Numerical Relativity of strongly gravitating systems L. Lehner

Perimeter Institute/CIFAR NR goal: understand ‘strongly gravitating’ regimes

“Instead of: If we think hard enough, we don’t need a computer, With the right resources we can simulate situations we can't even begin to think through, and thereby provide us with completely new and unexpected things to think about” [M. Choptuik]

• (M/L) ~ 1 ; (v/c) ~ 1 ; no `restrictive’ symmetries solve Einstein equations in full generality

• Overarching questions: – Gravitational waves & connection with behaviour of source: gravitational wave astronomy – Understanding of spacetime structure in relevant/interest scenarios – Explore conjectures – Provide intuition for `analytical modeling’, mathematical analysis – uncover surprises… What’s involved – express EEs in terms of an initial boundary value problem (formulation) – ensure a well posed problem is defined (eqns & BCs) adopt or rewrite eqns to ensure symmetric/strong hyperbolicity of evolution equations – discretize equations (to obtain an algebraic problem) which introduces a discrete length h (as h 0 one recovers continuum problem) adopt particular approximation methods/algorithms/discretization strategies – recognize discrete equations do not necessarily behave well (in continuum terms, eqns & data off the constraint surface and physical conditions) modify continuum eqns so that constraint surface is an attractor – Avoid singular region (through excision or slicing) – Implement eqns on sufficiently powerful computational infrastructure (software & hardware)

• Arguably < 1990s : head-on collision of black holes (Smarr’80) critical phenomena in GR (Choptuik 86-93). Things changed rapidly after that, especially from 2005 onwards • In particular: BBH problem has largely moved into the ``precision-physics’’ realm (i.e. ‘new-physics’/effort 0, but crucial for grav wave astronomy!) – Challenges in pamaterization of waveforms (and complete wavetrain), which method (EOB/Hybrid/EFT, anything else)?, ‘smart’ runs (ROM,SVD) – Connection with larger astrophysical context (effects on surrounding material/gas/plasma?) – Ultimately, is there a much smarter way to revisit the problem? (2-time scale approach?)

• Non-vacuum binaries: ‘global understanding of the problem’ OK, but much physics still to be explored and many opportunities to tie with already existing observations, analytic approaches and conjectures [TBD] • What is General Relativity isn’t correct? Few well defined options here! Use those as guidance for schemes like PPE and/or illustrate what new calculations to do in PN-based approaches [TBD] ? (sufficiently thorough? unlikely…) • But there is a much life beyond the 2-body problem! – Cosmology: (bubble collisions in multiverse, ekpyrotic scenarios, inhomogeneous cosmologies to ‘explain’ away the dark sector [BHs in lattices] – Core-collapse supernovae – AdS/CFT applications, or motivations for gravity studies [TBD] ; – higher dimensional gravity – Ellusive critical phenomena beyond spherical symmetry – BH evaporation qns, – etc….

Dirty-2-body problem in GR (NS/BH)

• Cauchy formulation of EEs: (generalized) Harmonic or: BSSN (ADM-augmented eqns for well posedness). [Note: both employing a rather rigid set of gauge conditions] .

• NS physics involves: Resistive, Relativistic Magneto Hydrodynamics; generalized equation of state; microphysics.

• Several scales: star size, micro-scales associated with magnetized matter dynamics, different local/global phenomena (neutrino production, neutrino difusssion, instabilities, turbulence, etc.). – Not all are equally relevant on the bulk dynamics for ‘short’ times, but can impact both observations and long-term behavior of the system 49 – Crucially, energy released from the system is ~ 10 ergs (MT/Msun ) ~ SN (M T/Msun ); temporarily a ‘ms magnetar pulsar’ is formed; a BH might form – (related interesting systems involving BHs and WDs, main seqn stars, etc: TDEs! But far harder to track numerically in full GR)

• PROMISELAND: gravity waves, EM waves, neutrinos: ‘multimessenger astronomy’ Ideally, we would like to have as much info as possible, and even better if it is during the cleanest stages.

– GWs are clean! (at least prior to merger depending on the system). But our available detectors only go so far in freqn. Especially relevant for NS-NS

– EM radiation is the complete opposite, but its existence in some systems will be key. [arxiv:1055.01607] Also, they can potentially provide crucial clues to help remove GW degeneracies after merger. Ie: possible magnetization levels? Provide evidence of possible deviations from General Relativity which might not be ‘observable’ through current grav. wave opportunities.

[arxiv:1410.0638] Connecting theory with possible observations: simulations What’s under the hood?

