Parameterized Post-Newtonian Analysis of Quadratic Gravity and Solar System Constraints

Abstract

This work systematically investigates the post-Newtonian behavior of general quadratic gravity in the weak-field regime. By extending the Einstein-Hilbert action to include quadratic curvature terms as $\mathcal{L}\propto R-\hat{I}»C^2+\hat{I}¼R^2$, the theory introduces two massive modes: a scalar mode and a ghost tensor mode. Using the post-Newtonian expansion method, we derive the explicit expressions for the metric for a general source up to 1.5PN order. Furthermore, for a point-mass source, we extend the solution to 2PN order and evaluate the effective Parameterized Post-Newtonian parameters ${\hat{I}}^3(r)$ and ${\hat{I}}^2(r)$. The results show that deviations from General Relativity are exponentially suppressed. The theory has the feature ${\hat{I}}^3(r)\equiv 1$ when $m_R=m_W$, and to ensure that gravity remains attractive, we have $m_W>m_R/4$. The leading correction to ${\hat{I}}^2(r)$ exhibiting a characteristic $\mathcal{O}(r\ln (r)e^{-mr})$ dependence. Based on the Solar System experiments, we derive preliminary constraints on the theory's parameters: $m_R,m_W\gtrsim 23{\mathrm{AU}}^{-1}$, corresponding to $\hat{I}»\lesssim 2.1\times 10^{19}{\mathrm{m}}^2$ and $\hat{I}¼\lesssim 7.1\times 10^{18}{\mathrm{m}}^2$. This study provides a theoretical foundation for future tests of quadratic gravity using pulsar timing arrays, gravitational-wave observations, and laboratory-scale short-range gravity experiments.

18 pages, 4 figures