Dark Matter and Energy
Rocky I: Dark Matter
Rocky II: Dark Energy
ISAPP
Rocky Kolb
September 2010
The University of Chicago
Radiation:
0.005%
Chemical Elements:
(other than H & He) 0.025%



Neutrinos:
0.17%
Stars:
0.8%
LCDM
H & He:
gas 4%
Cold Dark Matter:
(CDM) 25%
Dark Energy (L):
70%
Cosmological Constant (Dark Energy)
1917 Einstein proposed
cosmological constant, L.
1929 Hubble discovered
expansion of the Universe.
1934 Einstein called it
“my biggest blunder.”
1998 Astronomers found
evidence for it, and renamed
it “Dark Energy.”
Cosmological Constant (Dark Energy)
Einstein’s Equations: R  1 g  R  Lg   8 GT
2
Equation of State:
T   g p     p U U
Conservation of stress-energy: T  ;  0    a 31 w
1. If p    (w  1) then T  g  and L
8 G L
Cosmological constant behaves like fluid with w  1
2. Vacuum energy unchanged in expansion   a 0
Vacuum energy behaves like fluid with w  1
L
8 G L
p  w
The
TheCosmoillogical
Cosmological Constant
Constant
10–30 g cm3
So small, and yet not zero!
The Unbearable Lightness of Nothing
The Cosmoillogical Constant
Dark (and Useless) Energy
1 MeV liter1
The Cosmoillogical Constant
Illogical magnitude (what’s it related to?):
L 10 g cm
30
L  8 G L
-3
10
10
29
4
cm 
eV
2
 10
4
10
33
3
cm
eV 
2

4
The Cosmoillogical Constant
All fields: harmonic oscillators with zero-point energy
classical
E 0
quantum
E
1
2

The Cosmoillogical Constant
All fields: harmonic oscillators with zero-point energy
Photons: Lamb shift
Gravitons: Vacuum energy
e-
eg
g
e+
e+


all particles
LC
LC
LC
LC
  d 3k k 2  m2
 :
 M Pl :
 M SUSY :
 104 eV:
L   4
 L  M Pl4
4
 L  M SUSY
 L  Observed

all particles
LC
  dk k 3
 bad prediction
 1090 g cm 3
 1030 g cm 3
 1030 g cm 3
The Cosmoillogical Constant
Vf
hightemperature
lowtemperature
DV  L
f
GUT: 1074 g cm3
SUSY: 1030 g cm3
EWK: 1024 g cm3
CHIRAL: 1013 g cm 3
OBSERVED: 1030 g cm3
The Cosmoillogical Constant
The Cosmoillogical Constant
Illogical magnitude (what’s it related to?):
L 10 g cm
30
L  8 G L
10
-3
10
29
4
cm 
eV
2
 10
4
10
33
3
cm
eV 

4
2
Illogical timing (cosmic coincidence?):
M   R
GUT
EWK BBN
L
REC
TODAY
The Cosmoillogical Constant
Global warming, but universal cooling:
The Universe is cold and dark….and getting colder and darker!
(Dark Energy is now 700,000 ppm and will only increase!)
Cosmoillogical Constant (Dark Energy)
Do not directly observe
• acceleration of the universe
• dark energy
We infer acceleration/dark energy by comparing
observations
with the predictions of a
model
All evidence for dark energy/acceleration comes
from measuring the expansion history of the Universe
Edwin
Hubble
University of Chicago
1909 National Champions
s
Hubble’s Discovery Paper - 1929
v  H 0d
H 0  Hubble' s constant
Riess et al.
Expansion History of the Universe
distance: D  a
(cosmic scale factor)
velocity: a  H
(Hubble’s constant
deceleration
a0
  3p  0
time
scale factor a
scale factor a
acceleration: a   G (  3p)
(acceleration)
acceleration
a0
  3p  0
L: p   
time
apparent brightness of standard candle
Hubble Diagram
distant universe
past velocity
acceleration
1998–today
1929–1998
nearby universe
present velocity
H0
redshift of spectral lines
Expansion History of the Universe
Friedmann-Robertson-Walker metric
2

dr
2
2
2
2
2
2
2
2
ds  dt  a  t  
 r d  r sin  df 
2
1  kr

a(t) = cosmic scale factor
k  1, 0  spatial curvature constant: 3R  6k/a2(t)
Friedmann equation (G00  8 GT00)
2
k
a
3    3 2  8 G 
a
a
Expansion History of the Universe
Friedmann equation (G00  8 GT00)
k 8 G
H  2 

a
3
2
 i  t0 
i 
C
3H 02
critical density: C 
8 G
redshift: 1  z 
a0
a t 
Expansion History of the Universe
Friedmann equation (G00  8 GT00)
Hubble
constant
H
2
 z
 H
2
0
curvature
matter
radiation
2
3
4

 k 1  z   M 1  z   R 1  z  


• k   M  R  1
2
3
4
H 2  z   H 02  1  M  R 1  z   M 1  z   R 1  z  


• radiation contribution (R) small for z  103
H
2
 z
 H
2
0
2
3

 1  M 1  z   M 1  z  


• “All of observational cosmology is a search for two numbers.”
(H0 and M) — Sandage, Physics Today, 1970
Expansion History of the Universe
Many observables based on H(z) [ or  dz H1(z) ]
•
Luminosity distance
Flux = (Luminosity / 4 dL2)
•
Angular diameter distance
a  Physical size / dA
•
Volume (number counts)
N / V 1(z)
•
Age of the universe
•
Distances
Precision Cosmology
"How helpful to us is astronomy's pedantic accuracy,
which I used to secretly ridicule!"
Einstein’s statement to Arnold Sommerfeld on
December 9, 1915 (regarding measurements
of the advance of the perihelion of Mercury)
2.0
Hubble Diagram
Astier et al. (2006)
SNLS
1.5
L
1. Find standard candle (SNe Ia)
2. Observe magnitude & redshift
3. Assume a cosmological model
4. Compare observations & model
1.0
0.5
0
0
0.5
M
1.0
Einstein–de Sitter model
Expansion History of the Universe
Friedmann equation (G00  8 GT00)
Hubble cosmological
constant constant curvature
H
2
 z
 H
2
0
matter
radiation
0
2
3
4

 L 1  z   k 1  z   M 1  z   R 1  z  


• [Could add walls ( 1 z )1]
• 1  L  k   M   R
• radiation contribution (R) small for z  103
• k well determined (close to zero) from CMB
• M reasonably well determined
Expansion History of the Universe
Friedmann equation (G00  8 GT00)
dark
energy
curvature
matter
radiation
31 w
2
3
4
H 2  z   H 02   w 1  z 
  k 1  z    M 1  z    R 1  z  


