A unifying description of
Dark Energy
& Modified Gravity
David Langlois
(APC, Paris)
Outline
1.
2.
3.
4.
5.
Introduction
Main formalism
Linear perturbations
Horndeski’s theories and beyond
Link with observations
Based on
J. Gleyzes, D.L., F. Piazza & F. Vernizzi: 1304.4840, 1404.6495,1408.1952
J. Gleyzes, D.L. & F. Vernizzi: 1411.3712 (review + a few extensions)
Introduction & motivations
•  Plethora of dark energy models:
–  Dynamical dark energy: quintessence, K-essence
–  Modified gravity
•  Large amount of data from future large scale
cosmological surveys (LSST, Euclid, …)
•  Goal: effective description as a bridge between models
and observations.
•  Assumptions:
–
–
Single scalar field models
All matter fields minimally coupled to the same metric gµ⌫
ADM approach
•  The scalar field defines a preferred slicing
Constant time hypersurfaces = uniform field hypersurfaces
=
3
=
2
=
1
•  ADM decomposition based on this preferred slicing
Uniform scalar field slicing
•  Basic ingredients
–  Unit vector normal to the hypersurfaces
nµ =
p
1
X
X ⌘ g ⇢ r⇢ r
rµ ,
–  Projection on the hypersurfaces:
hµ⌫ = gµ⌫ + nµ n⌫
–  Intrinsic curvature tensor
(3)
–  Extrinsic curvature tensor
Kµ⌫ = hµ r n⌫
Rµ⌫
K = rµ nµ
ADM formulation
•  ADM decomposition of spacetime
ds2 =
Inverse metric
Hence
•
N 2 dt2 + hij dxi + N i dt
g
µ⌫
=
✓
2
1/N
N i /N 2
dxj + N j dt
j
hij
X ⌘ g µ⌫ rµ r⌫ = g 00 ˙ 2 =
2
N /N
N i N j /N 2
˙ 2 (t)
N2
Lagrangians of the form
Z
p
4
Sg = d xN h L(N, Kij , Rij , hij , Di ; t)
◆
Example: GR + quintessence
•  Consider a quintessence model
S=
Z
4
d x
p

MP2
g
R
2
1
@µ @ µ
2
V( )
•  For the Einstein-Hilbert term, one can use
(4)
R = Kµ⌫ K µ⌫
•  In the uniform
K 2 + R + 2rµ (Knµ
n⇢ r⇢ nµ )
slicing, this leads to the Lagrangian
2 ⇥
MPl
L=
Kij K ij
2
⇤
K2 + R +
˙ 2 (t)
2N 2
V ( (t))
Homogeneous evolution
•  FLRW metric:
ds2 =
•  Extrinsic curvature:
N 2 (t)dt2 + a2 (t)
Kji
ȧ
=
N̄ a
•  Homogeneous Lagrangian

ȧ
i
L̄(a, ȧ, N̄ ) ⌘ L Kj =
N̄ a
i
j
⌘H
i
i
,
R
j
j
ij dx
i
dxj
i
j
= 0, N = N̄ (t)
•  One can include matter by adding the Lagrangian for
matter, minimally coupled to the metric.
Friedmann equations
•  Variation of the action
S̄g =
Z
3
(
dtd x a
3
L̄ + N̄ LN
with
S̄m =
Z
3
S̄g =
dtd xN̄ a
3
✓
✓
3HF
@L
@Kij
⇢m
◆
bgd
Z
dt d3 x N̄ a3 L̄(a, ȧ, N̄ )
2
N̄ + 3a N̄
L̄
3HF
Ḟ
N̄
!
⌘ F h̄ij
N̄
a
+ 3pm
a
N̄
◆
h
1
Sm =
2
Z
d4 x
p
g T µ⌫ gµ⌫
•  Friedmann equations
L̄ + N̄ LN
3HF = ⇢m
a
)
L̄
3HF
Ḟ
=
N̄
pm
i
Simple example: K-essence
Armendariz-Picon et al. 00
•  K-essence + GR
2 ⇥
MPl
L=
Kij K ij
2
2
⇤
K + R + P (X, )
X=
•  Background Lagrangian and its derivatives:
2
LN = 2XPX
L̄ = P 3MPl
H2
@L
2
= MPl
K ij Khij
=)
F=
@Kij
2
2MPl
H
•  Friedmann equations
L̄ + N̄ LN
3HF = 3MP2 H 2 + P
L̄
Ḟ = MP2 (3H 2 + 2Ḣ) + P =
3HF
2XPX = ⇢m
pm
˙2
N2
Linear perturbations
•  Perturbations
N ⌘N
N̄ ,
Kij ⌘ Kij
H
j
i
•  Expand the Lagrangian up to quadratic order:
L(N, Kji , Rij , . . . )
with
L
(2)
= L̄ + LN
@L
@L
i
i
(2)
N+
K
+
R
+
L
+ ...
