Transcript BRANEWORLD COSMOLOGICAL PERTURBATIONS

Dark Energy and Modified Gravity
C
nn State
y 2008
Roy Maartens
ICG
Portsmouth
LCDM fits the high-precision data
galaxy distribution
cosmic microwave background
LCDM
SDSS
WMAP
M    K  1
8G i
i 
3H 02
supernovae
K  0
CMB
galaxies
3 independent
data sets
intersect
the improbable, mysterious
universe
there are
particle
physics
candidates
0.2
0.75
or Modified
Gravity?
LCDM fits the data well…
but we cannot explain it

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it’s the simplest model
compatible with all data up to now
no other model gives a better statistical fit
but …. theory cannot explain it

  obs 
~ H 02 M p2 ~ (10 33 eV) 2 (10 28 eV) 2 ~ (10 3 eV) 4
8G
  theory  vacuum energy ~ ( M new physics) 4  ( M susy ) 4 ~ (1 TeV) 4   
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why so small?
and … why
so fine-tuned?
obs
  ~ 0 : crucial for structure formation
but    a 0 while  m  a 3
‘coincidence’ problem
radiation ( 1/a4)
log 
matter ( 1/a3)
cosmological
constant
log a
Radiation
dominated
Matter
dominated
Dark energy
dominated
String theory and vacuum energy
G  8G(T  T ), T    vac g 
vac
 vac


~ (10 3 eV) 4
8G
string “landscape” and
multiverse to explain
fine-tuned small value?
speculative & controversial
 vac   / 8G
 vac   / 8G
 vac  0
vac
…. or from spacetime topology?
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“self-tuning” braneworld
the higher-dimensional vacuum energy is large,
as expected
(4  n )
4 n
 vac ~ (M new physics)

- but the 4D brane is protected from it
However: unstable
4D brane universe
(4+n)D spacetime
with a cut
Other quantum gravity approaches to
the vacuum energy

Loop Quantum Gravity:
ask Abhay and Martin
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Causal sets
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Others
“minimalist” attitude
LCDM is
the best model
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test this against data
wait for particle physics/QG to explain why
 vac  (103 eV) 4

focus on
* the best tests for w=-1
* the role of theoretical assumptions
e.g. w=const,
w(z) parametrizations,
curvature=0
… but we can do more
with the data
We can test alternatives
some alternatives to LCDM
Dynamical Dark Energy in General Relativity
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“quintessence”, coupled DE-dark matter,...
effective ‘Dark Energy’ via nonlinear effects of
structure formation?
‘Dark Gravity’ – Modify GR on large scales
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4D: scalar-(vector)-tensor theories [e.g. f(R)]
higher-D: braneworld models [e.g. DGP]
NB – all these alternatives require that the
vacuum energy does not gravitate:  vac  0
- they address the coincidence problem not the
vacuum energy problem
Dark Energy dynamics
dark
G  8GT  8GT
dark
T
 time - varying DE field
w
pDE
 DE

1
3
Modified Gravity dynamics
dark
G  G
 8GT
dark
G
 new scalar DOF
to induce accelerati on
quintessence
tracker scalar field, to solve the coincidence problem
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but parameters in the
potential must be
highly fine-tuned
more complicated dynamical models are poorly motivated or
suffer theoretical problems:
eg phantom scalar field (ghost - vacuum unstable)
k-essence (violates causality)
Chaplygin gas (what phenomenology?)
coupled quintessence

alternative approach to the coincidence problem:
* DM and DE only detected gravitationally
* unavoidable degeneracy
* there could be a coupling in the dark sector
 T()  Q    T( 
c) 
  3H (1  w )   Q   c  3H c 
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(coupling to SM fields strongly constrained)
intrinsic CDM bias – Euler equation violated
some models ruled out by instabilities
others lead to interesting features
eg w<-1 without ghosts
effective ‘DE’ from structure formation?
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more radical approach to the coincidence problem –
“structure formation implies acceleration”
  1 for   10h 1 Mpc but voids, walls  100h 1 Mpc
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nonlinear averaging/ backreaction?
voids dominate over filaments – accelerating effect?
averaging effects are real and important –
but probably too small to give acceleration
abandon Copernican principle?
Modified (dark) gravity
is GR wrong on large scales ?
* GR: acceleration via the anti-gravity of DE
(or perhaps via nonlinear effects)
* modified gravity:
acceleration via the weakening of gravity
on large scales
Challenge the standard theory?
Example from history:
Mercury perihelion
– Newton + ‘dark’ planet ?
no – modified gravity!
But – very hard to consistently modify GR in the IR
and – must pass local as well as cosmological tests
Key assumptions on MG theories:
metric theory
 energy-momentum conservation

 T   0
Key requirements
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on small nonlinear scales – must recover GR
on superhorizon scales – perturbations must evolve
compatibly with the background (‘separate
universe’)
On intermediate scales – Poisson equation is modified
GR = spin-2 graviton + minimal coupled matter
MG changes both features
Background
modified
Friedman:
H 2 (1  Adark ) 
8G

3
1 

H (1  Adark )  HAdark  4G (   p )
2
Adark  modification toGR
Examples:
f(R) modified gravity (R = Ricci scalar)
Lgrav  f ( R)
Adark