Equation of state parameter: w  p /  w  1 for L
if w  w(z):
31 w
1  z 
 z dz

 exp  3
1  w  z   
 0 z

parameterize: w(z)  w0  wa z / (1  z)
Cosmology is a search for two numbers (w0 and wa).
The Cosmoillogical Constant
confusing astronomical notation
related to supernova brightness
Einstein-de Sitter:
spatially flat, k 1,
matter-dominated model
(maximum theoretical bliss)
Astier et al. (2006)
SNLS
LCDM
supernova redshift z
The case for L:
1) Hubble diagram (SNe)
5) Galaxy clusters
6) Age of the universe
2) Cosmic Subtraction
3) Baryon acoustic oscillations 7) Structure formation
4) Weak lensing
The Cosmoillogical Constant
dynamics
lensing
x-ray gas
simulations
cmb
power
spectrum
TOTAL  1
CMB
M ~ 0.3
many methods
1.0  0.3  0.7  0
How We “Know” Dark Energy Exists
• Assume model cosmology:
– Friedmann-Lemaître-Robertson-Walker (FLRW) model
Friedmann equation: H2  8 G / 3  k/a2
– Energy (and pressure) content:   M  R  L + 
– Input or integrate over cosmological parameters: H0, B, etc.
• Calculate observables dL(z), dA(z), H(z), 
• Compare to observations
• Model cosmology fits with L, but not without L
• All evidence for dark energy is indirect : observed H(z) is not
described by H(z) calculated from the Einstein-de Sitter model
[spatially flat (from CMB) ; matter dominated (  M)]
Taking Sides!
• Can’t hide from the data – LCDM too good to ignore
– SNe
– Subtraction: 1.0  0.3  0.7
H(z) not given by
– Baryon acoustic oscillations
Einstein–de Sitter
– Galaxy clusters
– Weak lensing
–…
G00 (FLRW)  8 G T00(matter)
•
Modify right-hand side of Einstein equations (DT00)
1. Constant (“just” a cosmoillogical constant)
2. Not constant (dynamics described by a scalar field)
•
Modify left-hand side of Einstein equations
(DG00)
3. Beyond Einstein (non-GR)
4. (Just) Einstein (back reaction of inhomogeneities)
Tools to Modify the Right-Hand Side
1964 Austin-Healey Sprite
1974 Fiat 128
Tools to Modify the Right-Hand Side
scalar fields
(quintessence)
Duct Tape
anthropic principle
(the landscape)
Anthropic/Landscape/DUCTtape
• Many sources of vacuum energy
• String theory has many (10500 ?) vacua
• Some of them correspond to cancellations that yield a small L
• Although exponentially uncommon, they are preferred because …
• More common values of L results in an inhospitable universe
Quintessence/WD–40
• Many possible contributions.
• Why then is total so small?
• Perhaps unknown dynamics sets global
vacuum energy equal to zero……but we’re not there yet!
V(f)
Requires mf  1033 eV
L
0
f
Tools to Modify the Left-Hand Side
• Braneworld modifies Friedmann equation
Binetruy, Deffayet, Langlois
• Gravitational force law modified at large distance
Five-dimensional at cosmic distances
• Tired gravitons
Deffayet, Dvali
& Gabadadze
Gregory, Rubakov & Sibiryakov;
Dvali, Gabadadze & Porrati
Gravitons metastable - leak into bulk
• Gravity repulsive at distance R  Gpc
Csaki, Erlich, Hollowood & Terning
• n = 1 KK graviton mode very light, m  (Gpc)1
Kogan, Mouslopoulos,
Papazoglou, Ross & Santiago
• Einstein & Hilbert got it wrong
S  16 G 
1
d
4
f (R)
x g  R   R 
4
• “Backreaction” of inhomogeneities
Carroll, Duvvuri, Turner, Trodden
Räsänen; Kolb, Matarrese, Notari & Riotto;
Notari; Kolb, Matarrese & Riotto
Backreaction of Inhomogeneities
Homogeneous model
Inhomogeneous model
h
i  x 
a  Vh
a  Vi
3
h
H h  ah ah
3
i
H i  ai ai
 h  i  x   H h  H i ?
We think not!
(Buchert & Ellis)
Backreaction of Inhomogeneities
G (g)  G ( g)
Inhomogeneities–Example
Kolb, Matarrese, Notari & Riotto
• Perturbed Friedmann–Lemaître–Robertson–Walker model:
FLRW
G  x , t   G
 t    G  x , t 
G00FLRW  t    G00  x , t   8 GT00  x , t 
3
 a  8 G 





G
00 
 

a
3
8

G
 


2
. 2
• (a/a) is not 8 G /3
.
• (a/a is not even the expansion rate)
• Could  G00 be large, or is it 1010?
• Could  G00 play the role of dark energy?
Backreaction of Inhomogeneities
• The expansion rate of an inhomogeneous universe of average
density  need NOT be! the same as the expansion rate of a
homogeneous universe of average density  !
Ellis, Barausse, Buchert
• Difference is a new term that enters an effective Friedmann
equation — the new term need not satisfy energy conditions!
• We deduce dark energy because we are comparing to the wrong
model universe.
Célérier; Räsänen; Kolb, Matarrese, Notari & Riotto; Schwarz, …
Backreaction of Inhomogeneities
• Most conservative approach — nothing new
– no new fields (like 1033 eV mass scalars)
– no extra long-range forces
– no modification of general relativity
– no modification of gravity at large distances
– no Lorentz violation
– no extra dimensions, bulks, branes, etc.
– no anthropic/landscape/faith-based reasoning
• Magnitude?: calculable from observables related to  /
• Why now?: acceleration triggered by era of non-linear structure
• Possible attractor for effective L
Backreaction of Inhomogeneities
LCDM is the correct phenomenological model, but …
… there is no dark energy, gravity is not modified,
and the universe is not accelerating (in the usual sense).
Backreaction Causes Allergic Reaction
Acceleration From Inhomogeneities
• View scale factor as zero-momentum mode of gravitational field
• In homogeneous/isotropic model it is the only degree of freedom
• Inhomogeneities: non-zero modes of gravitational field
• Non-zero modes interact with and modify zero-momentum mode
Cosmology  scalar field theory analogue
cosmology
zero-mode
a
scalar-field theory
f  (vev of a scalar field)
non-zero modes
inhomogeneities
thermal/finite-density bkgd.
physical effect
modify a(t)
e.g., acceleration
modify f (t)
e.g., phase transitions
Acceleration From Inhomogeneities
Standard approach
• Model an inhomogeneous
Universe as a homogeneous
Universe model with   
Our approach
• Expansion rate of
inhomogeneous Universe 
expansion rate of homogeneous
• a(t) / V1/3 is the zeromode of
a homogeneous model
with   
Universe with   
• Inhomogeneities modify
zeromode [effective scale
factor is aD  VD1/3 ]
• Inhomogeneities only have a
local effect on observables
• Effective scale factor has a
(global) effect on observables
• Cannot account for observed
acceleration
• Potentially can account for
acceleration without
dark energy or modified GR
Subtraction
i  i/C
dynamics
C  3H02/ 8G
lensing
x-ray gas
simulations
cmb
power
spectrum
TOTAL  1 (CMB)
M  0.3
1  0.3  0.7
TOTAL  M  L
Subtraction
How can 1.0 = 0.3?
For a spatially flat FLRW universe H 2  8 G / 3
This is another way of stating   1.
This expression is not valid if FLRW is not valid
e.g., H 2 
8 G 
3




