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T E S T S O F G R AV I T Y W I T H G R AV I TAT I O N A L W AV E S & C O S M O L O G Y
T E S S A B A K E R , U N I V E R S I T Y O F O X F O R D
• The EFT of cosmological perturbations.
• The impact of GW170817.
• Constraints from cosmology & summary.
O U T L I N E
T H E E F T O F C O S M O L O G I C A L
Image: Tom Barrett
P E R T U R B AT I O N S
1. Let Θ be a vector of fields mediating gravity.
2. Taylor expand the gravitational Lagrangian in perturbations δΘ:
L ' L + L⇥a �⇥a +1
2L⇥a⇥b�⇥a�⇥b + . . .
Gives us linearised grav. field equations.
@L
@⇥a
ds
2 =� (1 + 2�) dt2 + 2@iB dx
idt
+ a
2(t) [(1� 2 ) �ij + 2@i@jE] dxidx
j
Scalar perturbations of metric:
L A G R A N G I A N E X PA N S I O N
3. Build Lagrangian containing all combinations of fields in Θ. Simplest case:
usual fluid matter sector +
�2L =
+ + +L 2 @i@
iBL @2B . . .
++L�� �2 � L� . . .
��2 . . .++ +L�� �� @i@iEL�@2E
�, , B, E, �.
~ 70 terms in total.
L A G R A N G I A N E X PA N S I O N
Enforce linear diff symmetry:
4. The action must be coordinate-invariant.
x
µ ! x
µ + ✏
µ
✏µ =�⇡, @i✏
�⇒
⇒ �2S ! �2S +terms linear in δφ, 𝛟, 𝛙 etc.
ii⇥ (⇡ or ✏)
must vanish
Invariance of action under a non-dynamical symmetry gives a set of constraint relations.
S Y M M E T R I E S A N D C O N S E Q U E N C E S
Invariance of action under a non-dynamical symmetry gives a set of constraint relations.
4. The action must be coordinate-invariant.
usual fluid matter sector +
�2L =
+ + +L 2 @i@
iBL @2B . . .
++L�� �2 � L� . . .
��2 . . .++ +L�� �� @i@iEL�@2E
S Y M M E T R I E S A N D C O N S E Q U E N C E S
Invariance of action under a non-dynamical symmetry gives a set of constraint relations.
4. The action must be coordinate-invariant.
L
+L�� L�
+L�� L�@2E
3H = 0
� 5 = 0 etc.
E.g. :
⇒ Solve system constraint equations.
⇒ Reduce to a handful of `true’ free functions.
S Y M M E T R I E S A N D C O N S E Q U E N C E S
T H E A L P H A PA R A M E T E R S
↵B(t) `braiding’ — mixing of scalar + metric kinetic terms.:
kinetic term of scalar field.↵K(t) :
speed of gravitational waves, . ↵T (t) : c2T = 1 + ↵T
↵M (t) =1
H
d ln M2(t)
dt: running of effective Planck mass.:
↵H(t) disformal symmetries of the metric.:
T H E A L P H A PA R A M E T E R SBellini & Sawicki,
1404.3713
V E C T O R - T E N S O R R E S U LT S
We can play the game over again with a vector field.
Aµ ⇠�A� �A0, @
i �A1
�
, , as for the scalar case.↵T (t) ↵K(t) ↵M (t) =1
H
d ln M2(t)
dt:
Results:
↵V (t) vector mass mixing ~ �A0 �A1
↵A(t) `auxiliary friction’ ~ �A0 �A1
↵C(t) conformal coupling excess ~ changes effective mass scale of theory
k4 �A1↵D(t) small-scale dynamics ~ �A21
Lagos+, 1711.09893
Fields # True free functions Example full theory
1 GR
5 Horndeski
7 Generalised Proca
4 Einstein-Aether
5 Massive Bigravity
12 Uber-case
gμν , φ
gμν
gμν , Aμ
gμν , Aμ, λ
gμν , qμν
gμν , φ, Aμ, qμν
R E S U LT S B R E A K D O W N
(Some restrictions imposed, e.g. 1 propagating d.o.f.)
