---
abstract: |
    The post-Newtonian formulation of a general class of $f(R)$ theories is set up to $3^{rd}$ order approximation.  It turns out that the information of a specific form of $f(R)$ gravity is encoded in the Yukawa potential, which is contained in the perturbative expansion of the metric components.  It is shown that the Yukawa potantial does appear in the $3^{rd}$ order expression of the effective refraction index of light, although it is cancelled in the $2^{nd}$ order expression.  Therefore the $f(R)$ theories are distinguishable from general relativity by gravitational lensing effect at the $3^{rd}$ order post-Newtonian approximation.  Our result opens the possibility to bring new insights into the issue of dark matter from $f(R)$ gravity. 

    Keywords

    :   $f(R)$ theory, post-Newtonion approximation, gravitational lens

    PACS numbers

    :   04.50.Kd, 04.25.Nx, 98.62.Sb

author:
- Hongguang Liu
- Xin Wang
- Haida Li
- Yongge Ma
bibliography:
- 'refs.bib'
title: 'Distinguishing $f(R)$ theories from general relativity by gravitational lensing effect'
---

Introduction
============

Recently modified gravity theories have received increasinged attention in issues related to “dark energy” [@riess2004type; @eisenstein2005detection; @astier2006supernova], “dark matter” [@page2007three; @zwicky1933spectral; @rubin1970rotation; @di2013dark], as well as non-trivial tests on gravity beyond general relativity (GR) [@Will:2014bqa].  Historically, Einstein’s GR is the simplest relativistic theory of gravity with correct Newtonian limit.  To pursue new physics, Weyl and Eddington even began to consider modifying GR just after it was established [@weylei; @eddington1920mathematical].  From the viewpoints of perturbutive quantum gravity, GR is non-renormalizable [@utiyama1962renormalization; @stelle1977renormalization; @buchbinder1992effective; @vilkovisky1992effective], while higher order gravity theories might alleviate the problem.  From the phenomenological viewpoints, there are many ways to modify GR, and some empirical approaches seem to have promising prospect, such as Dvali-Gabadadze-Porrati gravity [@dvali20004d], tensor-vector-scalar theory [@fujii2003scalar] and Einstein-Aether theory [@jacobson2001gravity].  Among such extended theories, particular attention has been devoted to the so-called $f(R)$-gravity.  This kind of theories is based on a generalization of the Einstein Hilbert Lagrangian to nonlinear functions $f(R)$ of the Ricci scalar [@sotiriou2010f].  $f(R)$-gravity covers a lot of characteristics of higher order gravity and is convenient to be operated.  Hence, $f(R)$ theories provide an ideal tool to study the possible extension of GR.  $f(R)$ theories of gravity can also be non-perturbatively quantized by loop quantum gravity approach [@Zhang:2011vi; @Zhang:2011qq].  To confront $f(R)$-gravity with observations in Solar System, one can get constraints on the theories from different measurements, such as the EötWash experiment [@kapner2007tests], the geodesic precession of gyroscopes measured by Gravity Probe B [@Everitt:2011hp] and the precession of the binary pulsars PSR J0737-3039 [@Breton:2008xy].  At cosmological scales, one would expect to employ $f(R)$ theories to account for the problems of “dark energy” [@nojiri2003modified; @capozziello2003curvature; @Huang:2009vw] and “dark matter” [@sobouti2007mathsf; @capozziello2009testing; @capozziello2012dark] needed in GR.  If $f(R)$ gravity could account for dark matter, besides matching the rotation curves of galaxy clusters, it should also match the measurements on gravitational lensing effect [@schneider1996detection].  However, it is shown in [@Lubini11] that, at $2^{nd}$ order post-Newtonian approximation, a rather general class of $f(R)$ theories is indistinguishable from GR in gravitational lensing effect.  Nevertheless, we will show in this paper that the class of $f(R)$ theories is indeed distinguishable from GR in gravitational lensing effect at $3^{rd}$ order post-Newtonian approximation.  In section \[ch:field\_equations\_fr\], we briefly review the field equations of metric $f(R)$-gravity.  In Sec.  \[ch:post\_newtonian\_expansion\] the post-Newtonian approximation of the class of $f(R)$ theories is formulated to the desired order.  In Sec.  \[ch:gravitational\_lensing\] we introduce the gravitational lensing in metric theories of gravity and show how the $f(R)$ gravity can be distinguishable from GR at $3^{rd}$-order post-Newtonian approximation.  Finally, conclusions and remarks are given in Sec.  \[ch:summary\].  Throughout the paper, the metric tensor $g_{\mu\nu}$ takes the signature $(-,+,+,+)$.