G
00 
3 
8 G

Lemaître–Tolman–Bondi
Célérier
Iguchi, Nakamura, Nakao
Moffat
Nambu and Tanimoto
Mansouri
Chang, Gu, Hwang
Alnes, Amarzguioui, Grøn
Mansouri
Apostolopoulos, Brouzakis, Tetradis, Tzavara
Garfinkle
Kai, Kozaki, Nakao, Nambu, Yoo
Marra, Kolb, Matarrese, Riotto
Mustapha, Hellaby, Ellis
Iguchi, Nakamura, Nakao
Vanderveld, Flanagan, Wasserman
Enqvist and Mattsson
Biswas, Mansouri, Notari
Marra, Kolb, Matarrese
Marra
Brouzakis, Tetradis, Tzavara
Biswas and Notari
Brouzakis and Tetradis
Alnes and Amarzguioui
Garcia-Bellido and Haugboelle
• Advantages:
– Solvable inhomogeneous model
– Can describe wide variety of
dynamics
• Disadvantages:
– Can’t encompass strong (volume)
backreaction (spherical symmetry)
– Generically have small dynamical
range before shell crossing
Lemaître–Tolman–Bondi
2

R
r, t  2

2
2
ds  dt 
dr  R 2  r , t  d 2
1  r 
Spherically symmetric metric
  d dr
 d dt
H 2  R R
Expansion rates
Spherically symmetric density
8 G   r , t  
R  r , t   ra  t 
FRW
R  r , t   a  t 
  r   kr 2
a  r   H 02  M r 3
H r2  R2 R2
a   r, t 
R 2  r , t  R  r , t 
Lemaître–Tolman–Bondi
• Spherical model
• Overall Einstein–de Sitter
• Inner underdense Gpc region
• Calculate dL(z)
Alnes at al.
• Compare to SNe data
• Fit with L  0!
• No local acceleration
(counterexample to no-go theorems)
Lemaître–Tolman–Bondi
• Possible to produce model with EXACT dL(z) and  (z) of LCDM
Célérier, et al. 2009
• Slight local overdensity.
Backgrounds and Backreactions
Can write ds2   (12) dt 2  a 2 (t) (12) dx 2 , but not with a(t)
from the underlying EdS model, but a(t) from a LCDM model.
How?
Give some thought to what is meant by a background solution.
Kolb, Marra, Matarrese
Backgrounds and Backreactions
Some thoughts on cosmological background solutions
Global Background Solution: FLRW solution generated using
   H, 3R  3RH (sub-H  Hubble volume average),
and the local equation of state (e.o.s.).
Average Background Solution: FLRW solution that describes
volume expansion of our past light cone. Energy content,
curvature, and e.o.s. that generates the ABS need not be  H,
3RH, nor local e.o.s. (Buchert formalism)
Phenomenological Background Solution: FLRW model that
best describes the observations on our light cone. Energy
content, curvature, and e.o.s. that generates the PBS need not be
 H, 3RH, and local e.o.s. (Swiss-cheese example)
Kolb, Marra, Matarrese
Backgrounds and Backreactions
Backreaction: the three backgrounds do not coincide
Strong Backreaction:
Global Background Solution does not describe expansion
history (hence does not describe observations)
(Buchert formalism)
Weak Backreaction:
Global Background Solution describes global expansion,
but Phenomenological Background Solution differs
(Swiss Cheese)
Kolb, Marra, Matarrese
“Backreaction” Causes Allergic Reaction
• We have been driven to consider some remarkable possibilities
– 10500 ground states in the landscape
– Modification of GR in the infrared
– Lorentz violation
– 1033 eV scalar fields
– Extra dimensions
• There should be some effort in rethinking some basic old things
– Is there a global background solution?
– Is LCDM just a phenomenological background solution?
– Could it revolutionize something in the early universe?
• Backreactions can potentially do three remarkable things
– Explain “why now”
– Express dark energy parameters in terms of observables
– Potentially predict L
Taking Sides
The expansion history of the universe is not described by the
Einstein-de Sitter model:
1. Well established: Supernova Ia
2. Circumstantial: subtraction, age, structure formation, …
3. Emergent techniques: baryon acoustic oscillations, clusters, weak lensing
Explanations:
1. Right-Hand Side: Dark energy
• Constant vacuum energy, i.e., a cosmoillogical constant
• Time varying vacuum energy, i.e., quintessence
2. Left-Hand Side
• Modification of GR
• Standard cosmological model (FLRW) not applicable
Phenomenology:
1. Measure evolution of expansion rate: is w  1?
2. Order of magnitude improvement feasible
Backgrounds and Backreactions
FLRW Assumption: global background solution follows from the
cosmological principle
Specify 3RH,  H, & local e.o.s.  Global Background Solution
describes a(t), H(t),
and all other observables
GBS  PBS if large peculiar velocities
Kolb, Marra, Matarrese
Backgrounds and Backreactions
Global Peculiar Velocities: velocities obtained after subtracting
the Hubble flow of the Global Background Solution
Averaged Peculiar Velocities: velocities obtained after
subtracting the Hubble flow of the Averaged Background Solution
Phenomenological Peculiar Velocities: obtained after
subtracting the Hubble flow of the Phenomenological Background
Solution
Background peculiar velocity not measured as a local effect
Kolb, Marra, Matarrese
Backgrounds and Backreactions
Bare Cosmological Principle: universe is homo/iso on
sufficiently large scales  can describe observable universe by a
mean-field description  Average Background Solution exists.
Bare Copernican Principle: no special place in the universe 
every observer can describe the universe by a mean-field
description  a Phenomenological Background Solution exists for
every observer (but not necessarily unique).
Kolb, Marra, Matarrese
Backgrounds and Backreactions
• Global Background Solution follows from
the FLRW assumption.
• Average Background Solution follows from
the Bare Cosmological Principle.
• Phenomenological Background Solution follows from
the Bare Copernican Principle (the success of LCDM).
Kolb, Marra, Matarrese
Dark Energy
"Nothing more can be done by the theorists. In this
matter it is only you, the astronomers, who can perform
a simply invaluable service to theoretical physics."
Einstein in August 1913 to Berlin astronomer
Erwin Freundlich encouraging him to mount
an expedition to measure the deflection of
light by the sun.
Observational Program
H(z)
dL(z)
supernova
dA(z)
clusters
baryon
osc.
strong
lensing
Growth of
structure
clusters
V(z)
weak
lensing
clusters
 k  2 H  k  4 G  k  0
weak
lensing
P(k,z)
Test gravity
solar
system
millimeter
scale
strong
lensing
accelerators
P(k,z)
source?
DETF* Experimental Strategy:
• Determine as well as possible whether the accelerating
expansion is consistent with being due to a cosmological
constant. (Is w  1?)
• If the acceleration is not due to a cosmological constant, probe
the underlying dynamics by measuring as well as possible the
time evolution of the dark energy. (Determine w(z).)
• Search for a possible failure of general relativity through
comparison of the effect of dark energy on cosmic expansion
with the effect of dark energy on the growth of cosmological
structures like galaxies or galaxy clusters. (Hard to quantify.)
* Dark Energy Task Force
DETF Cosmological Model
Parameterize dark-energy equation of state parameter w as:
w(a)  w0  wa (1  a)
• Today (a  1)
w(1)  w0
• In the far past (a → 0) w(0)  w0  wa
Standard eight-dimensional cosmological model:
w0 :
wa :
DE :
M :
B :
H0 :
z :
nS :
the present value of the dark-energy eos parameter
the rate of change of the dark-energy eos parameter
the present dark-energy density
the present matter density
the present baryon density
the Hubble constant
amplitude of rms primordial curvature fluctuations
the spectral index of primordial perturbations.
w(a)  w0 + wa(1a)
w0  present value
wa  early value
DETF figure of merit:
(area)1 of the error ellipse
0
wa
1
w0
Supernova Type Ia
• Measure redshift and intensity as function of time (light curve)
• Systematics (dust, evolution, intrinsic luminosity dispersion, etc.)
• A lot of information per supernova
• Well developed and practiced
• Present procedure:
– Discover SNe by wide-area survey (the “easy” part)
– Follow up with spectroscopy (the “hard” part)
(requires a lot of time on 8m-class telescopes)
– Photometric redshifts?
Photometric Redshifts
Traditional redshift
from spectroscopy
4000
5000
6000
7000
Photometric redshift
from multicolor
photometry
8000
Baryon Acoustic Oscillations
Post-recombination
• universe neutral
• photons travel freely
(decoupled from baryons)
• perturbations grow
(structure formation)
recombination
ionized
z » 1100
t » 380,000 yr
T ~ 3000 K
Time
neutral
Today
Big Bang
Pre-recombination
• universe ionized
• photons provide enormous
pressure and restoring force
• perturbations oscillate
(acoustic waves)
Eisenstein
Eisenstein
Baryon Acoustic Oscillations
• Each overdense region is an
overpressure that launches a
spherical sound wave
• Wave travels outward at c / 3
• Photons decouple, travel to us
and observable as CMB
acoustic peaks
WMAP
• Sound speed plummets,
wave stalls
• Total distance traveled 150 Mpc
imprinted on power spectrum
SDSS
Baryon Acoustic Oscillations
• Acoustic oscillation scale depends on M h2 and B h2
(set by CMB acoustic oscillations)
• It is a small effect (B h2 >> M h2)
• Dark energy enters through dA and H
Baryon Acoustic Oscillations
• Virtues
– Pure geometry.
– Systematic effects should be small.
• Problems:
– Amplitude small, require large scales, huge volumes
– Photometric redshifts?
– Nonlinear effects at small z, cleaner at large z ~ 23, but …
dark energy is not expected to be important at large z
Weak Lensing

