Running the oscillaton example - GRTLCollaboration/engrenage GitHub Wiki

Physical background

Familiar types of compact astrophysical objects like stars, white dwarfs and neutron stars are composed of baryonic matter, with their tendency to collapse due to gravity balanced by repulsive interactions between the baryonic particles composing them, in particular fermionic degeneracy pressure. If stable bosonic matter exists in the Universe, it too could form gravitationally bound objects that are detectable via their gravitational imprints.

Such "bosonic stars" are best described by a classical field or superfluid due to the high number density of bosons within them, and may have different properties depending on the particle spin. In such a description, the behaviour of the bosonic field or fluid is wave-like, and admits solitonic solutions - with the gravitational forces balanced by gradient pressure in the fluid, or self interactions between the particles.

New bosonic particles are motivated by several high energy extensions to the Standard model of Particle Physics. Whilst the Higgs is not sufficiently stable to be a candidate for such objects, its discovery confirms that scalar bosons exist in nature. If other, more stable bosonic particles exist, they must interact only weakly with baryonic matter, and as such would be natural candidates for a dark matter component of the universe.

Excitingly, in some regimes bosonic stars can mimic compact astrophysical objects. Collisions between them could therefore produce gravitational wave signals that are degenerate with those of black hole or neutron star collisions, resulting in important implications for gravitational-wave physics.

Studies of bosonic stars also aid our understanding of the true nature of astrophysical black holes, and the difficulty of observing their defining characteristic - the event horizon. They provide an ideal template for horizonless compact objects, and there is much to learn from contrasting their dynamical behaviour and observational imprints with those of black holes.

Beyond astrophysical applications, the interplay of classical fields with gravity and the formation of solitonic solutions is a rich field in mathematical relativity, and has proved invaluable in highlighting fundamental aspects of gravitational theory such as critical collapse.

In this example we study the real scalar field example of a bosonic star, also called an oscillaton. This is a quasi stable object which decays over a time period much longer than its oscillation period.

How to run

You should be able to execute the sections of the workbook by pressing shift and then enter. At each stage, try to understand what the code is doing - setting up initial data, integrating the solution, or plotting outputs.

Key things to observe

If you plot the field value at the centre you should see that it oscillates periodically with a fixed amplitude (the decay rate is too slow to be observed).

You can also plot the field value profile in space over time, which shows the shape of the oscillaton.

If you plot the BSSN variables, you will see some evolution. But this is not physical, but only gauge evolution since the solution provided is the stationary one. What you observe is the gauge evolving from the areal polar one in which the initial data is solved for, and the "puncture gauge" type slicing that the code uses.

Some exercises to try

  1. Review the features mentioned here. Find where are they implemented in the code and confirm they work as described.
  2. Look at the evolution of each of the variables and try to think of what physically is happening (refer to the lecture slides ADMtoBSSN.pdf to remind yourself what each quantity is).
  3. The main "diagnostic" is the Hamiltonian constraint. Simplify the expression in Eq. (13) of the Baumgarte paper for the case of spherical symmetry and find and match it to that in the implementation in the code.
  4. How small a constraint error is ok? The important thing is that the magnitude of the error should reduce as the resolution is increased - check that this does happens, both for the initial conditions and the evolution.
  5. Change the gauge evolution - set the lapse to 1 and the shift to zero (for all time - think about initial conditions and evolution). Does the simulation remain stable?
  6. Find and set G=0. What does this do physically? Does the code remain stable? Are the constraints satisfied?

Further extensions

You can convince yourself that this is only a gauge evolution if you study the energy in more detail, as suggested in Exercises to extend the code: level 2.

You can also amend the self interaction to destabilise the boson star and cause it to collapse into a BH, as in Exercises to extend the code: level 1.