Jekyll2019-08-19T18:30:16+00:00https://duetosymmetry.com/Leo C. SteinAssistant Professor @ Ole Miss. Specializing in gravity and general relativity.Leo C. Steinlcstein@olemiss.eduNumerical binary black hole collisions in dynamical Chern-Simons gravity2019-06-24T00:00:00+00:002019-06-24T00:00:00+00:00https://duetosymmetry.com/pubs/DCS-head-on<p class="align-right" style="width: 350px"><img src="https://duetosymmetry.com/images/DCS-head-on-degeneracy.png" alt="" /></p>
<blockquote>
<p>We produce the first numerical relativity binary black hole
gravitational waveforms in a higher-curvature theory beyond general
relativity. In particular, we study head-on collisions of binary
black holes in order-reduced dynamical Chern-Simons gravity. This is
a precursor to producing beyond-general-relativity waveforms for
inspiraling binary black hole systems that are useful for
gravitational wave detection. Head-on collisions are interesting in
their own right, however, as they cleanly probe the quasi-normal
mode spectrum of the final black hole. We thus compute the
leading-order dynamical Chern-Simons modifications to the complex
frequencies of the post-merger gravitational radiation. We consider
equal-mass systems, with equal spins oriented along the axis of
collision, resulting in remnant black holes with spin. We find
modifications to the complex frequencies of the quasi-normal mode
spectrum that behave as a power law with the spin of the remnant,
and that are not degenerate with the frequencies associated with a
Kerr black hole of any mass and spin. We discuss these results in
the context of testing general relativity with gravitational wave
observations.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduThe first numerical beyond-GR binary black hole merger simulation.Notes: A family of ramp functions and the Beta function2019-06-13T06:00:00+00:002019-06-13T06:00:00+00:00https://duetosymmetry.com/notes/a-family-of-ramp-functions-and-the-Beta-function<div id="box" class="jxgbox" style="width:500px; height:500px; margin-bottom:1em;"></div>
<div id="out"></div>
<p>Sometimes in a numerical method, you need to be able to continuously
turn a calculation on or off in space or time (here I will pretend
it’s in time). This can be easily accomplished if you have a function
that starts at a value of 0 before some first time <script type="math/tex">t_0</script>, and rises
up to a value of 1 by time <script type="math/tex">t_1</script>. Through an affine transformation
you can always map <script type="math/tex">[t_0, t_1] \to [0, 1]</script>. An example “ramp”
function is plotted above.</p>
<p>Now if this function appears in a differential equation, and you are
integrating it with an <script type="math/tex">n^{\textrm{th}}</script> order method, then it’s not
enough for the function to be continuous: you probably want the first
<em>n</em> derivatives to match (and thus vanish) at each endpoint.</p>
<p>Let’s go for a piecewise ramp function,</p>
<div>
\begin{align}
R_n(t) = \begin{cases}
0 & t < 0 \\
p_n(t) & 0 \le t \le 1 \\
1 & 1 < t
\end{cases}
\end{align}
</div>
<p>where <script type="math/tex">p_n(t)</script> is some polynomial in <em>t</em>.
Some counting tells you that these 2 endpoint values and <em>2n</em>
derivative conditions can be satisfied with a polynomial of degree
<em>2n+1</em>. Try changing the value of <em>n</em> above and see how the
smoothness changes.</p>
<p>Now for any value of <em>n</em>, it’s a straightforward algebra problem to
set up the polynomial and solve for the coefficients. You probably
want to know the answer for the general case, and a simple approach is
to do a few examples and look for the pattern. Here are the first
few:</p>
<div>
\begin{align}
p_0(t) &= t \\
p_1(t) &= t^2 (3-2t) \\
p_2(t) &= t^3 (10 - 15 t + 6 t^2) \\
p_3(t) &= t^4 (35 - 84 t + 70 t^2 - 20 t^3) \\
p_4(t) &= t^5 (126 - 420 t + 540 t^2 - 315 t^3 + 70 t^4)
\end{align}
</div>
<p>Can you spot the pattern? Don’t feel bad if you can’t, that’s why we
have the <a href="https://oeis.org/">OEIS</a>. If you search for the above
integers as a sequence, you’ll find
<a href="https://oeis.org/A091811">A091811</a>.</p>
<p>With this newfound knowledge, we can now write down the closed form
for the polynomial,</p>
<div>
\begin{align}
\label{eq:def}
p_n(t) = t^{n+1} \sum_{k=0}^n (-1)^k \binom{n+k}{k} \binom{2n+1}{n-k} t^k
\,.