observe
deflection
angle
b
4GM DLS
 
b DOS
dark energy
affects growth
rate of M
dark energy
affects geometric
distance factors
Weak Lensing
The signal from any single galaxy is very small,
but there are a lot of galaxies! Require photo-z’s?
Space vs. Ground:
• Space: no atmosphere PSF
• DES (2012)
– 1000’s of sq. degs.
deep multicolor data
• Space: Near IR for photo-z’s
• Ground: larger aperture
• LSST (2015)
– full hemisphere,
very deep 6 colors
• Ground: less expensive
• JDEM/Euclid (???)
Galaxy Clusters
Cluster redshift surveys measure
• cluster mass, redshift, and spatial clustering
Sensitivity to dark energy
• volume-redshift relation
• angular-diameter distance–redshift relation
• growth rate of structure
• amplitude of clustering
Problems:
• cluster selection must be well understood
• proxy for mass?
• need photo-z’s
Systematics Are The Key
The Power of Two (or 3, or 4)
Combined
Figure of merit  100
Technique A
Figure of merit  20
Technique Z
Figure of merit  20
Ongoing
Next step
FOM ~ 3× ongoing
Ultimate
FOM ~ 10 × ongoing
My guess of
future
progress
95% C.L.
What’s Ahead
2008
Lensing
2010
CFHTLS SUBARU
DLS SDSS ATLAS KIDS
BAO
FMOS
SDSS ATLAS
SNe
CSP ESSENCE
AMI
DUNE
DES, VISTA
Hyper suprime
Pan-STARRS
LSST
APEX SPT
SKA
JDEM
LAMOST DES, VISTA,VIRUS WFMOS LSST
Hyper suprime
JDEM
Pan-STARRS
SDSS CFHTLS
Clusters
2020
2015
DES
LSST
Pan-STARRS
JDEM
SKA
DES
XCS SZA AMIBA ACT
CMB WMAP 2/3
WMAP 5 yr
Planck
Planck 4yr
Roger Davies
“To me every hour of
the light and dark is
a miracle. Every
cubic inch of space
is a miracle.”
– Walt Whitman
Every cubic inch of
space is a miracle!
• cosmic radiation
• virtual particles
• Higgs potential
• extra dimensions
• dark matter
• dark energy
Radiation:
0.005%
Chemical Elements:
(other than H & He) 0.025%



