Bimetric Gravity and Dark Matter

Home , Bimetric gravity

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 Physics & General Relativity spinStandard 0: Model Higgs of boson Particle Physics & General Relativitywell-known spin 1/2: quarks, leptons a spin 0: Higgs boson a well-known spinSpin1 1/2/2:: quarks,leptons, leptonsquarks well-known9 consistent spinspinspin 1:1/2: 0:quarks, gluons, Higgs leptons photons, boson a 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⌫ > ;> ;> > ;>

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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 ...

...more copies of the known ﬁeld theories. ...new ﬁeld theories (i.e. higher spin).

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 ...

...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 Physics & General Relativitynew models are spinStandard 0: Model Higgs of boson Particle Physics & General Relativitywell-known spin 1/2: quarks, leptons a usually builtwell-known using spin 0: Higgs boson 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 a 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…;> > ;>

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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 ...

...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 Physics & General Relativity spinStandard 0: Model Higgs of boson Particle Physics & General Relativitywell-known spin 1/2: quarks, leptons a massless spin 0: Higgs boson 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 a 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 ! ;> > ;>

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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 ...

...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 Physics & General Relativity spinStandard 0: Model Higgs of boson Particle Physics & General Relativitywell-known spin 1/2: quarks, leptons a multiplets of spin 0: Higgs boson a well-known spinSpin1 1/2/2:: quarks,leptons, leptonsquarks gaugewell-known9 consistentgroups spinspinspin 1:1/2: 0:quarks, gluons, Higgs leptons photons, boson a 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…;> > ;>

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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 e g 1f n n n=0 Z X ⇣p ⌘ + d4xpg (g, ) 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⌫ = ⇤ g¯ µ⌫ maximally symmetric solutions: R¯ = ⇤g¯ of a self-interacting, massless spin-2 particle µ⌫ µ⌫ linear perturbation theory: gµ⌫ =¯gµ⌫ + gµ⌫: linear perturbation theory: gµ⌫ =¯gµ⌫ + gµ⌫: 2/14 ¯ ⇢ 1 µ⌫ g⇢ ⇤g gµ⌫ g¯µ⌫g =0 E ¯ ⇢2g ⇤ g 1 g¯ g =0 Eµ⌫ ⇢ g µ⌫ 2 µ⌫ equation for a massless spin-ﬁeld with 2 degrees of freedom ! equation for a massless spin-ﬁeld with 2 degrees of freedom ! 5 / 26 5 / 26 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 e g 1f n n n=0 Z X ⇣p ⌘ + d4xpg (g, ) 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 1 EinsteinEinstein’s equations: equations: Rµ⌫ 2 gµ⌫R + ⇤gµ⌫ =0 1 two derivatives Einstein equations: Rµ⌫ gµ⌫R + ⇤gµ⌫ =0 2 kinetic term ¯ maximallydescribes symmetric the two degrees solutions: of Rfreedomµ⌫ = ⇤ g¯ µ⌫ maximally symmetric solutions: R¯ = ⇤g¯ of a self-interacting, massless spin-2 particle µ⌫ µ⌫ linear perturbation theory: gµ⌫ =¯gµ⌫ + gµ⌫: linear perturbation theory: gµ⌫ =¯gµ⌫ + gµ⌫: 2/14 ¯ ⇢ 1 µ⌫ g⇢ ⇤g gµ⌫ g¯µ⌫g =0 E ¯ ⇢2g ⇤ g 1 g¯ g =0 Eµ⌫ ⇢ g µ⌫ 2 µ⌫ equation for a massless spin-ﬁeld with 2 degrees of freedom ! equation for a massless spin-ﬁeld with 2 degrees of freedom ! 5 / 26 5 / 26 General Relativity = unique description of self-interacting massless spin-2 field Mass Term … should not contain derivatives nor loose indices.

Examples: scalar (spin @0) @ µ m 2 2 vector (spin 1) (1) µ µ 2 2 µ⌫ 2 µ @µ@ m F F m A A (1) (2) µ⌫ µ F F µ⌫ m2AµA (2) µ⌫ µ

gµ⌫@ @ m22 (3) µ ⌫ µ⇢ ⌫ 2 µ⌫ This is a slideg titleµ⌫@ @ m22 g g F⇢Fµ⌫ m g AµA⌫ (3) (4) µ ⌫ gµ⇢g⌫F F µ⌫ m2gµ⌫A A (4) ⇢ µ ⌫

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1 Only way out: introduce second metric to contract indices

µ⌫ 1 µ⌫ 1 g fµ⌫ =Tr(g f) f gµ⌫ =Tr(f g)

Massive Gravity action is of the form

S [g]=S [g] d4xV(g, f) MG EH Z What determines fµ⌫?

µ 2 2 @µ@ m (1) µ 2 2 @µ@ m µ⌫ 2 µ (1) F Fµ⌫ m A Aµ (2) µ⌫ 2 µ Mass Term NonlinearF Massiveµ⌫F m GravityA Aµ (2) … should not contain derivatives nor loose indices.

Examples: scalarg (spinµ⌫@ @ 0) m 2 2 vector (spin 1) (3) µ ⌫ Mass term should have non-derivative interactions of gµ⌫, but µ⌫ 2 2 µ⇢ ⌫ 2 µ⌫ g @ @ m if we tryg tog contractF⇢Fµ⌫ them indicesg Aµ weA⌫ get: (3)(4) µ ⌫ µ⇢ ⌫ µ⌫ 2 µ⌫ g g F⇢F m g µA⌫ µA⌫ (4) For the spin-2 tensor contracting indices of the metric gives: g gµ⌫ =4 This is not a mass term. This is not a mass term. This is a slide title

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10 / 26

1 Only way out: introduce second metric to contract indices

µ⌫ 1 µ⌫ 1 g fµ⌫ =Tr(g f) f gµ⌫ =Tr(f g)

MassiveWhat Gravity determines action isf ofµ⌫ the? form

S [g]=S [g] d4xV(g, f) MG EH Z What determines fµ⌫?