Image: Tijana Mihajlovic
T H E I M PA C T O F G R AV I TAT I O N A L W AV E S
T H E A L P H A PA R A M E T E R S
↵B(t) `braiding’ — mixing of scalar + metric kinetic terms.:
kinetic term of scalar field.↵K(t) :
speed of gravitational waves, . ↵T (t) : c2T = 1 + ↵T
↵M (t) =1
H
d ln M2(t)
dt: running of effective Planck mass.:
↵H(t) disformal symmetries of the metric.:
G W 1 7 0 8 1 7 & G R B 1 7 0 8 1 7 A
c2T = 1 + ↵T
�t ' 1.7 s
) |↵T | . 10�15
!! c. f. |↵M |, |↵B | . O(1)!! c. f.
Image: LIGO-VIRGO Collaboration, ApJ 848:2 (2017).
G W 1 7 0 8 1 7 & G R B 1 7 0 8 1 7 A
1. No fine-tuned cancellation of intrinsic emission delay.
2. Propagation time dominated by cosmological regime.
3. No finely-tuned, protected functional cancellations in theory
Assumptions:Image: NASA Goddard
G W 1 7 0 8 1 7 & G R B 1 7 0 8 1 7 A
What does this mean for gravity theories?
Scalar case clearest; full theory is Horndeski gravity.
S =
Zd
4x
p�g ⌃5
i=2 Li + SM
L4 = G4R+G4,X
�(⇤�)2 �rµr⌫�rµr⌫�
L2 = K L3 = �G3 ⇤�
where Gi = Gi (�, X) and .X = �r⌫�r⌫�/2
L5 = G5Gµ⌫rµr⌫�� 1
6G5,X
�(r�)3
� 3rµr⌫�rµr⌫�⇤�+ 2r⌫rµ�r↵r⌫�rµr↵�
1710.06394 1710.05877 1710.05893 1710.05901
G W 1 7 0 8 1 7 & G R B 1 7 0 8 1 7 A
What does this mean for gravity theories?
Scalar case clearest; full theory is Horndeski gravity.
Linearised theory maps to alpha parameters:
) ↵T (t) =2X
M2⇤
h2G4,X � 2G5,� �
⇣�� �H
⌘G5,X
i
Barring fine-tuned cancellations, .) G4,X = G5,� = G5,X = 0
L4 = G4R+G4,X
�(⇤�)2 �rµr⌫�rµr⌫�
L5 = G5Gµ⌫rµr⌫�� 1
6G5,X
�(r�)3
� 3rµr⌫�rµr⌫�⇤�+ 2r⌫rµ�r↵r⌫�rµr↵�
G W 1 7 0 8 1 7 & G R B 1 7 0 8 1 7 A
What does this mean for gravity theories?
Scalar case clearest; full theory is Horndeski gravity.
Linearised theory maps to alpha parameters:
Barring fine-tuned cancellations, .) G4,X = G5,� = G5,X = 0
L4 = G4R+G4,X
�(⇤�)2 �rµr⌫�rµr⌫�
L5 = G5Gµ⌫rµr⌫�� 1
6G5,X
�(r�)3
� 3rµr⌫�rµr⌫�⇤�+ 2r⌫rµ�r↵r⌫�rµr↵�
= 0 by Bianchi identity
) ↵T (t) =2X
M2⇤
h2G4,X � 2G5,� �
⇣�� �H
⌘G5,X
i
G W 1 7 0 8 1 7 & G R B 1 7 0 8 1 7 A
What does this mean for gravity theories?
Scalar case clearest; full theory is Horndeski gravity.
L4 = G4R+G4,X
�(⇤�)2 �rµr⌫�rµr⌫�
L2 = K L3 = �G3 ⇤�L2 = K L3 = �G3 ⇤�
f(R)�RfR� = fR 0
f [R] gravity fits the template, so it survives.⇒
G W 1 7 0 8 1 7 & G R B 1 7 0 8 1 7 A
What does this mean for gravity theories?
