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Benasque 2012
Luca Amendola
University of Heidelberg
in collaboration with Martin Kunz, Mariele Motta, Ippocratis Saltas, Ignacy Sawicki
Horndeski Lagrangian:
too big to fail?
Benasque 2012
Classifying the unknown, 11. Cosmological constant2. Dark energy w=const3. Dark energy w=w(z)4. quintessence5. scalar-tensor models6. coupled quintessence7. mass varying neutrinos8. k-essence9. Chaplygin gas10. Cardassian11. quartessence12. quiessence13. phantoms14. f(R)15. Gauss-Bonnet16. anisotropic dark energy17. brane dark energy18. backreaction19. void models20. degravitation21. TeVeS22. oops....did I forget your model?
Benasque 2012
Classifying the unknown, 2
a) Lambda and w(z) models (i.e. change only the expansion)
b) modified matter (i.e. change the way matter clusters)
c) modified gravity (i.e. change the way gravity works)
d) non-linear effects (i.e. change the underlying symmetries)
Benasque 2012
Prolegomena zu einer jeden künftigen Metaphysik
©Kant
Observational requirements:
A) Isotropy
B) Large abundance
C) Slow evolution
D) Weak clustering
V
V
V
Physical requirements:
Scalar field
Prolegomena zu einer jeden künftigen Dark Energy physik
©Kant
Benasque 2012
First found by Horndeski in 1975 rediscovered by Deffayet et al. in 2011 no ghosts, no classical instabilities it modifies gravity! it includes f(R), Brans-Dicke, k-essence, Galileons, etc etc etc
4i matter
i
dx g L + L
The most general 4D scalar field theory with second order equation of motion
Theorem 1: A quintessential scalar field
Benasque 2012
equivalent to a Horndeski Lagrangian without kinetic terms easy to produce acceleration (first inflationary model) high-energy corrections to gravity likely to introduce higher-
order terms particular case of scalar-tensor and extra-dimensional theory
matterL+Rfgxd 4
The simplest Horndeski model which still producesa modified gravity: f(R)
Simplest MG: f(R)
Benasque 2012
Theorem 2: the Yukawa correction
Every Horndeski model induces atlinear level, on sub-Hubble scales, a Newton-Yukawa potential
/( ) (1 )rGMr e
r
where α and λ depend on space and time
Every consistent modification of gravitybased on a scalar field generates
this gravitational potential
Benasque 2012
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
… or to rule it out!
Benasque 2012
The great Horndeski Hunt
Let us assume we have only1) pressureless matter 2) the Horndeski field
and
Benasque 2012
Background: SNIa, BAO, …
Then we can measure H(z) and
Ω𝑘0
Then we can measure everything up to Ω𝑚 0
0 0
0 0
1( ) sinh( )
( )k
k
dzD z H
H zH
and therefore
Benasque 2012
Two free functions
)])(21()21[( 222222 dzdydxdtads At linear order we can write:
Poisson’s equation
anisotropic stress 0
( , )k a
The most general linear, scalar metric
𝛻2 Ψ=4𝜋 𝐺𝑌 (𝑘 ,𝑎) 𝜌𝑚 𝛿 𝑚
Benasque 2012
Modified Gravity at the linear level
scalar-tensor models
* 2
2,0
2
2
2( ' )( )
2 3 '
'( ) 1
'
cav
G F FY a
FG F F
Fa
F F
( , ) 1
( , ) 1
Y k a
k a
standard gravity
DGP1
( ) 1 ; 1 23
2( ) 1
3 1
c DEY a Hr w
a
f(R) 2 2
* 2 2
2 2,0
2 2
1 4( ) , ( ) 1
1 3 1 2cav
k km mG a R a RY a a
k kFGm m
a R a R
Lue et al. 2004; Koyama et al. 2006
Bean et al. 2006Hu et al. 2006Tsujikawa 2007
coupled Gauss-Bonnet see L. A., C. Charmousis, S. Davis 2006
( ) ...
( ) ...
Y a
a
Boisseau et al. 2000Acquaviva et al. 2004Schimd et al. 2004L.A., Kunz &Sapone 2007
Benasque 2012
Modified Gravity at the linear level
Every Horndeski model is characterized in the linear regime and for scales by the two functions
21 6 8
12 8 6
2a B B
a B B
de Felice, Tsujikawa 2011
𝑐𝑠𝑘≫1
Benasque 2012
Linear observables
If we could observe directly thegrowth rate…
−32𝑌 (𝑘 , 𝑧)Ω𝑚 ( 𝑓 =𝛿 ′
𝛿 )
Matter conservation equation
we could test the HL…
Benasque 2012
Reconstruction of the metric
massive particles respond to Ψ
massless particles respond to Φ-Ψ
2 2 2 2 2 2[(1 2 ) (1 2 )( )]ds a dt dx dy dz
Benasque 2012
Three linear observables
Amplitude
Σ=𝑌 (1+𝜂)
Redshift distortion
Lensing
clustering
lensing
Benasque 2012
Two model-independent ratios
Amplitude/Redshift distortion
Lensing/Redshift distortion
1
fP
b
02
mPf
Benasque 2012
Observing the HL
Lensing/Redshift distortion
02
mPf
If we can obtain an equation for P2
then we can test the HL
Benasque 2012
A consistency equation
Differentiating P2 and combining with the growth rate and lensing equations we obtain
a consistency relation for the HL valid for every kthat depends on 8 functions of z
02 ( , ) mP k z
f
=
Benasque 2012
A consistency equation
If we estimate P2(k,z) for many k’s wehave an overconstrained system of equations
) = 0) = 0) = 0) = 0……..
If there are no solutions, the HL is disproven!
𝑓 𝑖=¿
Benasque 2012
Reconstructing the HL ?We can estimate
𝑓 𝑗=¿
And therefore partially reconstruct the HLbut the reconstruction is not unique:
an infinite number of HL will give the same backgroundand linear dynamics
Benasque 2012
Conclusions
• The (al)most general dark energy model is the Horndeski Lagrangian
• It contains a specific prescription for how gravity is modified, the Yukawa term
• Linear cosmological observations constrain a particular combination of the HL functions
• Quantities like are unobservable with linear observations
• 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)
Ω𝑚 0, 𝑓 , 𝑓 𝜎8(𝑧)