From the Horndeski Lagrangian to observations
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Transcript From the Horndeski Lagrangian to observations
Horndeski Lagrangian:
too big to fail?
Luca Amendola
University of Heidelberg
in collaboration with Martin Kunz,
Mariele Motta, Ippocratis Saltas, Ignacy
Sawicki
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Observations are convergingβ¦
β¦to an unexpected universe
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Classifying the unknown, 1
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
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22.
Cosmological constant
Dark energy w=const
Dark energy w=w(z)
quintessence
scalar-tensor models
coupled quintessence
mass varying neutrinos
k-essence
Chaplygin gas
Cardassian
quartessence
quiessence
phantoms
f(R)
Gauss-Bonnet
anisotropic dark energy
brane dark energy
backreaction
void models
degravitation
TeVeS
oops....did I forget your model?
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Classifying the unknown, 2
a)
b)
c)
d)
Lambda and w(z) models (i.e. change only the expansion)
modified matter (i.e. change the way matter clusters)
modified gravity (i.e. change the way gravity works)
non-linear effects (i.e. change the underlying symmetries)
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Prolegomena zu einer
künftigen
jedenjeden
künftigen
DarkMetaphysik
Energy physik
©Kant
Observational requirements:
Physical requirements:
A) Isotropy
Scalar field
B) Large abundance
C) Slow evolution
D) Weak clustering
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Vπ
β π»2 0
Vπβ²
βͺπ
Vπβ²β²
βͺπ
Theorem 1: A quintessential scalar field
The most general 4D scalar field theory with second order equation of motion
ο©
οΉ
4
dx
ο
g
L
+
L
matter οΊ
οͺο₯ i
ο²
ο« i
ο»
οΌ First found by Horndeski in 1975
οΌ rediscovered by Deffayet et al. in 2011
οΌ no ghosts, no classical instabilities
οΌ it modifies gravity!
οΌ it includes f(R), Brans-Dicke, k-essence, Galileons, etc etc etc
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Simplest MG: f(R)
The simplest Horndeski model which still produces
a modified gravity: f(R)
ο² dx
4
g ο f ο¨R ο©+ Lmatter ο
οΌ equivalent to a Horndeski Lagrangian without kinetic terms
οΌ easy to produce acceleration (first inflationary model)
οΌ high-energy corrections to gravity likely to introduce higherorder terms
οΌ particular case of scalar-tensor and extra-dimensional theory
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Theorem 2: the Yukawa correction
Every Horndeski model induces at
linear level, on sub-Hubble scales, a Newton-Yukawa potential
GM
οr /ο¬
ο (r ) ο½ ο
(1 ο« ο‘ e )
r
where Ξ± and Ξ» depend on space and time
Every consistent modification of gravity
based on a scalar field generates
this gravitational potential
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The next ten years of DE research
Combine observations of background, linear
and non-linear perturbations to reconstruct
as much as possible the Horndeski model
β¦ or to rule it out!
