Munich, Oct 2013
Model-independent tests
of modified gravity models
or: Can we ever rule out dark energy?
Rap
hae
l, T
he
Sch
ool
of
Ath
ens,
Rom
e
Luca Amendola
University of Heidelberg
in collaboration with Martin Kunz, Mariele Motta,
Ippocratis Saltas, Ignacy Sawicki
1
Time view
matter
DE
radiation
BBN
CMB
Munich, Oct 2013 2
Munich, Oct 2013
The two main problems of
testing modified gravity
1) Problem of initial conditions e.g, how do we know if the shape of the power spectrum we observe is due to dark energy or to initial conditions?
2) Problem of design If our model parameter space is sufficiently large, we can design a model to fit any observation
3
Munich, Oct 2013
Prolegomena zu einer
jeden künftigen Metaphysik ©Kant
Observational requirements:
A) Isotropy
B) Large abundance
C) Slow evolution
D) Weak clustering
Prolegomena zu einer
jeden künftigen Dark Energy physik ©Kant
4
The past ten years of DE research
4 ,
,
1( )
2matterdx g R V L
4 ,
,
1( ) ( )
2matterdx g f R V L
4 ,
,
1( ) ( ) ( )
2matterdx g f R K V L
4 , , , ,
, ,
1 1( , ) ( ) ( )
2 2matterdx g f R G K V L
Munich, Oct 2013 5
• 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, clustering
DE etc etc etc
4
i matter
i
dx g L + L
The most general 4D scalar field theory with second order equation of motion
A quintessential scalar field
Munich, Oct 2013 6
• 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 produces a modified gravity: f(R)
Simplest modified gravity: f(R)
Munich, Oct 2013 7
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!
Munich, Oct 2013 8
The great Horndeski Hunt
Let us assume we have only 1) a perturbed FRW metric
2) pressureless matter 3) the Horndeski field
and
Munich, Oct 2013 9
Standard rulers
θ
0 0
0 0
1( ) sinh( )
( )k
k
R dzD z H
H zH
R
Munich, Oct 2013 10
Standard rulers
( )dz
H zR
R
Munich, Oct 2013 11
BAO ruler
Charles L. Bennett
Nature 440, 1126-1131(27 April 2006) Munich, Oct 2013 12
Background: SNIa, BAO, …
Then we can measure H(z) and
Then we can measure everything up to
0 0
0 0
1( ) sinh( )
( )k
k
dzD z H
H zH
and therefore
Munich, Oct 2013 13
Munich 2013
Two free functions
)])(21()21[( 222222 dzdydxdtads At linear order we can write:
Poisson’s equation
anisotropic stress
1
The most general linear, scalar metric
2 4 m mG
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
Munich, Oct 2013 15
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
DGP
1( ) 1 ; 1 2
3
2( ) 1
3 1
c DEY a Hr w
a
f(R)
2 2
* 2 2
2 2
,02 2
1 4
( ) , ( ) 1
1 3 1 2cav
k km m
G a R a RY a ak kFG
m ma R a R
Lue et al. 2004; Koyama et al. 2006
Bean et al. 2006 Hu et al. 2006 Tsujikawa 2007
massive bi-gravity
see L. A.,et al in prep. 2013 ( ) ....
( ) ....
Y a
a
Boisseau et al. 2000 Acquaviva et al. 2004 Schimd et al. 2004 L.A., Kunz &Sapone 2007
Munich, Oct 2013 16
Modified Gravity at the linear level
In the quasi-static limit, every Horndeski model is characterized at linear scales
by the two functions
2
42 2
5
2
51 2
3
1( , )
1
1( , )
1
k hk a h
k h
k hY k a h
k h
De Felice et al. 2011; L.A. et al.PRD, arXiv:1210.0439, 2012
Munich, Oct 2013 17
Modified Gravity at the linear level
De Felice et al. 2011; L.A. et al.,PRD, arXiv:1210.0439, 2012
Munich, Oct 2013 18
Rio de Janeiro 2013
Quasi-static approximation
From a wave equation:
To a “Poisson” equation:
2 2 2 2
sc k a H
Reconstruction of the metric
v Hv
dzperp )(
Non-relativistic particles respond to Ψ
Relativistic particles respond to Φ-Ψ
2 2 2 2 2 2[(1 2 ) (1 2 )( )]ds a dt dx dy dz
Munich, Oct 2013 20
Deconstructing the galaxy power spectrum
Redshift distortion
Galaxy
clustering
Present mass power
spectrum
Galaxy bias
Growth function
Line of sight angle
Munich, Oct 2013 21
Munich 2013
Three linear observables: A, R, L
μ=0 Amplitude
A
μ=1 Redshift distortion
R
Lensing L
clustering
lensing
(1 )Y
8 ,0( , ,0) ( )gal tk z Gb k A
8 ,0( , ,1) ( )gal tk z G f k R
2 2
8 ,0
3( ) ( )
2lens m tk k G k L
The only model-independent ratios
Redshift distortion/Amplitude
Lensing/Redshift distortion
1
R fP
A b
02
(1 )m YLP
R f
Redshift distortion rate
3
' 'R fP f
R f
How to combine them to test the theory?
