The Effective Field Theory approach to Dark Energy and
Modified Gravity phenomenology
Marco Raveri
Lecture for KICP Cosmology Class
Feb 24, 2017
Marco Raveri
24/02/17 14:53
Introduction
DE/MG
EFT approach
Conclusions
1. Why Dark Energy (DE)?
[ Everything You Always Wanted To Know About The Cosmological
Constant Problem (But Were Afraid To Ask), arXiv:1205.3365v1 ]
2. Why Modified Gravity (MG)?
[ Modified Gravity and Cosmology, arXiv:1106.2476 ]
3. Why both???
4. EFT approach:
• how to build the EFT of DE/MG
• conformal invariance
• why a scalar field?
• possible generalizations
• stability
• how to use the EFT (mapping other theories)
• cosmological phenomenology (EFTCAMB)
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Introduction
DE/MG
EFT approach
Conclusions
4. EFT approach:
• how to build the EFT of DE/MG
[ arXiv:1210.0201, arXiv:1211.7054,
arXiv:1304.4840, arXiv:1601.04064,
arXiv:1609.00716 ]
• stability
[ … + arXiv:1609.03599 ]
• how to use the EFT (mapping other theories)
[ … + arXiv:1601.04064 ]
• cosmological phenomenology (EFTCAMB)
[ … + arXiv:1312.5742, arXiv:1405.1022,
arXiv:1405.3590 ]
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Introduction
DE/MG
EFT approach
Conclusions
Invitation
One of the greatest problems of modern physics:
we developed our theory of gravity on the earth,
it works perfectly at the solar system level,
as soon as we consider the Universe it drastically fails.
Cosmic Acceleration Problem:
Cosmological Principle
Solar System GR
Ordinary matter + Dark matter
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Introduction
DE/MG
EFT approach
Conclusions
Invitation
One of the greatest problems of modern physics:
we developed our theory of gravity on the earth,
it works perfectly at the solar system level,
as soon as we consider the Universe it drastically fails.
d
es no
Cosmic AccelerationoProblem:
t wor
k!
Cosmological Principle
Solar System GR
Ordinary matter + Dark matter
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Introduction
DE/MG
EFT approach
Conclusions
26
fainter
24
22
Supernova Cosmology Project
High-Z Supernova Search
Is it a
cosmological
constant?
Calan/Tololo
Supernova Survey
25
20
18
0.2
24
0.4
0.6
1.0
magnitude
16
14
0.01
23
0.02
22
0.04
0.1
Accelerating
Universe
21
Decelerating
Universe
20
0.2
0.4
0.6
1.0
redshift
( Science “Breakthrough of the year” 1998; Perlmutter, Physics Today (2003); Discovery awarded with the Nobel Prize in 2011)
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Introduction
DE/MG
EFT approach
Conclusions
The Cosmological Constant problem
The cosmological constant can be there:
it is allowed by the requirements used to build General Relativity
(Lovelock’s theorem)
Gravitational field equations:
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Introduction
DE/MG
EFT approach
Conclusions
GR is a well defined “classical” theory
Take:
fit to the data, get a good fit (?), end of the story.
There is no “classical” CC problem
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Introduction
DE/MG
EFT approach
Conclusions
The problem arises when one takes into account quantum field
theory.
Why: in flat space-time the only invariant tensor is
Since the vacuum state must be the same for all observers, one
necessarily has
To curved
space-times
This does not necessarily guarantee that
is constant…
(Running Vacuum Cosmology)
[ A. Sakharov, Sov. Phys. Dokl. 12, 1040 (1968) ]
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Introduction
DE/MG
EFT approach
Conclusions
Now assume that vacuum gravitates:
rewrite:
where:
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Introduction
DE/MG
EFT approach
Measure
Conclusions
Predict
Adjust
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Introduction
DE/MG
EFT approach
Conclusions
Standard calculation
which turns out to be wrong…
wrong equation of state!
Is this a low energy problem?
