Weight of a tensor density Is there any freedom in choosing the weight of a tensor density?
I have seen in some papers that they introduce a tensor density made from metric with a special weight.
There is a tensor density with the weight of $\frac{-1}{2}$ :
$$\overset{\sim}{g_{ab}} = \sqrt{({\det g})} ^{\frac{-1}{2}}~~g_{ab}$$
In the paper The geometry of free fall and light propagation by Ehlers and his colleagues (Gen. Relativ. Gravit. 44 no. 6, pp. 1587–1609 (2012)), on page 1599 there is an example.
You can also look at the relation $2.4$ of this article.
I don't understand why this special weight is chosen for the tensor density constructed from metric.
 A: The weight would be the power of the Jacobian used in the transformation law. For an ordinary tensor the weight is then zero and for a tensor density it is one. In principle you can define quantities with any weight but you should call them something other than a tensor density. 
Note that I have not read your reference because it is behind a firewall.
A: In these particular cases, the authors are interested in the conformal structure, i.e. lightcone structure, of the manifold.  A conformal structure can be defined by an equivalence class of metrics, all of which are related to each other by a conformal transformation,
$$g_{ab}\sim e^{\omega(x)}\bar{g}_{ab}$$
A nice way to characterize a conformal structure is to say it is just the tensor density of weight $-1/2$ that you mention.  Under a conformal transformation, the metric determinant changes by
$$g\mapsto e^{4\omega}\bar{g},$$
so you can see that the density weight is chosen so that the tensor density $\tilde{g}_{ab}$ is invariant under the conformal transformation.  If you were in dimension other than $4$, you would choose the weight to be $-2/d$, to maintain this invariance under conformal transformations.  