• Einstein equations (gravity), also some alternative gravity theories

• Resistive MHD ([possibly magnetized] matter [and magnetosphere] behavior)

• Tabulated equations of state

• Neutrino treatment (leakage scheme, moment formulations, accounting for cooling & emission, timescales are OK) NS-NS: anatomy of merger

• Early on PN is enough • tidal effects visible ~ 10 5 (v/c) R kL, • nonlinear effects • then ‘bar’ structure. Strongly dependent on masses/EOS and more • How long does it last? So…. • Similarly, BH-NS is being tracked (with about the same amount of physics) • Quasi-circular and eccentric binaries • Gravitational waves & template encoding ongoing Much fun to be had • Electromagnetic counterparts! (also neutrino detection if close) – Tremendous ‘fixation’ on GRBs: BH & disk canonical idea • Interesting connection with dynamics and NS-NS vs BH-NS option • But, there can be EM counterparts even without a sGRB

– However, slowly, complementary options are being identified as well as alternatives. The problem is however *very* dirty • Baryon `poisoning’ of the system • Opacities of different processes involved • Uncertainties on what and how radiates

• We can take guidance on some systems we do know and measure: Pulsars and Supernovae Discriminating options • Concentrate on premerger stage: Emission Induced by magnetosphere’s dynamics as binary tightens. Pulsar guidance emission across many bands. High frequency ‘best’ understood (and sidesteps optical depth issues)

• Also, ‘kilonovae’ related to radiactive decay of r-processes. Distribution is tied to how neutron rich the material that powers the process is. Standard suspect: supernovae… but simulations point to ejecta not sufficiently neutron rich Also, one association with GRB 130603B [Tanvir etal Nature ‘13, Berger etal Astrophys J. Letters ‘13] ). infrared signal implies high neutron richness Pulsar guidance • NS isn’t in vacuum. [Goldreich- Julian] Magnetosphere induced by pair creation

• Charges shorts out E.B ‘force free’ condition

L ~ B 2 Ω4 R6 [1+sin(x) 2 ] [Spitkovsky 2006 ]

• Gaps, current sheet : zones where particle acceleration can take place

• Plasma arguments are ‘generic’ enough that should be applicable to compact binaries Pre -merger: PULSAR ON STEROIDS • As with pulsars, magnetosphere interactions with binaries can induce a strong Poynting flux scaling as L ~ B2 (1/a) 5 • Similar structure to pulsar magnetosphere: current sheet, gaps, closed/``open’’ field lines, polarity changes, etc. • In pulsar, radio emission poorly understood, but better handle on gamma/x-ray side [e.g. Bai- Spitkovski] • Could be seen to at least 10Mpc • Furthermore, collapse of merged will generate a burst of order sGRB’s energies What and how to see them?

[Spitkovsky-Bai] Gravity waves to guide the observation. Coincident signal and fairly non-collimated

[Green,Ortiz,Westernacher-Schneider ++, in prep] Aside: what if GR is not correct? GWs might tell us so, but we might need also EM waves

• Scalar-tensor theories [Fierz-Jordan-Brans-Dicke,Damour- Esposito-Farese,…] – Gravity mediated by usual tensor degrees of freedom + a non- minimally coupled scalar field – New phenomenology : • Dipole radiation • Spontaneous scalarization provides a non-trivial ‘scalar charge’ to compact stars • While significantly constrained by solar and pulsar tests, interesting parameter space remains • Non-linear interactions were largely unexplored more ‘generic’ scalarization possible [dynamics and/or induced scalarization] • Dipole radiation modifies dynamical behavior.

• Important deviations from GR behavior (eg separation and grav wave signals)

• Furthermore, non-monotonic behavior of corrections (implications for, e.g. PPE)

- Interaction between differently scalarized stars induces a dynamical readjustment of charges to become equal

- ALIGO will have a hard time digging this out for moderate and low mass binaries - EM signals can potentially save the day

[Barausse,Palenzuela,Ponce,LL 2013; Sampson etal 2014, also Shibata,Taniguichi, Okawa, Buonanno ‘14]

Fast -forward and outwards… Ejecta & properties

• Also, other ejecta from winds driven by the eventual accretion disk is possible, though this is less neutron rich [Fernandez etal ‘15] and expected signal would be in the optical. Summarizing multiple results & larger view • Amount & characteristics of ejecta quite tied to EoS.

– Very little < 0.001 M o for stiff EoS, velocity of ejecta ~ similar [0.1 – 0.4c] & Y e quite peaked at ~ 0.25

– On the other hand, enough [0.001-0.01 M o] for soft EoS [collision takes place deeper in the potential]. Y e peaked at ~ 0.2 with significant amounts in [0.1-0.2]

– Ye values? Y e in [0.25 – 0.4] optical signal [Fernandez,Kasen,Quataert ‘15], lower values, peaked in infrared [observation: 130603B] . Furthermore, nuclear

reaction networks point to Y e < 0.22 needed for explaining observed elements with A > 75 (and peaks at A ~130 and 190) [Kasen, Lippuner]

– Supernovae simulations do not seem to get Y e this low. Also, abundance of 244 Pu in deep-sea samples is 1/100 of what would be expected if supernovae does the job but consistent with binary merger rates [Wallner etal ’15]

• Interestingly: cosmic censorship, quite ‘robustly’ preserved by the relevant systems! Speculations & comments • IF GRB130603B is a `typical case’, and decay of r-processes elements indicate very neutron rich ejecta, then neutron star EoS lies on the softer side. [way out: low mass BHs and/or very high spins]

• IF so, then this makes it harder for BH-NS binaries to power sGRBS. (Tidal radius is smaller for softer EoS need for ISCO to be inside tidal radius requires higher values of spin than if the EoS is stiff).

