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

TL;DR

This paper analyzes the post-Newtonian behavior of quadratic gravity theories, deriving explicit metric solutions and PPN parameters, and constrains their parameters using Solar System experiments.

Contribution

It extends the post-Newtonian analysis of quadratic gravity to higher orders and provides constraints on the theory's parameters from Solar System tests.

Findings

Deviations from GR are exponentially suppressed in the PPN parameters.

When $m_R=m_W$, the parameter $eta(r)$ has a specific correction form.

Solar System constraints imply $m_R,m_W extgreater 23$ AU$^{-1}$.

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 $\mathcal{L}\propto R-\lambda C^2+\mu R^2$, 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 $\gamma(r)$ and $\beta(r)$. The results show that deviations from General Relativity are exponentially suppressed. The theory has the feature $\gamma(r)\equiv 1$ when $m_R=m_W$, and to ensure that gravity remains attractive, we have $m_W>m_R/4$. The leading correction to $\beta(r)$ 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: $m_R,m_W\gtrsim23~\mathrm{AU}^{-1}$, corresponding to $\lambda\lesssim2.1\times10^{19}~\mathrm{m}^2$ and $\mu\lesssim 7.1\times 10^{18}~\mathrm{m}^2$. 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.