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