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Introduction
DE/MG
Quantum Gravity cutoff
SUSY cut-off
EW phase transition
QCD phase transition
muon
electron
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(10
18
EFT approach
GeV )
4
4
(T eV )
4
(200 GeV )
Conclusions
fine tuning to 120 decimal places
fine tuning to 60 decimal places
fine tuning to 56 decimal places
4
(0.3 GeV )
4
(100M eV )
4
(M eV )
fine tuning to 44 decimal places
(meV )4
observed value
fine tuning to 36 decimal places
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Introduction
DE/MG
EFT approach
Measure
Conclusions
Predict
Adjust
Solar system orbits
Atomic levels
Too much!
…and not radiatively stable [ arXiv:1502.05296v1 ]
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Introduction
DE/MG
EFT approach
Conclusions
We do not know how to do this calculation…
What does the space of solutions look like?
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Introduction
DE/MG
EFT approach
Conclusions
We do not know how to do this calculation…
What does the space of solutions look like?
Second face of Lovelock’s theorem:
any extension of GR is not going to look like GR…
We do not know how quantum vacuum gravitates…
but
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Introduction
DE/MG
EFT approach
Conclusions
Dark energy
Save the geometrical structure of GR and add components
to the energy budget of the universe
For example add a simple scalar field:
Standard term
Marco Raveri
kinetic term
potential
17
Introduction
DE/MG
EFT approach
Conclusions
Dark energy
Save the geometrical structure of GR and add components
to the energy budget of the universe
For example add a simple scalar field:
Warning Weinberg’s no-go theorem
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Introduction
DE/MG
EFT approach
Conclusions
Modifications of gravity
Intriguing idea:
Realized in some models [ arXiv:1309.6562 ]
Give up (some of) the geometrical structure of GR and
try to source cosmic acceleration
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Introduction
DE/MG
EFT approach
Conclusions
There are other reasons why we might want to modify gravity
1
GR is not UV complete
At some point GR needs to be modified
Doomed by Lovelock’s theorem
+
the CC problem is a UV problem
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Introduction
DE/MG
EFT approach
Conclusions
There are other reasons why we might want to modify gravity
2
because we can
GR has been tested on solar system scales
other regimes: GW and black holes, cosmology
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Introduction
DE/MG
10
10
10
10
10
Curvature,
(cm
-2
)
10
10
10
10
10
10
10
10
10
10
10
10
10
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Conclusions
-11
Triple
-14
AdLIGO
LOFT +
Athena
-17
-20
-23
eLISA
Sgr A*
-26
Inv. Sq.
-29
-32
Atom
PPN
constraints
PTA
M87
-35
EHT
ELT S stars
-38
-41
-44
-47
-50
Tidal streams
(GAIA)
A
-53
P
-56
Planck
Facility
BAO
-59
DETF4
-62
10
[ arXiv:1412.3455v3 ]
EFT approach
-12
10
-10
10
-8
10
-6
10
-4
10
-2
10
0
Potential,
22
Introduction
DE/MG
10
10
10
10
10
Curvature,
(cm
-2
)
10
10
10
10
10
10
10
10
10
10
10
10
10
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Conclusions
-11
Triple
-14
AdLIGO
LOFT +
Athena
-17
SOLAR SYSTEM
-20
-23
eLISA
Sgr A*
-26
Inv. Sq.