\end{align}
</div>
<p>But was there a better way to find this than relying on the OEIS to
already have the result? But of course!</p>
<p>Rather than thinking about <script type="math/tex">p_n(t)</script> itself, let’s think about
<script type="math/tex">p_n^\prime(t)</script>. Since <script type="math/tex">p_n</script> is strictly increasing,
<script type="math/tex">p_n^\prime</script> is positive, while going to zero at the endpoints. In
fact it goes to zero like <script type="math/tex">t^n</script> at one endpoint, and <script type="math/tex">(1-t)^n</script> at
the other endpoint, because we wanted <em>n</em> derivatives to vanish at
each endpoint. Therefore we know the proportionality</p>
<div>
\begin{align}
p_n^\prime(t) \propto t^n (1-t)^n \,.
\end{align}
</div>
<p>The only thing to get right is the normalization, which we enforce by
asking that the integral of <script type="math/tex">p_n^\prime</script> is 1 at <script type="math/tex">t=1</script>. If you’ve
spent enough time on probability and statistics, then you’ll recognize
<script type="math/tex">p_n^\prime(t)</script> as a special case of the <a href="https://en.wikipedia.org/wiki/Beta_distribution">Beta
distribution</a>, with
shape parameters <script type="math/tex">\alpha = \beta = n+1</script>. So we know the
normalization,</p>
<div>
\begin{align}
p_n^\prime(t) = \frac{1}{B(n+1, n+1)} t^n (1-t)^n \,,
\end{align}
</div>
<p>where <script type="math/tex">B(a,b)</script> is the <a href="https://en.wikipedia.org/wiki/Beta_function">beta
function</a>, and we can
now call <script type="math/tex">p_n(t)</script> the <a href="https://dlmf.nist.gov/8.17#i">regularized incomplete beta
function</a>,
<script type="math/tex">p_n(t) = I_t(n+1,n+1)</script>.</p>
<p>The incomplete beta function has a <a href="https://dlmf.nist.gov/8.17#ii">representation in terms of the
Gauss hypergeometric function</a>,</p>
<div>
\begin{align}
B_x(a,b) = \frac{x^a}{a} F(a, 1-b; a+1; x) \,.
\end{align}
</div>
<p>The important fact for us is that we’re interested in <script type="math/tex">a=b=n+1</script>, in
which case one of the first two arguments is a non-positive integer,
and therefore <a href="https://dlmf.nist.gov/15.2#E4">the series will
terminate</a> as a finite-degree
polynomial. This way, you can prove Eq. \eqref{eq:def}!</p>
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</script>Leo C. Steinlcstein@olemiss.eduSometimes in a numerical method, you need to be able to continuously turn a calculation on or off in space or time.Surrogate models for precessing binary black hole simulations with unequal masses2019-05-22T00:00:00+00:002019-05-22T00:00:00+00:00https://duetosymmetry.com/pubs/NRSur7dq4<p class="align-right" style="width: 350px"><img src="https://duetosymmetry.com/images/NRSur7dq4-mismatch.png" alt="" /></p>
<blockquote>
<p>Only numerical relativity simulations can capture the full
complexities of binary black hole mergers. These simulations,
however, are prohibitively expensive for direct data analysis
applications such as parameter estimation. We present two new fast
and accurate surrogate models for the outputs of these simulations:
the first model, NRSur7dq4, predicts the gravitational waveform and
the second model, surfinBH7dq4, predicts the properties of the
remnant black hole. These models extend previous 7-dimensional,
non-eccentric precessing models to higher mass ratios, and have been
trained against 1528 simulations with mass ratios q≤4 and spin
magnitudes χ₁,χ₂≤0.8, with generic spin directions. The waveform
model, NRSur7dq4, which begins about 20 orbits before merger,
includes all ℓ≤4 spin-weighted spherical harmonic modes, as well as
the precession frame dynamics and spin evolution of the black
holes. The final black hole model, surfinBH7dq4, models the mass,
spin, and recoil kick velocity of the remnant black hole. In their
regime of validity, both models are shown to be more accurate than
existing models by at least an order of magnitude, with errors
comparable to the estimated errors in the numerical relativity