Field Equations of $f(R)$ theory {#ch:field_equations_fr}
================================

In metric $f(R)$ theories of gravity, the action of gravity coupled to matter fields is given by $$\label{eq:action}
S=\frac{1}{2\chi}{\int}f(R)\sqrt{-g}d^{4}x+S_{M}$$ where $g$ is the determinant of the metric tensor $g_{{\mu}{\nu}}$, $\chi=8{\pi}G/c^{4}$ with $c$ being the vacuum speed of light, $R=g^{{\mu}{\nu}}R_{{\mu}{\nu}}$ is the Ricci scalar, $f(R)$ is a nonlinear function and $S_{M}$ is the standard matter action.  The variation of action (\[eq:action\]) with respect to the metric $g_{\mu{\nu}}$ yields the Euler-Lagrange equations $$\label{eq:field_equations}
\begin{split}
&f'(R)R_{\mu \nu}-\frac{1}{2}f(R)g_{\mu \nu}-\nabla_{\mu}\nabla_{\nu}f'(R)+g_{\mu\nu}\square_gf'(R)={\chi}T_{\mu\nu}
\end{split}$$ where $\nabla_{\mu}$ is the covariant derivative for $g_{{\mu}{\nu}}$, $\square_g:=\nabla^{\mu}\nabla_{\mu}$, and $T_{{\mu}{\nu}}=(-2c/\sqrt{-g})(\delta{S}_{M}/{\delta}{g}^{{\mu}{\nu}})$ is the energy momentum tensor of matter.  Taking the trace of Eq. ( \[eq:field\_equations\]) we can get $$\label{eq:trace}
3\square_gf'(R)+f'(R)R-2f(R)={\chi}T$$ where $T$ is the trace of $T_{{\mu}{\nu}}$. Using Eq. ( \[eq:trace\]), we can rewrite Eq. ( \[eq:field\_equations\]) as $$\label{eq:field_equations_new}
\begin{split}
&R_{\mu \nu}=\frac{1}{f'(R)}(\frac{1}{3}g_{\mu\nu}f'(R)R-\frac{1}{6}f(R)g_{\mu \nu}+\nabla_{\mu}\nabla_{\nu}f'(R)+{\chi}(T_{\mu\nu}-\frac{1}{3}g_{\mu\nu}T))
\end{split}$$

