What you are trying to do is assign a signal -which is variable- to another signal during initialization. What do you expect to happen?
I.e. at the moment you define a signal, you can only initialize it. If you want to assign something to the signal, you have to write a declaration.
- definition -> initialization
- declaration -> assignment
So in this case you can define big a larger range, and fix the constant bits in initialization
signal big : std_logic_vector(31 downto 0) => (others => '1');
And when you want to assign sw to any part of big, do that after the begin.
big(31 downto 24) <= sw;
or
big(7 downto 0) <= sw;
etc. The bits you initialized as '1' will be overwritten by the assignment.