simulations.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduA surrogate model extending the parameter space range of fully precessing quasicircular inspiralsThe SXS Collaboration catalog of binary black hole simulations2019-04-10T00:00:00+00:002019-04-10T00:00:00+00:00https://duetosymmetry.com/pubs/SXS-catalog<p class="align-right" style="width: 350px"><img src="https://duetosymmetry.com/images/ParamsCorner_qchi.png" alt="" /></p>
<blockquote>
<p>Accurate models of gravitational waves from merging black holes are
necessary for detectors to observe as many events as possible while
extracting the maximum science. Near the time of merger, the
gravitational waves from merging black holes can be computed only
using numerical relativity. In this paper, we present a major update
of the Simulating eXtreme Spacetimes (SXS) Collaboration catalog of
numerical simulations for merging black holes. The catalog contains
2,018 distinct configurations (a factor of 11 increase compared to
the 2013 SXS catalog), including 1426 spin-precessing
configurations, with mass ratios between 1 and 10, and spin
magnitudes up to 0.998. The median length of a waveform in the
catalog is 39 cycles of the dominant <script type="math/tex">\ell=m=2</script> gravitational-wave
mode, with the shortest waveform containing 7.0 cycles and the
longest 351.3 cycles. We discuss improvements such as correcting for
moving centers of mass and extended coverage of the parameter
space. We also present a thorough analysis of numerical errors,
finding typical truncation errors corresponding to a waveform
mismatch of <script type="math/tex">\sim 10^{-4}</script>. The simulations provide remnant masses
and spins with uncertainties of 0.03% and 0.1% (90<script type="math/tex">^{\text{th}}</script>
percentile), about an order of magnitude better than analytical
models for remnant properties. The full catalog is publicly
available at <a href="https://www.black-holes.org/waveforms">https://www.black-holes.org/waveforms</a>.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduAccurate models of gravitational waves from merging black holes are necessary for detectors to observe as many events as possible while extracting the maximum scienceOn the Starts with a Bang Podcast!2019-03-25T06:00:00+00:002019-03-25T06:00:00+00:00https://duetosymmetry.com/news/on-the-starts-with-a-bang-podcast<iframe width="100%" height="300" scrolling="no" frameborder="no" allow="autoplay" src="https://w.soundcloud.com/player/?url=https%3A//api.soundcloud.com/tracks/595631034&color=%23ff5500&auto_play=false&hide_related=false&show_comments=true&show_user=true&show_reposts=false&show_teaser=true&visual=true"></iframe>
<p><a href="https://twitter.com/StartsWithABang">Ethan Siegel</a> is an
astrophysicist and science communicator who regular writes <a href="https://www.forbes.com/sites/ethansiegel/">a column
for Forbes</a>. When he’s not
busy writing, he also records the
<a href="https://soundcloud.com/ethan-siegel-172073460/tracks">Starts with a Bang
Podcast</a> which
has been running for 3 years. Ethan was kind enough to invite me on as
a guest for
<a href="https://soundcloud.com/ethan-siegel-172073460/starts-with-a-bang-42-black-holes-and-gravitation">Episode 42: Black Holes and
Gravitation</a>. We
talked about what you would see falling into a black hole, and what
the near future might hold for black hole astrophysics! Enjoy :)</p>Leo C. Steinlcstein@olemiss.eduI'm on Episode 42 of the Starts with a Bang PodcastTalk on the Latin American Webinar on Physics2019-03-13T06:00:00+00:002019-03-13T06:00:00+00:00https://duetosymmetry.com/news/lawphysics-webinar<iframe width="560" height="315" src="https://www.youtube.com/embed/7HO07-QtvMI" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen=""></iframe>
<p>Muchas gracias to <a href="https://acardenasavendano.org">Alejandro
Cárdenas-Avendaño</a> for inviting me to
give a webinar for the <a href="https://lawphysics.wordpress.com">Lating American Webinar on Physics (lawphysics)</a>.