• Softer EOS, makes it more difficult to extract NS radii at ALIGO/VIRGO without further tuning, but complementary EM info will help

• Rather robust qualitative understanding of outcome! Changing gears… • The AdS/CFT correspondence relates a (d)−QFT with a (d+1)−dimensional theory of gravity. – Any gravitational phenomena should have an equivalent CFT analog, and vice-versa. – A natural arena to study field theory open questions: transport properties in strongly coupled field theories, quantum turbulence, etc. – Plenty of applications. Most of which in equilibrium situations and in the probe limit (phase space analysis) (e.g. CMT applications) – Long list (and growing!) of efforts in dynamical settings [Chesler,Yaffe;Das,Nishioka,Takayangi,Basu;Bhattacharya,Minwalla;Romatschke, Bantilan,Gubser,Pretorius;Abajo,Aparicio,Lopez;Albash,Johnson,Ebrahim,Headri ck,Balasubramanian,Bernamonti,deBoer,Copland,Craps,Keski,Mueller,Shaffer,Shi gemor,Staessens,Galli,Schwelinger,Caceres,Kundu,Wi,Gauntlett,Simons,Wisema [Image: J. Santos] n,Sonner,Myers,Buchel,LL,vanNikerke,Abajo,daSilva,Lopez,Mas,Serantes,Dias,Sa ntos,Marolf, Horowitz ]

• Black holes have become the ‘harmonic oscillator of the 21 st century’ [A. Strominger] Holographic path to the promise land

• Goal: understand properties of out of equilibrium phenomena and its eventual thermalization. Is there a universal behavior?

• AdS/CFT offers a way into: strongly coupled field theories, provides a real-time analysis , allows for considering finite temperature setups and is amenable to general spacetime dimensions. From a gravity perspective, interesting excuse to push intuition

• Dynamical qns involve time-dependence ( solve PDEs) – Quark-gluon plasma: thermalization? Hydrodynamization? [Chesler,Yaffe,Heller,Romatschke,Mateos,vanderSchee,Fernandez,Bantilan,Gubser,Pretorius, ….] – ``Quantum quenches’’: universal properties as a response to fast and slow quenches in the Hamiltonian of a system (N=4 SYM <-> mass-deformed gauge theory [Balasubramanian etal; Buchel,Myers,vanNiekerk,LL;Das,Das….] • Out of equlibrium behavior from a given initial state will involve transfer of energy – How does it take place? – What’s its time scale? – How and where to does energy flow? – [Note: many problems are essentially the same in gravitational terms]

• Starting with a perturbation off a thermal state, thermalization time scale given by : perturbation time scale (adiabatic case), or a ~ scale consistent with the slowest –triggered- black hole QNM (abrupt case) – Perhaps perturbative arguments do provide the answer. In particular we can perturb BHs and analyze QNMs. Can also analyse perturbatively pure AdS and obtain relevant time scales. Is this all? • Perhaps not…. – Perturbative analysis & conclusions are notoriously delicate [e.g. abuse takes place all too often, phenomena might be obscured through an unfortunate choice of perturbative scheme. Also what a good scheme is might be a ``moving target’’] – QNMs aren’t a basis even in AdS [Warnick 2013] – QNMs for relevant cases might not yet be known (e.g. d=4,5 Kerr-AdS [Cardoso,Dias,Santos,Harnet,LL dec 2013])

• And worse yet… – Kerr-AdS is not known to be stable [in fact math arguments for the opposite]. Further, if not ‘linearly-stable’, we can’t use QNMs in a straightforward way – Math arguments for `pure’ AdS being unstable and specific illustrations. This is good ‘academically speaking’, but AdS/CFT demands more in regards to thermalization. Will all ``non-pure states in the CFT’’ yield configurations always leading to BH formation? If not, what’s the path to a thermal state? Turbulence (in hydrodynamics) some would say: “that phenomena you know is there when you see it’’ For Navier-Stokes (incompressible case):