Post-Newtonian Expansion {#ch:post_newtonian_expansion}
========================

The matter constituents in the universe are usually well approximated by a perfect fluid with mass density $\rho$ and pressure $p$ [@weinberg1972gravitation].  Hence we assume that the Newtonian potential $U$ of the mass distribution, the typical velocities $v$ and the pressure of the fluid obey such approximation respectively.  In the post-Newtonian approximation, we can further expand the dynamical variables in the field equations perturbatively in powers of $1/c$, since we have the the following order relation [@will1993theory; @clifton2008parametrized; @stabile2010post] $$\label{eq:expansion}
\frac{U }{{{c}^{2}}}\sim \frac{{{v}^{2}}}{{{c}^{2}}}\sim \frac{p}{\rho {{c}^{2}}}\sim \Pi \sim O({2})$$ where $\Pi$ is the ratio of the energy density to the rest-mass density.  We consider the case that the gravitational field is weak and assume that in absence of a gravitational field the background space-time is flat [@schneider1996detection].  We also assume that $f(0) = 0$, which neglects the contribution of a possible cosmological constant and excludes some form of $f(R)$ theories, e.g, $f(R)=1/R$. Note that actually the contribution of a possible cosmological constant can be equivalently substituted by the corresponding contribution of a energy-momentum tensor.  Moreover, the $f(R)$ form which is unable to get weak field solution is useless here.  In a weak field regime the metric tensor can be expanded about the Minkowski metric $\eta_{\mu\nu}$ in its Lorentzian coordinate system as $$g_{{\mu}{\nu}}=\eta_{\mu\nu}+h_{\mu\nu}$$ where $|h_{\mu\nu}|\ll1$. Up to $3^{rd}$ order the components of the metric tensor can be written as [@naf20101; @Lubini11; @will1993theory]: $$\label{eq:metric}
\begin{split}
{{g}_{00}}=&-1{{+}^{(2)}}{{h}_{00}}+O(4)\\
{{g}_{0i}}=&^{(3)}{{h}_{0i}}+O(5) \\
{{g}_{ij}}=&{{\delta }_{ij}}{{+}^{(2)}}{{h}_{ij}}+O(4) \\
\end{split}$$ where the left upper index (n) means the order $O(n)$. Using Eq. ( \[eq:metric\]) we can get the components of the Ricci tensor as $$\label{eq:ricci_tensor}
\begin{split}
{{R}_{00}}=&-\frac{1}{2} {{\nabla}^{2}}{^{(2)}}{h_{00}}+O(4)\\
{{R}_{0i}}=&\frac{1}{2}(-{{\nabla}^{2}}{^{(3)}}{h_{0i}}-\frac{1}{c}{^{(2)}h_{jj,0i}}+{^{(3)}h_{j0,ij}}+\frac{1}{c}{^{(2)}h_{ij,0j}})+O(5)\\
{{R}_{ij}}=&\frac{1}{2}(-{{\nabla}^{2}}{^{(2)}}{h_{ij}}+{^{(2)}h_{00,ij}}-{^{(2)}h_{kk,ij}}+{^{(2)}h_{ik,kj}}+{^{(2)}h_{kj,ki}})+O(4)\\
\end{split}$$ Assuming $f(R)$ to be analytic at $R = 0$, to the $2^{nd}$ order the Ricci scalar and thus $f(R)$ and $f'(R)$ read respectively as $$\label{eq:fr_expansion}
\begin{split}
R &=^{(2)}R+O(4) \\
f(R) &=f'{{(0)}^{(2)}}R+O(4) \\
f'(R) &=f'(0)+f''{{(0)}^{(2)}}R+O(4) \\
