravity: The authors analyze pure $R^2$ gravity, examining its behavior in various backgrounds. In four-dimensional spacetime, it is known that pure $R^2$ gravity is scale-invariant and devoid of propagating degrees of freedom in flat spacetime. The paper extends this analysis to higher dimensions, demonstrating that the absence of propagating modes in flat spacetime persists. However, for non-flat backgrounds, the inclusion of a positive cosmological constant is necessary when recasting the theory in its Einstein frame equivalent, which reveals one scalar and two tensor degrees of freedom.
2. Conformal Gravity: The second focal point of the paper is conformal gravity, distinguished by its Weyl-squared action. In four dimensions, conformal gravity is notorious for ghost-like tensor modes and lacks scalar propagating degrees. The transition to higher dimensions necessitates an adaptation to maintain conformal invariance, achieved through transitional metrics and auxiliary scalar fields. These modifications hint at a potentially richer spectrum of gravitational modes, including additional scalar and vector fields when the theory is examined in alternative frames.

Analytical and Numerical Insights

The investigation further segments into exploring conformal gravity in five-dimensional spacetimes, specifically within anisotropic and conformally flat backgrounds. By analyzing the Friedmann-like equations derived from the modified actions, the authors identify multiple exact and numerical solutions indicating accelerated expansion and varying cosmological epochs. For example, scenarios of exponential expansion followed by super-accelerated phases are meticulously elucidated, with convergence towards de Sitter-like behavior.

Implications and Consequences

Theoretical advancements in understanding higher-dimensional and higher-derivative gravities bear significant implications for unified field theories, including string theory and its various compactifications which inherently suggest the existence of extra spatial dimensions. The rich spectrum of potentially observable states in higher dimensions uncovered by this paper may offer new pathways for avoiding instabilities commonly associated with higher-derivative quantizations.

Importantly, this research suggests that special boundary conditions or alternative formulations (akin to Einstein frame transformations) might mitigate the ghost problem, extending the feasibility of these theories as effective descriptions of gravitational phenomena, possibly even influencing cosmological inflationary models.

Speculation on Future Directions

Given the sophisticated interplay of geometric and physical degrees of freedom illuminated in higher-dimensional settings, further studies might delve into implications for holographic dualities, quantum cosmology, and dark energy. Moreover, understanding the computation of anomalies and effective action in these frameworks remains an open question of considerable interest, particularly in the context of all pervasive quantum field theories both in both four and extra-dimensional settings. Such advancements could foreseeably shape the way theoreticians conceptualize the fundamental structure of gravity and spacetime.