Numerical Relativity Simulations of Gravitational-Wave Sources

Bruno GiacomazzoUniversity of Trento and INFN-TIFPA

www.brunogiacomazzo.org

General Relativity and Astrophysics

• Binary Black Hole Mergers• Binary Neutron Star Mergers• Neutron Star – Black Hole Mergers• Supernovae• Accretion Disks• Cosmology

In all these scenarios general relativity plays a fundamental role.

Developing a code that solves the full set of GR and (Magneto)Hydrodynamic equations is not an easy task.

Kawamura et al 2016

History of numerical relativity(see https://link.springer.com/article/10.1007/lrr-2015-1)

•1962 Arnowitt, Deser and Misner (ADM) 3+1 formulation•1966 May and White first 1D GR simulation of collapse to BH•1977 NR is born with coordinated efforts to evolve BHs•1985 Stark and Piran extract GWs from a simulation of rotating collapse to a BH in NR.•1992 Bona and Massó show that harmonic slicing has a singularity-avoidance property•1993 Anninos et al. first simulation of head-on collision of two BHs•1994 “Binary Black Hole Grand Challenge Project” is launched•1995-1998 BSSN formulation•1996 Brügmann mesh refinement simulation of BHs•1997 Cactus 1.0 is released•1997 Brandt & Brügmann “puncture” initial data

•2000 Brandt et al. simulate the first grazing collisions of BHs using a revised version of the Grand Challenge Alliance code

•2000 Shibata and Uryū first NS-NS merger simulation in GR•2003 Schnetter et al. “Carpet” driver for Cactus•2005 Pretorius first simulation of BH-BH inspiral and merger

History of numerical relativity(see https://link.springer.com/article/10.1007/lrr-2015-1)

Equations

Gµ⇥ = Rµ⇥ �12gµ⇥R = 8�Tµ⇥

Einstein Equations

Hydro Equations

Maxwell Equations

Gµ⇥ = Rµ⇥ �12gµ⇥R = 8�Tµ⇥

Einstein Equations

R � Rµµ

Rµ⇥ � R⇤µ⇤⇥

R⌅µ⇤⇥ ⇥ �⇤�⌅

µ⇥ � �⇥�⌅µ⇤ + �⌅

⇧⇤�⇧µ⇥ � �⌅

⇧⇥�⇧µ⇤

�⇤µ⇥ ⇥

12g⇤⌅

��µg⇥⌅ + �⇥gµ⌅ � �⌅gµ⇥

⇥

Ricci scalar

Ricci tensor

Riemann tensor

Space-Time Foliation

https:/

/arxiv.org/abs/gr-qc/0703035

ds2 = ���2 � ⇥i⇥

i⇥dt2 + 2⇥idxidt + ⇤ijdxidxj

BSSN schemehttps:/

/arxiv.org/abs/gr-qc/0703035

First Full BH-BH GW Signal

Pretorius 2005https://link.aps.org/doi/10.1103/PhysRevLett.95.121101

Credit: CITA/Scheel et al 2009, PRD 79, 024003

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GWs from BBH

Aylott et al 2009, CQG 26, 165008(NINJA project)

Several numrel groups computing BBH GW signals to be used for detection

Investigated mass ratios, spins (amplitude and orientation), eccentricity.

Einstein Toolkiteinsteintoolkit.org

• Set of publicly available tools for relativistic astrophysics

• Latest release on February 9 2018 (codename “Tesla”)

• More than 150 users on 6 continents

• Tested on several HPC infrastructures around the world

• Includes over 100 Cactus thorns, including:• McLachlan (space-time evolution)• GRHydro and IllinoisGRMHD (GRMHD equations)• Several initial data and analysis routines

• Data can be read and visualized by open source codes (e.g., Visit, PostCactus, yt)

References• Einstein Toolkit Webpage: http://einsteintoolkit.org

• Main Publications presenting the toolkit:• Loeffler et al 2012: http://arxiv.org/abs/1111.3344

• Moesta et al 2013: http://arxiv.org/abs/1304.5544

• Zilhao and Loeffler 2013: http://arxiv.org/abs/1305.5299

• Visualization Tools:• PostCactus & SimRep: https://bitbucket.org/DrWhat/pycactuset

• Visit: https://visit.llnl.gov/

• YT: http://yt-project.org/

• Every year workshops and (sometimes) schools are organized in EU and USA:• Atlanta, Georgia (USA, June 2018): http://et-workshop18.physics.gatech.edu/

• Lisbon, Portugal (EU, September 2018): https://centra.tecnico.ulisboa.pt/network/grit/einsteintoolkit2018/

• Open source framework

• Decentralized code development

• Active and friendly user community

• Module based approach

• Infrastructure modules

• Parameter file handling

• Parallelization (MPI + OpenMP)

• Adaptive Mesh refinement (Carpet)

• IO (ASCII and HDF5) + checkpointing

Figure by R. Haas

Adaptive Mesh Refinement Driver (Carpet)https://carpetcode.org/

• Carpet provides box-in-box mesh refinement• It is included in the toolkit

Visualization by Giacomazzo, Kaehler, Rezzolla

SimFactoryhttp://simfactory.org/

Command-line tools for setting up Cactus distribution and managing simulations on a variety of supercomputers, including PRACE machines.

