Why do we use only the first component scores to construct different indices? I have seen that in constructing indices by PCA the scores of only the first component is used. Although the first principle component explains the most variation contained in the data, often the VAF is too small (10% or even less). Then why do we use only the first component score instead of combining the scores of first few dimensions which would explain more variation when combined (I don't know if combining these score on different dimensions is legit)? 
 A: If I understand you correctly then the following is related:
https://stackoverflow.com/questions/12067446/how-many-principal-components-to-take
The short answer is that this is somewhat subjective. There are several things to keep in mind, such as why you are doing PCA in the first place. This is mentioned in the post I linked.
I would add a couple of things. Firstly, if the first PC has a particular interpretation then it will make sense to take it on its own. For example, if your data is multivariate returns on a broad basket of assets then the first PC will be associated with the influence of the market as a whole and can be a basis for the market portfolio.
Secondly, looking only at what proportion of the variance is explained by the first PC to gauge its information content may be misleading; you should also consider the dimension of the problem. To see why consider the theoretical limiting value of the largest eigenvalue of a Wishart matrix, $\mathbf{M} = \mathbf{X}'\mathbf{X} \sim W_d(n,\boldsymbol{\Sigma})$, where $\mathbf{X} \in \mathbb{R}^{n \times d}$. As $d,n \rightarrow \infty$ such that $\frac{d}{n} = y$ the Marcenko-Pastur law gives us that
\begin{equation}
\lambda_{max} \rightarrow (1+\sqrt y)^2.
\end{equation}
In other words, for purely random data, we expect that the amount of variance explained by the first PC will decrease as $d$ increases. So your empirical largest eigenvalue could still point to significant information content depending on how much larger it is than what you would expect in a purely random system. Note, again, that if you are doing dimensionality reduction then you would need more PCs anyways.
I hope this addresses your question.
