1. Parameterizing dark energy: a field space approach
Robert Crittenden
University of Portsmouth
Work with E. Majerotto, F. Piazza, L. Pogosian

2. Phenomenology of dark energy
Key questions from the theory-observation divide:
What are the variables we should use to describe dark energy?
What do observations tell us?
What are the theoretical priors on those variables?
How do we design experiments to address the questions do we want to answer?

3. Parameterizations of dark energy
Without a good theory, our choices for how we parameterize dark energy are fairly arbitrary.
Candidates: constant
linear
kink
density
acceleration, jerk
Any such parameterization amounts to a data compression and means potentially throwing away information.
What if interesting DE evolution is orthogonal to these parameterizations?
What does theory suggest might be interesting variables?

4. 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 we’ll eventually get higher frequency modes.
Very sensitive to assumptions about systematic errors!
Gaussian approximation to likelihood.
Crittenden & Pogosian
Huterer & Starkman
Huterer & Linder, Knox et al.

5. 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

6. Parameterizing dark energy
Why not report our constraints in the same way?
Choose a parameterization with plenty of degrees of freedom.
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!
See also Albrecht & Berstein (2007).

7. The importance of priors
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?
Occam’s 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!

8. Ruling out dynamical w(z)
Whatever the data are, there’s 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.

9. Phenomenology of dark energy
Whatever the data are, there’s 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 didn’t 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.

10. A phenomenological 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.

11. Quintessence priors
Quintessence uses a dynamical scalar field to produce acceleration.
In principle can reproduce any w(z), but that doesn’t 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

12. Thawing and freezing
Two generic classes of quintessence models (Caldwell & Linder 05):
Thawing models - fixed at early times and rolls when Hubble friction drops.
These start out at w=-1 and then increases as the field begins to roll.
Freezing models - field runs down steep (divergent) potential and stops when potential flattens out and friction becomes important.
These typically start with constant w > -1, and then naturally approach w=-1 as the field takes over driving the expansion.

13. Priors from quintessence
Can we use what we observe about dark energy to help us parameterize it?
Equations for a minimally coupled scalar field:
In inflation the slow roll approximation is usually used, but when cold dark matter is present, this isn’t usually justified:
Work with E Majerotto and F.Piazza

14. 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)?
Key assumption!

15. Small field displacement
Observationally we know the equation of state is close to w = -1, which indicates the field hasn’t 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.

16.  approximation If we expand the quantity
And keep the leading terms in (1+w), where
We can then analytically solve for (a) and the dark energy density:

17. Comparison to exact
This approximation gives an impressive fit to the full numerical evolution for thawing models.
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.

18. 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).

19. 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 doesn’t move and the rest of the potential is not relevant.
Uniform grid in (0,1)

20. 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 means the evidence for these models will be large.
This however makes it hard to rule out a large fraction of the possible models without greatly improving the error on (1+w).

21. 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.

22. Post-doc at Portsmouth
Soon to be advertised:
Post-doc in theoretical cosmology
Dark energy, inflation, brane worlds, etc.
STFC rolling grant
Proposed start date 1 January 2008

23. 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, giving a better
Another possibility is to minimize the projected chi-squared, which is the trace of the Fisher matrix.