µ 2 2 @µ@ m (1) µ 2 2 @µ@ m µ⌫ 2 µ (1) Nonlinear Massive Gravity F Fµ⌫ m A Aµ (2) µ⌫ 2 µ Mass Term NonlinearF Massiveµ⌫F m GravityA Aµ (2) … should not contain derivatives nor loose indices. Mass term should have non-derivative interactions of gµ⌫, but µ⌫ 2 2 Examples:if we try toscalar contractg (spin@µ the@ 0)⌫ indices m we get: vector (spin 1) (3) Mass term should have non-derivative interactions of gµ⌫, but µ⌫ 2 2 µ⇢ ⌫ 2 µ⌫ g @ @ m if we tryg tog contractF⇢Fµ⌫ them indicesg Aµ weA⌫ get: (3)(4) µ ⌫ µ⌫ g gµ⌫ µ=4⇢ ⌫ µ⌫ 2 µ⌫ g g F⇢F m g µA⌫ µA⌫ (4) For the spin-2 tensor contracting indices of the metric gives: g gµ⌫ =4 This is not a mass term. This is not a mass term. This is not a mass term. ThisSimplest is a slide Onlyway title wayout: out:Introduce introduce second second “metric” metric to contract to contract indices: indices

µ⌫ 1 µ⌫ 1 g fµ⌫ =Tr(g f) f gµ⌫ =Tr(f g)

Massive Gravity action is of the form

S [g]=S [g] d4xV(g, f) MG EH 3/14 Z

10 / 26

1 Only way out: introduce second metric to contract indices

µ⌫ 1 µ⌫ 1 g fµ⌫ =Tr(g f) f gµ⌫ =Tr(f g)

MassiveWhat Gravity determines action isf ofµ⌫ the? form

S [g]=S [g] d4xV(g, f) MG EH Z What determines fµ⌫?

µ⌫ 1 µ⌫ 1 g fµ⌫ =Tr(g f) f gµ⌫ =Tr(f g)

MassiveMassive gravity Gravity action: action is of the form

S [g]=S [g] kineticd4xV term(g, f)mass term MG EH 3/14 Z

10 / 26

1 Only way out: introduce second metric to contract indices

µ⌫ 1 µ⌫ 1 g fµ⌫ =Tr(g f) f gµ⌫ =Tr(f g)

MassiveWhat Gravity determines action isf ofµ⌫ the? form

S [g]=S [g] d4xV(g, f) MG EH Z What determines fµ⌫?

Nonlinear Massive Gravity

Mass term should have non-derivative interactions of gµ⌫, but if we try to contract the indices we get:

µ⌫ g gµ⌫ =4

This is not a mass term.

Only way out: introduce second metric to contract indices

µ⌫ 1 µ⌫ 1 g fµ⌫ =Tr(g f) f gµ⌫ =Tr(f g) µ 2 2 @µ@ m (1) µ 2 2 Massive@µ Gravity@ m action is ofµ the⌫ form2 µ (1) Nonlinear Massive Gravity F Fµ⌫ m A Aµ (2) Mass Term µ⌫ 2 µ … shouldNonlinear not containF Massiveµ⌫ Fderivativesm GravityA norAµ loose indices. (2) 4 Mass term shouldS haveMG[ non-derivativeg]=SEH[g interactions] d ofxVgµ⌫(,g, but f) µ⌫ 2 2 Examples:if we try toscalar contractg (spin@µ the@ 0)⌫ indices m we get: vector (spin 1) (3) Mass term shouldZ have non-derivative interactions of gµ⌫, but µ⌫ 2 2 µ⇢ ⌫ 2 µ⌫ g @ @ m if we tryg tog contractF⇢Fµ⌫ them indicesg Aµ weA⌫ get: (3)(4) µ ⌫ µ⌫What determinesdetermines f µ ⌫ ? ? g gµ⌫ µ=4⇢ ⌫ µ⌫ 2 µ⌫ g g F⇢F m g µA⌫ µA⌫ (4) For the spin-2 tensor contracting Shouldn’tindices of the itmetric be dynamicalgives: g gµ⌫ =4? This is not a mass term. 10 / 26 This is not a mass term. This is not a mass term. ThisSimplest is a slide Onlyway title wayout: out:Introduce introduce second second “metric” metric to contract to contract indices: indices

µ⌫ 1 µ⌫ 1 g fµ⌫ =Tr(g f) f gµ⌫ =Tr(f g)

MassiveMassive gravity Gravity action: action is of the form

S [g]=S [g] kineticd4xV term(g, f)mass term MG EH 3/14 Z

10 / 26

1 But there’s still a problem ...

Massive Gravity

nondynamical background metric fµ⌫, ﬁxed by hand 5 d.o.f. ! f Bimetricdynamical Theory µ⌫, determined by its equation of motion 5 + 2 = 7 d.o.f. [Rosen, 1940; Isham, Salam & Strathdee,1971/77] ! NonlinearBimetric action action: for two interacting tensors:

S [g, f]=m2 d4xpg R(g) 2⇤ b g Z ⇣ ⌘ + m2 d4x f R(f) 2⇤˜ d4xV(g, f) f Z p ⇣ ⌘ Z This is botha slide metrics title are dynamical and treated on equal footing

should describe massive & massless spin-2 field (5+2 d.o.f.)

4/14 But there’s still a problem ...

Massive Gravity

S [g, f]=m2 d4xpg R(g) 2⇤ b g Z ⇣ ⌘ + m2 d4x f R(f) 2⇤˜ d4xV(g, f) f Z p ⇣ ⌘ Z This is a slide title both metrics are dynamical and treated on equal footingLinear Massive Gravity

Fierz & Pauli (1939): equation for massive spin-2 ﬁeld

m2 ¯ ⇢g ⇤ g 1 g¯ g + FP g a g¯ g =0 Eµ⌫ ⇢ g µ⌫ 2 µ⌫ 2 µ⌫ µ⌫ should describe massive & massless spin-2 field (5+2 d.o.f.)propagates 5 degrees of freedom for a = 1

for a = 1 there is an additional propagating scalar mode that 6 gives rise to a ghost instability This looks good, but in general the theory has a ghost!