-29
-32
Atom
PPN
constraints
PTA
M87
-35
EHT
ELT S stars
-38
BH
-41
-44
We are
here
COSMOLOGY
-47
-50
Tidal streams
(GAIA)
A
-53
P
-56
Planck
Facility
BAO
-59
DETF4
-62
10
[ arXiv:1412.3455v3 ]
EFT approach
-12
10
-10
10
-8
10
-6
10
-4
10
-2
10
0
Potential,
23
Introduction
DE/MG
EFT approach
Conclusions
CMB-S4
WiggleZ
+ many more…
( CMB temperature, CMB lensing, galaxy weak lensing convergence and number counts fluctuations: the cosmological pie that I like )
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Introduction
DE/MG
EFT approach
Conclusions
The situation in model space:
Cosmological Constant
Kaluza-Klein Theories of
Gravity
Randall-Sundrum Gravity
Brane Cosmology
Higher Dimensional
Theories of Gravity
Dvali-Gabadadze-Porrati
Gravity
Higher Co-Dimension Brane
worlds
Einstein Gauss-Bonnet
Gravity
f(R) Theories
Models for Cosmic Acceleration
Modified Gravity
Horava-Lifschitz Gravity
Higher Derivative and Non-Local
Theories of Gravity
Covariant galileon
Galileons
DBI galileon
Multi-galileons
Scalar-Tensor Theories
Einstein-AEther Theories
Extra-Fields
Brans-Dicke theory
Horndeski
Rosen’s theory
Bimetric Theories
Drummond’s theory
Massive gravity
Dark Energy
Non-minimally coupled to
matter
Minimally coupled to matter
TeVeS Theories
Bigravity
Quintessence
K-essence
[ based on arXiv:1106.2476, by far not complete… ]
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Introduction
DE/MG
EFT approach
Conclusions
The EFT approach
The standard way:
Write the theory
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Get
equations of
motion
Do the limit
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Introduction
DE/MG
EFT approach
Conclusions
The standard way:
Write the theory
Get
equations of
motion
Do the limit
The EFT way:
Do the limit
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Write the
theory
Get
equations of
motion
27
Introduction
DE/MG
EFT approach
Conclusions
The standard way:
Write the theory
Get
equations of
motion
Do the limit
The EFT way:
Get
equations of
motion
Do the limit
Write a general
theory
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Introduction
DE/MG
EFT approach
Conclusions
The Cosmological Principle:
the universe is homogeneous and isotropic if seen by a comoving
observer
Relevant limit: cosmological perturbations
As a consequence it has a maximally symmetric 3+1 foliation
[ arXiv:1304.4840 ]
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Introduction
DE/MG
EFT approach
Conclusions
The ADM metric:
Normal vector, extrinsic and intrinsic curvature:
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Introduction
DE/MG
EFT approach
Conclusions
What are all the scalar quantities that we can define here:
The most general theory of gravity on the foliation
will depend on all of them:
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Introduction
DE/MG
EFT approach
Conclusions
Where to stop?
There are other quantities that we can add:
Lorentz violating terms:
[ arXiv:1409.1984v2, arXiv:1601.04064v2 ] they didn’t stop…
Higher order terms:
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Introduction
DE/MG
EFT approach
Conclusions
Now expand the EFT Lagrangian in perturbations:
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Introduction
DE/MG
EFT approach
Conclusions
Now expand the EFT Lagrangian in perturbations:
Average Lagrangian
First order action
Cosmological background
Second order action
Cosmological perturbations
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Introduction
DE/MG
EFT approach
Conclusions
Now expand the EFT Lagrangian in perturbations:
Derivatives evaluated on the background -> functions of time only
See why we excluded higher powers of operators?
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Introduction
DE/MG
EFT approach
Conclusions
After some algebraic reshuffling this can be casted in a more
familiar (?) form
only real trick is the Gauss-Codazzi relation
If interested in the details here’s the references…
[ arXiv:1304.4840, arXiv:1601.04064 ]
S=
Z
⇢
2
m
0
d4 x
g
[1 + ⌦(⌧ )] R + ⇤(⌧ ) a2 c(⌧ ) g 00
2
2
M̄
M24 (⌧ ) 2 00 2 M̄13 (⌧ ) 2 00 µ M̄22 (⌧ )
3 (⌧ )
µ 2
+
(a g )
a g Kµ
( Kµ )
K⌫µ Kµ⌫
2
2
2
2
a2 M̂ 2 (⌧ ) 00 (3)
+
g R + m22 (⌧ )(g µ⌫ + nµ n⌫ )@µ (a2 g 00 )@⌫ (a2 g 00 ) + Sm [gµ⌫ ]
2
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p
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Introduction
DE/MG
EFT approach
Conclusions
Where are we?