\end{split}$$ To the leading order the components of the energy-momentum tensor of matter fields read $$\begin{split}
&{\chi{T}^{00}}=\chi{^{(-2)}}{T^{00}}+O(4)\\
&{\chi{T}^{0i}}=\chi{^{(-1)}}{T^{0i}}+O(4)\\
&{\chi{T}^{ij}}=O(4)\\
\end{split}$$ Note that if $f''(0)=0$, Eq. ( \[eq:fr\_expansion\]) implies $f'(R)$ is constant.  Then Eq. ( \[eq:trace\]) yields at the $2^{nd}$ order $^{(2)}R=-\chi^{(2)}T_{00}/f'(0)$, which is consistent with the equation of GR at the same order.  Thus in this approach, GR is nothing else but the first term of the Taylor expansion of a more general $f(R)$ theory.  As one can see from the action (\[eq:action\]), we ask $f(R)$ to carry the same dimension as $R$’s.  Thus both $f'(R)$ and the term $f''(0) ^{(2)}R$ in Eq. ( \[eq:fr\_expansion\]) are dimensionless.  Since the term $f''(0) ^{(2)}R$ is required to be of order $O(2)$, the expansion of Eq. ( \[eq:fr\_expansion\]) will break down if $^{(2)}R {\geq}f'(0)/f''(0)$ [@Lubini11].  To derive neat equations in the post-Newtonian approximation, we impose the gauge conditions [@naf20101; @Lubini11] $$g_{ij,j}-\frac{1}{2}(g_{jj}-g_{00})_{,i}-\frac{f'(R)_{,i}}{f'(R)}=O(4)$$ $$g_{0j,j}-\frac{1}{2c}g_{jj,0}-\frac{f'(R)_{,0}}{cf'(R)}=O(5)$$ With the gauge conditions, we get from Eqs. ( \[eq:trace\]) and (\[eq:field\_equations\_new\]) $$\label{eq:dr}
{{\nabla }^{2(2)}}R=-\frac{1}{3f''(0)}{{\chi }^{(-2)}}{{T}^{00}}+{{{\frac{f'(0)}{3f''(0)}}}{^{(2)}}R} \\$$ $$\label{eq:dh00}
{{\nabla }^{2(2)}}{{h}_{00}}=-{\frac{4 \chi }{3f'(0)}}{^{(-2)}{T}^{00}}+\frac{1}{3}{^{(2)}}R \\$$ $$\label{eq:dhij}
{{\nabla }^{2(2)}}{{h}_{ij}}=-({\frac{2\chi }{3f'(0)}}{^{(-2)}{{T}^{00}}}+\frac{1}{3}{^{(2)}}R){{\delta }_{ij}}$$ $$\label{eq:dh0i}
\begin{split}
\nabla^2{^{(3)}{h_{0i}}}=&\frac{2\chi}{f'(0)}{^{(-1)}T^{0i}}-\frac{1}{2c}{^{(2)}h_{00,0i}}\\
\end{split}$$ For the sake of simplicity, we let $\alpha^2:=f'(0)/(3f''(0))$ and absorb the constant $1/f'(0)$ into the gravitational constant $G$. From Eq. ( \[eq:dr\]), we can get [@clifton2008parametrized; @naf20101] $$\label{eq:r}
{^{(2)}R}=\frac{2\alpha^2}{c^2}V(\bm{x},t)$$ with the Yukawa potential $$\label{eq:V}
V(\bm{x},t):=\frac{G}{c^2}\int{\frac{{^{(-2)}T^{00}(\bm{x}',t)}e^{-\alpha \vert \bm{x}-\bm{x}' \vert}}{\vert \bm{x}-\bm{x}' \vert}}d^3 x'$$ Note that the information of a specific form of $f(R)$ gravity is encoded in the parameter $\alpha$ in the potential $V$. It is easy to show by using Eq. ( \[eq:dr\]) that Eq. ( \[eq:dh00\]) can be written as $$\label{eq:dh00_new}
{{\nabla }^{2(2)}}{{h}_{00}}=-{\frac{\chi }{f'(0)}} {^{(-2)}{{T}^{00}}}+\frac{f''(0)}{f'(0)}{{\nabla }^{2(2)}}R$$ Thus we get $$\label{eq:dh00_1}
{{\nabla }^{2}}(^{(2)}{{h}_{00}}-\frac{1}{3\alpha{^2}}{^{(2)}}R)=-{\frac{\chi }{f'(0)}} {^{(-2)}{{T}^{00}}}$$ Using Eq. ( \[eq:r\]), the solution of Eq. ( \[eq:dh00\_1\]) can be given by $$\label{eq:h002}
