Dark energy phenomenology and the CMB

1
Dark energy phenomenology and the CMB

• Robert Crittenden

Work with S. Boughn, T. Giannantonio, E.
Majerotto, F. Piazza, L. Pogosian, N. Turok, R.
Nichol, P.S. Corasaniti, C. Stephan-Otto
2
Why use the CMB to study dark energy?

• Naively, dark energy is a late universe effect,
  while the CMB primarily probes the physics of the
  last scattering surface.
• Extrapolating backwards, the expected energy
  density of baryons/dark matter is a billion times
  higher at z1000, while the dark energy density
  is about the same.
• So we might not expect dark energy would much of
  an effect on the CMB!
• But despite this, it is a very useful tool in DE
  studies

Matter density
Dark energy
Radiation density
z1000
3
Ways the CMB is useful for DE

• Provides an inventory of virtually everything
  else in the Universe, so we can figure out what
  is missing.
• Acts as a standard ruler on the surface of last
  scattering with which we can measure the geometry
  of the Universe.
• Some CMB anisotropies are created very recently
• Integrated Sachs-Wolfe effect
• Non-linear effects like Sunyaev-Zeldovich
• In some dark energy models, like tracking models,
  the dark energy density can change significantly,
  so that it is important at z1000.

4
CMB as cosmic yardstick
Physical size of acoustic sound horizon
The time of recombination and the sound speed are
well determined, telling us what this important
scale is.
5
CMB as cosmic yardstick
Angular distance to last scattering surface
6
CMB as cosmic yardstick
If we have a small amount of curvature, this will
change the observed angular size. This is very
hard to distinguish from changes to the angular
distance to the last scattering.
WMAP figure
7
CMB as cosmic yardstick
The CMB is imprinted with the scale of the sound
horizon at last scattering. This characteristic
scale is shown by the Doppler peaks. Both the
curvature and the dark energy can change the
angular size of the Doppler peaks. Assuming a
cosmological constant, we get a constraint on
curvature. However, if we assume a flat universe,
we can find a constraint on the dark energy
density and its evolution.
WMAP compilation
Fix angular scale of sound horizon
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CMB as cosmic yardstick
WMAP3 results
Even including all the other data, a degeneracy
remains between the equation of state and a small
amount of curvature.
The CMB fixes a single integrated quantity and
has a fundamental degeneracy between the dark
energy density and its equation of state, which
can be broken by other observations.
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Ways the CMB is useful for DE

• Provides an inventory of virtually everything
  else in the Universe.
• Acts as a standard ruler on the surface of last
  scattering with which we can measure the geometry
  of the Universe.
• Some CMB anisotropies are created very recently
• Integrated Sachs-Wolfe (ISW) effect
• Non-linear effects like Sunyaev-Zeldovich
• In some dark energy models, like tracking models,
  the dark energy density can change significantly,
  so that it is important at z1000.

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ISW Outline

• What is the ISW effect?
• Why is it interesting?
• Detecting the ISW
• Challenges
• Present limits
• Future measurements
• Improving the detections
• Conclusions
• Tommaso will discuss most recent results next!

11
Two independent CMB maps
The CMB fluctuations we see are a combination of
two largely uncorrelated pieces, one induced at
low redshifts by a late time transition in the
total equation of state.
Late ISW map, zlt 4 Mostly large scale features
Requires dark energy/curvature
Early map, z1000 Structure on many scales Sound
horizon as yardstick
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Dark energy signature
The ISW effect is gravitational, much like
gravitational lensing, but instead of probing the
gravitational potential directly, it measures its
time dependence along the line of sight.
potential depth changes as cmb photons pass
through
gravitational potential traced by galaxy density
The gravitational potential is actually constant
in a matter dominated universe on large scales.
However, when the equation of state changes, so
does the potential, and temperature anisotropies
are created.
13
What can the ISW do for us?