Neutrinos:
0.17%
Stars:
0.8%
LCDM
H & He:
gas 4%
Cold Dark Matter:
(CDM) 25%
Dark Energy (L):
70%
I must reject fluids and ethers of all kinds, magnetical,
electrical, and universal, to whatever quintessential
thinness they may be treble-distilled and (as it were)
super-substantiated.
Samuel Taylor Coleridge
Theory of Life (1816)
Dark Matter and Energy
Rocky I: Dark Matter
Rocky II: Dark Energy
ISAPP
Rocky Kolb
September 2010
The University of Chicago
Dark Energy Task Force Charge
“The DETF is asked to advise the agencies on the optimum† near
and intermediate-term programs to investigate dark energy and, in
cooperation with agency efforts, to advance the justification,
specification and optimization of LST# and JDEM‡.”
1. Summarize existing program of funded projects
2. Summarize proposed and emergent approaches
3. Identify important steps, precursors, R&D, …
4. Identify areas of dark energy parameter space existing or
proposed projects fail to address
5. Prioritize approaches (not projects)
Optimum  minimum (agencies); Optimum  maximal (community)
# LST  Large Survey Telescope
‡ JDEM  Joint Dark Energy Mission
†
Dark Energy Task Force Report
Context
The issue: acceleration of the Universe
Possibilities: dark energy (L or not), non-GR
Motivation for future investigations
Goals and Methodology
Goal of dark energy investigations
Methodology to analyze techniques/implementations
Findings
Techniques & implementations (largely from White Papers)
Systematic uncertainties
What we learned from analysis
Recommendations
A Dark Energy Primer
DETF Fiducial Model and Figure of Merit
Staging Stage IV from the Ground and/or Space
DETF Technique Performance Projections
Dark Energy Projects (Present and Future)
Context
1. Conclusive evidence for acceleration of the Universe.
Standard cosmological framework  dark energy (70% of mass
2. Possibility: Dark Energy constant in space & time (Einstein’s L).
3. Possibility: Dark Energy varies with time (or redshift z or a  (1z
4. Impact of dark energy can be expressed in terms of “equation of
w(a)  p(a) / (a) with w(a)  1 for L.
5. Possibility: GR or standard cosmological model incorrect.
6. Not presently possible to determine the nature of dark energy.
Context
7. Dark energy appears to be the dominant component of the
physical Universe, yet there is no persuasive theoretical
explanation. The acceleration of the Universe is, along with
dark matter, the observed phenomenon which most directly
demonstrates that our fundamental theories of particles and
gravity are either incorrect or incomplete. Most experts
believe that nothing short of a revolution in our understanding
of fundamental physics will be required to achieve a full
understanding of the cosmic acceleration. For these reasons,
the nature of dark energy ranks among the very most
compelling of all outstanding problems in physical science.
These circumstances demand an ambitious observational
program to determine the dark energy properties as well as
Goals and Methodology
1. The goal of dark-energy science is to determine the very
nature of the dark
energy .
Toward this goal, our observational program must:
a. Determine whether the accelerated expansion is due to a
cosmological constant.
b. If it is not due to a constant, probe the underlying dynamics
by
measuring as well as possible the time evolution of dark
energy, for
example by measuring w(a).
c. Search for a possible failure of GR through comparison of
cosmic
expansion with growth of structure.
Because the field is so new, it has
suffered from a lack of standardization
which has made it very difficult to
compare directly different approaches.
To address this problem we have done
our own modeling of the different
techniques so that they could be
compared in a consistent manner.
These quantitative calculations form the
basis of our extensive factual findings, on
which our recommendations are based
Goals and Methodology
4.
Goals of dark-energy observational program through
measurement of
expansion history of Universe [dL(z) , dA(z) , V(z)], and through
measurement of growth rate of structure. All described by w(a).
If failure of
GR, possible difference in w(a) inferred from different types of
data.
5.
To quantify progress in measuring properties of dark energy
we define
dark energy figure-of-merit from combination of uncertainties
in w0 and wa.
The DETF figure-of-merit is the reciprocal of the area
of the
error ellipse enclosing the 95% confidence limit in the
Goals and Methodology
7.
We made extensive use of statistical (Fisher-matrix)
techniques
incorporating CMB and H0 information to predict future
performance
(75 models).
8. Our considerations follow developments in Stages:
I. What is known now (1/1/06).
II. Anticipated state upon completion of ongoing projects.
III. Near-term, medium-cost, currently proposed projects.
IV. Large-Survey Telescope (LST) and/or Square Kilometer
Array (SKA),
and/or Joint Dark Energy (Space) Mission (JDEM).
Eighteen Findings
1.
2.
Four observational techniques dominate White Papers:
a. Baryon Acoustic Oscillations (BAO) large-scale surveys
measure features in distribution of galaxies. BAO: dA(z)
and H(z).
b. Cluster (CL) surveys measure spatial distribution of
galaxy clusters. CL: dA(z), H(z), growth of structure.
c. Supernovae (SN) surveys measure flux and redshift of
Type Ia SNe. SN: dL(z).
d. Weak Lensing (WL) surveys measure distortion of
background images due to gravitational lensing. WL:
dA(z), growth of structure.
Different techniques have different strengths and weaknesses
and sensitive in different ways to dark energy and other
cosmo. parameters.
Eighteen Findings
4.
Four techniques at different levels of maturity:
a. BAO only recently established. Less affected by
astrophysical uncertainties than other techniques.
b. CL least developed. Eventual accuracy very difficult to
predict. Application to the study of dark energy would have
to be built upon a strong case that systematics due to
non-linear astrophysical processes are under control.
c. SN presently most powerful and best proven technique. If
photo-z’s are used, the power of the supernova technique
depends critically on accuracy achieved for photo-z’s. If
spectroscopically measured redshifts are used, the power
as reflected in the figure-of-merit is much better known,
with the outcome depending on the ultimate systematic
uncertainties.
d. WL also emerging technique. Eventual accuracy will be
Systematics: none, optimistic, pessimistic
Eighteen Findings
5.
A program that includes multiple techniques at Stage IV can
provide more than an order-of-magnitude increase in our
figure-of-merit. This would be a major advance in our
understanding of dark energy.
6.
No single technique is sufficiently powerful and well
established that it is guaranteed to address the order-ofmagnitude increase in our figure-of-merit alone.
Combinations of the principal techniques have substantially
more statistical power, much more ability to discriminate
among dark energy models, and more robustness to
systematic errors than any single technique. Also, the case
for multiple techniques is supported by the critical need for
confirmation of results from any single method.
Co
m
bi
Technique #2
na
tio
n
Technique #1
Eighteen Findings
7.
Results on structure growth, obtainable from weak lensing or
cluster observations, are essential program components in
order to check for a possible failure of general relativity.
8. In our modeling we assume constraints on H0 from current
data and constraints on other cosmological parameters
expected to come from measurement of CMB temperature
and polarization anisotropies.
a. These data, though insensitive to w(a) on their own,
contribute to our knowledge of w(a) when combined with
any of the dark energy techniques we have considered.
b. Increased precision in a particular cosmological parameter
9. Improvements
in dark
cosmological
parameters
tend
not to
may improve
energy constraints
from
a single
improve
knowledge
of dark
energy from
a multi-technique
technique,
valuable
for comparing
independent
methods.
10. program
Setting spatial curvature to zero greatly helps SN, but modest
impact on other techniques. Little difference when in
Eighteen Findings
s (H 0): 8 km s 1 Mpc1  4 km s1 Mpc1
k  0 prior
Eighteen Findings
12. Our inability to forecast reliably systematic error levels is the
biggest impediment to judging the future capabilities of the
techniques. We need
a.
b.
c.
d.
BAO– Theoretical investigations of how far into the non-linear regime the data
can be modeled with sufficient reliability and further understanding of galaxy
bias on the galaxy power spectrum.
CL– Combined lensing and Sunyaev-Zeldovich and/or X-ray observations of
large numbers of galaxy clusters to constrain the relationship between galaxy
cluster mass and observables.
SN– Detailed spectroscopic and photometric observations of about 500
nearby supernovae to study the variety of peak explosion magnitudes and
any associated observational signatures of effects of evolution, metallicity, or
reddening, as well as improvements in the system of photometric calibrations.
WL– Spectroscopic observations and multi-band imaging of tens to hundreds
of thousands of galaxies out to high redshifts and faint magnitudes in order to
calibrate the photometric redshift technique and understand its limitations. It
is also necessary to establish how well corrections can be made for the
intrinsic shapes and alignments of galaxies, removal of the effects of optics
(and from the ground) the atmosphere and to characterize the anisotropies in
Eighteen Findings
13. Six types of Stage III projects have been considered:
a. a BAO survey on an 8-m class telescope using
spectroscopy
b. a BAO survey on an 4-m class telescope using photo-z’s
c. a CL survey on an 4-m class telescope using photo-z’s for
clusters detected in ground-based SZ surveys
d. a SN survey on a 4-m class telescope using spectroscopy
from a 8-m class telescope
e. a SN survey on a 4-m class telescope using photo-z’s
f. A WL survey on a 4-m class telescope using photo-z’s
a.
These projects are typically projected by proponents to cost
in the range of 10s of $M.
Eighteen Findings
14. Our findings regarding Stage-III projects are
1. Only an incremental increase in knowledge of dark-energy
parameters is likely to result from a Stage-III BAO project