6 / 26 4/14 The Ghost-Free Theory Based on

ASM, M. von Strauss 1412.3812

S.F. Hassan, ASM, M. von Strauss Linear Massive Gravity Fierz & PauliGhost-free (1939): equation for massive spin-2 field interaction potential m2 ¯ ⇢ 1 FP 1208.1515 g ⇤ g g¯ g + g a g¯ g =0 Eµ⌫ ⇢ Ghost-freeg µ⌫ 2 µ⌫ 2 µ⌫ µ⌫ interaction potential propagates 5 degrees of freedom for a = 1 1203.5283 for a = 1 there is an additional propagating scalar mode that 6 gives rise to a ghost instability Ghost-free interaction potential3 de Rham, Gabadadze, Tolley (2010); - free Bimetric◆V (g, Theory f)=m4 g 3 e g 1f ⇣ ◆ 4 p n n Hassan,1 Rosen⇣ (2011) S.F.V Hassan,(g, f)= R.m Rosen,pg n ASMen g f Ghost-free interaction potential6 / 26 n=1 nX=1 ⇣p ⌘ [de Rham,1109.3230 Gabadadze & Tolley, 2010; HassanX2& Rosen,4 2011;⇣ Hassan,p ASM⌘& Rosen, 2011] Sb[g, f]=m d 3xpgR(g) [de Rham,✓◆ Gabadadze & Tolley, 2010; Hassan g& Rosen, 2011; Hassan, ASM & Rosen, 2011]⌘ ⇣ ✓V (g, f)=m4 Zg e g ⌘1f 2p 4 n n 4 The potential involves3 ... The potential involves ... + mf dn=1x fR(f) d xV(g, f) ◆ 4 1 ⇣X ⇣ ⌘ V (g, f)=m pg n en g f Z p Z [de Rham, Gabadadze & Tolley, 2010; Hassan & Rosen,p 2011; Hassan, ASM & Rosen, 2011] ...an arbitraryn=1 mass scale m, 3 free parameters n ...an arbitrary✓ mass⇣ scale4 ⌘m, 3 free parameters4 4n ⌘ X 4 p 1 4 1 [de Rham, Gabadadze & Tolley, 2010; HassanV (g,& fRosen,)= 2011;m p Hassan,◆g ASMn&eRosen,n 2011]g4f = m f 4 n1en ⇣f g (1) ...the↵ elementary symmetricV (g, f)= polynomialsm pg n en g f ✓ The... potentialthe elementary involves symmetric...n=0 polynomials⌘ n=0 X ✓q ◆ p X ✓q ◆ n=0 ⇣ ⌘ The potential involves ... 2 X p 2 ⌫⌫1...... ⌫⌫nn+1...... 4 µ µ1 µ µn ee...n((SSan)=)=arbitrary arbitrary spin-2 mass✏✏µ mass...µ scale scale ...m ✏,✏ 31 freen n+1 parameters4 SS1 ⌫...S...Sn n ⌫ n ⌦ nn!(4!(4 n)!✓µ11...µnnnn+1+1...44 ⌫1 1 ⌫n⌘n ...an arbitrary mass scale3m interaction, 3 free parameters parameters n 1 2 2 ...the elementarye1(S)=Tr[ symmetricS] polynomialse2(S)= 2 (Tr[S]) Tr[S ] ...... aa square-root square-rootsquare-root matrixmatrix SS defined defineddefined throughthrough through SS2 2==g g1f1f ...the elementary symmetric polynomials ⇣ ⌘ e2 (S)= 1 (Tr[S])3 3Tr[S2]Tr[S]+2Tr[S3] 5/14 3 6 ⌫1...⌫nn+1...4 µ1 µn en(S)= ✏µ1...µnn+1...4 ✏ S ⌫1 ...S ⌫n 2 n!(4⌫1...⌫nn)!n+1...4 µ1 µn e (S)= ✏ ✏ ⇣ S ...S 13⌘ / 26 n n!(4 n)! µ1...µnn+1...4 Consistent theory⌫1 for massless⌫n & massive spin-213 / 26 ) 2 1 ...a square-root matrix S defined through S = g f 2 1 ...a square-root matrix S defined through S = g f

13 / 26 13 / 26 Based on

ASM, M. von Strauss 1412.3812

S.F. Hassan, ASM, M. von Strauss Linear Massive Gravity

Fierz & Pauli (1939): equation for massive spin-2 ﬁeld

m2 ¯ ⇢ 1 FP 1208.1515 g ⇤ g g¯ g + g a g¯ g =0 Eµ⌫ ⇢ g µ⌫ 2 µ⌫ 2 µ⌫ µ⌫ propagates 5 degrees of freedom for a = 1 1203.5283 for a = 1 there is an additional propagating scalar mode that 6 gives rise to a ghost instability

de Rham, Gabadadze, Tolley (2010); - free Bimetric Theory S.F. Hassan, R. Rosen, ASM Hassan, Rosen (2011)

6 / 26 2 4 1109.3230 Sb[g, f2]=4 m d xpgR(g) Sb[g, f]=mg d xpg gR(g) Z Z + m2 d4x fR(f) d4xV(g, f) + m2 d4x f fR(f) d4xV(g, f) f Z Z Z p Z p 4 4 ◆V (g, f)=m4 4 g 4 e g 1f ⇣ 4 1 4p n n 1 V (g, f)=m p◆g n en g4f = m f 4 n1en ⇣f g (1) ↵ V (g, f)=m pg n enn=0 g f n=0 ✓q ◆ Xn=0 ⇣p ✓q ⌘ ◆ X n=0 p X ✓ X ⇣p ⌘ ⌘ 1 2 2 elementary ✓ e1(S)=Tr[S] e2(S)= (Tr[S]) ⌘Tr[S ] ⌦ 2 2 symmetric polynomials: 1 31 2⇣ 2 3 ⌘ e1(S)=Tr[e (SS])= e2(Tr[(S)=S]) 2 3Tr[(Tr[SS])]Tr[S]+2Tr[Tr[S ] S ] 3 6 1 3⇣ 2⇣ 3 ⌘ ⌘ e3(S)= 6 Consistent(Tr[S]) theory3Tr[S for]Tr[ masslessS]+2Tr[& massiveS ] spin-2 5/14 ) ⇣ ⌘ Consistent theory for massless & massive spin-2 ) Physics of Massive Spin-2 Fields Bimetric mass spectrum [Hassan, ASM & von Strauss, 2012] [Hassan, ASM & von Strauss, 2012] ProportionalProportional backgrounds backgrounds [Hassan, ASM & von Strauss, 2012] Hassan, ASM, von Strauss (2012) Mass spectrum perturb bimetric equations around proportional backgrounds: 2 ParticularlyParticularly important important solution solution to tog equationsµ equations⌫ =¯gµ⌫ + of ofg motion: motion:µ⌫ fµ⌫ = c g¯µ⌫ + fµ⌫ Bimetric mass spectrum [Hassan, ASM & von Strauss, 2012] Bimetric mass spectrum [Hassan, ASM & von Strauss,¯ 2012]¯ 22 Maximally symmetric solutions: ffµµ⌫⌫==cﬂuctuationsc g¯g¯µµ⌫⌫ withwithwith diagonalizablec == const.const. into mass eigenstates:

