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					    Dark Matter and Energy

             Rocky I: Dark Matter

             Rocky II: Dark Energy

ISAPP                              September 2010
Rocky Kolb                The University of Chicago
Radiation:   Chemical Elements:
0.005%       (other than H & He) 0.025%
                      Neutrinos:

LCDM                            H & He:
                                gas 4%

                       Cold Dark Matter:
                       (CDM) 25%

             Dark Energy (L):
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
 Equation of State:     T  g p     p UU

 Conservation of stress-energy: T  ;  0    a31w    p  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

   Vacuum energy behaves like fluid with w  1

                           L    8 G L
   The Cosmological Constant
   The CosmoillogicalConstant
            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?):
                            10               10                  
                                             4                          4
        L 10 g cm
              30     -3          4
                                        eV               3

                      10        cm         10         eV 
                                        2                      2
        L  8 G L         29                     33
     The Cosmoillogical Constant
All fields: harmonic oscillators with zero-point energy

     classical                        quantum
       E 0                           E   1
                                           2   
        The Cosmoillogical Constant
All fields: harmonic oscillators with zero-point energy

    Photons: Lamb shift                                 Gravitons: Vacuum energy
                             e-                                                      e-
g                                                          g
                 e+                                                             e+

                    
                  all particles
                                    d 3k k 2  m2        
                                                       all particles
                                                                         dk k 3

            LC    :                 L   4           bad prediction
            LC    M Pl :              L  M Pl
                                                         10 90 g cm 3
            LC    M SUSY :            L  M SUSY
                                                         10 30 g cm 3
            LC    104 eV:            L  Observed     1030 g cm 3
The Cosmoillogical Constant




                 DV  L

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?):
                                10               10                  
                                                 4                          4
        L 10 g cm
              30         -3          4
                                            eV               3

                          10        cm         10         eV 
                                            2                      2
        L  8 G L             29                     33

Illogical timing (cosmic coincidence?):

                          M  R

                    GUT                     EWK BBN           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
            with the predictions of a

All evidence for dark energy/acceleration comes
from measuring the expansion history of the Universe

University of Chicago   1909 National Champions
    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
                  acceleration: a   G (  3p)              (acceleration)

                                                                  3p  0
scale factor a

                                             scale factor a
                        deceleration                            L: p   
                            3p  0

                                   time                                            time
                                         Hubble Diagram

apparent brightness of standard candle                 distant universe
                                                           past velocity


                                          nearby universe
                                          present velocity
                                          redshift of spectral lines
Expansion History of the Universe
         Friedmann-Robertson-Walker metric
                     dr 2                       2
  ds  dt  a  t  
    2    2   2
                               r d  r sin  df 
                                 2  2   2   2

                    1  kr

 a(t) = cosmic scale factor

 k  1, 0  spatial curvature constant: 3R  6k/a2(t)

         Friedmann equation (G00  8 GT00)
                   a     k
                 3    3 2  8 G 
                   a    a
Expansion History of the Universe
        Friedmann equation (G00  8 GT00)
                    k 8 G
                H  2
                   a   3
                        i  t0 
                   i 
                         3H 02
  critical density: C 
                         8 G

        redshift: 1  z 
                            a t 
      Expansion History of the Universe
                    Friedmann equation (G00  8 GT00)

        constant        curvature       matter      radiation

 H 2  z   H 02   k 1  z    M 1  z    R 1  z  
                                 2              3              4

                                                                

• k   M  R  1
  H 2  z   H 02  1   M   R 1  z    M 1  z    R 1  z  
                                              2              3              4

                                                                             
• radiation contribution (R) small for z  103
  H 2  z   H 02  1   M 1  z    M 1  z  
                                        2              3

                                                        
• “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)
             Astier et al. (2006)
                                           Hubble Diagram

                                          1. Find standard candle (SNe Ia)
                                          2. Observe magnitude & redshift
                                          3. Assume a cosmological model

 1.0                                      4. Compare observations & model


         0             0.5          1.0
                      M                       Einstein–de Sitter model
      Expansion History of the Universe
                   Friedmann equation (G00  8 GT00)

        Hubble cosmological
       constant constant curvature                     matter      radiation

    H 2  z   H 02   L 1  z    k 1  z    M 1  z    R 1  z  
                                    0              2              3              4

                                                                                  

• [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)

                        energy                curvature        matter    radiation

                             31 w
H 2  z   H0  w 1  z          k 1  z   M 1  z   R 1  z  
             2                                    2             3             4

                                                                               

Equation of state parameter: w  p /  w  1 for L
                            31 w          z dz                  
if w  w(z):     1  z               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

