Metric $f(R)$ Instability

Question

I was reading this manuscript about Metric \(f(R)\) instability, could anyone explain why the value of \(\mu^4\) creates strong instability in Eq. (4)?

One piece I don't understand is where it says "Thus, the \(T^{3}/6\mu^{4}\) dominates the coefficient in front of \(R_1\) term in Eq. (6) and leads to strong instability" just below Eq.(7). Why is the value of the coefficient of \(R_1\) may create instabilities.

Consider the field equation:\(D^{2}R-3\frac{D_{a}R D_{a}R}{R}+\frac{R^{4}}{6{\mu}^{4}}-\frac{R^{2}}{2}=-\frac{T R^{3}}{6{\mu}^{4}}\), why the value of \(\mu^4\), whether positive or negative, may create instabilities?

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Comments

Consider the field equation: \(D^{2}R-3\frac{D_{a}R D_{a}R}{R}+\frac{R^{4}}{6{\mu}^{4}}-\frac{R^{2}}{2}=-\frac{T R^{3}}{6{\mu}^{4}}\), why the value of \(\mu^4\), whether positive or negative, may create instabilities?

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A comment by David Zaslavasky has been omitted from repost, since the linked manuscript is merely 4 pages long.  

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I am extremely sorry @physicsnewbie, it seems you are correct. I found from a meta.SE discussion that SE "redistributes [deleted content] to 10k+rep users and moderators", and whether it redistributes content at all is a choice it makes, but it still owns its content. 

Answer 1

I actually dealt this model some in my thesis work.  (Shameless plug: Stability of spherically symmetric solutions in modified theories of gravity.)  The basic picture in the Dolgov-Kawasaki paper is that they start with a background solution where the Ricci scalar is equal to the trace of the stress-energy tensor, \(R_0 = T\) (inside a star, say). This solution should match up to some exterior solution with \(R_0 \approx \mu^4\) (one of the cosmological models that were motivating Carroll, Duvvuri, Trodden, & Turner);  however, if \(\mu^4 < 0\), this turns out to be impossible.  (Basically, the Ricci scalar tends to diverge at large distances from the star.)  

Having dispensed with the \(\mu^4 < 0\) case, they look at at the \(\mu^4 > 0\) case.  In this case, one can obtain a well-behaved solution in which \(R_0 \approx T\) inside the star and \(R_0 \approx \mu^4\) asymptotically.  The next step is to look at the behavior of perturbations about this solution;  the linearized perturbation equation is eq. (5) in their paper.  The equation of motion for the first-order perturbations grows exponentially with time, since the last term on the left-hand side of eq. (5) is so large in magnitude and negative.

I should mention that while Dolgov & Kawasaki's proof isn't iron-clad (see my comments at the start of Section III.B in the paper I linked to above), it does persist in a more rigorous analysis.  You might also take a look at The Large Scale Structure of f(R) Gravity, which obtains a similar result.