2 perturb bimetric equations around– gives proportional two copies of backgrounds: Einstein’s equationsmassless (↵↵ Gmmµf⌫/m/mg) gµ⌫ + ↵ fµ⌫ perturb bimetric equations– gives around two proportional copies of Einstein’s backgrounds: equations ( ⌘ f /g) Perturbations around proportional backgrounds: ⌘M f c2g 2 1 massive µ⌫ µ⌫ µ⌫ gµ⌫ =¯gµ⌫ + gµ⌫ fµ⌫ = c g¯µR2⌫µ+⌫(¯gf)µ⌫ 1 g¯µ⌫R(¯g)+⇤g(↵, n,c)¯gµ⌫ =0/ gµ⌫ =¯gµ⌫ + gµ⌫ fµ⌫ = Rc µg¯⌫µ(¯⌫g+) f2µ2g¯⌫µ⌫R(¯g)+⇤g(↵, n,c)¯gµ⌫ =0 1 Rµ⌫(¯g) 12 g¯µ⌫R(¯g)+⇤f (↵, n,c)¯gµ⌫ =0 ﬂuctuations diagonalizable into mass eigenstates:Rµ⌫(¯g) linear2 g¯µ⌫R equations:(¯g)+⇤f (↵, n,c)¯gµ⌫ =0 ﬂuctuationsCan be diagonalizablediagonalised into into mass mass eigenstates: eigenstates:) – consistency condition:¯ ⇢ ⇤g(↵, n,c)=⇤f (↵1, n,c) determines c – consistency condition:2 2µ⌫ ⇤Gg⇢(↵, ⇤ng,c)=Gµ⌫⇤f (↵2 g¯,µ⌫n,cG) =0determines c masslessmasslessGµG⌫ µ⌫ gµ⌫gµ+⌫ ↵+ ↵fEµ⌫fµ⌫ massless (2 d.o.f.) / / m2 2 2¯ ⇢ 1 FP massivemassiveMµM⌫ µ⌫ fµ⌫fµ⌫ c cgµµ⌫g⌫µ⌫massiveM⇢ ⇤ g (5 d.o.f.)Mµ⌫ 2 g¯µ⌫M + 2 (Mµ⌫ g¯µ⌫M)=0 / /maximally symmetricE backgrounds with Rµ⌫(¯g)=⇤gg¯µ⌫ )maximally symmetric backgrounds with Rµ⌫(¯g)=⇤gg¯µ⌫ ) with Fierz-Pauliwith mass mass mFP= mFP(↵, n,c) linearlinear equations: equations: )) 6/14¯ ⇢ ⇢ 1 1 17 / 26 µ⌫¯ G⇢G ⇤g⇤GµG⌫ g¯µ⌫g¯GG=0=0 16 / 26 E µ⌫ ⇢ g µ⌫ 2 2 µ⌫ E 16 / 26 ⇢ m2 2 ¯ ⇢ 1 1 FPmFP µ⌫¯µ⌫M⇢M ⇤g⇤ MµM⌫ g¯µ⌫g¯MM+ + (M(µM⌫ g¯µ⌫g¯M)=0M)=0 E E ⇢ g µ⌫ 2 2 µ⌫ 2 2 µ⌫ µ⌫ withwith mass massmFPmFP=m= FPmFP(↵,(↵n, ,cn),c)

17 / 2617 / 26 Ghost-free bimetric theory = unique description of massless + massive spin-2 field What is the physical metric ?

How does matter couple to the tensor fields ? Which metric represents gravity? Yamashita, de Felice, Tanaka; Matter Coupling de Rham, Heisenberg, Ribeiro (2015) Absence of ghost: only one of the metrics couples to matter

2 2 4 4 2 4 Sb[g,S fgf]== mg mgd xpd gRxp(gRg)+(g)mf d x fR(f) Z Z Z p 4 2 4 4 4 4 1 + mf d x fR(fm) d xdpgxV(g,n fen) g f n=0 Z p Z Z X ⇣p ⌘ + d4xpg (g, ) 4 Lmatter ◆ 4 Z 1 ⇣ V (g, f)=m pg n en g f n=0 only known coupling thatX does not re-introduce⇣p ⌘ the ghost Absence! of ghosts: only one metric can couple to matter! ✓ ⌘ g µ ⌫ is gravitational the gravitational metric metric ! 1 2 2 e1(S)=Tr[S] e2(S)= (Tr[S]) Tr[S ] The gravitational metric is not2 massless! ⇣ ⌘ 7/14 e (S)= 1 (Tr[S])3 3Tr[S2]Tr[S]+2Tr[S3] 3 6 ⇣ ⌘ 18 / 26 Consistent theory for massless & massive spin-2 ) Physical interpretation [Hassan, ASM & von Strauss, 2012]

Bimetric Theory = Gravity + massive spin-2

Baccetti, Martin-Moruno, Visser (2012); Hassan, ASM, von Strauss (2012/14); Mass EigenstatesProportionalCan rewrite backgrounds bimetric action[Hassan, in terms ASM & ofvon gravitational Strauss, 2012] metric gµ⌫ and massive spin-2 ﬁeld MAkrami,µ⌫: Hassan, Koennig, ASM, Solomon (2015)