j
j
i
i
@Kj
@Rj
1
1 @2L
1 @2L
i
k
i
k
2
K
K
+
R
R
= LN N N +
j
l
2
2 @Kji @Klk
2 @Rji @Rlk j l
2
2
@2L
@
L
@
L
i
k
i
i
+
K
R
+
N
K
+
N
R
j
l
j
j + ...
i
i
i
k
@N @Kj
@N @Rj
@Kj @Rl
Linear perturbations
•  The coefficients of the quadratic Lagrangian are
evaluated on the homogeneous background, e.g.
@2L
i k
i k
ik
⌘
Â
+
A
+
K j l
K
jl
l j
j
l
@Ki @Kk
@2L
@Rij
@Rkl
! (ÂR , AR )
@2L
@Kij
@Rkl
ˆ C)
! (C,
...
•  For simplicity, assume the three conditions
1
ˆ
ÂK + 2AK = 0 , C + C = 0 , 4ÂR + 3AR = 0
2
to ensure that the EOM are 2nd order, then the quadratic
action depends on only five time-dependent functions.
Linear perturbations
•  Action at quadratic order
S
(2)
=
Z
M
dx dt a
2
3
3
2

Kji Kij
K 2 + ↵K H 2 N 2 + 4 ↵B H K N
✓p ◆
h
+ (1 + ↵T ) 2
R + (1 + ↵H )R N
a3
where the alpha’s [Bellini & Sawicki’s notation] are explicitly given in
terms of the derivatives of the Lagrangian.
-
-
-
-
-
GR: M = MP , ↵i = 0
Quintessence, K-essence: ↵K 6= 0
Brans-Dicke, F(R): M = M (t)
Kinetic braiding: ↵B 6= 0
Horndeski ( ↵H = 0) and beyond Horndeski ( ↵H 6= 0 )
Linear degrees of freedom
hij = a2 (t) e2⇣
•  Scalar & tensor perts:
ij
+
TT
ij
•  Quadratic action for the true degrees of freedom:
S
(2)
L⇣˙⇣˙ ⌘
1
=
2
Z

(@i ⇣)2
M2 2
dx dt a L⇣˙⇣˙ ⇣ + L@⇣@⇣
+
˙ ij
2
a
4
3
2
2 ↵K + 6↵B
M
(1 + ↵B )2
3
,
˙2
L@⇣@⇣ ⌘ 2M 2
(
1 + ↵T
1 + ↵H ⇣
1 + ↵M
1 + ↵B
M2
(@k ij )2
(1 + ↵T )
4
a2
Ḣ ⌘
H2
1 d
H dt
✓
1 + ↵H
1 + ↵B
•  Stability
–  No ghost:
M2 > 0
L⇣˙⇣˙ > 0 ,
–  No gradient instability:
c2s ⌘
L@⇣@⇣
> 0,
L⇣˙⇣˙
c2T ⌘ 1 + ↵T > 0
◆)
Horndeski theories
Horndeski 74; Nicolis et al. 08; Deffayet et al. 09 & 11
•  Most general action for a scalar field leading to at most second order
equations of motion (Horndeski ‘74)
Combination of the following four Lagrangians
LH
2 = G2 ( , X)
with
µ⌫
LH
3 = G3 ( , X) ⇤
(4)
LH
R
4 = G4 ( , X)
(4)
LH
Gµ⌫
5 = G5 ( , X)
µ⌫
2G4X ( , X)(⇤ 2
1
µ⌫
+ G5X ( , X)(⇤ 3
3
•  ADM formulation ?
⌘ r⌫ r µ
µ⌫ )
3⇤
µ⌫
µ⌫
+2
µ⌫
µ
⌫
)
Horndeski into ADM form
Goal: translate the Lagrangian that depend on scalar field
derivatives into an ADM expression.
p
X nµ
µ =
p
1
p
X(Kµ⌫ nµ a⌫ n⌫ aµ )
nµ n⌫ n r X
µ⌫ =
2
X
with
a µ ⌘ n r nµ
Gauss-Codazzi relations are also useful.