S=
Z
⇢
2
m
0
d4 x
g
[1 + ⌦(⌧ )] R + ⇤(⌧ ) a2 c(⌧ ) g 00
2
2
M̄
M24 (⌧ ) 2 00 2 M̄13 (⌧ ) 2 00 µ M̄22 (⌧ )
3 (⌧ )
µ 2
+
(a g )
a g Kµ
( Kµ )
K⌫µ Kµ⌫
2
2
2
2
a2 M̂ 2 (⌧ ) 00 (3)
+
g R + m22 (⌧ )(g µ⌫ + nµ n⌫ )@µ (a2 g 00 )@⌫ (a2 g 00 ) + Sm [gµ⌫ ]
2
p
here, along with all other matter species
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Introduction
DE/MG
EFT approach
Conclusions
What are the assumptions about the matter sector?
In writing this we implicitly assume that matter is minimally and
universally coupled to the metric tensor
This means that particles of the m specie would travel on
geodesics of g in absence of non-gravitational forces
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Introduction
DE/MG
EFT approach
Conclusions
Connection to the Einstein frame
The Einstein frame is a conformal frame where the
EH part of the action is not modified
Is GR conformal invariant?
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Introduction
DE/MG
EFT approach
Conclusions
After an appropriate conformal transformation
details here: arXiv:1210.0201
notice difference in notation…sigh…
Is the structure of the EFT unchanged?
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Introduction
DE/MG
EFT approach
Conclusions
Where is the trick?
We have broken time translations
does not leave invariant the action
Stuckelberg trick
Now under an infinitesimal coordinate transformation:
is invariant
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Introduction
DE/MG
EFT approach
Conclusions
Apply this to the EFT action: transform all pieces and expand
details here [ arXiv:1211.7054 ]
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Introduction
DE/MG
EFT approach
Conclusions
Apply this to the EFT action
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Introduction
DE/MG
EFT approach
Conclusions
Why a scalar field?
It’s the Goldstone boson of time translations
FRW background itself breaks time translations
everything that respects the FRW background is going
to have at least this form
( connection with unitary gauge )
Is there any difference between DE and MG?
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Introduction
DE/MG
EFT approach
Conclusions
Possible extensions
The ultimate limit of the idea of the EFT of DE/MG
is the cosmological principle
•
Additional scalar, vector and tensor fields [ arXiv:1604.01396 ]
•
Non-universal couplings [ arXiv:1504.05481, arXiv:1609.01272 ]
•
Higher order terms in derivatives [ arXiv:1409.1984, arXiv:1601.04064 ]
•
Higher order in perturbation theory [ ? ]
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Introduction
DE/MG
EFT approach
Conclusions
How to use the EFT
Two ways to approach the EFT
1
fix the form of the EFT functions with a model
2
study the EFT functions
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Introduction
DE/MG
EFT approach
Conclusions
Map models into the EFT framework
1
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quintessence
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Introduction
DE/MG
EFT approach
Conclusions
Map models into the EFT framework
2
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f(R) gravity
48
Introduction
DE/MG
EFT approach
Conclusions
Map models into the EFT framework
3
Galileons/Horndeski
4
Beyond Horndeski
5
Horava gravity
…
…many more…
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Introduction
DE/MG
EFT approach
Conclusions
Stability of perturbations
Positive Newtonian constant
No ghost instabilities
No gradient instabilities
Positive mass of the scalar
Priors for the model
Very technical (and complicated) discussion.
Some of it in all EFT of DE/MG papers.
Especially [ arXiv:1609.03599v1 ].
