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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.
Andrea Marchini Testing modi�ed gravity through cosmology
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.
Andrea Marchini Testing modi�ed gravity through cosmology
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
Andrea Marchini Testing modi�ed gravity through cosmology
Cosmological constant
Data suggest the existence of a new components with w
extremely close to -1.
The simplest explanation could be the presence of a
cosmological constant Λ in the Einstein equations.
Andrea Marchini Testing modi�ed gravity through cosmology
Cosmological constant
Data suggest the existence of a new components 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
Andrea Marchini Testing modi�ed gravity through cosmology
Cosmological constant
Data suggest the existence of a new components 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?
In order to solve these issues a plethora of models have been
proposed.
Andrea Marchini Testing modi�ed gravity through cosmology
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.
Andrea Marchini Testing modi�ed gravity through cosmology
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.
Andrea Marchini Testing modi�ed gravity through cosmology
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.
Andrea Marchini Testing modi�ed gravity through 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)
Andrea Marchini Testing modi�ed gravity through cosmology
E�ect on the Cosmic Microwave Background
Andrea Marchini Testing modi�ed gravity through cosmology
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.
Andrea Marchini Testing modi�ed gravity through cosmology
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?
Andrea Marchini Testing modi�ed gravity through cosmology
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.
Andrea Marchini Testing modi�ed gravity through cosmology
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.
Andrea Marchini Testing modi�ed gravity through cosmology