S [g, f]=2m2 d44x gR(g) 2 4 4 b mg g d xpgR(g)+mf d xK(g, M)+ d xV(g, M) ParticularlyZ important solution to equations of motion: + mZ2 d4x fR(f) d4xVZ (g, f) Z f 4 Z p 2Z + d xpg matter(g, ) f¯µ⌫ = c g¯µ⌫ with c = const. L 4 Z ◆ 4 1 ⇣ (linearised) Vgravitational(g, f)=m p gmetric:n en g f n=0 ⇣ 2⌘ – givesRecall: two copiesgµ⌫X of Einstein’sGµ⌫p ↵ equationsMµ⌫ (↵ m /m ) / f g ✓ massless massive⌘ ⌘ e (S)=Tr[forS] smalle (S↵)==1m1(Tr[f /mS])g2 (i.e.Tr[S weak2] gravity!), the massive spin-2 1 ) Rµ⌫2(¯g) 2 2 g¯µ⌫R(¯g)+⇤g(↵, n,c)¯gµ⌫ =0 ﬁeld interacts only ⇣ weakly with matter:⌘ e (S)= 1 (Tr[S])3 3Tr[S21]Tr[S]+2Tr[S3] 3 6 Rµ⌫(¯g) 2 g¯µ⌫R(¯g)+⇤f (↵, n,c)¯gµ⌫ =0 ↵ ⇣ 0 is the General Relativity⌘ limit of Bimetric Theory Consistent! theory for massless & massive spin-2 ) – consistency condition: ⇤g(↵, n,c)=⇤f (↵, n,c) determines c 19 / 26

8/14 maximally symmetric backgrounds with R (¯g)=⇤ g¯ ) µ⌫ g µ⌫

16 / 26 Physical interpretation [Hassan, ASM & von Strauss, 2012]

Bimetric Theory = Gravity + massive spin-2

S [g, f]=2m2 d44x gR(g) 2 4 4 b mg g d xpgR(g)+mf d xK(g, M)+ d xV(g, M) ParticularlyZ important solution to equations of motion: + mZ2 d4x fR(f) d4xVZ (g, f) Z f 4 Z p 2Z + d xpg matter(g, ) f¯µ⌫ = c g¯µ⌫ with c = const. L 4 Z ◆ 4 1 ⇣ (linearised) Vgravitational(g, f)=m p gmetric:n en g f n=0 ⇣ 2⌘ – givesRecall: two copiesgµ⌫X of Einstein’sGµ⌫p ↵ equationsMµ⌫ (↵ m /m ) / f g ✓ massless massive⌘ ⌘ e (S)=Tr[forS] smalle (S↵)==1m1(Tr[f /mS])g2 (i.e.Tr[S weak2] gravity!), the massive spin-2 1 ) Rµ⌫2(¯g) 2 2 g¯µ⌫R(¯g)+⇤g(↵, n,c)¯gµ⌫ =0 ﬁeld interacts only ⇣ weakly with matter:⌘ e (S)= 1 (Tr[S])3 3Tr[S21]Tr[S]+2Tr[S3] 3 6 Rµ⌫(¯g) 2 g¯µ⌫R(¯g)+⇤f (↵, n,c)¯gµ⌫ =0 ↵ ⇣ 0 is the General Relativity⌘ limit of Bimetric Theory Consistent! theory for massless & massive spin-2 The gravitational) metric is not massless but a superposition of mass eigenstates. – consistency condition: ⇤g(↵, n,c)=⇤f (↵, n,c) determines c 19 / 26 Max, Platscher, Smirnov (2017): analysis of gravitational wave oscillations 8/14 maximally symmetric backgrounds with R (¯g)=⇤ g¯ ) µ⌫ g µ⌫

16 / 26 Physical interpretation [Hassan, ASM & von Strauss, 2012]

Physical interpretation [Hassan, ASM & von Strauss, 2012] Bimetric Theory = Gravity + massive spin-2

Bimetric Theory = Gravity + massiveBaccetti, spin-2 Martin-Moruno, Visser (2012); Hassan, ASM, von Strauss (2012/14); Mass EigenstatesProportionalCan rewrite backgrounds bimetric action[Hassan, in terms ASM & ofvon gravitational Strauss, 2012] metric gµ⌫ and massive spin-2 field MAkrami,µ⌫: Hassan, Koennig, ASM, Solomon (2015) Can rewrite bimetric action in terms of gravitational metric gµ⌫ and massive spin-2 field Mµ⌫: S [g, f]=2m2 d44x gR(g) 2 4 4 b mg g d xpgR(g)+mf d xK(g, M)+ d xV(g, M) ParticularlyZ important solution to equations of motion: 2 4 + mZ2 d4x 2fR(f)4 d4xVZ (g, f) 4 Z m d xpgR(gf )+m d xK (g, M)+ d xV(g, M) 4 g Z f Z + d x g (g, ) p ¯ 2 p matter Z fZµ⌫ = c g¯µ⌫ withZ c = const. L 4 Z ◆ ⇣ 4 (linearised) Vgravitational(g, f)=m4p gmetric: e g 1f + d xpg (g, ) n n Lmatter n=0 ⇣ 2⌘ – givesRecall: two copiesgµ⌫X of Einstein’sGµ⌫p ↵ equationsMµZ⌫ (↵ m /m ) / f g ✓ massless massive⌘ ⌘ g forG small ↵↵2=1Mm1 /m 2 (i.e. weak2 gravity!), the massive spin-2 Recall: e1(S)=Tr[µ⌫ S] Rµ⌫e2(¯(Sg)=) 2 (Tr[µg¯f⌫ SR])g(¯g)+Tr[S⇤] (↵, ,c)¯g =0 )/ µ⌫ 2 µ⌫ g n µ⌫ field1 interacts3 only21⇣ weakly with3 matter:⌘ e3(S)= (Tr[S]) 3Tr[S ]Tr[S]+2Tr[S ] forfor small small ↵ 6= m Rf /m µ ⌫ (¯ g g (i.e. )gravity weak2 g¯µ ⌫isR gravity!),dominated(¯g)+⇤f the(↵ by, massive then,c )¯masslessgµ spin-2⌫ =0 mode ) ↵ ⇣ 0 is the General Relativity⌘ limit of Bimetric Theory field interactsConsistent only! theory weakly for massless with matter:& massive spin-2 the massive–) consistency spin-2 field condition: interacts⇤g only(↵, weaklyn,c)= with⇤f (↵ matter, n,c) determines c ↵ 0 is the General Relativity limit of Bimetric Theory 19 / 26 ! 8/14 19 / 26 maximally symmetric backgrounds with R (¯g)=⇤ g¯ ) µ⌫ g µ⌫

16 / 26 Physical interpretation [Hassan, ASM & von Strauss, 2012]

Bimetric Theory = Gravity + massive spin-2 Physical interpretation [Hassan, ASM & von Strauss, 2012]