Current List of HPC machines supported

McLachlan

• Produced using the KRANC code: http://kranccode.org

• Can solve Einstein Equations using different formulations (BSSN and Z4)

• Implement finite difference schemes up to 8th order

• Implements singularity-avoiding slicing conditions

GW170814

LIGO/Caltech/MIT/Leo Singer (Milky Way image: Axel Mellinger)

GRMHD equations

GRMHD APPLICATIONS

Moesta et al 2014Gold et al 2014

Paschalidis et al 2013Kawamura et al 2016

The system of equations is written in a conservative form (Valencia formulation, Anton et al 2006):

where U is the vector of conserved variables, Fi the fluxes, and S the source terms.They can then be solved using HRSC methods using approximate Riemann solvers.To these one has to add the equations for the evolution of the magnetic field:

GRMHD equations

GRHYDRO

Moesta et al 2013: http://arxiv.org/abs/1304.5544

• First publicly available fully GRMHD code

• Based on the public version of the Whisky code

• Fully embedded in the Einstein Toolkit

• Uses Valencia formulation (as Whisky)

IllinoisGRMHD

http://math.wvu.edu/~zetienne/ILGRMHD/index.html

• First publicly available fully GRMHD code using the vector potential as evolution variable

• Very robust code for GRMHD in AMR• Limited EOS support (only ideal fluid up to now)• Now included in the Einstein Toolkit

WhiskyMHD (private)

Kawamura et al 2016

• Currently maintained and developed in Trento (Giacomazzo)

• It makes use of the Einstein Toolkit

• Main applications are• NS-NS mergers (GWs and SGRBs)• Accretion onto Supermassive BH-BH mergers

• It supports piecewise-polytropic EOS

• Current developments include implementation of neutrino emission and finite temperature nuclear EOSs

Credit: NSF/LIGO/Sonoma State University/A. Simonnet

Gamma-Ray Bursts

• Discovered in 1967• Two types:

• Short (<2 s)• Long (>2 s)

• Long GRBs are due to Supernovae explosions (discovered in 1998)

• Short GRBs have been quite a mystery

Credit: NASA GSFC & Caltech/MIT/LIGO Lab

GW170817: the first NS-NS detection

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Kawamura et al 2016

sim. & vis.: Wolfgang Kastaun

NS-NS

SMNS

STABLE NS + Torus

BH + TORUSNS-BH

HMNS

<~1s

<~1 hour

Low-Mass NS-NS

High-Mass NS-NS

BNS

MER

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GW: EFFECT OF EOS

Ciolfi et al 2017

SMNS NS HMNS

Dietrich et al 2017

GW: EFFECT OF EOS AND MASS RATIO

Dietrich et al 2017

GW: EFFECT OF EOS AND MASS RATIO

Using GW and GRB to Infer Maximum NS Mass

Ruiz, Shapiro, and Tsokaros 2018, PRD 97, 021501(R)

SMNS HMNS BH

In order to produce a jetit seems necessary tohave an HMNS phasefollowed by BH collapse.

This would constrain themaximum mass toMmax~2.15-2.28 in orderto explain GW170817and GRB 170817A

CONCLUSIONS• Numerical Relativity can handle several astrophysical phenomena with quite detailed physics

• Open source codes are now available and programming skills are not anymore a requirement to work in this field

• Suggested reviews on Numerical Relativity:• Cardoso, Gualtieri, Herdeiro, Sperhake 2015, LRR 18, 1: https://link.springer.com/article/10.1007/lrr-2015-1• Font 2008, LRR 11, 7: https://link.springer.com/article/10.12942/lrr-2008-7• Gourgoulhon 2007, Lecture Notes: https://arxiv.org/abs/gr-qc/0703035

• Suggested reviews on NS Binary Mergers: • Shibata + Taniguchi 2011, LRR 14, 6: https://link.springer.com/article/10.12942/lrr-2011-6• Faber + Rasio 2012, LRR 15, 8: https://link.springer.com/article/10.12942/lrr-2012-8• Paschalidis 2017, CQG 34, 084002: https://doi.org/10.1088/1361-6382/aa61ce