                           related to supernova brightness
                          confusing astronomical notation

                                                                                           (maximum theoretical bliss)
                                                                                             matter-dominated model
                                                                                               spatially flat, k 1,
                                                                                               Einstein-de Sitter:
   Astier et al. (2006)

                                                                    supernova redshift z

The case for L:
  1) Hubble diagram (SNe)         5) Galaxy clusters
  2) Cosmic Subtraction           6) Age of the universe
  3) Baryon acoustic oscillations 7) Structure formation
  4) Weak lensing
      The Cosmoillogical Constant

 dynamics                lensing              x-ray gas

cmb                                            power

            TOTAL  1             M ~ 0.3
             CMB              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
   – Baryon acoustic oscillations        H(z) not given by
   – Galaxy clusters                     Einstein–de Sitter
   – 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

      Duct Tape

anthropic principle
 (the landscape)
• 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
• 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!


                        Requires mf  1033 eV

     Tools to Modify the Left-Hand Side
• Braneworld modifies Friedmann equation                             Binetruy, Deffayet, Langlois

• Gravitational force law modified at large distance                              Deffayet, Dvali
                                                                                  & Gabadadze
   Five-dimensional at cosmic distances
• Tired gravitons                                                Gregory, Rubakov & Sibiryakov;
   Gravitons metastable - leak into bulk                             Dvali, Gabadadze & Porrati

• 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                  f (R)
                                 x g  R   R
                                                                Carroll, Duvvuri, Turner, Trodden
    S  16 G 
                             4             4

• “Backreaction” of inhomogeneities                    Räsänen; Kolb, Matarrese, Notari & Riotto;
                                                                Notari; Kolb, Matarrese & Riotto
  Backreaction of Inhomogeneities
  Homogeneous model                 Inhomogeneous model

       h                           i  x 
  a  Vh
   h                                a  Vi
H h  ah ah                    H i  ai ai
            h  i  x   H h  H i ?
                    We think not!
                                               (Buchert & Ellis)
Backreaction of Inhomogeneities

     G (g)  G ( g)
                                                     Kolb, Matarrese, Notari & Riotto

• Perturbed Friedmann–Lemaître–Robertson–Walker model:
   G  x , t   G  t    G  x , t 

   G00  t    G00  x , t   8 GT00  x , t 

    a  8 G 
                     3         
                    G00 
   a    3        8 G       
    . 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               scalar-field theory

zero-mode                a              f  (vev of a scalar field)

non-zero modes     inhomogeneities      thermal/finite-density bkgd.

physical effect      modify a(t)             modify f (t)
                   e.g., acceleration     e.g., phase transitions
    Acceleration From Inhomogeneities
    Standard approach                       Our approach

• Model an inhomogeneous           • Expansion rate of
  Universe as a homogeneous          inhomogeneous Universe 
                                     expansion rate of homogeneous
 Universe model with   
                                     Universe with   
                                   • Inhomogeneities modify
• a(t) / V1/3 is the zeromode of
  a homogeneous model                zeromode [effective scale
  with                          factor is aD  VD1/3 ]

• Inhomogeneities only have a      • Effective scale factor has a
  local effect on observables        (global) effect on observables

• Cannot account for observed      • Potentially can account for
  acceleration                       acceleration without
                                     dark energy or modified GR
            i  i/C            C  3H02/ 8G
 dynamics                  lensing                   x-ray gas

cmb                                                  power

            TOTAL  1 (CMB)             M  0.3
                         1  0.3  0.7
                   TOTAL  M  L
                         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
              8 G       3         
e.g., H 2                   G00 
               3       8 G       
Iguchi, Nakamura, Nakao
Nambu and Tanimoto
Mansouri                                       • Advantages:
Chang, Gu, Hwang
Alnes, Amarzguioui, Grøn
                                                  – Solvable inhomogeneous model
Mansouri                                          – Can describe wide variety of
Apostolopoulos, Brouzakis, Tetradis, Tzavara
Garfinkle                                           dynamics
Kai, Kozaki, Nakao, Nambu, Yoo
Marra, Kolb, Matarrese, Riotto
Mustapha, Hellaby, Ellis
Iguchi, Nakamura, Nakao
                                               • Disadvantages:
Vanderveld, Flanagan, Wasserman                    – Can’t encompass strong (volume)
Enqvist and Mattsson
Biswas, Mansouri, Notari                             backreaction (spherical symmetry)
Marra, Kolb, Matarrese
                                                   – Generically have small dynamical
Brouzakis, Tetradis, Tzavara                         range before shell crossing
Biswas and Notari
Brouzakis and Tetradis
Alnes and Amarzguioui
Garcia-Bellido and Haugboelle
Spherically symmetric metric                      R2  r , t  2
     d dt       d dr            ds 2  dt 2               dr  R 2  r , t  d 2
                                                  1  r 