The vector-tensor equivalent of Horndeski is Generalised Proca:
L4 = G4R
+G4,X
⇥(rµA
µ)2 + c2r⇢A�r⇢A� � (1 + c2)r⇢A�r�A⇢⇤
L5 = G5Gµ⌫rµA⌫
� 1
6G5,X [(rµA
µ)3 � 3d2rµAµr⇢A�r⇢A� + similar ]
where and .X = �1
2AµA
µGi = Gi(X)
L1 = �1
4Fµ⌫F
µ⌫ L2 = G2 L3 = G3rµAµ
Heisenberg+ 1605.05565
G W 1 7 0 8 1 7 & G R B 1 7 0 8 1 7 A
What does this mean for gravity theories?
The vector-tensor equivalent of Horndeski is Generalised Proca:
L4 = G4R
+G4,X
⇥(rµA
µ)2 + c2r⇢A�r⇢A� � (1 + c2)r⇢A�r�A⇢⇤
L5 = G5Gµ⌫rµA⌫
� 1
6G5,X [(rµA
µ)3 � 3d2rµAµr⇢A�r⇢A� + similar ]
↵T =A2
M2⇤
h2G4,X � (HA� A)G5,X
i) G4,X = G5,X = 0
Heisenberg+ 1605.05565
G W 1 7 0 8 1 7 & G R B 1 7 0 8 1 7 A
What does this mean for gravity theories?
The vector-tensor equivalent of Horndeski is Generalised Proca:
L4 = G4R
+G4,X
⇥(rµA
µ)2 + c2r⇢A�r⇢A� � (1 + c2)r⇢A�r�A⇢⇤
L5 = G5Gµ⌫rµA⌫
� 1
6G5,X [(rµA
µ)3 � 3d2rµAµr⇢A�r⇢A� + similar ]
↵T =A2
M2⇤
h2G4,X � (HA� A)G5,X
i) G4,X = G5,X = 0
by Bianchi identity again.= 0
L4 = G4R
+G4,X
⇥(rµA
µ)2 + c2r⇢A�r⇢A� � (1 + c2)r⇢A�r�A⇢⇤
G W 1 7 0 8 1 7 & G R B 1 7 0 8 1 7 A
What does this mean for gravity theories?
The vector-tensor equivalent of Horndeski is Generalised Proca:
L1 = �1
4Fµ⌫F
µ⌫ L2 = G2 L3 = G3rµAµ
mg . 10�32 eV
G W 1 7 0 8 1 7 & G R B 1 7 0 8 1 7 A
What does this mean for gravity theories?
For bimetric theories, get a bound on graviton mass:
mg . 10�22 eV
This is not competitive with existing Solar System bounds:
(from Lunar Laser Ranging & Earth-Moon precession)
de Rham, 1606.08462
Image: Yicai Global
C O S M O L O G I C A L O B S E R VAT I O N S
Image: Allen Cai
T H E A L P H A PA R A M E T E R S
↵B(t) `braiding’ — mixing of scalar + metric kinetic terms.:
kinetic term of scalar field.↵K(t) :
speed of gravitational waves, . ↵T (t) : c2T = 1 + ↵T
↵M (t) =1
H
d ln M2(t)
dt: running of effective Planck mass.:
↵H(t) disformal symmetries of the metric.:
C O D E S
Zumalacarregui, Bellini, Sawicki & Lesgourgues (2016).
Hu, Raveri, Frusciante & Silvestri (2014).
