The static gravitational potential at fifth order Andreas Maier - PowerPoint PPT Presentation

The static gravitational potential at fifth order Andreas Maier Los Angeles, 10 December 2019 J. Blümlein, A. Maier, P . Marquard arXiv:1902.11180 J. Blümlein, A. Maier, P . Marquard, G. Schäfer, C. Schneider arXiv:1911.04411

GRAVITATIONAL-WAVE TRANSIENT CATALOG- 1 512 512 FREQUENCY [HZ] 128 z] equency [H 128 z] requency [H 32 GW151012 GW170104 GW150914 GW151226 32 r F F 512 FREQUENCY [HZ] 128 32 GW170608 GW170729 GW170814 GW170809 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 512 FREQUENCY [HZ] 128 32 GW170823 GW170818 GW170817 : BINARY NEUTRON STAR 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 10 20 30 40 TIME [SECONDS] TIME [SECONDS] TIME [SECONDS] LIGO-VIRGO DATA : HTTPS://DOI.ORG/10.7935/82H3-HH23 EINSTEIN’S THEORY S. GHONGE, K. JANI | GEORGIA TECH WAVELET (UNMODELED) 2 / 26

Gravitational waves [LIGO Scientific Collaboration and Virgo Collaboration 2016] ringdown inspiral merger 3 / 26

Gravitational waves [LIGO Scientific Collaboration and Virgo Collaboration 2016] ringdown inspiral merger 4 / 26

Compact binary systems Power counting r r s – • Masses comparable: m ≡ m 1 ∼ m 2 Generalisation to different masses straightforward • Nonrelativistic system: v ≪ 1 • Virial theorem: mv 2 ∼ Gm 2 r Post-Newtonian (PN) expansion : p Combined expansion in v ∼ Gm=r ≪ 1 5 / 26

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Post-Newtonian expansion Theory status Conservative dynamics, no spin: complete results up to 4PN ( v 8 ) • ADM Hamiltonian formalism [Damour, Jaranowski, Schäfer 2016] • Fokker Lagrangian in harmonic coordinates [Bernard, Blanchet, Bohé, Faye, Marchant, Marsat 2017] • Non-relativistic effective field theory [Foffa, Mastrolia, Porto, Rothstein, Sturani, Sturm 2017–2019] Partial results at 5PN [Foffa, Mastrolia, Sturani, Sturm, Torres Bobadilla 2019; Blümlein, Maier, Marquard 2019] [Bern, Cheung, Roiban, Shen, Solon, Zeng 2019; Bini, Damour, Geralico 2019] Method in this talk: non-relativistic effective field theory [Goldberger, Rothstein 2004] 8 / 26

General Relativity • Here: point-like objects • No spin • No finite-size effects (neutron stars: 5PN, black holes: 6PN) • Harmonic gauge fixing: @ — ( √− gg —� − ” —� ) = 0 g = det( g —� ) • Dimensional regularisation: d = 3 − 2 › S GR = S EH + S GF + S pp Z d d +1 x √− gR 1 S EH = 16 ıG Z d d +1 x √− g Γ — Γ — 1 S GF = − 32 ıG s Z Z @x — @x � X X i i S pp = − dfi i = − − g —� m i m i dt @t @t i i R = g —� R —� Γ — = g ¸˛ Γ — ¸˛ 9 / 26

Non-relativistic effective theory [Goldberger, Rothstein 2004] Similar to non-relativistic QCD [Caswell, Lepage 1985; Pineda, Soto 1997; Luke, Manohar, Rothstein 2000; . . . ] Full theory: Effective theory: General relativity NRGR R dt 1 2 m i v 2 i + Gm 1 m 2 S GR = S EH + S GF + S pp S NRGR = + : : : − → r potential gravitons: classical potentials r ;~ k 0 ∼ v k ∼ 1 r radiation gravitons: radiation gravitons k 0 ∼ v r ;~ k ∼ v r 10 / 26

Potential matching Expansion of action p Expand S GR in v ∼ Gm=r ≪ 1 , e.g. s Z @x — Z @x � dt √− g 00 + O ( v i ) X X i i S pp = − − g —� @t = − m i dt m i @t i i Coupling to spatial components of metric suppressed Temporal Kaluza-Klein decomposition [Kol, Smolkin 2010] ! − 1 A j g —� = e 2 ffi e − 2 d − 1 d − 2 ffi ( ‹ ij + ff ij ) − A i A j A i v 0 v 2 v · · · ff ij ffi A i 11 / 26

Potential matching Diagrammatic expansion Equate amplitude in effective and full theory: − iV + 1 + 1 + : : : q 2! 3! = + + + + + · · · All momenta potential, p 0 ∼ v r ≪ p i ∼ 1 r , → expand propagators: p 2 + p 2 1 = 1 p 4 + O ( v 4 ) 0 p 2 − p 2 ~ ~ ~ 0 12 / 26

