Sapienza - Università di Roma · Sapienza - Università di Roma February 20, 2013 Andrea Marchini...

Testing modi�ed gravity through cosmology

Andrea Marchini

Sapienza - Università di Roma

February 20, 2013

Andrea Marchini Testing modi�ed gravity through cosmology

Cosmic acceleration

ds2 = −dt2 + a2(t)dΣ2

Gµν = 8πGTµν

(Flat FLRW metric)

(Einstein equations)

a

a= −4πG

3(ρ+ 3P) (II Friedmann equation)

Equation of state: P = wρ.

Only a �uid with w < −13can be responsible for a cosmic

acceleration.

For matter (w = 0) and radiation (w = 1/3) it's impossible to

generate a phase of accelerated expansion.


Discovery of the cosmic acceleration

The luminosity distance depends on the evolution a(t).

We can de�ne a deceleration parameter q0 = −H−20

(a

a

)t=t0

.

dL =(

L

4πF

) 12 = 1

a

∫ 1a

da

a2H(a)= z

H0

[1 + 1

2(1− q0) z

]+ O(z3)

The cosmological redshift is de�ned as z = a0a− 1.


Cosmological constant

Data suggest the existence of a new component with w

extremely close to -1.

The simplest explanation could be the presence of a

cosmological constant Λ in the Einstein equations.

Issues with Λ

Fine tuning: interpreting ρΛ as the vacuum energy is hard because

its value is quite small compared to the prediction of particle

physics: ρΛ/ρvac ∼ 10−123.

Coincidence: why are the matter and vacuum energy densities

approximately equal today?

a

a



Data suggest the existence of a new components with w










Issues with Λ






a

a







Issues with Λ






In order to solve these issues a plethora of models have been

proposed.


Modi�ed gravity models

The alternative models are mainly of two kinds:to modify Tµν adding new components with the properfeatures;to modify the lagrangian of General Relativity.

Modi�ed gravity models are quite interesting because they can

explain the cosmic acceleration avoiding the presence of exotic

components.

f(R) theories

S =∫d4x√−g(R

2κ + Lm)⇒ S =

∫d4x√−g(f (R)2κ + Lm

)(1 + fR(R))Rµν − 1

2(f (R)− R) gµν + (gµν�−∇µ∇ν) fR(R) = kTµν

These theories have to satisfy constraints of local gravity tests

in the solar system and cosmological constraints.


What can we measure cosmologically?

Perturbed metric

ds2 = −(1 + 2Ψ (t,~x))dt2+a2(t)(1− 2Φ (t,~x))d~x2

Expansion history: a(t)

Non-relativistic dynamics (growth

of structure): Ψ

Relativistic dynamics (weak

lensing, Sachs�Wolfe e�ect):

Ψ + ΦAndrea Marchini Testing modi�ed gravity through cosmology

Parametrization

For the GR we have: Φ = Ψ, Ψ = − a2ρ∆2k2M2

P

.

We can parametrize the e�ect of modi�ed gravity as:

Φ =γ (a, k)Ψ, Ψ = −µ (a, k) a2ρ∆2k2M2

P

.

This way the background is �xed to that of the standard case

but the evolution of matter perturbations can be di�erent.

Bertschinger-Zukin parametrization (B. & Z., PRD 78, 024015, 2008)

µ(a, k) =1+β1λ21 k

2as

1+λ21 k2as

, γ(a, k) =1+β2λ22 k

2as

1+λ22 k2as

For f (R) gravity only the lengthscale λ1 is not �xed by the

theory. Usually it is expressed as the present lengthscale in

units of the horizon scale: B0 = 2λ21H20/c

2.


Cosmic Microwave Background

The CMB is a thermal radiation that decoupled from the

matter in the early Universe.

Its power spectrum is the principal observable in cosmology.


E�ects on the Cosmic Microwave Background

The shape of the CMB power spectrum depends on the Ψ and

Φ potentials.

Integrated Sachs-Wolfe e�ect

(∆T

T

)ISW

=∫dη[Ψ (η,~x)− Φ (η,~x)

]

CMB lensing

∆T (n)→ ∆T (n + d (n))d (n) = ∇ψ (n,Ψ,Φ)


E�ect on the Cosmic Microwave Background


Lensing

The amplitude of the lensing e�ect can be parametrized by AL.

This is just an e�ective parameter that characterize the

amount of the CMB lensing.

In case of standard expected signal AL = 1, with no lensing

AL = 0.

In the framework of f(R) theories the value of AL is larger than

the standard case.

Measuring an e�ective amplitude AL > 1 could be a signature

of a modi�cation of GR.


SPT and ACT

Recently the South Pole Telescope and the Atacama

Cosmology Telescope measured the CMB spectrum at high

multipoles.

The analysis is performed combining SPT and ACT with

WMAP data (lmax ' 1000).

The experiments reported two di�erent constraints for AL:

ACT: AL = 1.70± 0.38 at 68% c.l. (not consistent with AL = 1)

SPT: AL = 0.85± 0.15 at 68% c.l.

Evidence for modi�ed gravity?


Results

Constraints on B0 (Marchini et al.,arXiv:1302.2593 submitted to PRD):

ACT: B0 < 0.90 at 95% c.l.SPT: B0 < 0.14 at 95% c.l.

Previous constraint using CMB data is B0 < 0.4 at 95% c.l:

T. Giannantonio et al., JCAP 1004, 030 (2010)A. Hojjati et al., JCAP 005, 1108 (2011)L. Lombriser et al., PRD 85, 124038 (2012)

SPT gives signi�cantly stronger constraints than ACT.

The di�erence it's due to the anomalous lensing signal in the

ACT TT spectrum.


Future goals

To constrain B0 using the lensing power spectrum (not only

the TT spectrum).

To perform a similar analysis with the Planck data that will be

released in March 2013.

Forecast for the e�ect of MG in the weak lensing observed in

large galaxy survey performed by experiments as Euclid.

To constrain other models of modi�ed gravity.


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