Expansion rates                                H  R R
                                                                       H r2  R2 R2

                                                                          a   r, t 
Spherically symmetric density                  8 G   r , t  
                                                                    R 2  r , t  R  r , t 

                               R  r , t   ra  t 
                           R  r , t   a  t 
                                   r   kr 2
                                a  r   H 02  M r 3
• 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)
• 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.


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

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

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

   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

       dL(z)                         dA(z)                          V(z)

                        baryon       strong      weak                   strong
supernova   clusters                                        clusters
                         osc.       lensing     lensing                lensing

                                  Growth of
                                                 k  2 H  k  4 G  k  0

                                    lensing            P(k,z)              source?

                                 Test gravity
                solar     millimeter
                                        accelerators       P(k,z)
               system       scale
          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 :    the present value of the dark-energy eos parameter
   wa :    the rate of change of the dark-energy eos parameter
   DE :   the present dark-energy density
   M :    the present matter density
   B :    the present baryon density
   H0 :    the Hubble constant
   z :    amplitude of rms primordial curvature fluctuations
   nS :    the spectral index of primordial perturbations.
     w(a)  w0 + wa(1a)
wa      w0  present value
        wa  early value

                     DETF figure of merit:
                     (area)1 of the error ellipse

                   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   8000

       Photometric redshift
           from multicolor
             Baryon Acoustic Oscillations
Pre-recombination                   Post-recombination
   • universe ionized                  • universe neutral
   • photons provide enormous          • photons travel freely
     pressure and restoring force        (decoupled from baryons)
   • perturbations oscillate           • perturbations grow
     (acoustic waves)                    (structure formation)

Big Bang

           ionized          z » 1100             neutral
                         t » 380,000 yr
                           T ~ 3000 K

          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    WMAP
  and observable as CMB
  acoustic peaks

• Sound speed plummets,
  wave stalls
• Total distance traveled 150 Mpc   SDSS
  imprinted on power spectrum
        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

                              b

 observe          4GM DLS              dark energy
deflection                       affects geometric
  angle            b DOS             distance factors
                dark energy
               affects growth
                  rate of M
                      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:                        • DES (2012)
• Space: no atmosphere PSF                  – 1000’s of sq. degs.
                                              deep multicolor data
• Space: Near IR for photo-z’s
                                         • LSST (2015)
• Ground: larger aperture                   – 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
   • 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              Technique A
Figure of merit  100   Figure of merit  20

                           Technique Z
                        Figure of merit  20
Ongoing            Next step            Ultimate
                 FOM ~ 3× ongoing   FOM ~ 10 × ongoing
   My guess of

                                          95% C.L.
                          What’s Ahead
                2008               2010              2015               2020

Lensing     CFHTLS SUBARU          DES, VISTA      DUNE     LSST        SKA
      DLS SDSS ATLAS KIDS            Hyper suprime          JDEM

           SDSS ATLAS             Hyper suprime        JDEM

SNe           CSP ESSENCE          DES                   LSST
           SDSS CFHTLS             Pan-STARRS            JDEM

Clusters      AMI   APEX SPT       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:   Chemical Elements:
0.005%       (other than H & He) 0.025%
                      Neutrinos:

LCDM                            H & He:
                                gas 4%

                       Cold Dark Matter:
                       (CDM) 25%

             Dark Energy (L):
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)
                               Samuel Taylor Coleridge
                                   Theory of Life (1816)
    Dark Matter and Energy

             Rocky I: Dark Matter

             Rocky II: Dark Energy

ISAPP                              September 2010
Rocky Kolb                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
       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
       Techniques & implementations (largely from White Papers)
       Systematic uncertainties
       What we learned from analysis
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)
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.
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
      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
      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

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)
       incorporating CMB and H0 information to predict future
       (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.    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.
2.    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
     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-of-
     magnitude 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.
  m                    Technique #2

            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
       may improve cosmological parameters tend not to
9. Improvements in dark energy constraints from a single
       technique, valuable for energy from a multi-technique
    improve knowledge of darkcomparing 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.   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.
   b.   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.
   c.   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.
   d.   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
   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
   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
   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-of-
     merit, 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-of-
    merit, 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
                     DETF Legacy
I.     Standardization
      1. Parameterize dark energy as w0 – wa
      2. Eight-parameter cosmological model
      3. Priors
      4. Figure of merit
II.   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
                   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