Potential matching Diagrammatic expansion V = i log 1 + + + ! + + + · · · ! = i + + + : : : | {z } 1 PN 13 / 26

Potential matching Known results Confirmation of previous results: • 1PN: [Goldberger, Rothstein 2004] • 2PN: [Gilmore, Ross 2008] • 3PN: [Foffa, Sturani 2011] • 4PN: • “static” contribution v = 0 : [Foffa, Mastrolia, Sturani, Sturm 2016; Damour, Jaranowski 2017] • v � = 0 : [Foffa, Sturani 2019; Foffa, Porto, Rothstein, Sturani 2019] [Blümlein, Maier, Marquard 20XX] New: • 5PN static contribution: [Foffa, Mastrolia, Sturani, Sturm, Torres Bobadilla 27 Feb 2019; Blümlein, Maier, Marquard 28 Feb 2019 ] 14 / 26

Potential matching Diagram generation • 4PN including velocities: #loops QGRAF source irred no source loops no tadpoles 0 3 3 3 3 1 70 70 70 70 2 1770 1770 1770 1468 3 13400 9792 9482 5910 4 11822 5407 4685 1815 • Static case: #loops QGRAF source irred no source loops no tadpoles sym 0 1 1 1 1 1 1 2 2 2 2 1 2 19 19 19 15 5 3 360 276 258 122 8 4 10081 5407 4685 1815 50 5 332020 128080 101570 27582 154 15 / 26

Potential matching Static 5PN calculation − iV S 5PN = + + + + + + + + + + + : : : 16 / 26

Potential matching Feynman rules i = − p p 2 2 c d ~ i j 1 j 2 = − ` ´ i 1 i 2 ‹ i 1 j 1 ‹ i 2 j 2 + ‹ i 1 j 2 ‹ i 2 j 1 + (2 − c d ) ‹ i 1 i 2 ‹ j 1 j 2 p 2 p 2 ~ = − i m i m i . . . m n n Pl p 1 i 1 i 2 = i c d ( V i 1 i 2 ffiffiff + V i 2 i 1 ffiffiff ) 2 m Pl p 2 p 2 ‹ i 1 i 2 − 2 p i 1 V i 1 i 2 1 p i 2 ffiffiff = ~ p 1 · ~ 2 p 1 j 1 j 2 c d ( V i 1 i 2 ;j 1 j 2 + V i 2 i 1 ;j 1 j 2 + V i 1 i 2 ;j 2 j 1 + V i 2 i 1 ;j 2 j 1 = i ffiffiffff ) ffiffiffff ffiffiffff ffiffiffff 16 m 2 Pl p 2 i 1 i 2 p 2 ( ‹ i 1 i 2 ‹ j 1 j 2 − 2 ‹ i 1 j 1 ‹ i 2 j 2 ) − 2( p i 1 2 ‹ j 1 j 2 + p j 1 V i 1 i 2 ;j 1 j 2 1 p i 2 1 p j 2 2 ‹ i 1 i 2 ) + 8 ‹ i 1 j 1 p i 2 1 p j 2 = ~ p 1 · ~ 2 ffiffiffff 17 / 26

Potential matching Feynman rules i 1 i 2 p 1 k 1 k 2 = i V i 1 i 2 ;j 1 j 2 ;k 1 k 2 V i 2 i 1 ;j 1 j 2 ;k 1 k 2 ( ˜ + ˜ ) ffffff ffffff p 2 32 m Pl j 1 j 2 V i 1 i 2 ;j 1 j 2 ;k 1 k 2 ˜ = V i 1 i 2 ;j 1 j 2 ;k 1 k 2 + V i 1 i 2 ;j 2 j 1 ;k 1 k 2 + V i 1 i 2 ;j 1 j 2 ;k 2 k 1 + V i 1 i 2 ;j 2 j 1 ;k 2 k 1 ffffff ffffff ffffff ffffff ffffff p 2 p 2 “ V i 1 i 2 ;j 1 j 2 ;k 1 k 2 − ‹ j 1 j 2 ` 2 ‹ i 1 k 1 ‹ i 2 k 2 − ‹ i 1 i 2 ‹ k 1 k 2 ´ = ( ~ 1 + ~ p 1 · ~ p 2 + ~ 2 ) ffffff ‹ i 1 j 1 ` 4 ‹ i 2 k 1 ‹ j 2 k 2 − ‹ i 2 j 2 ‹ k 1 k 2 ´ − ‹ i 1 i 2 ‹ j 1 k 1 ‹ j 2 k 2 ˜” + 2 ˆ n p k 2 1 p i 2 2 − p i 2 1 p k 2 ‹ i 1 j 1 ‹ j 2 k 1 + 2 4 ` ´ 2 p i 1 1 + p i 1 p i 2 2 ‹ j 1 k 1 ‹ j 2 k 2 − p k 1 1 p k 2 2 ‹ i 1 j 1 ‹ i 2 j 2 ˜ + 2 ˆ` ´ 2 p k 1 1 p k 2 p k 2 1 p i 2 2 − p i 2 1 p k 2 p i 1 1 + p i 1 p i 2 + ‹ j 1 j 2 ˆ 2 ‹ i 1 i 2 + 2 ‹ i 1 k 1 − 2 ‹ k 1 k 2 ˜ ` ´ ` ´ 2 2 + p j 2 4 p i 2 1 ‹ i 1 k 1 ‹ j 1 k 2 + p j 1 “ 2 ‹ i 1 k 1 ‹ i 2 k 2 − ‹ i 1 i 2 ‹ k 1 k 2 ´ ` 2 1 p i 2 1 ‹ k 1 k 2 − 2 p k 2 − p k 2 ‹ i 1 j 1 ` 1 ‹ i 2 k 1 ´ 1 ‹ i 1 i 2 ‹ j 1 k 1 ˜” ˆ + 2 + p j 2 p j 1 − 4 p i 2 “ 2 ‹ i 1 k 1 ‹ i 2 k 2 − ‹ i 1 i 2 ‹ k 1 k 2 ´ 2 ‹ i 1 k 1 ‹ j 1 k 2 ` 1 1 p k 2 2 p k 2 2 ‹ i 2 k 1 − p i 2 2 ‹ i 1 i 2 ‹ j 1 k 1 + ‹ i 1 j 1 ` 2 ‹ k 1 k 2 ´˜”o ˆ + 2 d − 1 √ c d = 2 ; m Pl = 1 = 32 ıG d − 2 18 / 26

