I have just seen in physics forums the following paper:
Conformal Gravity Challenges String Theory
I had no previous knoledge of these theory and now I have no time to search in google references for it so I just expose it without any claim about how good or flawed it could be.
The author, Philip D. Manhein, semmengly a refuted cosmologist, reviews the genesis of general relativity and isolates two parts, kinematics, background independence, and dynamics, Einstein equations. The point is to search an alternative dynamics. The ultimate reason for Einstein equations is a fenomenológical law, the Newton potential V=1/r. These can be shown to be a solution to the Poisson equation. But if we want to allow small variations compatible with actual observations we could go to V=b/r + c.r with a very small c. A solution of these kind can be shown to be compatible with a fourth order derivative Poisson equation:
In background independent terms we can make a theroy based in the Weyl (conformal) tensor Cuvnk form which we can derivate equations of motion of the form:
These equations have Schwarschild type solutions and some other aspects coincident with Einstein theory. Even thought the most important concern, of course, are de diferences with Einstein which are basically 3:
1)At galactic scales the mass distribution deduced from the apropiate equations fits the observed distribution without requiring dark matter (here I would point that recently seemengly there have been indirect observations of dark matter so maybe these could be a problem afther all).
2) At cosmologic scales an equation equivalent to the Friedman-Robertson-Walker (with an apropiate energy moment tensor) can be formulated and we obtain a solution whic describes an aceleratedly expanding universe withouth a cosmological constant (which is forbiden in that theory because of conformal invariance).
3) It is a power counting renormalizable theory. That means that if we construct a perturbative quantum gravity from it it could be renormalizable, i.e. fully consistent. And it would be a 4 dimensional theory, no need for extra dimensions.
Until now all seems very correct.But beeing a relativelly easy theory (as compared to string theory for example) one could think that there is some sublety involved and in fact that was the case. If one calculates the propagator for the gravity sector one finds a term which from knowledge of quantization of gauge theories seems to be associated to a ghost state which without some apropiate way to handle it would remove the unitarity of the theory. Well, the author in these paper, and these is the important development here, claims to have resolved that problem, which seemed to be a generic problem for theories with fourth order derivatives.
I´ll try to read the article more carefullly, and of course also wait for posible reactions in the physic comunity. All it sounds very interesting but suposedly one would be carefull with "fundamental" theories developed by a cosmologist. Well, the paper was brief so in the worst of the cases it didn´t mean an excesive mess of time.
P.S. String theorists claim that there are two factors which seem to indicate that stringn theory must be the "only game in town". One of them is that a quantum gravity would be a fixed point in the renormalization flow defined by the beta funtions of the theory if it going to advoid the need of an innite number of parameters. You can read a carefull exposition of the argument in Jackes Distler blog, concretelly here. Well, conformal theories have ultraviolet fixed points and these is a conformal theory so it would fit the requirement (in that link Distler claims that string theory escapes the problem by a diferent method, even though string theory seen as a conformal theory on the world sheet fits the requirement of ultraviolete fixed point, I am not sure if I am mising some point with these two appearences of UV fixed point from two slightly diferents viewpoints)
I am also aware that something called Poisson deformations (or something similar) seems to indicate some uniquiniess of string theory. I dón´t know how the result is obtained and the secenaries it covers so I can´t judge it´s relevance for the present theory.
Update Afther a bit of search I have found that there articles which have the details of the calculations:
http://arxiv.org/abs/astro-ph/0505266 (classical part)
http://arxiv.org/abs/0706.0207 (quantum part)
You can read another opinion about the article here:
The theory was presented in the recent Pascos 2007 conferences and seemengly it had a good aceptation and now people is studying the details of the math in search of possible faliures. If not found seems it definitivelly looks like a promising theory.