                                     DETF FoM  (area of ellipse)1

                                               errors in w0 and wa
                                                 are correlated

                           0                        wa

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 (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
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
2. Specify cosmological parameters of fiducial cosmological
   model (including parameterization of dark energy).
Not much controversy in non-dark energy parameters (we use

Parameterize dark energy as a function of redshift or scale factor
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
   • 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
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
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

• 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
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
5. Quantify this information into a “figure of merit”

 Discuss DETF figure of merit

 Discuss where FoMSWG differs
                     DETF FoM
         (w0, wa )  (wp, wa)                   excluded
wp   DETF FoM  (area of ellipse)1
                                      sw0          swp)   w
                 [s(wp)s(wa)]1                            1

                                            0     zp              z

                                            errors in wp and wa
                                             are uncorrelated

                        0                       wa
          “… 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
proposal should be free to argue for their own figure of merit
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

• Generally, errors in different wi

  are correlated (like errors in w0 and wa)

                                                0          wa
• Expand w(a) in a complete
set of

  orthogonal eigenvectors ei(a)
         1  w(a )   a     
  with eigenvalues iaeii  a(like wp
                     i 0
and wa)
                                                0          wa

• 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

• 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

• 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

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 (1  a)
5. Calculate s(wp), s(wa), and zp
II. Determine the departure of the growth of structure from the
result of the fiducial model to probe dark energy and test
Calculate fully marginalized s(Dg)
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.
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
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
   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 ?       light-cone average

• Don’t see it in Newtonian limit.                    not Newtonian

• Don’t see it in perturbation theory.              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
  with   1.                                             red herring

• If this is a large effect one would expect
  to see large velocities.         with respect to which background?
• 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  ui; j  1 hik hkj  Q i j  s i j
                    2                            (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   6 s          2                                          Hirata & Seljak;

                                                   2 G   0
    q                                        2
           Q                2
                                                                            Ishibashi & Wald

• However must coarse-grain over some finite domain:

        Q       
                                h Q d 3x

                                h d 3x

• Evolution and smoothing do not commute:                           Q   D
                                                                              Q
                   
    Q        Q                  Q            Q         Q
                                      2              2                      Buchert & Ellis;
        D                D                D          D          D           Kolb, Matarrese & Riotto

•   Q   D
             Q                     Can have q  0 but qD  0 (“no-go” goes)
       Inhomogeneities and Smoothing
• Define a coarse-grained scale factor:                                      Kolb, Matarrese & Riotto
   aD  VD VD 0                 VD   d 3 x h
                  1/3                                                        Buchert & Ellis

• Coarse-grained Hubble rate:
        aD 1
   HD     3 Q D
• Effective evolution equations:
   aD    4 G                                                  R         3
             eff  3 peff       eff     D
                                                                D                  not
   aD     3                                         16 G 16 G                     described
    aD  8 G
                                                3QD       R                         by a simple
            eff                3 peff                 D
    aD   3                                   16 G 16 G

• Kinematical back reaction:          QD     2
                                              3      Q2
                                                                Q
                                                                     D   2 s      2
       Inhomogeneities and Smoothing
• Kinematical back reaction:      QD    2
                                         3      Q2
                                                               Q
                                                                      D   2 s    2

• For acceleration:               eff  3 peff                         0
                                                                      4 G

                                  a Q                   a                  
                                                                             
• Integrability condition (GR):     6
                                    D   D        a   4
                                                                      R           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    20 2 2      23 4 2
                      f            f  2f mean of  2f  0
            H       9            54
                  130 2 ,i      4 4
                         f f,i 
                                        2f  2f  f ,ij f,ij    
                   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?
                        1                                   a
•        f f             dk k T  k 
      2             2                     2               5
                   A 2 2                                 10
                      a H   0
•    4
           f f
           2   2    2
                   A 4 4      dk k
                                          T   2
                                                  k          0
                                                          10  
                     a H        0                             a0 

– Individual Newtonian terms large, i.e., hr2fr2fi  1                 Räsänen

– But total Newtonian term vanishes hr2fr2fi  hf,,ijf,ij i

– 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):
        R          
                 aD2   QD  0  no acceleration

• In general, gradient expansion gives         Notari; Kolb, Matarrese, & Riotto
                                2n
      R   rn a  n 3
                          rn    2n derivatives  f 
            n 1               mn                     
                                2n
       QD   qn a n 3   qn    2n derivatives  f m 
            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?

 • 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.”

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