using photo-z’s. The primary benefit would be in exploring
photo-z uncertainties.
2. A modest increase in knowledge of dark-energy
parameters is likely to result from Stage-III SN project
using photo-z’s. Such a survey would be valuable if it
were to establish the viability of photometric determination
of supernova redshifts, types, and evolutionary effects.
3. A modest increase in knowledge of dark-energy
parameters is likely to result from any single Stage-III CL,
WL, spectroscopic BAO, or spectroscopic SN survey.
4. The SN, CL, or WL techniques could, individually, produce
factor-of-two improvements in the DETF figure-of-merit, if
DETF Projections
Combination of all techniques from a Stage-III photometric survey
Stage II
Stage III-p
Stage III-o
Eighteen Findings
15. Four types of next-generation (Stage IV) projects have been
considered:
a. an optical Large Survey Telescope (LST), using one or
more of the four techniques
b. an optical/NIR JDEM satellite, using one or more of four
techniques
c. an x-ray JDEM satellite, which would study dark energy by
the cluster technique
d. a Square Kilometer Array, which could probe dark energy
by weak lensing and/or the BAO technique through a
hemisphere-scale survey of 21-cm emission
Each of these projects is in the $0.3-1B range, but dark
energy is not the only (in some cases not even the primary)
science that would be done by these projects. According to
the White Papers received by the Task Force, the technical
Eighteen Findings
17. The Stage IV experiments have different risk profiles:
a. SKA would likely have very low systematic errors, but
needs technical advances to reduce its cost. The
performance of SKA would depend on the number of
galaxies it could detect, which is uncertain.
b. Optical/NIR JDEM can mitigate systematics because it will
likely obtain a wider spectrum of diagnostic data for SN,
CL, and WL than possible from ground, incurring the usual
risks of a space mission.
c. LST would have higher systematic-error risk, but can in
many respects match the statistical power of JDEM if
systematic errors, especially those due to photo-z
measurements, are small. An LST Stage IV program can
be effective only if photo-z uncertainties on very large
samples of galaxies can be made smaller than what has
Eighteen Findings
18. A mix of techniques is essential for a fully effective Stage IV
program. The technique mix may be comprised of elements
of a ground-based program, or elements of a space-based
program, or a combination of elements from ground- and
space-based programs. No unique mix of techniques is
optimal (aside from doing them all), but the absence of weak
lensing would be the most damaging provided this technique
proves as effective as projections suggest.
DETF Projections
Combination of all techniques from Stage-IV ground-based survey
Stage II
Stage IV-p
Stage IV-o
DETF Projections
Combination of all techniques from Stage-IV space-based survey
Stage II
Stage IV-p
Stage IV-o
DETF Projections
Six Recommendations
I.
We strongly recommend that there be an aggressive
program to explore dark energy as fully as possible,
since it challenges our understanding of fundamental
physical laws and the nature of the cosmos.
II.
We recommend that the dark energy program have
multiple techniques at every stage, at least one of which
is a probe sensitive to the growth of cosmological
structure in the form of galaxies and clusters of galaxies.
III. We recommend that the dark energy program include a
combination of techniques from one or more Stage III
projects designed to achieve, in combination, at least a
factor of three gain over Stage II in the DETF figure-ofmerit, based on critical appraisals of likely statistical and
Six Recommendations
IV. We recommend that the dark energy program include a
combination of techniques from one or more Stage IV
projects designed to achieve, in combination, at least a
factor of ten gain over Stage II in the DETF figure-ofmerit, based on critical appraisals of likely statistical and
systematic uncertainties. Because JDEM, LST, and SKA
all offer promising avenues to greatly improved
understanding of dark energy, we recommend continued
research and development investments to optimize the
programs and to address remaining technical questions
and systematic-error risks.
V.
We recommend that high priority for near-term funding
should be given as well to projects that will improve our
understanding of the dominant systematic effects in dark
Six Recommendations
VI. We recommend that the community and the funding
agencies develop a coherent program of experiments
designed to meet the goals and criteria set out in these
recommendations.
DETF Legacy
I.
II.
Standardization
1. Parameterize dark energy as w0 – wa
2. Eight-parameter cosmological model
3. Priors
4. Figure of merit
Importance of combinations
We have a website with library of Fisher matrices &
combiner programs (All power to the people!)
III. DETF Technique Performance Projections
1. Thirty-two (count ‘em, 32!) data models
2. Optimistic & pessimistic projections
3. Four techniques, two stages, five platforms
IV. Use DETF Technique Performance Projections as a
FoMSWG
JOINT DARK ENERGY MISSION
SCIENCE WORKING GROUP
STATEMENT OF TASK
June 2008
The purpose of this SWG is to continue the work of the
Dark Energy Task Force in developing a quantitative
measure of the power of any given experiment to advance
our knowledge about the nature of dark energy. The
measure may be in the form of a “Figure of Merit” (FoM) or
an alternative formulation.
DETF FoM
 DE   DE (today) exp {3[1  w (a) ] d ln a}
LCDM:
w (a)  1
w(a) = w0  wa(1  a)
w  w0 today & w  w0  wa in the far past
w0 Marginalize over all other parameters and find uncertainties in
LCDM value
1
DETF FoM  (area of ellipse)1
errors in w0 and wa
are correlated
0
wa
FoMSWG
From DETF:
The figure of merit is a quantitative guide; since the nature of dark
energy is poorly understood, no single figure of merit is
appropriate for every eventuality.
FoMSWG emphasis!
FoMSWG
FoMSWG (like DETF) adopted a Fisher (Information) Matrix
approach toward assessing advances in dark energy science.
1 f f
observe b quantities
ij   2 bi bj
observable b function of parameter pi : f (pi) b s b p p
FoMSWG
1. Pick a fiducial cosmological model.
Not much controversy: LCDM [assumes Einstein gravity (GR)].
1. ns
scalar spectral index
2. M
present matter density M h2
3. B
present baryon density
4. k
present curvature density
5. DE
present dark energy density
6. D g
departure of growth from GR prediction caused by dar
7. DM
SNe absolute magnitude
8. G0
departure of growth during linear era (unity if GR)
9. ln D2S(k*) primordial curvature perturbation amplitude
FoMSWG
2. Specify cosmological parameters of fiducial cosmological
model (including parameterization of dark energy).
Not much controversy in non-dark energy parameters (we use
WMAP5).
Parameterize dark energy as a function of redshift or scale factor
FoMSWG
2. Specify cosmological parameters of fiducial cosmological
model (including parameterization of dark energy).
Issue #1: parameterization of w(a)
(want to know a function—but can only measure
parameters)
• DETF: w(a)  w0  wa(1  a) w  w0 today & w  w0  wa in
the far past
– advantage: (only) two parameters
– disadvantages: can’t capture more complicated behaviors
of w
– FoM based on excluding w  1 (either w0  1 or wa  0)
• FoMSWG: w(a) described by 36 piecewise constant values wi
defined in bins between a  1 and a  0.1
–advantage: can capture more complicated behaviors
–disadvantage: 36 parameters (issue for presentation, not
FoMSWG
2. Specify cosmological parameters of fiducial cosmological
model (including parameterization of dark energy).
Issue #2: parameterization of growth of structure (testing gravity)
• DETF discussed importance of growth of structure, but
offered no measure
• Many (bad) ideas on how to go beyond Einstein gravity—no
community consensus on clean universal parameter to test
for modification of gravity
• FoMSWG made a choice, intended to be representative of
Growth
of Structure  Growth of Structure (GR) + Dg ln M(z)
the trends
Dg : one-parameter measure of
departure from Einstein gravity
FoMSWG
3. For pre-JDEM and for a JDEM, produce “data models”
including systematic errors, priors, nuisance parameters, etc.
• Most time-consuming, uncertain, controversial, and critical
aspect
• Have to predict* “pre-JDEM” (circa 2016) knowledge of
cosmological parameters, dark energy parameters, prior
information, and nuisance parameters
• Have to predict how a JDEM mission will perform
• Depends on systematics that are not yet understood or
We made “best guess” for pre-JDEM
completely quantified
Strongly recommend don’t reopen this can of worms
* Predictions are difficult, particularly about the future
FoMSWG
4. Predict how well JDEM will do in constraining dark energy.
This is what a Fisher matrix was designed to do:
• can easily combine techniques
• tool (blunt instrument?) for optimization and comparison
Technical issues, but fairly straightforward
FoMSWG
5. Quantify this information into a “figure of merit”
Discuss DETF figure of merit
Discuss where FoMSWG differs
DETF FoM
w
DETF FoM  (area of ellipse)1
 [s(wp)s(wa)]1
excluded
sw0
swp)
w
1
excluded
0
1
wp
(w0, wa )  (wp, wa)
zp
errors in wp and wa
are uncorrelated
0
wa
z
FoMSWG
“… no single figure of merit is appropriate …”
… but a couple of graphs and a few numbers can convey a l
I. Determine the effect of dark energy on the expansion
history of the universe by determining w(a), parametrized as
described above (higher priority)
II. Determine the departure of the growth of structure from the
result of the fiducial model to probe dark energy and test
gravity
proposal should be free to argue for their own figure of merit
FoMSWG
I. Determine the effect of dark energy on the expansion
history of the universe by determining w(a), parametrized as
described above (higher priority)
1. Assume growth of structure described by GR
2. Marginalize over all non-w “nuisance” parameters
3. Perform “Principal Component Analysis” of w(a)
4. Then assume simple parameterization w(a) = w0  wa (1  a)
and calculate s(wp), s(wa), and zp
FoMSWG
w0
1
• Generally, errors in different wi
are correlated (like errors in w0 and wa)
wa
0
wa
1
• Expand w(a) in a complete
set of
orthogonal eigenvectors
ei(a)
35
1  w(a)   a iaeii  a(like