{^{(2)}h_{00}(\bm{x},t)}=\frac{2}{c^2}(U(\bm{x},t)+\frac{1}{3}V(\bm{x},t))$$ where the Newtonian potential $U$ reads $$\label{eq:U}
U(\bm{x},t):=\frac{G}{c^2}\int{\frac{{^{(-2)}T^{00}(\bm{x}',t)}}{\vert \bm{x}-\bm{x}' \vert}}d^3 x'$$ It is obvious that the Newtonian potential $U$ remains unchanged for different forms of $f(R)$ gravity.  Similarly, the solution of Eq. ( \[eq:dhij\]) reads $$\label{eq:hij2}
{^{(2)}h_{ij}(\bm{x},t)}=\frac{2\delta_{ij}}{c^2}(U(\bm{x},t)-\frac{1}{3}V(\bm{x},t))$$ From Eqs. ( \[eq:dh00\]) and (\[eq:dhij\]), we have $${\nabla ^2 }(^{(2)}{h}_{00}-2^{(2)}{{h}_{ii}})=-\frac{4 \chi }{3f'(0)} {^{(-2)}{T}^{00}}+\frac{1}{3}{^{(2)}}R+2({\frac{2\chi }{3f'(0)}} {^{(-2)}{T}^{00}}+\frac{1}{3}{^{(2)}}R)={^{(2)}R}$$ Using Eqs. ( \[eq:h002\]) and (\[eq:hij2\]), it is easy to get $${\nabla ^2 }(^{(2)}{h}_{00}-2^{(2)}{{h}_{ii}})=\frac{2}{c^2}{\nabla ^2 }(V-U)$$ Thus from Eq. ( \[eq:r\]) we have $$V=\frac{1}{{\alpha}^2}{\nabla ^2 }(V-U)$$ Then Eq.(\[eq:h002\]) can be written as $$\label{eq:h002_new}
{^{(2)}h_{00}(\bm{x},t)}=\frac{1}{c^2}{\nabla ^2 }(-\psi+\frac{2}{{3\alpha}^2}(V-U))$$ where the potential $\psi$ is defined as $$\begin{split}
\psi(\bm{x},t):=&-\frac{G}{c^2}\int{{^{(-2)}T^{00}(\bm{x}',t)}\vert \bm{x}-\bm{x}' \vert}d^3x'\\
\end{split}$$ such that ${\nabla ^2 }\psi=-2U$. Hence Eq. ( \[eq:dh0i\]) can be written as $$\label{eq:dh0i_new}
\nabla^2({^{(3)}h_{0i}}+\frac{1}{2c^3}(-\psi+\frac{2(V-U)}{3\alpha{^2}})_{,0i})=\frac{2{\chi}}{f'(0)}{^{(-1)}T^{0i}}$$ The solution of Eq. ( \[eq:dh0i\_new\]) reads $$\label{eq:h0i}
{^{(3)}h_{0i}(\bm{x},t)}=\frac{1}{c^3}(-4 Y_i(\bm{x},t)+\frac{1}{2}\psi(\bm{x},t)_{,0i}+Z(\bm{x},t)_{,0i})$$ where $$\label{eq:y_z}
\begin{split}
Y_i(\bm{x},t):=&\frac{G}{c}\int{\frac{{^{(-1)}T^{0i}}(\bm{x}',t)}{\vert \bm{x}-\bm{x}' \vert}}d^3x'\\
Z(\bm{x},t):=&\frac{1}{3\alpha{^2}}(U(\bm{x},t)-V(\bm{x},t))\\
\end{split}$$ So up to $3^{rd}$ order post-Newtonian approximation, the final form of the metric components reads $$\label{eq:fr_metric}
\begin{split}
{g_{00}}&=-1+\frac{2}{c^2}U+\frac{2}{3c^2}V\\
{g_{0i}}&=\frac{1}{c^3}(-4 Y_i+\frac{1}{2}\psi_{,0i}+Z_{,0i})\\
{g_{ij}}&=(1+\frac{2}{c^2}U-\frac{2}{3c^2}V){\delta_{ij}}\\
\end{split}$$ In contrast, the metric components to the same order approximation in GR reads [@will1993theory] $$\label{eq:gr_metric}
\begin{split}
{g_{00}}&=-1+\frac{2}{c^2}U\\
{g_{0i}}&=\frac{1}{c^3}(-4 Y_i+\frac{1}{2}\psi_{,0i})\\
{g_{ij}}&=(1+\frac{2}{c^2}U){\delta_{ij}}\\
\end{split}$$ Hence the difference between the $f(R)$ gravity and GR comes from the Yukawa potential $V$. In the limit $f''(0) \to 0$, we get $\alpha \to \infty$ and $V \to 0$. Then the solution (\[eq:fr\_metric\]) goes back to (\[eq:gr\_metric\]) of GR. 