• Differential measurement of structure evolution
• Only arises when matter domination ends!
• Independent evidence for dark energy
• Matter dominated universe in trouble
• Direct probe of the evolution of structures
• Do the gravitational potentials grow or decay?
• Negative correlations ruled out!
• Constrain modified gravity models?
• Structure formation on the largest scales
• Measure dark energy clustering
• (Bean Dore, Weller Lewis, Hu Scranton)

14
Modified gravity
Modified gravity theories might have very
different structure growth. Thus, they lead to
very different predictions for ISW even with the
same background expansion! Extra dimensional
changes typically affect largest scales the most.
This is where the predictions are most uncertain.
Lue, Scoccimarro, Starkman 03
Caldwell, Cooray, Melchiorri 07
15
DGP model
On small scales, there is an Anzatz (Koyama
Maartens) for solving for the growth of
structure, but things are still uncertain for
large scales. In the ISW, this leads to different
predictions, particularly at high redshifts where
a higher signal could be generated. The signal
at low l (l lt 20) is still uncertain, though a
new Anzatz has recently been proposed. (Song,
Sawicki Hu).
Song, Sawicki Hu 06
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What can the ISW do for us?

• Differential measurement of structure evolution
• Only arises when matter domination ends!
• Independent evidence for dark energy
• Matter dominated universe in trouble
• Direct probe of the evolution of structures
• Do the gravitational potentials grow or decay?
• Negative correlations ruled out
• Constrain modified gravity models?
• Structure formation on the largest scales
• Measure dark energy clustering
• Potentially discriminate d.e. sound speeds at
  3?
• (Bean Dore, Weller Lewis, Hu Scranton)

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How do we detect ISW map?

• The typical scale is the horizon size, because
  smaller structures tend to cancel out.
• On linear scales positive and negative effects
  equally likely.
• Difficult to measure directly
• Same frequency dependence.
• Small change to spectrum.
• Biggest just where cosmic variance is largest.
• But we can see it if we look for correlations of
  the CMB with nearby (z lt 2) matter!

RC N. Turok 96 SDSS H. Peiris D. Spergel
2000
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Cross correlation spectrum
The gravitational potential determines where the
galaxies form and where the ISW fluctuations are
created! So the galaxies and the CMB should be
correlated, though its not a direct
template. Most of the cross correlation arises on
large or intermediate angular scales (gt1degree).
The CMB is well determined on these scales by
WMAP, but we need large galaxy surveys.
Can we observe this? Yes, but its difficult!
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Fundamental problem
While we see the CMB very well, the usual signal
becomes a contaminant when looking for the
recently created signal. Effectively we are
intrinsically noise dominated and the only
solution is to go for bigger area. But we are
fundamentally limited by having a single sky.
Noise!
Signal
ISW map, zlt 4
Early map, z1000
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Difficult to measure

• Small signal to noise!
• Possible foreground contaminations
• Galactic foregrounds
• Clustered extra-galactic sources emitting in
  microwave
• Sunyaev-Zeldovich effect
• Consistency tests
• insensitive to level of galactic cuts
• insensitive to point source cuts
• comparable signal in both hemispheres
• correlation on large angular scales
• independent of CMB frequency channel

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Other interesting effects

• Other effects can source large angular cross
  correlations which are also frequency
  independent.
• Magnification effect
• Structure at low redshift can appear in high
  redshift maps, making high redshift maps
  apparently correlated with CMB from ISW effect
  (LoVerde et al 06).
• No additional anisotropies, but could bias
  interpretation of high redshift correlations
• Doppler effect from rescattering
• On large scales, rescattering after reionisation
  can produce further anisotropies via the Doppler
  (or kinetic SZ) effect.
• Largest when probability for rescattering changes
  quickly.
• Some anisotropies produced at relatively low
  redshifts and can also bias ISW interpretation.
  (T. Giannantonio, RC).

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Correlations seen in many frequencies!