Bimetric Theory = GravityCan rewrite + massive bimetric spin-2 action in terms of gravitational metric gµ⌫ and massive spin-2 ﬁeld Mµ⌫: Bimetric mass spectrum [Hassan, ASM & von Strauss, 2012]

Can rewrite bimetric action2 in terms4 of gravitational2 metric4 gµ⌫ and 4 mg d xpgR(g)+m d xK(g, M)+ d xV(g, M) massive spin-2 field Mµ⌫: f perturb bimetricZ equations around proportionalZ backgrounds:Z Baccetti, Martin-Moruno, Visser (2012); 2 4 Mass Eigenstates 2 4 Hassan,2 g µASM,⌫ =¯4 gvonµ⌫ +Straussgµ⌫ (2012/14);fµ⌫ =4 c g¯µ⌫ + fµ⌫ + d xpg matter(g, ) mg d xpgR(g)+mf d xK(g, M)+ d xV(g, M) L Akrami, Hassan, Koennig, ASM, Solomon (2015) Z Z fluctuationsZ diagonalizable intoZ mass eigenstates: 4 2 4 + d x2pg matter(g, ) Sb[g, f]=mg d xpgR(g) Recall: gµ⌫ Gµ⌫ ↵ MLµ⌫2 Z massless Gµ⌫ gµ⌫ + ↵ fµ⌫ 2 4 4 / Z + m d x fR(f) d xV(g, f) / 2 f M f c g Z Z massive µ⌫ µ⌫ µ⌫ p for2 small ↵ = mf /m/ g (i.e. weak gravity!), the massive spin-2 Recall: gµ⌫4 Gµ⌫ )↵ Mµ⌫ ◆ 4 / field1 ⇣ interacts only weakly with matter: V (g, f)=m pg nlinearen g equations: f n=0) for small X↵ = m⇣fp/m↵g⌘(i.e.0 weakis the gravity!), General the Relativity massive spin-2 limit of Bimetric Theory ) ⇢ 1 ✓ ¯µ⌫ G⇢ !⇤g⌘Gµ⌫ g¯µ⌫G =0 fieldis the interacts General Relativity only weakly limit withof bimetric matter: 2theory E1 2 2 19 / 26 e1(S)=Tr[S] e2(S)= (Tr[S]) Tr[S ] 2 2 ⇢ 1 mFP ↵ 0 is the¯ Generalµ⌫ M⇢ Relativity⇤g Mµ⌫ limitg¯µ of⌫ BimetricM + Theory(Mµ⌫ g¯µ⌫M)=0 1 3 2⇣ 3 ⌘ 2 2 e3(S)= !(Tr[(whenS]) all3Tr[ EotherS ]Tr[ Sparameters]+2Tr[S ] are fixed, 6 this⇣ makes the spin-2with mass mass m FP ⌘ =infinitelymFP(↵ large), n,c) 19 / 26 Consistent theory for massless & massive spin-2 )

17 / 26

8/14 Ghost-free bimetric theory = General Relativity + additional tensor field Structure of Vertices (bimetric action expanded in mass eigenstates)

Quadratic (Fierz-Pauli)

what about higher orders?

This is a slide title

9/14 Babichev, Marzola, Raidal, ASM, Structure of Vertices Urban, Veermäe, von Strauss (2016)

Quadratic (Fierz-Pauli) Cubic (suppressed by )

This is a slide title

9/14 Babichev, Marzola, Raidal, ASM, Structure of Vertices Urban, Veermäe, von Strauss (2016)

Quadratic (Fierz-Pauli) Cubic (suppressed by )

This is aself-interactions slide title of massless spin-2 sum up to General Relativity

no vertices giving rise to decay of massive into massless spin-2 massive spin-2 particle gravitates like baryonic matter self-interactions of massive spin-2 are enhanced in the GR limit

9/14 Could Dark Matter have spin-2 ? Cosmology The cosmological cake

25% Dark Matter

This70% is your Dark Energy

5% normal matter

10/14 The cosmological cake

25% ? Dark Matter This70% is your Dark Energy

5% normal matter

10/14 25% ? Dark Matter This70% is your Dark Energy

Viable cosmology with 5% normal self-accelerating Akrami, Hassan, Könnig, ASM, Solomon (2015); matter solutions Könnig, Patil, Amendola (2014); Akrami, Koivisto, Mota, Sandstad (2013); Volkov; von Strauss, ASM, Enander, Mörtsell, Hassan; Comelli, Crisostomi, Nesti, Pilo (2011) 10/14 Symmetries?

“partial masslessness”

Apolo, Hassan (2016) Hassan, von Strauss, ASM (2012/15) 25% Deser, Waldron (2001) ? Dark Matter This70% is your Dark Energy

“partial masslessness” massive spin-2 ?

Apolo, Hassan (2016) Hassan, von Strauss, ASM (2012/15) 25% Babichev, Marzola, Raidal, ASM, Deser, Waldron (2001) Dark Matter Urban, Veermäe, von Strauss (2016); ? Aoki, Mukohyama (2016); This70% is your Dark Energy

Physical interpretation Bimetric[Hassan, ASM & Theoryvon Strauss, 2012] = Gravity + massive spin-2

Physical interpretation [Hassan, ASM & von Strauss, 2012] Bimetric Theory = Gravity + massive spin-2 Can rewrite bimetric action in terms of gravitational metric gµ⌫ and Bimetric Theory = Gravitymassive + massive spin-2 field spin-2Mµ⌫: Can rewrite bimetric action in terms of gravitational metric gµ⌫ and massive spin-2 field Mµ⌫: Can rewrite bimetric action2 in terms4 of gravitational2 metric4 g and 4 mg d xpgR(g)+mf d xK(µg,⌫ M)+ d xV(g, M) 2massive4 spin-2 field 2Mµ⌫:4 4 mg d xpgR(g)+mf d ZxK(g, M)+ d xV(Zg, M) Z 4 Z Babichev, Marzola,Z Raidal, ASM, Z + d xpg matter(g, ) m2 d4x gR(g)+m2 d4xK(g, M)+4 d4xV(g, M) L Spin-2 Dark Matter g Urban,p Veermäe, von fStrauss (2016) + d xpg matter(g, ) L Z Z Z Z Z 4 + d x2pg matter(g, ) Recall: gµ⌫ Gµ⌫ ↵ MLµ⌫ Recall the (linearised) gravitational metric:Recall: g G ↵2M / Z µ⌫ / µ⌫ µ⌫ massless massive for small ↵ = mf /mg (i.e. weak gravity!), the massive spin-2 for small ↵ = m /m (i.e. weak2 gravity!), the massive spin-2 )Recall: gµ⌫ fGµg⌫ )↵ Mµ⌫ field interacts only/ weaklyfield with matter: interacts only weakly with matter: and the General Relativity limit of bimetric theory:for small ↵ = mf /m↵g (i.e.0 weakis the gravity!), General the Relativity massive spin-2 limit of Bimetric Theory ↵ )0 is the General Relativity limit of Bimetric Theory !field interacts only weakly! with matter: This is a slide title 19 / 26 19 / 26 ↵ 0 is the General Relativity limit of Bimetric Theory ! gravity is weak because the physical Planck mass is large 19 / 26 ( )