0
HE
HExpansion rate
Munich, Oct 2013 23
Summarizing….
Matter conservation equation independent of gravity theory
02
(1 )m YLP
R f
3
' 'R fP f
R f
0
HE
H
Observables
Munich, Oct 2013 24
The anisotropic stress is directly observable
A unique combination of model independent observables
Observables
Theory
3
2
2
3
3 (1 )1
'2 ( 2 )
P z
EE P
E
Munich, Oct 2013 25
Testing the entire Horndeski Lagrangian
A unique combination of model independent observables
L.A. et al. 1210.0439
Observables
Theory
3 2
2 42 2
2 53
3 (1 ) 11
' 12 ( 2 )
P z k hh
E k hE P
E
Munich, Oct 2013 26
Horndeski Lagrangian: not too big to fail
L.A., M. Motta, I. Sawicki,
M. Kunz, I. Saltas, 1210.0439
2
, , ,2 3( ) 0k kkk kkg g g
2
2
( ) '( , )
REag z k
LEa
If this relation is falsified, the Horndeski theory is rejected*
Munich, Oct 2013 27
Beyond the quasi-static condition
General consistency relation
M. Motta, L.A, I. Sawicki,
M. Kunz, I. Saltas, 2013
Munich, Oct 2013 28
Euclid Surveys – Simultaneous (i) visible imaging (ii) NIR photometry (iii) NIR spectroscopy
– 15,000 square degrees
– 100 million redshifts, 2 billion images
– Median redshift z = 1
– PSF FWHM ~0.18’’
– >1000 peoples, >10 countries
Euclid in a nutshell
Euclid satellite
arXiv Red Book 1110.3193 Munich, Oct 2013 29 arXiv Theory Review 1206.1225
Euclid forecasts...
L.A, M. Kunz, A. Vollmer,
A. Trilleras, S. Fogli, work in progress, 2013
Munich, Oct 2013 30
Euclid forecasts...
Combining galaxy clustering, weak lensing and SN....
Munich, Oct 2013 31
100k SN (LSST)
Results..
L.A, M. Kunz, A. Vollmer,
A. Trilleras, S. Fogli, work in progress, 2013
Munich, Oct 2013 32
Model 1: η constant for all z, k
Error on η around 5%
Model 2: η varies in z
Error on η
Results..
L.A, M. Kunz, A. Vollmer,
A. Trilleras, S. Fogli, work in progress, 2013
Munich, Oct 2013 33
Model 3: η varies in z and k
Error on η
Importance of k-binned data
Munich, Oct 2013 34
growth data compilation
Macaulay et al 2013
Under the carpet..
Munich, Oct 2013 35
Problem of non-linearity:
screening effects mix linear and non-linear scales
same density contrast
different physics
Three Messages
1
If DE is not a Horndeski field, then
we don’t know what could be
Munich, Oct 2013 36
2
k-binned data are crucial!
e.g. growth factor, redshift distortion parameter
3
Only by combining galaxy clustering and lensing
can DE be constrained (or ruled out!) in a model-independent way
Munich, Oct 2013
Cambridge
University
Press
37
Dark Force
Schlamminger et al 2008
/
2 2
(1 )
''( )
( )
rGM GMe
r r
m V
r
Limits on Yukawa coupling are strong but local!
Munich, Oct 2013 38
The Yukawa correction
Every Horndeski model induces at linear level, on sub-Hubble scales, a Newton-Yukawa potential
/( ) (1 )rGMr e
r
where α and λ depend on space and time
Munich, Oct 2013 39
Quasi-static approximation
From a wave equation:
To a “Poisson” equation:
2 2 2 2
sc k a H
Munich, Oct 2013 40
Final remarks
Munich, Oct 2013 41
Few things I learned
Munich, Oct 2013 42
• String theory is not dead…and can save cosmology
• SN Ia progenitors still a mystery
• Euclid will fly in a different landscape…
• Industrial approach to model comparison
One thing I am trying to learn
Munich, Oct 2013 43
How to make a perfect Bavarian toast
Few things to improve
Munich, Oct 2013 44
• Faster N-body simulations
• New standards:
GW sirens, gal ages, AGN, real-time observables
• Effects of inhomogeneities and anisotropies on
parameter estimation
Few things to understand
Munich, Oct 2013 45
• Does vacuum energy gravitate?
• Is the coincidence problem(s) to be taken seriously?
• Does a “linear regime” really exists?
And can it be identified by observations alone?
Few wishes for the future
Munich, Oct 2013 46
• More theory-lite observational results
• More observation-lite theoretical results
• Meet again soon for deSitter III !
Munich, Oct 2013 47