Parametrized Post-Newtonian Lensing Framework

• The PPN lensing framework is a systematic expansion of gravitational metrics in inverse powers of the speed of light, linking weak lensing observables to underlying gravitational potentials.
• It extends traditional post-Newtonian theory to include second-order and nonlinear corrections such as lens-lens coupling and gravitomagnetic effects in both GR and modified gravity models.
• This approach provides a theory-agnostic tool to detect departures from GR and control non-Gaussianities, ensuring robust mass mapping in cosmological weak lensing analyses.

A parametrized post-Newtonian (PPN) lensing framework systematically extends the post-Newtonian expansion of gravity to cosmological and astrophysical settings, enabling the analysis of weak gravitational lensing—including its second-order and nonlinear corrections—in both general relativity (GR) and broad classes of modified gravity models. The PPN formalism introduces order-by-order metric expansions with theory-agnostic parameters (notably γ, β, ε), allowing precise mapping between weak-field gravitational effects and lensing observables. This framework plays a critical role in distinguishing departures from GR and quantifying potential systematics in cosmological lensing analyses.

1. Foundations of the Parametrized Post-Newtonian (PPN) Lensing Formalism

The PPN lensing framework originates by recasting the gravitational metric as a perturbative series in inverse powers of the speed of light (1/c), with each order associated with a distinct contribution to the gravitational field and, hence, to light propagation. The fundamental metric in this expansion is expressed, at up to second post-Newtonian order (2PN), as: $$ds^2 = -[1 - (2U/c^2) + (2\beta U^2/c^4)]c^2 dt^2 + [1 + (2\gamma U/c^2) + (3\epsilon U^2/(2c^4))] d\vec{x}^2$$ where $U$ is the Newtonian potential, while $\gamma$, $\beta$, and $\epsilon$ are PPN parameters distinguishing linear ($U$) and nonlinear ($U^2$) corrections (with generalizations for time-dependent and vector perturbations).

In GR, the canonical values $\gamma = \beta = \epsilon = 1$ are fixed, but in scalar–tensor and other modified gravity theories, these parameters may take different values, directly affecting the gravitational dynamics experienced by light rays. This formalism captures both first-order (Born approximation) and nonlinear contributions (e.g., lens-lens coupling, gravitomagnetic corrections).

2. Deflection Angle and Second-Order Corrections in Weak Lensing

Within the framework, the calculation of photon trajectories is based on the null geodesic equation: $$\frac{dk^\alpha}{d\lambda} = -\Gamma^\alpha_{\mu\nu} k^\mu k^\nu$$ yielding the observed deflection angle at linear order: $$\alpha_i^{(1)}(\vec{\theta}, w) = \frac{1+\gamma}{c^2} \int_0^w dw' \left( \frac{w - w'}{w} \right) \partial_i U[w'\vec{\theta}, w']$$ Second-order (2PN) deflections encapsulate Born corrections (displacing the path within the potential) and explicit nonlinear metric corrections. The 2PN term contains the key combination

$(6 - 4\beta + 4\gamma + 3\epsilon - 2\gamma^2)$

which determines the amplitude of second-order, non-Gaussian features in the convergence field and allows explicit tracking of departures from GR.

Gravitomagnetic effects (from vector potentials $V_i$) and higher-order scalar potentials (such as $\Phi_2$, $\epsilon'$, $\beta'$) are also included in the most general cosmological parametrization, rendering the framework flexible for alternative gravity theories.

3. Convergence, Distortion Tensor, and Non-Gaussianity Diagnostics

After obtaining deflection angles $\alpha_i$, the distortion tensor is constructed as: $$\psi_{ij} = \frac{\partial \alpha_i}{\partial \theta_j}$$ The convergence is then

$\kappa(\vec{\theta}) = \frac{1}{2} \psi_{ii}$

which is decomposed as: $$\kappa(\vec{\theta}) = \kappa^{(1)}(\vec{\theta}) + \kappa^{(2)}(\vec{\theta}) + \kappa^{(\mathrm{GM})}(\vec{\theta})$$ with the leading-order (linear) term

$$\kappa^{(1)}(\vec{\theta}) = \frac{1+\gamma}{c^2} \int_0^w dw' g(w, w') \nabla^2 U[w'\vec{\theta}, w']$$

where $g(w, w') = w'(w - w') / w$ is the lensing kernel. The second-order correction, $\kappa^{(2)}$, includes products of $U$ and its gradients, notably a term proportional to

$$(6 - 4\beta + 4\gamma + 3\epsilon - 2\gamma^2) U \nabla^2 U$$

and a “lens-lens” coupling of the form $(1+\gamma)^2 (\partial_{ijk} U \ldots)$. The amplitude of extra non-Gaussianity in the observed convergence is quantified by an effective parameter: $$\Delta f_{\mathrm{NL}} = \frac{1}{8}(1 - 4\beta + 3\epsilon) = \frac{1}{8}(3\gamma^2 + \gamma - 4)$$ Deviations from GR amplify lensing-induced non-Gaussian statistics, potentially biasing mass reconstructions if non-negligible.