massive spin-2 field decouples from matter, interacts only with gravity

11/14 8 Constraints

12 α

10 Decay 14

Log Perturbativity - Production

ratio of 18 Planck masses

20 1 2 3 4 5 6 7

12/14 Log10mFP/GeV Spin-2 mass Features

heavy spin-2 field automatically resembles dark matter when gravity resembles general relativity interactions with baryonic matter are suppressed by the Planck mass

spin-2 mass and interaction scale are on the order of a few TeV

This is a slide title

13/14 Features

heavy spin-2 field automatically resembles dark matter when gravity resembles general relativity interactions with baryonic matter are suppressed by the Planck mass

spin-2 mass and interaction scale are on the order of a few TeV

This is a slide title Chu & Garcia-Cely (2017): may be lowered to MeV by taking into account self-interactions of massive spin-2

Gonzalez, ASM, von Strauss (2017): interesting new effects for more than one massive spin-2 field

13/14 Features

heavy spin-2 field automatically resembles dark matter when gravity resembles general relativity interactions with baryonic matter are suppressed by the Planck mass

spin-2 mass and interaction scale are on the order of a few TeV

This is a slide title no need for extra fields, artificial symmetries or fine tuning

bimetric theory could explain dark matter in the context of gravity

massive spin-2 field is a natural addition to the Standard Models

13/14 Summary

Massive spin-2 fields… review: ASM, Mikael von Strauss; 1512.00021

provide one of the few known consistent modifications of GR

are uniquely described by ghost-free bimetric theory

could be a dark matter candidate whose coupling to baryonic matter is suppressed by the Planck scale

Larger theoretical framework: String Theory ? Additional symmetries ? Quantum gravity ? Can we detect/observe the massive spin-2 ?

14/14 Linear Massive Gravity

Fierz & Pauli (1939): equation for massive spin-2 ﬁeld

m2 ¯ ⇢g ⇤ g 1 g¯ g + FP g a g¯ g =0 Eµ⌫ ⇢ g µ⌫ 2 µ⌫ 2 µ⌫ µ⌫ propagates 5 degrees of freedom for a = 1 for a = 1 there is an additional propagating scalar mode that 6 gives rise to a ghost instability Thank you for your attention!

6 / 26 Back-up slides Linear Massive Gravity Linear Massive Gravity

Fierz & Pauli (1939): equation for massive spin-2 ﬁeld Fierz & Pauli (1939): equation for massive spin-2 ﬁeld

2 2 ⇢ 1 mFPLinear Constraints ⇢ mFP ¯ g⇢ ⇤g gµ⌫ g¯µ⌫g + gµ⌫ a g¯µ⌫g =0 ¯ g + g a g¯ g =0 Eµ⌫ 2 Linear2 Massive Gravity Eµ⌫ ⇢ 2 µ⌫ µ⌫ Fierz & Pauli (1939): equation for massive spin-2 ﬁeld propagates 5 degrees of freedom for a = 1 Divergencepropagates of equation 5 degrees gives of freedom for a = 1 m2 ¯ ⇢g ⇤ g Trace1 g¯ givesg + FP g a g¯ g =0 for a = 1 there is an additionalE propagatingµ⌫ ⇢ scalarg µ mode⌫ 2 thatµfor⌫ a = 1 there2 isµ⌫ an additionalµ⌫ propagating scalar mode that 6 Linear Massive Gravity 6 gives rise to a ghost instability 5 contraintsgives riseon 10 to acomponents, ghost instability Fierz & Pauli (1939): equation for massive spin-2 ﬁeld

equationm2 propagates 5 degrees of freedom ¯ ⇢g ⇤ g 1 g¯ g + FP g a g¯ g =0 Eµ⌫ ⇢ g µ⌫ 2 µ⌫ 2 µ⌫ µ⌫ propagates 5 degrees of freedom for a = 1 propagates 5 degrees of freedom for a = 1

for a = 1 there is an additional propagating scalar mode that for a = 1 there6 is anTrace additional equation propagating contains two scalar derivatives, mode not that a constraint 6 gives rise to a ghost instability gives rise to a ghost instability

6 / 26

6 / 26 6 / 26

6 / 26 [Hassan, ASM & von Strauss, 2012] Proportional backgrounds [Hassan, ASM & von Strauss, 2012] Proportional backgrounds[Hassan, ASM & von Strauss, 2012] ProportionalProportionalProportional backgrounds backgrounds [Hassan,[Hassan, ASM ASM && von Strauss, 2012] 2012]