• Breaks symmetry (recovered only in a ‘statistical sense’) • Exponential growth of (some) modes [not linearly-stable] • Global norm ( non-driven case ): Exponential decay possibly followed by power law, then exponential • Energy cascade (direct d>=3, inverse/direct d=2) • E(k) ~ k -p (5/3 and 3 for 2+1 ) ‘Turbulence’ in gravity? • Perhaps there isn’t… (arguments against it, mainly in 4d) – Perturbation theory (e.g. QNMs) – Numerical simulations (e.g. ‘scale’ bounded) – (hydro has shocks/turbulence, GR no shocks) • Perhaps there is… – AdS/CFT <-> AdS/Hydro ( turbulence?! [Van Raamsdonk 08] ) – Applicable if LT >> 1 L ( ρ/ν) >> 1 L (ρ/ν) v = Re >> 1 – (membrane paradigm? / Blackfolds ) • List of questions… • Tension in the correspondence or gravity? • Reconcile with QNMs expectation? (and perturb theory?) • If there is, does it have similar properties to hydro case? • What’s the analogue `gravitational’ Reynolds number? If there is turbulence…. • Multiple scales would ``pop up’’ dynamically

• Linearized analysis is insufficient

• Self similarity of spacetime fractal structure

• Spectra of energy might leave particular relics in, e.g. grav waves, matter/energy structure, etc.

• Can play a role as a ‘virtual’ censor depending on decay properties

• Can help understand turbulent behavior in hydro

• Out of equilibrium behavior might show clearly spacetime dimensionality, etc… • AdS/CFT gravity/fluid correspondence (definition?) [Bhattacharya,Hubeny,Minwalla,Rangamani; VanRaamsdonk; Baier,Romatschke,Son,Starinets,Stephanov]

[Carrasco,LL,Myers,Reula,Singh 2012] • Using correspondence on can reconstruct the spacetime

• Spacetime describes gravitational ‘tornadoes’ connecting boundary with horizon.

• Also, 3+1 hydro with conformal eos leads to direct cascade. Bulk & holographic calculation

[Adams,Chesler,Liu PRL 2014] [Green,Carrasco,LL, PRX 2013] observations • Inverse cascade carries over to relativistic hydro and so, gravitational turbulence in 3+1 and 4+1 move energy in opposite directions

(…warning for particular studies imposing symmetries that can eliminate relevant phenomena) .

• Consequently 4+1 gravity (relative to QNM differences) equilibrates more rapidly ( direct cascade dissipation at viscous scales which does not take place in 3+1 gravity)

• Note 1: GR-Hydro correspondence established in the regime where slow QNMs dominate. How is the transition to such regime?

• Note 2… there are always limits to numerical solns! • From a hydro standpoint: geometrization of hydro in general and turbulence in particular: – Provides a new angle to the problem, might give rise to scalings/Reynolds numbers in relativistic case, etc. Answer long standing questions from a different direction . However, to actually do this we need to understand things from a purely gravitational standpoint. Obvious first targets:

– What mediates vortices merging/splitting in 2 vs 3 spatial dims? – Can we interpret how turbulence arises within GR? – Can we predict global solns on hydro from geometry considerations? (e.g. Oz-Rabinovich ’11) – Can we indentify what triggers this phenomena ‘outside’ the long-wavelength regime assumption? – Is AdS really needed? Hydro analysis? • Must go beyond linear level. Obtain linear modes (sound, shear), then write Navier Stokes eqn:

• κ( k,p,q) determine the couplings. These satisfy: • conservation of energy

• and if: cons. of enstrophy

Inverse cascade in 2+1 dimensions

The above is a non-perturbative analysis, ‘typical’ pert analysis will miss turbulence unless taken to high enough order

pure AdS and a path to thermalization

• What if we don’t start with a BH?. Consider an out-of-equilibrium scenario in a CFT. One path to thermalization, from a holographic perspective, is through the formation of a black hole.

• Thus, studying the dynamics of ‘pure-AdS’ perturbed by suitable fields provides a way to probe (in a suitable limit) how this can be achieved.

• Bizon-Rostworowski revisited Choptuik’s problem in (spherically symmetric) AdS [numerics + perturbation theory] . Dias-Horowitz-Santos for gravitational wave case [perturbation theory].

BUT! [Balasubramanian,Buchel,Green,LL,Liebling] also [Craps,Evnin,Vanhoof] [Dimikatropulos,Freivogel,Lippert,Yang] Taking a step back [Fang,Green,Yang,LL] A lot more is going on • Astrophysical connection: grav waves, role in energetic phenomena (eg. GRBs), grav waves in alternative theories, black hold ‘balding’, etc.

• Black hole evaporation

• Cosmology: bubble collisions in multiverse, ekpyrotic scenarios,…

• Higher dimensional gravity: black rings, Myers-Perry bhs…

• AdS/CFT motivated work: instability of AdS and ‘equilibration’ questions

• Surprises in gravity: turbulence in the gravitational field, fractal behavior of horizons

• ‘Analytical’ perturbation methods are key to extend/expand what NR is saying