• X-ray background (Boughn RC)
• SDSS quasars (Giannantonio, RC, et al.)
• Radio galaxies
• NVSS confirmed by Nolta et al (WMAP
  collaboration)
• Wavelet analysis shows even higher significance
  (Vielva et al. McEwan et al.)
• FIRST radio galaxy survey (Boughn)
• Infrared galaxies
• 2MASS near infrared survey (Afshordi et al.)
• Optical galaxies
• APM survey (Folsalba Gaztanaga)
• Sloan Digital Sky Survey (Scranton et al., FGC,
  Cabre et al.)
• Band power analysis of SDSS data (N. Pamanabhan,
  et al.)

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Detections of ISW

• Correlations seen at many frequencies, covering a
  wide range in redshift.
• All consistent with cosmological constant model,
  if a bit higher than expected. This has made them
  easier to detect!
• Relatively weak detections, and there is
  covariance between different observations!
• Correlations shown at 6 degrees to avoid
  potential small angle contaminations (e.g. SZ).
  (Gaztanga et al.)

APM
SDSS
2mass
X-ray/NVSS
New!
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Scale with comoving distance
APM
SDSS
2mass
X-ray/NVSS
QSO
Signal declines and moves to smaller scales at
higher redshift.
We plot the observations for a fixed projected
distance.
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What does it say about DE?

• Thus far constraints are fairly weak from ISW
  alone.
• Consistent with cosmological constant model.
• Can rule out models with much larger or negative
  correlations.
• Very weak constraints on DE sound speed.

Corasantini, Giannantonio, Melchiorri
05 Gaztanaga, Manera, Multamaki 04
26
Parameter constraints

• A more careful job is needed!
• Quantify uncertainties
• Bias - usually estimated from ACF consistently.
  How much does it evolve over the samples?
  Non-linear or wavelength dependent?
• Foregrounds - incorporate them into errors.
• dN/dz - how great are the uncertainties?
• Understand errors
• To use full angular correlations, we need full
  covariances for all cross correlations.
• Monte Carlos needed with full cross correlations
  between various surveys.

27
Extended covariance matrix

• To combine them, we must understand whether and
  how the various experiments could be correlated
• Overlaps in sky coverage and redshift.
• Magnification bias.
• First efforts have begun to combine (Ryan
  Scranton TG)
• NVSS
• SDSS, LRG QSO
• 2MASS
• Preliminary results indicate gt 5? total signal!

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How good will it get?
For the favoured cosmological constant the best
signal to noise one can expect is about 7-10.
This requires significant sky coverage, surveys
with large numbers of galaxies and some
understanding of the bias. The contribution to
(S/N)2 as a function of multipole moment. This
is proportional to the number of modes, or the
fraction of sky covered, though this does depend
on the geometry somewhat. Of course, this
assumes we have the right model-- It might be
more!
RC, N. Turok 96 Afshordi 2004
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Future forecasts

• Ideal experiment
• Full sky, to overcome noise
• 3-D survey, to weight in redshift (photo-z ok)
• z 2-3, to see where DE starts
• 107 -108 galaxies, to beat Poisson noise
• Unfortunately, z1000 noise limits the signal
  to the 7-10? level, even under the best
  conditions.
• Realistic plans
• Short term - DES, Astro-F (AKARI)
• Long term - LSST, LOFAR/SKA

Pogosian et al 2005 astro-ph/ 0506396
30
Getting rid of the noise
Is there any way to eliminate the noise from the
intrinsic CMB fluctuations? Suggestion from L.
Page use polarization!
The CMB is polarized, and this occurs before ISW
arises, either at recombination or very soon
after reionization! Can we use this to subtract
off the noise? To some extent, yes!
31
The polarized temperature map
Suppose we had a good full sky polarization map
(EE) and a theory for the cross correlation (TE).
We could use this to estimate a temperature map
(e.g. Jaffe 03) that was 100 correlated with
the polarization. Subtracting this from the
observed map would reduce the noise somewhat,
improving the ISW detection! Only a small
effect at the multipoles relevant for the ISW,
but could improve S/N by 20.
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ISW Conclusions

• ISW effect is a useful cosmological probe,
  capable of telling us useful information about
  nature of dark energy.
• It has been detected in a number of frequencies
  and a range of redshifts, providing independent
  confirmation of dark energy.
• Many measurements are higher than expected, but
  what is the significance?
• There is still much to do
• Fully understanding uncertainties and covariances
  to do best parameter estimation.
• Using full shape of probability distributions.
• Finding new data sets.
• Reducing noise with polarization information.

