Bimetric Gravity and Dark Matter Angnis Schmidt-May Invisibles Workshop Karlsruhe Institute of Technology, 06.09.18 Motivation Massless & Massive Spin-2 Fields Navigation The Ghost-Free Theory Physics of Massive Spin-2 Fields Cosmology Summary Motivation “Beyond known physics" can mean ... ...more copies of the known field theories. ...new field theories (i.e. higher spin). “Beyond known physics" can mean ... “Beyond known... physics"more copies can mean of the... known field theories. ...new field theories (i.e. higher spin). ...more copies of the known field theories. ...new field theories (i.e. higher spin). “Beyond known physics" can mean ... ...more copies of the known field theories. ...new field theories (i.e. higher spin). Going beyond known physics ConsistentGoing Field beyond Theories knownStandard physics Model of Particle Physics GoingGoing beyond beyond known known physics physics & General Relativity Standard Model of Particle Physics & General Relativity Standard Model of Particle Physics & General Relativity StandardspinSpin 0: 0: Model Higgs Higgs of Particle boson boson φ Physics & General Relativity spinStandard 0: Model Higgs of boson Particleφ Physics & General Relativitywell-known spin 1/2: quarks, leptons a spin 0: Higgs boson φ a well-known spinSpin1 1/2/2:: quarks,leptons, leptonsquarks well-known9 consistent spinspinspin 1:1/2: 0:quarks, gluons, Higgs leptons photons, boson φa W- & Z-bosons Aµ 9> consistent spin 1: gluons, photons, W-a & Z-bosons Aµ9 consistent>=> well-knowntheories Thisspin is a slide 2:1 title/2: graviton gµ⌫ > theories spinspinSpinspin 1: 2: 1: gluons, graviton gluons,quarks, photons, photon,g leptons W- W- & & Z-bosons Z-boson A µ > = µ⌫ => theories9 consistent spin spin 2: 1: graviton gluons,gµ photons,⌫ W- & Z-bosons Aµ >> >=>;> theories Spinspin 2: 2: graviton graviton gµ⌫ > ;> ;> > ;> 1/14 4 / 26 4 / 264 / 26 4 / 26 “Beyond known physics" can mean ... ...more copies of the known field theories. ...new field theories (i.e. higher spin). “Beyond known physics" can mean ... “Beyond known... physics"more copies can mean of the... known field theories. ...new field theories (i.e. higher spin). ...more copies of the known field theories. ...new field theories (i.e. higher spin). “Beyond known physics" can mean ... ...more copies of the known field theories. ...new field theories (i.e. higher spin). Going beyond known physics ConsistentGoing Field beyond Theories knownStandard physics Model of Particle Physics GoingGoing beyond beyond known known physics physics & General Relativity Standard Model of Particle Physics & General Relativity Standard Model of Particle Physics & General Relativity StandardspinSpin 0: 0: Model Higgs Higgs of Particle boson boson φ Physics & General Relativity spinStandard 0: Model Higgs of boson Particleφ Physics & General Relativitywell-known spin 1/2: quarks, leptons a spin 0: Higgs boson φ a well-known spinSpin1 1/2/2:: quarks,leptons, leptonsquarks well-known9 consistent spinspinspin 1:1/2: 0:quarks, gluons, Higgs leptons photons, boson φa W- & Z-bosons Aµ 9> consistent spin 1: gluons, photons, W-a & Z-bosons Aµ9 consistent>=> well-knowntheories Thisspin is a slide 2:1 title/2: graviton gµ⌫ > theories spinspinSpinspin 1: 2: 1: gluons, graviton gluons,quarks, photons, photon,g leptons W- W- & & Z-bosons Z-boson A µ > = µ⌫ => theories9 consistent spin spin 2: 1: graviton gluons,gµ photons,⌫ W- & Z-bosons Aµ >> >=>;> theories Spinspin 2: 2: graviton graviton gµ⌫ > ;> ;> + Supersymmetry> ;> 1/14 4 / 26 4 / 264 / 26 4 / 26 “Beyond known physics" can mean ... ...more copies of the known field theories. ...new field theories (i.e. higher spin). “Beyond known physics" can mean ... “Beyond known... physics"more copies can mean of the... known field theories. ...new field theories (i.e. higher spin). ...more copies of the known field theories. ...new field theories (i.e. higher spin). “Beyond known physics" can mean ... ...more copies of the known field theories. ...new field theories (i.e. higher spin). Going beyond known physics ConsistentGoing Field beyond Theories knownStandard physics Model of Particle Physics GoingGoing beyond beyond known known physics physics & General Relativity Standard Model of Particle Physics & General Relativity Standard Model of Particle Physics & General Relativity StandardspinSpin 0: 0: Model Higgs Higgs of Particle boson boson φ Physics & General Relativitynew models are spinStandard 0: Model Higgs of boson Particleφ Physics & General Relativitywell-known spin 1/2: quarks, leptons a usually builtwell-known using spin 0: Higgs boson φ a more copies of spinSpin1 1/2/2:: quarks,leptons, leptonsquarks well-known9 consistent spinspinspin 1:1/2: 0:quarks, gluons, Higgs leptons photons, boson φa W- & Z-bosons Aµthese9> particlesconsistent spin 