0
2000
4000
6000
DTT` [µK2]
�0.50.00.5�DTT
` /DTT` [%]
0
10
20
30
40
50
DEE` [µK2]
⇤CDM↵K,B
↵K,M
↵K,T + w
↵K,B,M,T + w
�0.50.00.5
�DEE` /DEE
` [%]
0.0
0.5
1.0
1.5
2.0
2.5
3.0
107 ⇥ D��` [µK2]
101 102 103
Multipole `
�0.50.00.5�D��
` /D��` [%]
�100
�50
0
50
100
DTE` [µK2]
2 1000 2000 3000Multipole `
�0.50.00.5
�DTE` [µK2]
100
101
102
103
104
105
P(k)[(h�1 Mpc)3]
z=0⇤CDM↵K, B
↵K, M
↵K, T + w
↵K, B, M, T + w
�0.50.00.5�P(k)/P(k) [%]
100
101
102
103
104
105
P(k)[(h�1 Mpc)3]
z=0.5
�0.50.00.5
�P(k)/P(k) [%]
100
101
102
103
104
105
P(k)[(h�1 Mpc)3]
z=1
10�4 10�3 10�2 10�1 100 101
k[h/Mpc]
�0.50.00.5�P(k)/P(k) [%]
100
101
102
103
104
105
P(k)[(h�1 Mpc)3]
z=2
10�3 10�2 10�1 100 101
k[h/Mpc]
�0.50.00.5
�P(k)/P(k) [%]
E F F E C T S O N O B S E R VA B L E S
Bellini+, 1709.09135
CMB temperature power spectrum.
CMB lensing power spectrum
δM(z)
MPl= M0a
β
αB(z) = αB0 aξ
T H E C U R R E N T S TAT E O F P L AY
−1.2 −0.8 −0.4 0.0
αB0
4 8 12 16
ξ0.0 0.2 0.4 0.6
M0
0.9 1.2 1.5 1.8
β
4
8
12
16
ξ
0.0
0.2
0.4
0.6
M0
0.9
1.2
1.5
1.8
β
Planck + BKP
+lens + mpk + BAO + RSD
Kreisch & Komatsu, 1712.02710
SDSS (galaxy survey) + Planck CMB + BOSS BAOs & RSDs + lensing data.
δM(z)
MPl= M0a
β
αB(z) = αB0 aξ
T H E C U R R E N T S TAT E O F P L AY
−1.2 −0.8 −0.4 0.0
αB0
4 8 12 16
ξ0.0 0.2 0.4 0.6
M0
0.9 1.2 1.5 1.8
β
4
8
12
16
ξ
0.0
0.2
0.4
0.6
M0
0.9
1.2
1.5
1.8
β
Planck + BKP
+lens + mpk + BAO + RSD
Kreisch & Komatsu, 1712.02710
Caution: stability conditions lead to non-trivial contours.
O N G O I N G & F U T U R E E X P E R I M E N T S
2020
now
2023
2021
T H E N E W T H E O R Y L A N D S C A P E
T H E N E W T H E O R Y L A N D S C A P E
Quintic Galileons
Quartic Galileons
TeVeS
SVT
Bigravity
Massive Gravity
Quintessence
K-essence
KGB
Cubic Galileon
Fab Four
Generalised Proca
Horava-Lifschitz
Einstein-Aether
Brans-Dicke
Uncertain? multiscalar-tensor, nonlocal gravity, Chaplygin gases.
DHOST
(Beyond) Horndeski
f(R)
T H E N E W T H E O R Y L A N D S C A P E
Quintic Galileons
Quartic Galileons
TeVeS
SVT
Bigravity
Massive Gravity
Quintessence
K-essence
KGB
Cubic Galileon
Fab Four
Generalised Proca
Horava-Lifschitz
Einstein-Aether
Brans-Dicke
Uncertain? multiscalar-tensor, nonlocal gravity, Chaplygin gases.
DHOST
f(R)
(Beyond) Horndeski
C O N C L U S I O N S
1. The EFT of cosmological perturbations — agnostic & efficient tests of the gravity model landscape.
2. Framework can be linked directly to recent GW events with powerful results.
3. Goal: EFT used by next-generation experiments as standard format for dark energy constraints.
References:1604.01396 1710.063941711.09893