Gravitational Lensing {#ch:gravitational_lensing}
=====================

A gravitational lens refers to a distribution of plates (such as a cluster of galaxies) between a distant source (a background galaxy) and an observer, that is capable of bending the light from the source, as it travels towards the observer.  The lensing effect can magnify and distort the image of the background source [@schneider1992gravitational].  According to Fermat’s principle, the world line of a light should extremize its arrival time $T$ with respect to an observer under the variation of $\gamma$. In metric theories of gravity, this principle implies that the world line of a light coincides with a null geodesic in the spacetime.  In the Lorentzian coordinate system of the flat background spacetime, let $dl^2 = \delta_{ij}d x^i d x^j$ be the spatial Euclidean line element.  Up to a constant, the travel time of light on a null geodesic $\gamma$ is given by $$\label{eq:light_path}
T=\int{_{\gamma }dt}\text{=}\int{_{\gamma }\frac{dt}{dl}dl}\text{=}\frac{1}{c}\int{_{\gamma }ndl}$$ where we defined the effective refraction index of light as $$\label{eq:n_def}
n:=c\frac{dt}{dl}$$ Then Eq.(\[eq:light\_path\]) takes the form similar to that of the propagation of a light through a medium in Newtonian’s space and time. 

$2^{nd}$ Order Expansion
------------------------

At the $2^{nd}$ order post-Newtonian approximation, the only nonzero perturbative metric components are $h_{00}$ and $h_{ij}$ in Eqs. ( \[eq:h002\]) and (\[eq:hij2\]).  For a null geodesic, we have [@schneider2006gravitational; @Lubini11; @blanchet2001general] $$\label{eq:ds_2}
d{{s}^{2}}=-(1-{}^{(2)}{{h}_{00}}){{c}^{2}}\text{d}{{t}^{2}}+({{\delta }_{ij}}+{}^{(2)}{{h}_{ij}})d{{x}^{i}}d{{x}^{j}}=0$$ Hence we can get $${{c}^{2}}d{{t}^{2}}=\frac{{{\delta }_{ij}}+^{(2)}{{h}_{ij}}}{1{{-}^{(2)}}{{h}_{00}}}d{{x}^{i}}d{{x}^{j}}=\frac{1+^{(2)}h}{1-^{(2)}{{h}_{00}}}d{{l}^{2}}$$ where $h$ is defined such that Eq. ( \[eq:hij2\]) can be written as $^{(2)}{h}_{ij} = ^{(2)}h{\delta}_{ij}$. Using Eq. ( \[eq:n\_def\]), we can obtain the effective index of refraction as $$\label{eq:102}
n=c\frac{dt}{dl}=\sqrt{\frac{1+(2)h}{1-(2){{h}_{00}}}}$$ which at the $2^{nd}$ order reads $$\label{eq:102}
n=1+\frac{1}{2}{{(}^{(2)}}{{h}_{00}}{{+}^{(2)}}h)$$ By Eqs. ( \[eq:h002\]) and (\[eq:hij2\]), it equals to $$\label{eq:102}
n=1+\frac{2}{c^2}U(x,t)$$ which illustrates that at $2^{nd}$ order, the effective refraction index $n$ is only determined by the Newtonian potential $U$. As shown in Eqs. ( \[eq:fr\_metric\]) and (\[eq:gr\_metric\]), the difference between $f(R)$ gravity and GR comes from the potential $V$ rather than $U$. Hence one can not distinguish $f(R)$ theories from GR at $2^{nd}$ order approximation by the gravitational lensing effect [@Lubini11]. 