ProportionalProportional backgrounds backgrounds [Hassan,[Hassan, ASM ASM& &vonvon Strauss, Strauss, 2012] 2012] Hassan, ASM, von Strauss (2012) ParticularlyParticularly important important solution solution to to equations of of motion: motion: ProportionalParticularlyParticularlyParticularly solutions important important solution solution solution to to to equations equations equations of of motion: motion: motion: ¯¯ 22 ParticularlyParticularly important important¯ solution solutionAnsatz:22 to to equations equationsffµµ⌫⌫ == ofcc of motion:g¯g¯ motion:µµ⌫⌫ withwith c == const.const. ff¯µµ⌫⌫ ==c2c g¯g¯µµ⌫⌫ withwithcc==const.const. f¯µ⌫ = c g¯µ⌫ with c = const. ¯ 2 2 fµ⌫f¯µ=⌫ c=g¯cµ⌫g¯µ⌫ withwithc c==const.const. –– gives gives two two copies copies of of Einstein’s Einstein’s equations equations ( (↵↵ mmf /m/mg)) –– gives givesgives two two two copies copies of of of Einstein’s Einstein’s Einstein’s equations equationsequations ( ↵ ( ↵ m m f /m f /m g ) g :) ⌘ f g – gives two copies of Einstein’s equations (↵ ⌘⌘mf /mg) ⌘ – gives– gives two two copies copies of Einstein’s of Einstein’s equations equations (↵ (↵ mmff/m/mgg)) 11 ⌘ 11 Rµ⌫(¯g) 2 g¯µ⌫⌘⌘R(¯g)+⇤g(↵, n,c)¯gµ⌫ =0 Rµ⌫(¯(¯gg)) 1 g¯g¯µµ⌫⌫RR(¯(¯ggR)+)+µ⌫⇤(¯⇤gg)(g↵(↵, ,2ng¯,cnµ,c⌫)¯Rg)¯µg(¯⌫µg⌫=0)+=0⇤g(↵, n,c)¯gµ⌫ =0 R (¯g) 12g¯2 1 R(¯g)+⇤ (↵, 1 ,c)¯g =0 Rµµ⌫⌫R(¯µg⌫)(¯g)2 g¯µµ⌫⌫g¯Rµ⌫(¯Rg(¯)+gR)+⇤(¯g⇤gg(g)↵(↵, ,1nn,cng¯,c)¯g)¯Rgµµµ⌫(¯⌫g=0)+=0⇤ (↵, ,c)¯g =0 R (¯g) 211g¯2 R(¯gR)+µ⌫⇤(¯g)(↵, 2g¯,cµ⌫)¯Rg (¯g)+=0⇤f (↵, n ,c)¯gµ⌫ =0 Rµ⌫ (¯g) 1122 g¯1µµ⌫⌫R(¯g)+µ⌫ ⇤ff (↵,2n nµ,c⌫ )¯µg⌫µ⌫ =0f n µ⌫ RRµµ⌫⌫R(¯(¯gg))(¯g) g¯g¯µµ⌫⌫g¯RR(¯(¯Rgg)+(¯g)+⇤f⇤f((↵↵(↵,,,nn,c,c,c)¯)¯g)¯ggµµ⌫⌫=0=0 µ⌫ 22 2 µ⌫ f n µ⌫ – consistency condition: ⇤g(↵, n,c)=⇤f (↵, n,c) determines c –– consistency consistency condition:– consistency⇤⇤gg((↵↵,,n condition:n,c,c)=)=⇤⇤f (f↵(⇤↵,g,(n↵,cn,,c)n)determines,cdetermines)=⇤f (↵c, cn,c) determines c –– consistency consistency–consistency consistency condition: condition: condition: condition:⇤⇤ g g ⇤(( ↵ ↵g ( ,, ↵ ,n ,c n ,c )= )= ⇤⇤ ⇤f f (f ( ↵(↵ ↵ , , , n nn ,c ,c ) )) determinesdeterminesdetermines cc c

maximally symmetric backgrounds with Rµ⌫(¯g)=⇤gg¯µ⌫ maximally symmetric backgrounds with Rµ⌫(¯g)=⇤gg¯µ⌫ maximallymaximallymaximally symmetric symmetric symmetric)maximally backgrounds backgrounds backgrounds symmetric with with withR backgroundsRµRµ⌫⌫µ(¯(¯⌫gg(¯)=)=g)=⇤⇤gg⇤g¯µg with⌫g¯µ⌫ Rµ⌫(¯g)=⇤gg¯µ⌫ )))maximally)Maximally symmetric symmetric) backgrounds backgrounds withwith Rµ⌫(¯g)=⇤gg¯µ⌫ )

16 / 26 16 / 26 1616 / /26 2616 / 26 16 / 26 16 / 26 Recovery of GR Max, Platscher & Smirnov (2017)

ratio of Planck masses

Spin-2 mass DM Production

standard freeze-out does not work: no thermal equilibrium, expansion rate always dominates over interaction rate

gravitational production does not work: required DM mass is too large and violates perturbativity bound

freeze-in mechanism works: gives lower bound on DM mass This is a slide title Decay Bounds Nonlinear Massive Gravity

Mass term should have non-derivative interactions of gµ⌫, but if we try to contract the indices we get: Which metricPhysical represents interpretation gravity?[Hassan, ASM & von Strauss, 2012] µ⌫ g gµ⌫ =4 Bimetric Theory = Gravity + massive spin-2

This is not a mass term. Absence of ghost: only one of the metrics couples to matter Can rewrite bimetric action in terms of gravitational metric gµ⌫ and M Only way out: introduce second metric to contract indices massive2 4 spin-2 ﬁeld µ⌫: 2 4 Sgf = mg d xpgR(g)+mf d x fR(f) µ⌫ 1 µ⌫ 1 2Z 4 2 4Z 4 4 g fµ⌫ =Tr(g f) f gµ⌫ =Tr(f g) mg d xpgR(g)+mf d xK(g,p M)+ d xV(g, M) 4 4 1 Z Zm d xpg Z nen g f 4 Massive Gravity action is of the form n+=0 d xpg matter(g, ) Z X ⇣pL ⌘ Hassan, ASM, von Strauss4 (2013); Z Higher-Derivative Action + d xpg matter(g, ) 4 Gording & ASM (2018) L SMG[g]=SEH[g] d xV(g, f) Z2 Recall: gµ⌫ Gµ⌫ ↵ Mµ⌫ Z / What Equations determines for f ? can be solved perturbativelyfor small in ↵ = mf /mg (i.e. weak gravity!), the massive spin-2 µ⌫ only known) coupling that does not re-introduce the ghost ! ﬁeld interacts only weakly with matter: 10 / 26 Higher-derivative action for g :is the↵ gravitational0 is the General metric Relativity limit of Bimetric Theory ! µ⌫ ! 19 / 26 The gravitational metric is not massless! { { 18 / 26 general relativity (Weyl tensor) 2 + total derivative

curvature corrections to GR capture effects of heavy spin-2 field Hinterbichler & Rosen (2012) Vierbein Formulation

Einstein-Hilbert terms:

interaction terms:

equivalent to bimetric theory and ghost-free only if

existence of square-root and intersection of light cones (Hassan & Kocic, 2017)

natural generalization to other spacetime dimensions Hassan & ASM (2018) General interaction graph j i

used to be a (nondynamical) which has been integrated out