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Phenomenology of dark energy

• Key questions from the theory-observation divide
  
• What are the variables we should use to describe
  dark energy?
• What are the theoretical priors on those
  variables?
• What should we go after experimentally?
• What questions do we want to answer with the
  data?

34
Parameterizing dark energy

• Without a good theory, our choices for how we
  parameterize dark energy are fairly arbitrary.
• Candidates constant
• linear
• kink
• density
• acceleration, jerk
• What if interesting DE evolution is orthogonal
  to these parameterizations?
• Can they show what experiments are capable of
  seeing?
• Why limit ourselves to a few degrees of freedom?

35
What can observations tell us?

• Principal components
• Parameterize with enough bins in red shift to
  allow significant freedom in w(z) (e.g. 30 )
• Find the eigenvectors of the Fisher matrix to see
  what could be measured with future data
• By combining data we may eventually be able to
  learn about 4-5 parameters, starting with low
  frequency, but well eventually get higher
  frequency modes.
• Very sensitive to assumptions about systematic
  errors!

Crittenden Pogosian Huterer Starkman Huterer
Linder, Knox et al.
36
Phenomenology of dark energy

• Spectra of eigenvalues from future experiments

Most informative
Higher ones are best determined 1/?2 Where do
we draw the line? It depends on what we think
we already know. In the absence of any prior
information, they are all informative. But we
always know something.
Least informative
37
Parameterizing dark energy

• Why not report our constraints in the same way?

Use a parameterization with plenty of degrees of
freedom. (See also talk by Albrecht?) Report the
best determined eigenmodes, their amplitudes and
eigenvalues of the likelihood. This would allow
us test any w(z) we wanted, not missing any
potential useful high frequency information. We
can always project to any particular
parameterization later using this information.
We end up however with lots of poorly
constrained modes, so its useful to have a prior
on this broad function space.
38
Figures of merit

• We have to choose something to optimize to decide
  what experiments to build, which is usually
  called the figure of merit.
• Often the volume of the error ellipsoid is
  minimized, which is related to the determinant of
  the Fisher matrix. This could lead to squeezing
  in only one dimension at the expense of the
  others.
• An alternative is related to the trace of the
  inverse Fisher matrix, which is simply the mean
  squared error
• This is dominated by the modes which have the
  greatest errors. Using it will tend to spread
  what we learn over a large number of independent
  modes.
• Which ever Figure of Merit you choose, when
  working with a large number of parameters it is
  essential to put in the priors to ensure we
  measure what is most interesting.

39
Comparing models

• Bayesian evidence comparison
• To compare models, we integrate the likelihood of
  the data over the possible model parameters
• Key questions

What fraction of the parameter volume improves
the fit? Occams razor Prior parameter
distribution
How much better does this model fit the
data? Best fit likelihood
The prior plays a key roll in comparing models,
particularly if the fit is not dramatically
better. But it is generally unknown!
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An explicit prior on w(z)
Rather than implicitly putting hard priors by the
choice of parameters, we can put in soft priors
explicitly. One way to do this is to treat w(z)
as a random field described by a correlation
function This is independent of binning choice
and has the effect of preferring smooth w(z)
histories over quickly changing ones. Strong
long range correlations will reproduce the
constant or linear prior. But if the data are
strong enough to overcome the priors, then higher
frequency modes could be seen.

