Compact objects are highly dense objects in the universe where general relativistic effects cannot be ignored. They are characterized by extremely small radius (~>10 km) but very massive (~> the mass of the Sun). These objects include black holes, neutron stars, and even quark stars (stars made of quarks).

There are two major motivations to study these compact objects:

- First, these objects are laboratories for testing general relativity, since they are highly relativistic.
- Second, measuring the inner structures of these objects can constrain the equation of states at very high densities.

My Master research helps to achieve these goals through the following ways:

- Oscillations (pulsations) of compact objects, e.g. neutron stars or quark stars, which can generate gravitational waves (analogous to playing drums: the membrane oscillates and generates sound waves);
- Universalities between the fundamental oscillations, moment of inertia, quadrupole moment, and tidal Love number, e.g. the I-Love-Q relation, which are useful for disentangling the effects of the equations of state and the alternative theories of gravity.

For the first part, we have derived an accurate Cowling approximation (that ignores perturbations in spacetime) for fundamental modes in fully relativistic equations. We have applied Chandrasekhar’s variational formula for the fundamental modes in incompressible stars to other kinds of compact stars, and found a very good agreement with numerical calculations.

For the second part, we have found that the incompressible limit is the underlying reason for those universalities and have derived their analytic approximations through the post Newtonian and post Minkowskian expansions. We also found a universal relation (i.e. independent of the equation of state) for the frequency of fundamental mode, tidal love number and moment of inertia.

**The significances of these universal relations are:**

**A. **Given these universal relations, compact stars’ properties can be inferred from a few measurements, e.g. from tidal deformation to moment of inertia and fundamental mode frequency;

**B.** Some of the properties of compact stars do not follow these universal relations, e.g. the radius of neutron stars. Measurements of these properties can help to differentiate various *equations of state*;

**C. **The violations of these universal relations can indicate *unusual **equations of state* in compact stars and/or *gravitational physics*.

Our list of publications on these topics.