Calculate execution time from linear speedup
The term you're looking for is called parallel efficiency, which is a measure of how efficiently an algorithm uses its parallel resources. And you are correct that it is:

E = parallel speedup / p


Where p is the number of processors (or cores).

Your equation for t(c_2) is still valid, when the parallel efficiency is the same for c_1 and c_2, as it's equivalent to:

t(c_1) / c_1 = t(c_2) / c_2


A parallel efficiency greater than one is truly possible. While it is observed sometimes, this is because of NUMA effects which accompany the access to more processors / cores (i.e., more of the problem fits in caches).

Also see:
https://computing.llnl.gov/tutorials/parallel_comp