1: gluons, photons, W-a & Z-bosons Aµ9 consistent>=> well-knowntheories Thisspin is a slide 2:1 title/2: graviton gµ⌫ > theories spinspinSpinspin 1: 2: 1: gluons, graviton gluons,quarks, photons, photon,g leptons W- W- & & Z-bosons Z-boson A µ > = µ⌫ => theories9 consistent spin spin 2: 1: graviton gluons,gµ photons,⌫ W- & Z-bosons Aµ >> >=>;> theories Spinspin 2: 2: graviton graviton gµ⌫ > ;> less understood…;> > ;> 1/14 4 / 26 4 / 264 / 26 4 / 26 “Beyond known physics" can mean ... ...more copies of the known field theories. ...new field theories (i.e. higher spin). “Beyond known physics" can mean ... “Beyond known... physics"more copies can mean of the... known field theories. ...new field theories (i.e. higher spin). ...more copies of the known field theories. ...new field theories (i.e. higher spin). “Beyond known physics" can mean ... ...more copies of the known field theories. ...new field theories (i.e. higher spin). Going beyond known physics ConsistentGoing Field beyond Theories knownStandard physics Model of Particle Physics GoingGoing beyond beyond known known physics physics & General Relativity Standard Model of Particle Physics & General Relativity Standard Model of Particle Physics & General Relativity StandardspinSpin 0: 0: Model Higgs Higgs of Particle boson boson φ Physics & General Relativity spinStandard 0: Model Higgs of boson Particleφ Physics & General Relativitywell-known spin 1/2: quarks, leptons a massless spin 0: Higgs boson φ a well-known spinSpin1 1/2/2:: quarks,leptons, leptonsquarks well-known&9 massiveconsistent spinspinspin 1:1/2: 0:quarks, gluons, Higgs leptons photons, boson φa W- & Z-bosons Aµ 9> consistent spin 1: gluons, photons, W-a & Z-bosons Aµ9 consistent>=> well-knowntheories Thisspin is a slide 2:1 title/2: graviton gµ⌫ > theories spinspinSpinspin 1: 2: 1: gluons, graviton gluons,quarks, photons, photon,g leptons W- W- & & Z-bosons Z-boson A µ > = µ⌫ => theories9 consistent spin spin 2: 1: graviton gluons,gµ photons,⌫ W- & Z-bosons Aµ >> >=>;> theories Spinspin 2: 2: graviton graviton gµ⌫ > ;> MASSLESS ! ;> > ;> 1/14 4 / 26 4 / 264 / 26 4 / 26 “Beyond known physics" can mean ... ...more copies of the known field theories. ...new field theories (i.e. higher spin). “Beyond known physics" can mean ... “Beyond known... physics"more copies can mean of the... known field theories. ...new field theories (i.e. higher spin). ...more copies of the known field theories. ...new field theories (i.e. higher spin). “Beyond known physics" can mean ... ...more copies of the known field theories. ...new field theories (i.e. higher spin). Going beyond known physics ConsistentGoing Field beyond Theories knownStandard physics Model of Particle Physics GoingGoing beyond beyond known known physics physics & General Relativity Standard Model of Particle Physics & General Relativity Standard Model of Particle Physics & General Relativity StandardspinSpin 0: 0: Model Higgs Higgs of Particle boson boson φ Physics & General Relativity spinStandard 0: Model Higgs of boson Particleφ Physics & General Relativitywell-known spin 1/2: quarks, leptons a multiplets of spin 0: Higgs boson φ a well-known spinSpin1 1/2/2:: quarks,leptons, leptonsquarks gaugewell-known9 consistentgroups spinspinspin 1:1/2: 0:quarks, gluons, Higgs leptons photons, boson φa W- & Z-bosons Aµ 9> consistent spin 1: gluons, photons, W-a & Z-bosons Aµ9 consistent>=> well-knowntheories Thisspin is a slide 2:1 title/2: graviton gµ⌫ > theories spinspinSpinspin 1: 2: 1: gluons, graviton gluons,quarks, photons, photon,g leptons W- W- & & Z-bosons Z-boson A µ > = µ⌫ => theories9 consistent spin spin 2: 1: graviton gluons,gµ photons,⌫ W- & Z-bosons Aµ >> >=>;> theories Spinspin 2: 2: graviton graviton gµ⌫ > ;> just one field…;> > ;> 1/14 4 / 26 4 / 264 / 26 4 / 26 How do we make a spin-2 field massive ? Can several spin-2 fields interact ? Massless & Massive Spin-2 Fields Which metric represents gravity? Absence of ghost: only one of the metrics couples to matter 2 4 2 4 Sgf = mg d xpgR(g)+mf d x fR(f) Z Z p 4 m4 d4xpg β e g 1f − n n − n=0 Z X ⇣p ⌘ + d4xpg (g, φ) Lmatter Z Massless Theory Massless Gravity General Relativity only known coupling that does not re-introduce the ghost Massless Gravity ! General Relativity= classical with nonlinear Einstein-Hilbert field theory action forfor metricmetricg tensor gµ⌫ is the gravitational metric µ⌫ ! General Relativity with Einstein-Hilbert action for metric g The gravitationalµ⌫ metric is not massless! S [g]=M 2 d4xpg R(g) 2⇤ EH P − Einstein-Hilbert action: Z S [g]=M 2 d4xpg R(g) 2⇤ EH ⇣ P ⌘ − Z ⇣ ⌘ 18 / 26 This is a slide title EinsteinEinstein’s equations: equations: R 1 g R + ⇤g =0 µ⌫ − 2 µ⌫ µ⌫ Einstein equations: R 1 g R + ⇤g =0 µ⌫ − 2 µ⌫ µ⌫ ¯ maximallydescribes symmetric the two degrees solutions: of Rfreedomµ⌫