So here it is, our newest paper:
It's so new that we don't even have an arXiv.org number at the time I write this blog post. It'll come out on arXiv on Sunday and we'll submit the paper to Phys. Rev. D in a few days. You can get an exclusive sneak peak now.
This paper is special to me for a couple of reasons.
First, in case you haven't noticed: There are two women and two men on the author list! 50% women, 50% men. That's how physics should be done. I am very happy about this. This is the first time in my life that I am a co-author on a paper with gender-balanced authorship. Hopefully, this will happen more in the future as we work to fight the gender-imbalance in physics/astrophysics. There is a lot of work that's left to do, unfortunately.
Second, the paper is about rotating stellar collapse to neutron stars and connects back to the very first paper that I wrote back in the summer of 2003 (TEN years ago; I am getting old!). It's great to get back to my roots once in a while. Back then, I was a student at Heidelberg and writing my Diplom thesis about rotating core collapse simulations that I conducted with Adam Burrows at Arizona.
Third, the paper wasn't even my idea! Ernazar Abdikamalov is a postdoc in TAPIR working with me on a variety of projects in core-collapse supernova theory and radiation transport. Ernazar suggested the project to me and identified it as a great project in which we could involve an undergraduate researcher. Some background on Ernazar: He received his PhD in 2009 from SISSA in Trieste and was a postdoctoral scholar at the Center for Computation and Technology before he joined my group at Caltech.
Alex DeMaio from Rutgers joined us this past summer on a LIGO REU Summer Undergraduate Research Fellowship (SURF). Alex worked with Ernazar on carrying out and analyzing the simulations. She did a great job and contributed tremendously to this paper. Alex is now a junior at Rutgers. She has continued to work with us and helped complete the paper.
What is this paper about? Some Background.
Here is the thing: We don't know how massive stars spin deep inside. Optical Astronomers can measure their surface velocities and it's possible to use asteroseismology to probe the structure and spin of less massive (smaller) stars, but the internal structure and rotation of massive stars that explode in core-collapse supernovae remains a mystery. This is a pity, since rotation may have a big effect on the way the supernova explosion develops once the core has collapsed to a hot newborn proto-neutron star.
When the next core-collapse supernova happens in the Milky Way (this happens every 50-100 years), we may have a way to probe how the cores of massive stars rotate: Gravitational waves. Gravitational waves are, quite literally, ripples in spacetime. LIGO is hunting for them, and they are emitted by fast large scale asymmetric mass-energy motions that change on short timescales. This is precisely what happens in a rotating massive star: The centrifugal force, which is strongest at the equator deforms the core into an oblate shape, making an oblate proto-neutron star:
This picture shows the proto-neutron star 12 milliseconds after its birth. It's quite aspherical! The contour lines show isodensity contours and the labels are in units of 10x g/ccm. The colors show specific entropy, and yellowish/red regions have been shocked when the neutron star was made. The core of the proto-neutron star is very dense, wasn't shocked and has very low entropy.
At the end of core collapse, the inner core of a massive star moves with about a third of the speed of light. When the core reaches densities comparable to those in an atomic nucleus, the nuclear force suddenly increases the pressure support in the core. This decelerates the inner core within only about a millisecond! It's useful to think of the inner core as a basket ball that falls to the ground (collapse) and elastically bounces back. "Core Bounce" is what supernova theorists call the moment when the inner core reaches nuclear density, gets decelerated, and rebounds into the outer stellar core.
If the core is not spherically symmetric (rotation!), core bounce gives off a burst of gravitational waves:
What did we do?
It has been known for a while (see, e.g., my review from 2009) that the shape of the gravitational wave signal depends quite sensitively on the spin of the inner core. Various authors in the past hinted at the possibility of reading off the core's rotation from the gravitational wave signal, but nobody had demonstrated that it's really possible. In this paper, we quantitatively demonstrated that it is possible to extract information about the core's spin from a gravitational wave signal observed by LIGO.
Ernazar and Alex performed more than a hundred simulations of rotating core collapse with an axisymmetric general-relativistic hydrodynamics code. From these simulations, they generated a gravitational waveform catalog. Sarah turned this catalog into what gravitational-wave astronomers call a numerical template bank for matched filtering. Matched filtering is a method for extracting a signal from noisy data by cross-correlating the data stream (that is, LIGO output in our case) with a known signal model (template). This method is normally used to look for gravitational waves from coalescing binaries of black holes and neutron stars. It was Sarah's idea to try it for rotating massive star collapse and it turns out to work very well! This is because the signal shape is very regular and determined by relatively few parameters. Sarah injected trial signals that weren't part of the template bank into simulated random LIGO noise and used her matched filtering pipeline to find the best matching template. Sarah then recorded the rotation parameters (e.g., total angular momentum, degree of differential rotation) of the best matching template as the "measured" quantities.
Here is an example result showing how well we can extract the total angular momentum of the inner core:
The x-axis has the true angular momentum of the inner core that corresponds to the injected trial wave signal. The top panel compares this to the measured angular momentum and the bottom panel gives the relative measurement error. Don't worry about the various different types of symbols plotted here (check out the paper if you are interested). The main point here is that we can extract the total angular momentum of the inner core within ~20%! The blue-triangle outlier at very low angular momentum comes from the fact that in slowly spinning cores the wave signal is not dominated by rotation, but by convective motions after bounce. These are stochastic in nature and cannot be predicted precisely.
Detecting the precollapse degree of differential rotation is harder. First of all, it's more difficult to parameterize, since there are many ways in which angular momentum could be distributed inside a massive star's core. We assumed that the core is rotating before collapse with constant angular velocity on cylindrical shells, because the Poincare-Wavre theorem suggests that the electron-degenerate cores of massive stars should rotate in this way (but it is not clear that this is true!). This is a standard assumption in rotating core collapse and we used a rotation law that reduces the angular velocity as a function of cylindrical radius and a single differential-rotation parameter A. We showed that our method can "measure" the correct value of A, provided the core is very rapidly spinning. This is because differential rotation does not affect the wave signal unless the core is fast spinning.
What are the limitations?
Every work has limitations and it's important to mention them:
The biggest potential systematic problems outside of what we control are uncertainties in the equation of state of nuclear matter and the detailed composition of the inner core (the relevant compositional variable is the number of leptons [neutrinos, electrons] per baryon). Variations in these can lead to waveform changes that can spoil the accuracy with which we can extract information about core rotation. Our simulations also do not include magnetic fields, but others (e.g., Obergaulinger et al. 2006) have shown that the effects of realistic magnetic fields are unlikely to alter the dynamics and gravitational wave signal in the phases that we study.
On the gravitational-wave astronomy side, we considered only a single LIGO detector and assumed optimal source-detector orientation. In reality there will be multiple detectors taking data in coincidence, but a real signal will also be weaker.