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

Figures (4)

• Figure 1: Allowed parameter space in the $(m_W, m_R)$ plane constrained by Experiment 1. The black line represents $m_R=m_W$.
• Figure 2: Allowed parameter space in the $(m_W, m_R)$ plane constrained by all of the experiments. Here, we employ the second-order approximation for $\beta(r)$ from Eq. \ref{['eq:betaO2']}. The blue "line" is the region constrained by Experiment 1 shown in Fig. \ref{['fig:ratio']}.
• Figure 3: A zoomed-in view of Fig. \ref{['fig:combined']}
• Figure 4: Allowed parameter space in the $(m_W, m_R)$ plane constrained by Experiment 1 and 2, where the first-order approximation, the second-order approximation, and the full expression of $\beta(r)$ are used, respectively. The region constrained by the second-order approximation of $\beta(r)$ almost perfectly overlaps with that of the full expression.