4. Application to Scalar–Tensor and Alternative Gravity Theories

The PPN lensing framework fundamentally enables the analysis of a general class of modified gravity models. When applied to scalar–tensor theories, such as Jordan–Brans–Dicke, $f(R)$ gravity, and chameleon/symmetron models, the second-order coefficients show a remarkable redundancy: $$3\epsilon - 4\beta = -5 + \gamma + 3\gamma^2 \quad \text{(for many scalar–tensor theories)}$$ This implies that the entire family of nonlinear corrections—which could, in principle, have produced significant observable systematics—depend only on the gravitational slip parameter, $\gamma$. Because astrophysical and Solar System tests confine $|\gamma - 1| \ll 1$, the magnitude of $\Delta f_{\mathrm{NL}}$ is suppressed, with $|\Delta f_{\mathrm{NL}}| < 1$ for $\gamma$ in the range $1/2 \leq \gamma \leq 1$.

A crucial implication is that higher-order corrections do not generically introduce non-Gaussianities that would compromise weak lensing as a direct tracer of underlying mass, provided the gravity theory passes standard local tests ($\gamma \approx 1$).

5. Robustness of Weak Lensing in the PPN Cosmological Context

The cosmological adaptation of the PPN formalism preserves the key features required to extend Solar System–tested gravity into the cosmological, statistically inhomogeneous regime. With the inclusion of time-dependent background, vector potentials, and general scalar–tensor sector parameters, the framework links the observable lensing fields directly to both the underlying matter distribution and to possible gravitational extensions. The preservation of “clean” weak-lensing mass mapping holds for all theories in which the higher-order PPN corrections revert to near-GR values in the presence of strong astrophysical constraints on $\gamma$.

Therefore, weak lensing remains robust against non-Gaussian contamination or systematic bias induced by commonly studied scalar–tensor modifications and accurately tracks the distribution of cosmic mass even in the presence of second-order metric nonlinearities.

6. Key Formulas and Observational Implications

A synopsis of central formulas for practical lensing calculations within this framework:

Context	Key Expression	Description
2PN Metric	$$ds^2 = -[1 - (2U/c^2) + (2\beta U^2/c^4)] c^2dt^2 + [1 + (2\gamma U/c^2) + (3\epsilon U^2/(2c^4))] d\vec{x}^2$$	Metric with explicit PPN parameters at 2PN order
1st Order $\kappa$	$$\kappa^{(1)}(\vec{\theta}) = \frac{1+\gamma}{c^2} \int_0^w dw' g(w,w') \nabla^2 U[w'\vec{\theta}, w']$$	Linear-order convergence as a function of PPN $\gamma$
Effective non-Gaussianity	$$\Delta f_{\mathrm{NL}} = \frac{1}{8}(3\gamma^2 + \gamma - 4)$$	Modulation of lensing-induced non-Gaussianity by gravity theory parameters

Observationally, this underpinning ensures mass reconstructions and cosmological parameter inferences from weak lensing surveys are not contaminated by additional gravitationally-sourced non-Gaussianity—unless future constraints reveal small, but nonstandard, values for $\gamma$ or associated PPN parameters.

7. Synthesis and Theoretical Significance

The parametrized post-Newtonian lensing framework provides a comprehensive, order-controlled route to analyzing weak lensing and its corrections in GR and alternative theories. For a broad class of scalar–tensor theories, the nonlinear corrections to gravitational lensing convergence and induced non-Gaussianities are entirely controlled by the gravitational slip parameter $\gamma$, which is observationally constrained to be close to unity. Consequently, 2PN and associated higher-order modifications are strongly suppressed and do not threaten the utility of weak lensing as a direct, unbiased tracer of mass on cosmological scales.

This formalism, as implemented for both model discrimination and practical data analysis, supplies a theory-agnostic infrastructure for future empirical investigations of gravity, mass mapping, and cosmological inference (Vanderveld et al., 2011).