$3^{rd}$ Order Expansion
------------------------

We now consider the $3^{rd}$ order post-Newtonian approximation which is needed in dealing with light rays in spacetime [@will1993theory].  At the $3^{rd}$ order expansion, the line element of the metric can be written as $$\begin{split}
d{{s}^{2}}=&-(1-{}^{(2)}{{h}_{00}}){{c}^{2}}\text{d}{{t}^{2}}+({{\delta }_{ij}}+^{(2)}{{h}_{ij}})d{{x}^{i}}d{{x}^{j}}+2^{(3)}{{h}_{0i}}cdtd{{x}^{i}}\\
\end{split}$$ For a null geodesic, by using Eq. ( \[eq:n\_def\]) we have $$\begin{split}
0=&-(1-{}^{(2)}{{h}_{00}}){n}^2+({{\delta }_{ij}}+^{(2)}{{h}_{ij}})\frac{d{{x}^{i}}d{{x}^{j}}}{{dl}^2}+2^{(3)}{{h}_{0i}}\frac{d{{x}^{i}}}{dl}n\\
\end{split}$$ Thus we obtain $$\label{eq:n_3}
n=-\frac{{^{(3)}h_{0i}}}{-1+{^{(2)}h_{00}}}\frac{dx^i}{dl}
\pm{F({^{(2)}h_{00}},{^{(2)}h},{^{(3)}h_{0i}})}$$ where $$\label{eq:F}
\begin{split}
F:=&\sqrt{\frac{({^{(3)}h_{0i}}dx^i)^2}{(1-{^{(2)}h_{00}})^2{dl}^2}
-\frac{(1+{^{(2)}h})({dx^i}^2)}{(-1+{^{(2)}h_{00}}){dl}^2}}=\sqrt{\frac{({^{(3)}h_{0i}}dx^i)^2}{(1-{^{(2)}h_{00}})^2{dl}^2}
+\frac{1+{^{(2)}h}}{1-{^{(2)}h_{00}}}}\\
\end{split}$$ At $3^{rd}$ order approximation, Eq. ( \[eq:F\]) can be expressed as $$\begin{split}
F({^{(2)}h_{00}},{^{(2)}h},{^{(3)}h_{0i}})=&F(0,0,0)+\frac{\partial F({^{(2)}h_{00}},{^{(2)}h},{^{(3)}h_{0i}})}{\partial {^{(2)}h_{00}}}|_{(0,0,0)}{^{(2)}h_{00}}+\frac{\partial F({^{(2)}h_{00}},{^{(2)}h},{^{(2)}h})}{\partial {^{(2)}h}}|_{(0,0,0)}{^{(2)}h}\\
&+\frac{\partial F({^{(2)}h_{00}},{^{(2)}h},{^{(3)}h_{0i}})}{\partial {^{(3)}h_{0i}}}|_{(0,0,0)}{^{(3)}h_{0i}}+{^{(4)}E(^{(2)}h_{00},{^{(2)}h})}+O(5)\\
=&1+\frac{1}{2}
{^{(2)}h_{00}}+\frac{1}{2}{^{(2)}h}+O(4)
\end{split}$$ where $^{(4)}E({^{(2)}h_{00}},{^{(2)}h})$ represents the expansion terms at fourth order. 