with eigenvalues
wp
i 0
and wa)
wp
0
• Have 36 principal components
– Errors s (a i) are uncorrelated
– Rank how well principal components are measured
• Can do this for each technique individually & in combination
FoMSWG
• Graph of principal components as function of z informs on
redshift sensitivity of technique [analogous to z p] (may want first fe
• Desirable to have reasonable redshift coverage
FoMSWG
• Graph of s for various principal components informs on
sensitivity to w  1 [analogous to s(wa) and s(wp)]
• If normalize to pre-JDEM, informs on JDEM improvement over pre
FoMSWG
1. Assume growth of structure described by GR
2. Marginalize over all non-w parameters
3. Perform “Principal Component Analysis” of w(a)
4. Then assume simple parameterization w(a) = w0  wa (1DETF
 a)
analysis
5. Calculate s(wp), s(wa), and zp
FoMSWG
II. Determine the departure of the growth of structure from the
result of the fiducial model to probe dark energy and test
gravity
Calculate fully marginalized s(Dg)
FoMSWG
proposal should be free to argue for their own figure of merit
Different proposals will emphasize different methods, redshift
ranges, and aspects of complementarity with external data. There
is no unique weighting of these differences. Proposers should
have the opportunity to frame their approach quantitatively in a
manner that they think is most compelling for the study of dark
energy. Ultimately, the selection committee or project office will
have to judge these science differences, along with all of the other
factors (cost, risk, etc). The FoMSWG method will supply one
consistent point of comparison for the proposals.
FoMSWG
Judgment on ability of mission to determine departure of Dark Energ
1. Graph of first few principal components
for individual techniques and combination
•
•
Redshift coverage
Complementarity of techniques
2. Graph of how well can measure modes
•
Can easily compare to pre-JDEM
(as good as data models)
•
Relative importance of techniques (trade offs)
3. Three numbers: s(wp), s(wa), and zp
•
Consistency check
FoMSWG
Conclusions:
1. Figure(s) of Merit should not be the sole (or even most
important) criterion
1. Systematics
2. Redshift coverage
3. Departure from w   1 must be convincing!
4. Ability to differentiate “true” dark energy from modified
gravity is important
5. Multiple techniques important
6. Robustness
2. Crucial to have common fiducial model and priors
3. Fisher matrix is the tool of choice
1. FoMSWG (and DETF) put enormous time & effort into data
models
2. Data models can not be constructed with high degree of
“Backreaction” Causes Allergic Reaction
• No compelling argument that backreactions are the answer
– We don’t know necessary or sufficient conditions
– Just because some unrealistic model seems to give SNe
dL(z) doesn’t mean that backreactions are the answer
• No proof that backreactions are not the answer
– Physics is littered with discarded no-go theorems
– Just because some unrealistic model doesn’t give SNe dL(z)
doesn’t mean that backreactions are not the answer
Strong Allergic Reaction
• No-go theorem: local deceleration parameter positive. irrelevant
• Why take spatial average at fixed time ?
• Don’t see it in Newtonian limit.
• Don’t see it in perturbation theory.
light-cone average
not Newtonian
not perturbative
• Even with large non-linear perturbations, can write metric in
perturbed Newtonian form ds 2   (12) dt 2 + a2(t) (12) dx 2
red herring
with   1.
• If this is a large effect one would expect
with respect to which background?
to see large velocities.
Inhomogeneities–Cosmology
• For a general fluid, four velocity u  (1,0)
(local observer comoving with energy flow)
• For irrotational dust, work in synchronous and comoving gauge
ds  dt  hij ( x, t )dx dx
2
2
i
j
• Velocity gradient tensor
Qi j  u i; j  12 hik hkj  Q i j  s i j
(s i j is traceless)
• Q is the volume-expansion factor and s ij is the shear tensor
(shear will have to be small)
• For flat FLRW, hij(t)  a2(t)ij
Q  3H and s ij = 0
What Accelerates?
• No-go theorem: Local deceleration parameter positive:
3Q  Q 