Rµ⌫ = ((4)Rµ⌫ )k + (n n⇢
R = (4)R + K 2
(4)
Rµ
Kµ⌫ K µ⌫
⌫⇢ )k
KKµ⌫ + Kµ K
2rµ (Knµ
aµ )
⌫
Horndeski into ADM form
GLPV 1304
Combination of the Lagrangians
L2 =A2
L 3 = A3 K
L4 =A4 (K 2
L5 =A5 (K
with
3
Kij K ij ) + B4 R
ij
3KKij K + 2Kij K
p
Z
ik
K jk )
G3
p
dX ,
A2 =G2
X
2
X
Z
p
p
X dX 2
XG4 ,
A3 =
G3X
A4 =
G4 + 2XG4,X +
A5 =
3
1
( X) 2 G5,X
3
X
G5, ,
2
+ B5 K
ij
⇥
Rij
p
Z
⇤
hij R/2
G
p 5 dX ,
B4 =G4 +
X
4
X
Z
p
X dX
B5 =
G5X
A4 =
B4 + 2XB4 X
A5 =
XB5 X /3
Beyond Horndeski
GLPV 1404 & 1408
•  The Lagrangians
L2 =A2
L 3 = A3 K
L4 =A4 (K 2
L5 =A5 (K
3
Kij K ij ) + B4 R
ij
3KKij K + 2Kij K
ik
K jk )
+ B5 K
ij
⇥
Rij
⇤
hij R/2
with generic functions A4 , B4 , A5 and B5 do not lead
to any additional degree of freedom.
No Ostrogradski instabilities !
Beyond Horndeski
•  In covariant form, we get extra terms
L4 = G4 ( , X) (4)R
+ F4 ( , X)✏
µ⌫⇢
2G4X ( , X)(⇤
✏
µ0 ⌫ 0 ⇢0
L5 = G5 ( , X) (4)Gµ⌫
µ⌫
0 0 0
+ F5 ( , X)✏µ⌫⇢ ✏µ ⌫
⇢
2
µ µ0 ⌫⌫ 0 ⇢⇢0
µ⌫
,
1
+ G5X ( , X)(⇤
3
0
µ µ0 ⌫⌫ 0 ⇢⇢0
µ⌫ )
3
3⇤
µ⌫
µ⌫
+2
µ⌫
µ
0
which do not belong to Horndeski class.
•  Lagrangians with F4 = 0 or with
can be
F5 = 0
connected to Horndeski, via disformal transformations
g̃µ⌫ = gµ⌫ + ( , X) @µ @⌫
GLPV 1408
⌫
)
Perturbations in an arbitrary gauge
•  Description in an arbitrary slicing ?
t
= const
•  Transformation
t ! t + ⇡(t, ~x)
Perturbations in an arbitrary gauge
•  Stueckelberg trick: t ! t + ⇡(t, ~x)
•  The new quadratic action can be derived via the substitutions:
f
!
g 00
!
Kij
!
K
!
Rij
!
(3)
(3)
R
!
1
f + f˙⇡ + f¨⇡ 2 ,
2
g 00 + 2g 0µ @µ ⇡ + g µ⌫ @µ ⇡@⌫ ⇡ ,
Kij
Ḣ⇡hij
@i @j ⇡ ,
1 2
@ ⇡,
2
a
(3)
Rij + H(@i @j ⇡ + ij @ 2 ⇡) ,
4
(3)
R + 2 H@ 2 ⇡ .
a
K
3Ḣ⇡
Note: the 3-dim quantities
on the right are defined
with respect to the new
time hypersurfaces.
Perturbations in the Newtonian gauge
•  Perturbed metric
ds2 =
(1 + 2 )dt2 + a2 (t) (1
•  Modified Einstein’s equations
or
Gµ⌫ = M
pD =
2 2
+ 3 ↵B
k̃
( ⇢D
2
2
1 + ↵B k̃
1 2
2
D
=
2
a
2k 2
"
2 2
1 ↵T + 8 ↵B k̃
(
2
2
1 + ↵B k̃
2 )
ij
dxi dxj
Gmod
µ⌫ = M
matter
D
Tµ⌫
+ Tµ⌫
3HqD ) +
⇢D
2
matter
Tµ⌫
with
2 2
+ 5 ↵B
k̃
HqD +
2
2
1 + ↵B k̃
1 4
9 k̃
3HqD ) +
1
+
7
⇢m
2
2 k̃ 2
↵B
HqD + ↵T ⇢m
#
Testing deviations from GR
•  In the quasi-static approximation for sub-horizon
scales (time derivatives are neglected), one can express
1.  the effective Newton constant
2.  the slip parameter
k2
a2
⌘ 4⇡ Ge↵ (t, k) ⇢m
⌘ (t, k)
in terms of the coefficients ↵i .
•  Full system of equations can be implemented in a
modified numerical code.
m
Conclusions
•  Unified treatment of dark energy and modified gravity
models, based on ADM formalism.
–
–
–
–
Easy comparisons between models
Identification of degeneracies
Observational data can constrain many models simultaneously
Explore unchartered territories (e.g. theories beyond Horndeski)
Theoretical models
Effective description
Observations
•  Very general and efficient way to describe linear
perturbations in scalar-tensor theories with only five
time-dependent functions.