Potential matching Diagram families Algebraic manipulations ( FORM [Vermaseren et al.] ) → massless propagators: , Z d d l 1 ı d= 2 · · · d d l 5 N ( q; l 1 ; : : : ; l 5 ) P f ( q ) = p 2 a 1 p 2 a 10 ı d= 2 ~ · · · ~ 1 10 19 / 26

Potential matching Master integrals Reduction to master integrals ( crusher ): [Chetyrkin, Tkachov 1981; Laporta 2000] V S 5PN = c 0 + c 1 + c 2 + O ( › ) + c 3 c j : Laurent series in › = 3 − d 2 , polynomials in m 1 ; m 2 ; r − 1 ; G − 1 a q Master integrals factorise into and known b [Lee, Mingulov 2015; Damour, Jaranowski 2017] 20 / 26

Potential matching Result N = − G V S r m 1 m 2 1PN = G 2 V S 2 r 2 m 1 m 2 ( m 1 + m 2 ) » 1 – 2PN = − G 3 V S 2( m 2 1 + m 2 r 3 m 1 m 2 2 ) + 3 m 1 m 2 » 3 – 3PN = G 4 ` ´ V S m 3 1 + m 3 r 4 m 1 m 2 + 6 m 1 m 2 ( m 1 + m 2 ) 2 8 » 3 – 4PN = − G 5 + 31 + 141 ` ´ ` ´ V S m 4 1 + m 4 m 2 1 + m 2 4 m 2 1 m 2 r 5 m 1 m 2 3 m 1 m 2 2 2 2 8 " # 5PN = G 6 5 2 ) + 91 2 ) + 653 V S 16( m 5 1 + m 5 6 m 1 m 2 ( m 3 1 + m 3 6 m 2 1 m 2 r 6 m 1 m 2 2 ( m 1 + m 2 ) 21 / 26

Potential matching Velocity corrections Full corrections include velocities and higher time derivatives : » 1 – L 2PN = + G 3 2( m 2 1 + m 2 r 3 m 1 m 2 2 ) + 3 m 1 m 2 » 15 – a 2 − 1 + Gm 1 m 2 r 8 ~ a 1 ~ 8( ~ a 1 ~ r )( ~ a 2 ~ r ) + ( terms depending on ~ v 1 ; ~ v 2 ) Can be eliminated using • Total time derivatives ‹ L ∝ d dt F ( ~ r; ~ v 1 ; ~ v 2 ) “ ” “ ” a 1 + Gm 2 a 2 − Gm 1 • Equations of motion ‹ L ∝ ~ r 3 ~ r ~ r 3 ~ r » 1 – L 2PN = − G 3 2 ) + 5 4( m 2 1 + m 2 r 3 m 1 m 2 4 m 1 m 2 + ( terms depending on ~ v 1 ; ~ v 2 ) 22 / 26

Post-Newtonian expansion Energy 0.00 BH - BH - 0.05 E /( μ c 2 ) 4PN / 2PN 3PN - 0.10 NS - NS 1PN N Equal mass case - 0.15 0.00 0.05 0.10 0.15 0.20 0.25 0.30 [( rs Ω )/( 2c )] 2 / 3 23 / 26