It is obvious that the “$-$” sign in Eq. ( \[eq:n\_3\]) should be neglected, since otherwise the refraction index $n$ would become negative.  Hence Eq. ( \[eq:n\_3\]) becomes $$\label{eq:n_3_new}
\begin{split}
n=&-\frac{{^{(3)}h_{0i}}\frac{dx^i}{dl}}{-1+{^{(2)}h_{00}}}+(1+\frac{1}{2}
{^{(2)}h_{00}}+\frac{1}{2}{^{(2)}h})={^{(3)}h_{0i}}\frac{dx^i}{dt}\frac{n}{c}+(1+\frac{1}{2}
{^{(2)}h_{00}}+\frac{1}{2}{^{(2)}h})
\end{split}$$ It is easy to see that Eq. ( \[eq:n\_3\_new\]) can be solved as $$\begin{split}
n&=1+\frac{1}{2}
{^{(2)}h_{00}}+\frac{1}{2}{^{(2)}h}+\frac{1}{c}{^{(3)}h_{0i}}\frac{dx^i}{dt}=n_2+\frac{1}{c}{^{(3)}h_{0i}}{u}^{i}
\end{split}$$ where ${u}^{i}=dx^i/dt$ is the components of the coordinate speed of light, and $n_2:=1+2U(x,t)/{c^2}$ is the refraction index at $2^{nd}$ order.  Therefore, at $3^{rd}$ order post-Newtonian approximation, the effective refraction index of light is obviously dependent on the $3^{rd}$ order metric components $h_{0i}$. From Eqs. ( \[eq:y\_z\]), (\[eq:fr\_metric\]) and (\[eq:gr\_metric\]) one can see that, in contrast to the case of GR, in $f(R)$ gravity $h_{0i}$ is effected also by the Yukawa potential $V$. Hence, $f(R)$ theories are distinguishable from GR by gravitational lensing effect at $3^{rd}$ order post-Newtonian approximation. 

Concluding Remarks {#ch:summary}
==================

In this paper, the post-Newtonian approximation of a general class of $f(R)$ theories is formulated up to $3^{rd}$ order.  At the $3^{rd}$ order expansion, the metric components contain not only the Newtonial potential $U$ but also the Yukawa potential $V$ together with the third order potentials.  The information of a specific form of $f(R)$ theories is contained in the Yukawa potential.  While the Yukawa potential does not show in the $2^{nd}$ order expression of the effective refraction index $n$ of light, it does appear in the $3^{rd}$ order expression of $n$. Therefore, the $f(R)$ theories are distinguishable from GR by the gravitational lensing effect at the $3^{rd}$ order post-Newtonian approximation.  It should be noted that the conclusion that $f(R)$ theories can lead to the gravitational lensing effect different from that of GR can also be obtained by the approach of Minkowski functionals [@Chenxiaoji:2014mxa].  One of the motivations for develping modified gravity theories is to account for the observed mass profiles in galaxies as well as clusters of galaxies without the inclusion of dark matter.  The existence of dark matter in GR is confirmed by the observational data not only from the dynamical analysis, such as rotation curves in spiral galaxies [@Sofue:2000jx] and velocity dispersions in early-type systems [@Napolitano:2008ft; @romanowsky2009mapping], but also from gravitational lensing observations [@Auger:2009hj; @Tortora:2010sy].  Observations indicate that we need to take into account almost the same large amount of dark matter to explain the gravitational lensing effect as that for the dynamical data like the velocity dispersion or the temperature profile of the X-ray emitting intracluster medium [@DeFilippis:2004vy; @DeFilippis:2005hx] in galaxy clusters or spiral galaxies.  Up to now, certain $f(R)$ theories are tested by the dynamical data in galaxy clusters and spiral galaxies [@nucamendi2001alternative; @sobouti2007mathsf; @Martins:2007uf; @Capozziello:2008ny; @capozziello2009testing; @capozziello2012dark].  It is thus desirable to further test $f(R)$ theories by the gravitational lensing observations.  Our results in this paper show that $f(R)$ theories are indeed distinguishable from GR by the gravitational lensing effect.  It is interesting to further study whether the dark matter content in GR, which is needed to explain the gravitational lensing, can be replaced by certain $f(R)$ theories.  Thus our results indicate the possibility to find certain forms of $f(R)$-gravity to explain gravatational lensing obsevation as well as dynamic data without the presence of dark matter. 

This work is supported in part by the NSFC (grant nos.11235003 and 11475023), the Research Fund for the Doctoral Program of Higher Education of China, and the National Undergraduate Training Programs for Innovation of China.