Can a scalar field model gravity? How accurate would be the results? Are there any difficulties with such a model? Newtonian gravity can be described by the equation:
$$ \nabla^2 \phi = 4 \pi \rho G $$
where $\rho$ is the mass density, $\phi$ is the gravitational potential, and G is the universal gravitational constant.
Of course, one of the shortcomings is that it is not consistent with special relativity. General relativity handles this shortfall but is a tensor theory. However,the above equation can be modified as follows:
$$ (\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{{\partial t}^2}) \phi = 4 \pi \rho G $$
This would be consistent with special relativity. Would this model give results close to experiment? Would it give correct results in some observed situations where Newtonian gravity fails? Why or why not?
 A: There are several ways that scalar gravity fails, but the most dramatic one is that a scalar theory of gravity does not predict that light will bend in a gravitational field.
A: There is indeed a scalar field model of gravity, in fact Einstein originally tried that before settling on a spin 2 description. Scalar gravity is called Einstein-Nordstrom gravity, here is a link to wikipedia: http://en.wikipedia.org/wiki/Nordstr%C3%B6m%27s_theory_of_gravitation. At the nonlinear level it amounts to using $R$ in Einsteins equations instead of $G_{\mu\nu}$.
What you wrote down was indeed guessed. The problem is that $\rho$ actually isn't relativistically invariant--energy and momentum get mixed under boosts--so you really need to use $T$, the trace of the stress energy tensor. You also need to have the gravitaional sector to have nonlinear interactions, because gravity carries energy and so it couples to itself. So you can generalize what you wrote, that is the Einstein-Nordstrom theory.
While scalar gravity does reproduce the Newtonian limit, the Newtonian limit is easy to get. The problems all amount to the fact that the graviton is a spin 2 particle, not a spin 0 particle.
For example, scalar gravity cannot couple to light. This is because a scalar (spin-0) can only couple to the trace of the stress energy tensor $T$, but Maxwell's equations are famously conformally invariant at the classical level and so $T=0$. This violates Einstein's equivalence principle (which one reason why Einstein wouldn't have liked it). It also is empirically ruled out (which is a great reason for us to rule it out, though Einstein didn't have those experiments when he was developing GR).
Scalar gravity also has completely different properties for gravitational waves: it has one helicity 0 polarization instead of 2 helicity 2 ones. This would change the output of radiation from a binary pulsar system, for example.
Another consequence is that Birkhoff's theorem is no longer true. A scalar mode can be sensitive to overall changes of scale in an object--a spherically symmetric object with time varying radius $R(t)$ will radiate in scalar gravity, but will definitely not radiate in GR.
