A student is investigating the properties of a simple pendulum in a physics laboratory. The student pulls a small sphere attached to a string of length \( L \) to an initial starting angle \( \theta_0 \) and releases it from rest, measuring a period of oscillation \( T_0 \). The student then repeats the experiment with the same sphere and string but pulls it to a new initial starting angle of \( 2\theta_0 \). Assuming both angles are small enough that the small-angle approximation is valid, what is the ratio of the new period to the original period \( T_0 \)?