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The great Horndeski Hunt
Let us assume we have only
1) pressureless matter
2) the Horndeski field
and
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Background: SNIa, BAO, β¦
Then we can measure H(z) and
1
dz
D( z ) ο½
sinh( H 0 οο k 0 ο²
)
H ( z)
H 0 οο k 0
and therefore Ξ©π0
Then we can measure everything up to Ξ©π0
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Two free functions
The most general linear, scalar metric
ds 2 ο½ a 2 [(1 ο« 2ο)dt 2 ο (1 ο« 2ο)(dx 2 ο« dy 2 ο« dz 2 )]
At linear order we can write:
ο§ Poissonβs equation
π» 2 Ξ¨ = 4ππΊ π π, π ππ πΏ π
ο§ anisotropic stress
ο ο«ο
ο
ο¨ (k0, aο½) ο½ ο
οο
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Modified Gravity at the linear level
ο§ standard gravity
ο§ scalar-tensor models
ο§ f(R)
ο§ DGP
ο§ coupled Gauss-Bonnet
Y (k , a) ο½ 1
ο¨ (k , a) ο½ 1
G* 2( F ο« F '2 )
Y (a) ο½
FGcav ,0 2 F ο« 3F '2
F '2
ο¨ (a) ο½ 1 ο«
F ο« F '2
k2
1 ο« 4m 2
*
G
a R,
Y (a) ο½
k2
FGcav ,0
1 ο« 3m 2
a R
Boisseau et al. 2000
Acquaviva et al. 2004
Schimd et al. 2004
L.A., Kunz &Sapone 2007
k2
a2 R
ο¨ (a) ο½ 1 ο«
k2
1 ο« 2m 2
a R
m
1
; ο’ ο½ 1 ο« 2 Hrc wDE
3ο’
2
ο¨ (a) ο½ 1 ο«
3ο’ ο 1
Bean et al. 2006
Hu et al. 2006
Tsujikawa 2007
Y (a) ο½ 1 ο
Y (a) ο½ ...
ο¨ (a) ο½ ...
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Lue et al. 2004;
Koyama et al. 2006
see L. A., C. Charmousis,
S. Davis 2006
Modified Gravity at the linear level
Every Horndeski model is characterized in the linear regime
and for scales ππ π β« 1 by the two functions
a1 ο½ 2 B6 B8ο2
a2 ο½ B8 B6ο1
de Felice, Tsujikawa 2011
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Linear observables
Matter conservation equation
If we could observe directly the
growth rateβ¦
3
β π(π, π§)Ξ©π
2
we could test the HLβ¦
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πΏβ²
π=
πΏ
Reconstruction of the metric
ds2 ο½ a2[(1 ο« 2ο)dt 2 ο (1 ο« 2ο)(dx2 ο« dy 2 ο« dz 2 )]
massive particles respond to Ξ¨
massless particles respond to Ξ¦-Ξ¨
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Linear observables
πΏπππ π, π§ = πΊ(π, π§) π(π, π§) π8 πΏπ‘,0 (π)
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Three linear observables
Amplitude
Redshift distortion
clustering
lensing
Ξ£ = π(1 + π)
Lensing
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Two model-independent ratios
Amplitude/Redshift distortion
Lensing/Redshift distortion
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f
P1 ο½
b
οm0ο
P2 ο½
f
Observing the HL
Lensing/Redshift distortion
οm0ο
P2 ο½
f
If we can obtain an equation for P2
then we can test the HL
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Two model-independent ratios
We combine now growth rate and lensing
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A consistency equation
οm0ο
P2 ( k , z ) ο½
f
Differentiating P2 and
combining with the growth rate and lensing equations we obtain
a consistency relation for the HL valid for every k
that depends on 8 functions of z
πΆ π; ππ π§ =
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A consistency equation
ππ =
If we estimate P2(k,z) for many kβs we
have an overconstrained system of equations
πΆ(π1 ; ππ (π§)) = 0
πΆ(π2 ; ππ (π§)) = 0
πΆ(π3 ; ππ (π§)) = 0
πΆ(π4 ; ππ (π§)) = 0
β¦β¦..
If there are no solutions, the HL is disproven!
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Reconstructing the HL ?
We can estimate
ππ =
And therefore partially reconstruct the HL
but the reconstruction is not unique:
an infinite number of HL will give the same background
and linear dynamics
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Conclusions
β’ The (al)most general dark energy model is the Horndeski
Lagrangian
β’ It contains a specific prescription for how gravity is
modified, the Yukawa term
β’ Linear cosmological observations constrain a particular
combination of the HL functions
β’ Quantities like Ξ©π0 , π, ππ8 (π§) are unobservable with linear
observations
β’ In principle, observations in a range of scales and redshifts
can rule out the HL
The HL is not too big to fail!
(but it is too big to be reconstructed)
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