Want to hear me talk about numerical relativity and theories beyond
GR? Well, you’re in luck, because the transmission is on YouTube for
you to watch! It was really fun, and viewers had great questions.</p>
<p>Listen above or click through to the lawphysics site. Talk
information:</p>
<p><em>Title</em>: Testing Einstein with numerical relativity: the precision
frontier, and theories beyond general relativity</p>
<p><em>Abstract</em>: Advanced LIGO and Virgo have already detected black holes
crashing into each other ten times. With their upgrades we anticipate
a rate of about 1 gravitational-wave detection per week. More signals
and higher precision will take the dream of testing Einstein’s theory
of gravity, general relativity, and make it a reality. But would we
know a correction to Einstein’s theory if we saw it? How do we make
predictions from theories beyond GR? And do current numerical
relativity simulations have enough precision that we could be
confident in any potential discrepancy between observations and
predictions?</p>Leo C. Steinlcstein@olemiss.eduWant to hear me talk about numerical relativity and theories beyond GR?The binary black hole explorer: on-the-fly visualizations of precessing binary black holes2018-11-18T00:00:00+00:002018-11-18T00:00:00+00:00https://duetosymmetry.com/pubs/binaryBHexp<p>Check out the <a href="https://vijayvarma392.github.io/binaryBHexp/">project demo page</a>!</p>
<p class="align-right" style="width: 350px"><img src="https://duetosymmetry.com/images/precessing_m2000.png" alt="" /></p>
<blockquote>
<p>Binary black hole mergers are of great interest to the astrophysics
community, not least because of their promise to test general
relativity in the highly dynamic, strong field regime. Detections
of gravitational waves from these sources by LIGO and Virgo have
garnered widespread media and public attention. Among these sources,
precessing systems (with misaligned black-hole spin/orbital angular
momentum) are of particular interest because of the rich dynamics
they offer. However, these systems are, in turn, more complex
compared to nonprecessing systems, making them harder to model or
develop intuition about. Visualizations of numerical simulations of
precessing systems provide a means to understand and gain insights
about these systems. However, since these simulations are very
expensive, they can only be performed at a small number of points in
parameter space. We present <em>binaryBHexp</em>, a tool that makes use of
surrogate models of numerical simulations to generate on-the-fly
interactive visualizations of precessing binary black holes. These
visualizations can be generated in a few seconds, and at any point
in the 7-dimensional parameter space of the underlying surrogate
models. With illustrative examples, we demonstrate how this tool can
be used to learn about precessing binary black hole systems.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduReal-time interactive visualizations of merging black holes in seconds!High-accuracy mass, spin, and recoil predictions of generic black-hole merger remnants2018-09-26T00:00:00+00:002018-09-26T00:00:00+00:00https://duetosymmetry.com/pubs/surfinBH<p>This work was featured in <a href="http://www.caltech.edu/news/when-black-holes-collide-85110">Caltech
News</a>, <a href="https://news.olemiss.edu/making-better-predictions-black-hole-smash-ups/">U
of MS
News</a>,
and was picked up by
<a href="https://phys.org/news/2019-01-physicists-supercomputers-ai-accurate-black.html">phys.org</a>
and <a href="https://aps.altmetric.com/details/53748662">other news outlets</a>.</p>
<p class="align-right" style="width: 350px"><img src="https://duetosymmetry.com/images/interaction.png" alt="" /></p>
<blockquote>
<p>We present accurate fits for the remnant properties of generically
precessing binary black holes, trained on large banks of
numerical-relativity simulations. We use Gaussian process regression
to interpolate the remnant mass, spin, and recoil velocity in the
seven-dimensional parameter space of precessing black-hole binaries
with mass ratios q≤2, and spin magnitudes χ₁, χ₂≤0.8. For precessing
systems, our errors in estimating the remnant mass, spin magnitude,
and kick magnitude are lower than those of existing fitting formulae
by at least an order of magnitude (improvement is also reported in
the extrapolated region at high mass ratios and spins). In addition,
we also model the remnant spin and kick directions. Being trained
directly on precessing simulations, our fits are free from
ambiguities regarding the initial frequency at which precessing
