The Einstein-Hilbert action leading to Einstein's equations is $$S\sim\int R \sqrt{-g}\, {\rm d}^4 x$$ There is a class of modifications of Einstein's relativity called $f(R)$ theories of gravity which takes the action as $$S\sim \int f(R) \sqrt{-g}\, {\rm d}^4 x$$ where $f(R)$ is some function. $f(R)$ theories are considered one of the simplest extensions of Einstein's relativity which could be used to explain dark energy or dark matter.

I have read in a few sources that $f(R)$ theories cannot explain the lensing mass in the many colliding galaxies such as the famous bullet cluster. The reconstruction of mass from gravitational lensing in unmodified relativity yields the following image (unaccounted matter blue, known matter red)

Bullet cluster

But the inability of $f(R)$ to explain this seems curious since the theory contains an effectively massive scalar mode which can travel as a longitudinal wave at subluminal speeds.

My naive intuition would be that the "massive" scalar degree of freedom of the curvature decouples from the baryonic matter in the collision and travels on it's own a bit further to then deflect light.

Is that a possible scenario? Can the extra degree of freedom in $f(R)$ theories cause the deflection of light?

EDIT: As Arthur Suvorov notes, the answer to the original question is yes. But it is actually a "yes, but", because one is able to construct the appropriate theory for the case of the collision itself but still has to constrain it with other observations. Probably the most relevant ones would be the motion of celestial bodies in the solar system, the rotation curves of galaxies and cosmological constraints.

So the question is, how do these constraints collide? Could somebody summarize the successes and failures of the attempts to explain dark matter and especially the mentioned lensing with $f(R)$ gravity?

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    $\begingroup$ As for the bullet cluster specifically, I can't comment. But certainly some work has gone into these kinds of thinking; see for instance this paper by Bohmer et al arxiv.org/abs/0709.0046v3. More generally, you can always think of any f(R) gravity as being a scalar-tensor (Brans-Dicke) theory by looking at conformal transformations (I know Salvatore Capozziello has some nice papers on this, which I cannot find specifically at the moment); the Brans-Dicke theory has a free parameter $\omega$ which molds light deflection. In short, I am sure the answer is a resounding 'yes' $\endgroup$ Jan 17, 2015 at 13:55
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    $\begingroup$ The question then comes to: does the functional form for $f$ you found in order to explain the deflection properties match with other astrophysical bits of evidence as to the true theory of gravity? Hulse-Taylor pulsar, absence of gravitational waves, cosmology, black holes,... . The scalar freedom would mean objects emit dipole type radiation (as well as the usual quadrupolar gravitational radiation), which also has not been observed yet (phys.ufl.edu/ireu/IREU2012/pdf_reports/…). So the $f(R)$ scalar mode cannot be `too' strong. $\endgroup$ Jan 17, 2015 at 13:57

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This paper by Lubini et al. suggests that the answer to your question is "no": arxiv.org/abs/1104.2851v2. Lubini et al. consider the weak-field limit of $f(R)$ gravity where matter that contributes to spacetime curvature is assumed to be non-relativistic (Lubini et al. say at one point that they are assuming the spacetime is stationary. But this is not necessary; it is only necessary to assume that the matter contributing to the stress-energy tensor has velocities small compared to c). These assumptions should provide a good approximation for clusters and cluster collisions including the Bullet Cluster which has an impact velocity of "only" about 1% of the speed of light (see arxiv.org/abs/1307.0982).

The field equation resulting from varying the metric in the $f(R)$ action involves $f(R)$ and $f'(R)$. Lubini et al. assume that $f(R)$ in analytic in $R$ at $R=0$ so that in the weak field limit $$f(R) \approx f(0) + f'(0)R$$ and $$f'(R) \approx f'(0) + f''(0)R.$$ Lubini et al. show that in the weak-field limit the metric is given by (setting $G=c=1$) $$ds^2 = -(1-h_{00})dt^2 + \delta_{ij}(1 + h)dx^idx^j$$ where $$h_{00}({\bf x},t) = -2\Bigl(\Phi({\bf x},t) + \frac{1}{3}\Psi({\bf x},t)\Bigr),$$ $$h({\bf x},t) = -2\Bigl(\Phi({\bf x},t) - \frac{1}{3}\Psi({\bf x},t)\Bigr),$$ $\Phi({\bf x},t)$ is the Newtonian potential given by $$\Phi({\bf x},t) = -\int\frac{\rho({\bf x}',t)}{|{\bf x} - {\bf x}'|}d^3x',$$ and $\Psi({\bf x},t)$ is a "massive" scalar field given by the Yukawa form $$\Psi({\bf x},t) = -\int\frac{\rho({\bf x}',t)}{|{\bf x} - {\bf x}'|}e^{-a|{\bf x} - {\bf x}'|} d^3x'$$ where $$a^2 = \frac{f'(0)}{3f''(0)}.$$

$\Psi({\bf x},t)$ is the effectively massive scalar mode contained in $f(R)$ theories that is not contained in general relativity. The motion of a massive, non-relativistic particle is determined in terms of $h_{00}$ which depends on the additional scalar field $\Psi({\bf x},t)$. However, the motion of massless particles (e.g., photons) is determined by the combination $h_{00} + h$ which according to the above equations is independent of $\Psi({\bf x},t)$. Lubini et al. show that the motion of photons depends only on $h_{00} + h$ in the weak-field limit by calculating an effective refractive index. This is done by noting that according to classical geometric optics the trajectory of a photon extemizes the quantity $t = \int n dl$ where $n$ is the refractive index and $l$ is a spatial length along the path of the photon. The above metric implies that along a null geodesic $$dt^2 = \frac{1+h}{1-h_{00}}dl^2 \approx (1 + h + h_{00})dl^2$$ and therefore in the weak-field limit the effective refractive index is $$n = 1 + \frac{1}{2} (h + h_{00}).$$ Since this does not depend on $\Psi({\bf x},t)$, gravitational lensing is not affected by $\Psi({\bf x},t)$ in the weak-field limit and is given by the standard general relativity result in this limit.

The deflection of light can also be determined using the geodesic equation. This is done by Stabile and Stabile: arxiv.org/abs/1108.4721v2. Stabile and Stabile confirm Lubini et al.'s results for $f(R)$ theories and they also consider $f(R,R^{ab}R_{ab})$ theories. They conclude that $f(R,R^{ab}R_{ab})$ theories can alter the general relativity prediction for gravitational lensing, but that the lensings is suppressed so that even more dark matter would be needed to explain the observed lensing.

So, in summary, it seems unlikely that the additional scalar mode of $f(R)$ theories could explain gravitational lensing without dark matter since in the leading order correction to general relativity, the trajectory of photons is not affected by the additional scalar mode.

  • $\begingroup$ Thanks, this was exactly the missing piece of the puzzle I was looking for. It is interesting that at least to linear order the metric is conformally equivalent (by a $1+2 \psi/3$ factor) to a weak-field Einstein metric. This is certainly not true in general because of the Einstein frame and all that jazz, but I am wondering under what conditions and to what order the "conformal-Einsteinness" applies in $f(R)$. $\endgroup$
    – Void
    Jan 28, 2015 at 12:21

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