q
 6 s
Q
2
2
2
Hirata & Seljak;
Flanagan;
Giovannini;
Ishibashi & Wald
 2 G    0
• However must coarse-grain over some finite domain:
Q
D



h Q d 3x
D
D
h d 3x
• Evolution and smoothing do not commute:
•
Q

Q

D
D

 Q
 Q
D
D
 Q
2
D
 Q
2
D
 Q
D
Q

D
 Q
D
Buchert & Ellis;
Kolb, Matarrese & Riotto
Can have q  0 but qD  0 (“no-go” goes)
Inhomogeneities and Smoothing
• Define a coarse-grained scale factor:
aD  VD VD 0 
1/3
Kolb, Matarrese & Riotto
astro-ph/0506534;
Buchert & Ellis
VD   d 3 x h
D
• Coarse-grained Hubble rate:
aD 1
HD 
3 Q D
aD
• Effective evolution equations:
aD
4 G

 eff  3 peff 
aD
3
3
 eff
2
 aD  8 G
eff
  
3
 aD 
• Kinematical back reaction:
3 peff
R
QD
D
  D

16 G 16 G
3
R
3QD
D


16 G 16 G

QD  23 Q2
D
 Q
2
D

not
described
by a simple
pw
2 s2
D
Inhomogeneities and Smoothing
• Kinematical back reaction:

QD  23 Q2
• For acceleration:
eff  3 peff
• Integrability condition (GR):
a Q 
6
D
D

D
 Q
2
D

2 s2
D
QD
  D
0
4 G
a
4
D
a
2
D
3
R
D


0
• Acceleration is a pure GR effect:
– curvature vanishes in Newtonian limit
– QD will be exactly a pure boundary term, and small
• Particular solution: 3QD   h3RiD  const.
– i.e., Leff  QD (so QD acts as a cosmological constant)
Any Indication in Perturbation Theory?
Kolb, Notari, Matarrese & Riotto (KNMR)
• 2nd-order perturbation theory in fx) (Newtonian potential):
QH
H
20 2 2
23 4 2

f 
 f  2f mean of  2f  0
9
54
130 2 ,i
4 4

f f,i 
 2f  2f  f ,ij f,ij
27
27

Post-Newtonian

Newtonian
– Each derivative accompanied by conformal time  = 2/aH
– Each factor of  accompanied by factor of c.
– Highest derivative is highest power of  / c : “Newtonian”
– Lower derivative terms / cn : “Post-Newtonian”
– f and its derivatives can be expressed in terms of  /
Any Indication in Perturbation Theory?
• 
• 
2
4
f f
 f f
2
2
1
2
A 2 2
a H
1
A 4 4
a H
2
kH
 dk k T  k 
2
0
kH
 dk k
0
3
T
2
k 
a
10
a0
5
a
10  
 a0 
2
0
– Individual Newtonian terms large, i.e., hr2fr2fi  1
Räsänen
– But total Newtonian term vanishes hr2fr2fi  hf,,ijf,ij i
KNMR
– Post-Newtonian: hrf ¢ rf i  105) huge! (large k2/a2H2)
Any Indication in Perturbation Theory?
DH / a3(2f)n1 (f)2 ~ a3(k/aH)2nf n1
• f  A  2 £ 105
• (aH)2n  a02nH02n (a0 /a)n
• H01  3000h1 Mpc
• (k/aH)2nf n1 » (3£103)2n (k/h Mpc1)2n (2£105)n1
– n  1:
4£103 (k/h Mpc1)2 (a0/a)2
: curvature
– n  2:
6£101 (k/h Mpc1)4 (a0/a)1
:?
– n  3:
9£101 (k/h Mpc1)6 (a0/a)0
:L
• Of course have to include transfer function, integrate over k, etc.
Any Indication in Perturbation Theory?
• First term in gradient expansion (2 spatial derivatives):
3
R
D
 aD2
QD  0  no acceleration
• In general, gradient expansion gives
Notari; Kolb, Matarrese, & Riotto

2n

3
n 3
m
R   rn a
 rn    2n derivatives  f 
D
n 1
mn



2n

n 3
m
QD   qn a
q

2
n
derivatives
f
 
 n 
n2
mn


• Newtonian terms, (2f)n » (k/aH)2nf n, individually are large,
but only appear as surface terms, hence small in total
• Post-Newtonian terms, (f)2n ~ (k/aH)2nf 2n, individually are small,
but do not appear as surface terms
• Dominant term is combination: (2f)n1 (f)2 ~ (k/aH)2nf n+1
Any Indication in Perturbation Theory?
• Lowest-order term to make big contribution is n  3 (6 derivatives)
• Notice n  3 contributes to QD and h3RiD terms / a0, i.e.,
expansion as if driven by a cosmological constant !!!
• But why stop at n = 3 ?????
• We have developed a RG-improved calculation (still inadequate)
Many issues:
• non-perturbative nature
• how are averaged quantities related to observables?
• comparison to observed LSS
• gauge/frame choices
• physical meaning of coarse graining?
Program:
• can inhomogeneities change effective zero mode?
• how does it affect observables?
• can one design an inhomogeneous universe that “accelerates”?
• could it lead to an apparent dark energy?
• can it be reached via evolution from usual initial conditions?
• does it at all resemble our universe?
• large perturbative terms resum to something harmless?
• is perturbation theory relevant?
Thinking Forward about Backreactions
• Can the effect be large for many smaller ( H01) voids?
• Can one large void be compatible with observations?
• Is the spherical symmetry of LTB a bug or a feature?
• Voids caustics, walls, coherent structures not in P(k)!
• Must be able to express backreactions in terms of w(a).
– eventually we must make a prediction!
• If backreactions are important, i.m.o., it must be an effect that is
– non-Newtonian
– non-perturbative
• Someone please solve the “cosmological constant problem.”