quantities are defined. We also construct a model for remnant
properties of aligned-spin systems with mass ratios q≤8, and spin
magnitudes χ₁, χ₂≤0.8. As a byproduct, we also provide error
estimates for all fitted quantities, which can be consistently
incorporated into current and future gravitational-wave
parameter-estimation analyses. Our model(s) are made publicly
available through a fast and easy-to-use Python module called
<a href="https://pypi.org/project/surfinBH/">surfinBH</a>.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduModeling black holes remnants directly from numerical relativityMeasuring stochastic gravitational-wave energy beyond general relativity2018-07-09T00:00:00+00:002018-07-09T00:00:00+00:00https://duetosymmetry.com/pubs/stochastic-GWs-beyond-GR<p class="align-right" style="width: 350px"><img src="https://duetosymmetry.com/images/stochastic-relations.png" alt="" /></p>
<blockquote>
<p>Gravity theories beyond general relativity (GR) can change the
properties of gravitational waves: their polarizations, dispersion,
speed, and, importantly, energy content are all heavily theory
dependent. All these corrections can potentially be probed by
measuring the stochastic gravitational-wave background. However,
most existing treatments of this background beyond GR overlook
modifications to the energy carried by gravitational waves, or rely
on GR assumptions that are invalid in other theories. This may lead
to mistranslation between the observable cross-correlation of
detector outputs and gravitational-wave energy density, and thus to
errors when deriving observational constraints on theories. In this
article, we lay out a generic formalism for stochastic
gravitational-wave searches, applicable to a large family of
theories beyond GR. We explicitly state the (often tacit)
assumptions that go into these searches, evaluating their generic
applicability, or lack thereof. Examples of problematic assumptions
are as follows: statistical independence of linear polarization
amplitudes; which polarizations satisfy equipartition; and which
polarizations have well-defined phase velocities. We also show how
to correctly infer the value of the stochastic energy density in the
context of any given theory. We demonstrate with specific theories
in which some of the traditional assumptions break down:
Chern-Simons gravity, scalar-tensor theory, and Fierz-Pauli massive
gravity. In each theory, we show how to properly include the
beyond-GR corrections, and how to interpret observational results.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduGravity theories beyond general relativity (GR) can change the properties of gravitational waves: their polarizations, dispersion, speed, and, importantly, energy content are all heavily theory-dependent. All these corrections can potentially be probed by measuring the stochastic gravitational-wave backgroundBlack hole scalar charge from a topological horizon integral in Einstein-dilaton-Gauss-Bonnet gravity2018-05-08T00:00:00+00:002018-05-08T00:00:00+00:00https://duetosymmetry.com/pubs/EDGB-BH-charge<p class="align-right" style="width: 350px"><img src="https://duetosymmetry.com/images/spacetime-standalone.png" alt="" /></p>
<blockquote>
<p>In theories of gravity that include a scalar field, a compact
object’s scalar charge is a crucial quantity since it controls
dipole radiation, which can be strongly constrained by pulsar timing
and gravitational wave observations. However in most such theories,
computing the scalar charge requires simultaneously solving the
coupled, nonlinear metric and scalar field equations of motion. In
this article we prove that in linearly-coupled
Einstein-dilaton-Gauss-Bonnet gravity (which admits a shift symmetry
of the dilaton), a black hole’s scalar charge is completely
determined by the horizon surface gravity times the Euler
characteristic of the bifurcation surface, without solving any
equations of motion. Within this theory, black holes announce their
horizon topology and surface gravity to the rest of the universe
through the dilaton field. In our proof, a 4-dimensional topological
density descends to a 2-dimensional topological density on the
bifurcation surface of a Killing horizon. We also comment on how our
proof can be generalised to other topological densities on general
G-bundles.</p>
</blockquote>Leo C. Steinlcstein@olemiss.eduIn theories of gravity that include a scalar field, a compact object’s scalar charge is a crucial quantity since it controls dipole radiation, which can be strongly constrained by pulsar timing and gravitational wave observations