R f
H 
R

  f R  11  2   f RR
2
6H
H
 H 
DGP modified gravity (braneworld model)
1
Adark 
rc H
Geometric tests
(eg supernovae, BAO)
probe the background
expansion history
general feature
geometric tests on their own cannot distinguish
modified gravity from GR
why?
geometric tests are based on the comoving distance
r ( z)  
z
0
dz '
H ( z' )
- the same H(z) gives the same expansion history
we can find a GR model of DE
to mimic the H(z) of a modified gravity theory:
8G
(    DE )
3
8G
2
dark gravity H (1  Adark ) 

3
3H 2 ( z )
choose  DE ( z ) 
Adark ( z )
8G
then
rGR ( z )  rDG ( z )
GR  DE
and
H2 
wGR ( z )  wDG ( z )
how to distinguish DG and DE models that both fit
the observed H(z)?
they predict different rates of growth of structure
structure formation is suppressed by acceleration in
different ways in GR and modified gravity:
* in GR – because DE dominates over matter
* in MG – because gravity weakens
Geff  G
δ/a
Geff  G
  2 H  4Geff  
eg f ( R)
DE : Geff  G
MG : Geff  G   increases
Geff  G   decreases
(G determined
by local physics)
Geff  G
Geff  G
eg DGP
Geff  G
Distinguish
DE from MG
via growth of
structure
DE + MG models
LCDM
DE and MG with
the same H(z)
MG model (modification to GR)
DE model (GR)
LCDM
rates of growth of
structure differ
f 
d ln 
d ln a
f
(bias evolution?)
Y Wang
L Guzzo et al
CMB photons carry the signature of the
effect of DE or MG on structure formation
integrated Sachs-Wolfe effect
R Caldwell
Lensing also carries a signature
of the effect of DE or MG
complication: linear to
nonlinear transition
(need N-body simulations)
f(R) gravity
simplest scalar-tensor gravity:
Lgrav,GR  R  Lgrav  f ( R)
implies a new light scalar degree of freedom in gravity
4
eg.
at low energy,
f ( R)  R 
,  ~ H0
R
1/R dominates
This produces late-time self-acceleration
 but the light scalar strongly violates solar system/
binary pulsar constraints
 all f(R) models have this problem
 Possible way out: ‘chameleon’ mechanism,
i.e. the scalar becomes massive in the solar system
- too contrived?
Modified gravity from braneworlds?
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new massive graviton modes
new effects from higher-D fields and other branes
perhaps these could dominate at low energies
different possibilities
our brane
* ‘bulk’ fields as effective
DE on the brane
extra dimension
(eg ekpyrotic/ cyclic)
shadow
brane
* effective 4D gravity on
the brane modified on
large scales
(eg DGP)
gravity
matter
+ dilaton,
form fields…
DGP – the simplest example
Friedman on the
brane
H 8G
H 


rc
3
2
1
late time :   0  H 
rc
early time : H  rc
1
8G
H 

3
2
early universe – recover GR dynamics
late universe – acceleration without DE
gravity “leaks” off the brane
therefore gravity on the brane weakens
passes the solar system test: DGP
GR
The background is very simple – like LCDM
… too good to be true
analysis of higher-D perturbations shows
- there is a ghost in the scalar sector of the
gravitational field
This ghost is from higher-D gravity
* It is not apparent in the background
* It is the source of suppressed growth
  2 H  4Geff (t ) 
Geff  GBrans-Dicke with   0
The ghost makes the quantum vacuum unstable
Can DGP survive as a classical toy model?
The simplest models fail
f(R) and DGP – simplest in their class
– simplest modified gravity models
 both fail because of their scalar degree of freedom:
f(R) strongly violates solar system constraints
DGP has a ghost in higher-D gravity
Either GR is the correct theory on large scales
Or
Modified gravity is more complicated

THEORY: find a ghost-free generalized DGP or
find a ‘non-ugly’ f(R) model – or find
a new MG model?
PHENOMENOLOGY: model-independent tests
of the failure of GR ?
structure formation
2
ds  (1  2)dt  (1  2)a dx
2
2
2
k2
Poisson equation
  4Geff  
2
a
k2
d


(a m )
Euler equation
2
a
dt
k2
8
(



)

Geff   dark stress constraint
2
a
3
GR: Geff  G,  dark  0
MG: modified gravity strength + ‘dark’ anisotropic stress
examples
DGP : Geff  G,  dark  
f(R) : Geff  G,  dark 
f RR
fR
Testing for MG
In principle:
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
Total density perturbation gives
Geff
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Galaxy velocities give 

   (   )ds

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Lensing gives   

Then    determines  dark
We can also derive a consistency test for GR vs MG:


k 2 (  )  a m  H m
 ,
2
4Ga 
GR :   1, MG :   1
Song & Koyama
MG versus Coupled DE?
Coupled DE in GR introduces complications
MG: all fields feel modified gravity equally, so
equivalence principle is not violated
Coupled DE: CDM breaks EP because of the coupling

Poisson equation is the same

But Euler equation
is modified

baryons
CDM
d
( a b )  k 2 
dt
d
(a b )  k 2  (1   )
dt
This can be detected in principle via peculiar
velocities
some conclusions
observations imply acceleration
theory did not predict it – and cannot yet explain it
GR with dynamical DE – no natural model
modifications to GR – theory gives no natural model
simplest models fail [f(R), DGP]
 Observations cannot ‘find’ a theory
 Too many models to test each one
 Need model-independent approaches
 key questions:
1. is Λ the dark energy?
2. if not, is it GR+dynamical DE – or Dark Gravity?
 In principle: expansion history + structure formation
test can answer 1+2
 As a by-product – we understand GR and gravity
better
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