41
Quintessence priors
Quintessence in principle can reproduce any w(z),
but that doesnt mean all are equally likely.
Ideally we would like to know the probability
distribution for the various DE histories based
on theoretical prejudice, mapping priors on V and
initial conditions into w(z). Unfortunately, we
have yet to agree on which models should be
included (or their relative weightings), much
less how the parameters of a given model should
be distributed. This makes them hard to falsify!
Weller Albrecht
42
Priors from quintessence

• In principle, a scalar field can produce any w(z)
  behavior desired from the right potential V(?),
  but some behaviors are more likely than others,
  particularly when w close to -1.
• Minimally coupled scalar field
• Typically in inflation the slow roll
  approximation is used, but when cold dark matter
  is present, this isnt usually justified

Work with E Majerotto and F.Piazza
43
Smoothness of potential

• Constraints on the potential
• Observed density
• Still evolving today
• If we assume,
• If f and its derivatives are of order 1, the
  constraints suggest a smoothness scale of order
• What does this smoothness mean for w(z)?

44
Small field displacement

• Observationally we know the equation of state is
  close to w -1, which indicates the field hasnt
  moved in recent times
• The field displacement is thus small compared to
  the typical smoothness scale of the potential, so
  it should be reasonable to expand the potential
  around its present value.

45
? approximation

• If we expand the quantity
• And keep the leading terms in (1w), where
• We can analytically solve for ?(a) and the dark
  energy density

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Comparison to exact

• This approximation gives an impressive fit to the
  full numerical evolution.
• Blue - full field dynamics Red - ?
  approximation Black - linear parameterization
• These are fit to the same w and derivative today,
  and may be improved by fitting in the middle of
  the range of interest.
• Gives another two parameter description of DE and
  gives us a field space measure on DE models.

47
Thawing and freezing

• Two generic classes of quintessence models
  (Caldwell Linder 05)
• Thawing models - fixed at early times and rolls
  when Hubble friction drops.
• Freezing models - field runs down steep
  (divergent) potential and stops when potential
  flattens out and friction becomes important.
• Our approximation is only appropriate for thawing
  models, because the diverging potential is
  inconsistent with our assumptions.

48
Likelihoods

• We can compare to observations using SN, CMB and
  BAO data.
• Top - linear parameterization
• Bottom - ? parameterization, matching w, w
  today.
• Similar likelihood curves show differences in
  evolution not well constrained with present data.
• Shows a focusing of the models near w -1,
  excluded regions require large change in ?
  (Scherrer 06).

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Priors on w(z)

• The previous curves show only the likelihood,
  without accounting for the prior probability of
  the models.
• The analytic solution allows us to relate the
  probability of potential to probability of w(z)
  via the Jacobian
• Reflects the fact that if the potential is
  locally flat, the field doesnt move and the rest
  of the potential is not relevant.

Uniform grid in (?0,?1)
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Posterior

• We can fold the prior with the likelihood from
  the data to find the final posterior
  distribution.
• A large volume of the models live near the best
  fit data, which is good for the evidence
  calculations.
• This however makes it hard to rule out a large
  fraction of the possible models without greatly
  improving the error on (1w).

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Conclusions

• Priors on DE are impossible to avoid they are
  necessary to discriminate between models and to
  decide what we choose to measure.
• Priors are implicit in how we choose to
  parameterize DE, so we might be better off
  allowing a large degree of freedom and making the
  priors explicit.
• Thus far little has been done to relate the
  priors on dark energy parameters to more
  fundamental parameters.
• In quintessence we have made an attempt to do
  this, which shows a focusing of models near w-1
  and also provides a simple template for thawing
  models.

52
Phenomenology of dark energy

• Whatever the data are, theres bound to be some
  dynamical DE model which fits better than a
  cosmological constant.
• Whether its interesting or not depends on our
  priors.

These data are quite consistent with a
cosmological constant, but there could be a
better fit.
53
Phenomenology of dark energy

• Whatever the data are, theres bound to be some
  dynamical DE model which fits better than a
  cosmological constant.
• Whether its interesting or not depends on our
  priors.

An oscillating function might be a better fit,
which would be missed if the chosen
parameterization didnt allow that freedom.
Because of the size of the errors in this case,
we would likely prefer w-1 unless we had a
model that predicted this precise
behavior. However, if the errors were smaller,
the improvement in the fit might justify a more
complex theory.