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a viable candidate for dark energy?

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Luca Amendola

INAF/Osservatorio Astronomico di Roma

Is higher-order gravity

- Collaboration: D. Polarski, S. Tsujikawa, R. Gannouji, C. Charmousis, S. Davis, E. Magliaro

- Firenze GGI 2006

…to an unexpected universe

Nucleosynthesis (z~109)

CMB (z~1000)

Standard candles (z~1)

Perturbations (z~0-1000)

A) Lambda & friends

B) scalar field

C) modified gravity

D) non-linear effect

A) Lambda & friends

B) scalar field

C) modified gravity

D) non-linear effect

1) Models which are almost impossible

to rule out

2) Models which are easy to rule out

A) Lambda & friends

B) scalar field

C) modified gravity

D) non-linear effect

A) Lambda & friends

B) scalar field

C) modified gravity

D) non-linear effect

- It is more general
- Scalars are predicted by fundamental theories

Observational requirements:

Slow evolution

Light mass

V()

Compton wavelength = Hubble length

- But beside the potential there can be also a coupling…

T(m)= CT(m)

T= -CT(m)

coupled conservation laws :

First basic

property:

C2/G = scalar-to-tensor ratio

Jordan frame

Einstein frame

Newtonian limit: the scalar interaction generates an attractive extra-gravity

in real space

Yukawa term

α

λ

NO DE

uninteresting

Chameleon models

Coupled DE....

Make mφvery large

Make βvery small

Make them density dependent

Make them species dependent

A species-dependent interaction

T(cdm)= CT(cdm)

T= -CT(cdm)

T(bar)= 0

T(rad)= 0

today

No coupling

mat

rad

= 0

a ~ tp

p = 2/3

MDE:

field

coupling

rad

mat

= /9

a ~ tp

p = 6/(42+9)

MDE:

field

MDE

kinetic phase, indep. of potential!

Assume

V(f) = f-a

today

mat

rad

Dominated by

potential energy

α

Dominated by

kinetic energy

β

field

The equation of state w=p/r

depends on bduring fMDE and on aduring tracking:

we = 4b2/9: past value (decelerated)

wf = -2/(a+2): present value (accelerated)

cl)

Planck:

Scalar force 100 times weaker than gravity

A) Lambda & friends

B) scalar field

C) modified gravity

D) non-linear effect

Higher order gravity !

Simplest case:

Capozziello,Turner,

Carroll, Odintsov…

L.A., S. Capozziello, F. Occhionero, 1992

Simplest case:

Extra-dim.

degrees of freedom

Higher order

gravity

Coupled scalar field

Scalar-tensor gravity

is in fact a scalar-tensor model in the Jordan frame

and a coupled dark energy model

in the Einstein Frame

In Einstein Frame

Coupled dark energy with a strong coupling !

is a scalar-tensor theory with Brans-Dicke

parameter ω=0 or β=1/2

Therefore if

and f(R) is not yet ruled out

see eg. Nojiri & Odintsov 2003; Brookfield et al. 2006

In Einstein Frame

today

mat

rad

field

rad

mat

field

MDE

Jordan F:

Caution:

Plots in the

Einstein frame!

More exactly

today

mat

rad

During this phase,

field

rad

mat

and therefore

field

MDE

Caution:

Plots in the

Einstein frame!

L.A., S. Tsujikawa, D. Polarski 2006

Plots by R. Gannouji, U. of Montpellier

An autonomous dynamical system

For all f(R) theories:

The wrong matter era

(the t1/2 behavior) exists always

The good matter era

(the t2/3 behavior)exists

only if m(-1)=0

This immediately rules out many cases:

Gallery of Failed Cosmologies

Plots by R. Gannouji, U. of Montpellier

a) Dark energy domination

is always preceded by

the wrong matter era (if any), except...

b) …when the acceleration is driven

by a Lambda term as in

we find that an exact matter era exists only

for very special combination of parameters

so probably most of these models are ruled out.

Is this the most general second-order scalar-tensor theory ?

No!

This is a Coupled Gauss Bonnet model. Can we put

constraints on it without specifying the potential/couplings?

Yes, if the scalar field is a DE field

Everything depends on

Observable deviations

from linearized

Newtonian gravity

Observable deviations

from linearized

Newtonian gravity

in the slow-rolling and

small-coupling limit

Direct observables

Growth of matter fluctuations

ISW effect

L.A., C. Charmousis,S. Davis,

astro-ph/0506306

Quanto a figure perfette o nobili,

credo che per murare le quadre

sieno più perfette che le sferiche,

ma per ruzzolare o condurre i carri

stimo più perfette le tonde che le

triangolari. (Il Saggiatore, 1623)

As to the perfection or nobility

of geometric figures, I think that

for works in masonry square figures

are more perfect than spherical ones,

but to roll or drive a carriage I consider

the circular figures more perfect

than the triangular ones.

(Il Saggiatore, 1623)

Hubble size

Galactic size

Adopting a PGB

potential

Abundance

Mass

L.A. & R. Barbieri 2005