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Introduction
DE/MG
-0.5
-0.6
EFT approach
Conclusions
-0.80
-0.996
-0.85
-0.997
-0.8
-0.90
-0.998
-1.0
-0.95
-0.999
-1.2
-1.00
-1.000
-1.4
-1.5
-1.05
-0.5 -0.3
-0.1 0.0 0.1
0.3
0.5
minimally coupled 5e
on CPL background
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-1.001
-0.025 0.0
0.025
0.05 0.075
linear Omega
on wCDM background
-4.5
-3.5
-2.75
-2.0
-1.0
designer f(R) on
wCDM background
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Introduction
DE/MG
EFT approach
Conclusions
Phenomenology
Cosmological background
Linear scalar perturbations
Linear tensor perturbations
Non-linear perturbations
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Introduction
DE/MG
EFT approach
Conclusions
Cosmological background
Complete freedom at the background level
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Introduction
DE/MG
EFT approach
Conclusions
Linear scalar perturbations
Behavior is strongly model dependent.
Very difficult (if not impossible) to get non-approximate
and general analytic results
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Introduction
DE/MG
EFT approach
Conclusions
Modification of the ISW effect
Modified CMB lensing
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Introduction
DE/MG
EFT approach
Conclusions
Modified growth of perturbations:
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Introduction
DE/MG
EFT approach
Conclusions
The price of modifying lensing…
no GW’s here…
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Introduction
DE/MG
EFT approach
Conclusions
10!1
104
c = 1.5
c = 1.0
c = 0.5
2
T
2
T
2
T
103
102
10!2
c = 1.5
c = 1.0
c = 0.5
10!3
5
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10
2
T
2
T
2
T
10
50 100
l
1000
5
10
1
50 100
l
1000
10!1
2
l!l"1" CTT
l # 2Π $ ΜK %
2
l!l"1" CBB
l # 2Π $ ΜK %
Linear tensor perturbations
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Introduction
DE/MG
EFT approach
Conclusions
Non-linear perturbations
[ arXiv:1611.07966 ]
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Introduction
DE/MG
EFT approach
Conclusions
EFTCAMB
Patch of CAMB/CosmoMC: full perturbative Boltzmann-Einstein
equations;
No approximations: NO quasi-static, NO sub-horizon;
Complete set of perturbative equations: include all second order EFT
operators;
Various expansion histories: LCDM, wCDM, CPL +…
Pure EFT mode: study the underlying DE/MG theory through a
parametrization of the EFT functions;
Mapping EFT mode: study a single model. Built-in f(R), minimally
coupled quintessence, Horava gravity. More to come very soon!
Exploration of parameter space and comparison with cosmological
datasets with a built in theoretical viability check.
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Introduction
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DE/MG
EFT approach
Conclusions
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Introduction
DE/MG
EFT approach
Conclusions
From the cosmology of a model to data constraints
5
10
50
100
(a)
1000
10
50
100
1000
(b)
-3.0
P (arbitrary normalization)
5
(c)
-4.5
-6.0
-7.5
-10
-8
-6
-4
-2
-9.0
-10
PLC
PLC, BG
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2
P (arbitrary normalization)
2
-8
-6
-4
PLC, CMBL
PLC, WiggleZ
-2 -9.0 -7.5
-6.0
-4.5
-3.0
All combined
Viability Prior
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Introduction
DE/MG
EFT approach
Conclusions
CosmicFish
Standalone code to perform Fisher matrix forecasts;
Most of the cosmological observables of interest;
State of the art algorithm and strongly optimized;
Modern software design: fully modular, unit testing, complete
documentation, …;
Uses EFTCAMB to ensure wide coverage of DE/MG models;
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Introduction
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DE/MG
EFT approach
Conclusions
64
Introduction
DE/MG
EFT approach
Conclusions
…to the future of a model
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Introduction
DE/MG
EFT approach
Conclusions
Join THE DARK SiDE
rule the universe
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Introduction
DE/MG
EFT approach
Conclusions
Modern approaches to DE and MG phenomenology;
Unify in a single box these two types of models;
Possibly capture phenomenological features of models in
the space of solutions to the CC problem;
Use it to test gravity on cosmological scales if not possible;
Gear them up to be stand the experimental challenge of the
next decade.
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