Parameterizing Dark Energy Z. Huang, R. J. Bond, L. Kofman

Canadian Institute of Theoretical Astrophysics

w - Dark Energy Equation of State

Constant w=w0 w(a)=w0+wa(1-a) Principal components

Data: CMB + SN + LSS + WL + Lya Code: modified cosmomc

w0 = -0.98 ± 0.05

s1=0.12 s2=0.32 s3=0.63

What we know about w

Phenomenologically• w is close to –1 at low redshift• Rich information at z<2, weak information at

2<z<4, almost no information at z>4• Results depend on parametrization. Need

theoretical priors.

What if we start from physics?

Use physics to solve problems

When Canadian plug does not fit UK socket…

Dynamics of Quintessence/Phantom

we do need assumptions

simplicity of w(a).

simplicity of V(φ).

Dynamics of Quintessence/Phantom

define

Field equation + Friedmann Equations

Popular story of quintessence: Fast rolling (large eV) in early universe (scaling regime); Slow rolling (small eV) in late universe.

Simplicity assumptions

1. V(f) is monotonic. 2. Quintessence rolls down (dV/dt<0). Phantom rolls up

(dV/dt>0).3. w is close to -1 at low redshift. Quantitatively, |1+w|<0.4 at

0<z<1.

4. V(f) is a “simple” function. Quantitatively, |hV| is less than or of the same order of either Planck scale or eV.

examples, V(f) = V0 exp(-lf)V(f) = V0 + V1 fV(f) = V0 fn (n=0,±1, ±2, ±3,…)

In the relevant redshift range (e.g. 0<z<4),

1-parameter parametrization• Additional assumption: slow-roll at 0<z<10

(initial velocity Hubble-damped at low redshift).

where

es>0, quintessence

es=0, cosmological constant

es<0, phantom

CMB + SN + LSS + WL + Lya

“average slope” es=<ev>

Initial velocity is damped by Hubble friction

Time variation of eV is not important

Given solution w(a) and eV(a), define trajectory variables:

• es= eV uniformly averaged at 1/3<a<1.

• ew = (1+w)/f(a/aeq). (remind: 1-param formula is wfit=-1+ es f(a/aeq))

Constraint Equation

• Define w0=w|

a=1,

wa=-dw/da|a=1

w0 and wa are functions of (es, Ωm).

Numerical fitting yields

binned SNe samples (192 samples)

Some w0-wa mimic cosmological constant

2-parameter parametrization• Assumption: slow-roll at 0<z<2 (with possible non-damped

velocity).

Hubble damping termCMB + SN + LSS + WL +

Lya

2-parameter parametrization - residual velocity at low redshift.

3-parameter parametrizationIn general eV varies. Assuming no oscillation , we model

Other corrections can only be numerically fitted:

•redefine aeq.

•O(θ3) term numerical fitting.

•as- es power suppression (if es and as are both large, the power of Hubble damping term would be suppressed).

When all smoke clears

3-parameter parametrization

3-parameter fitting

• Perfectly fits slow-to-moderate roll.

Fit wild rising trajectories

Measuring 3 parameters• Use 3-parameter for 0<z<4. Assume w(z>4)=wh

(free parameter).

Comparing 1-2-3-parameter1-parameter: use 1-param formula for all redshift.

2-parameter: use 2-param formula for 0<z<2, assuming w(z>2)=wh (free constant).

3-parameter: use 3-param formula for 0<z<4, assuming w(z>4)=wh (free constant).

Conclusion: all the complications are irrelevant, now only can measure es

CMB + SN + WL + LSS +Lya

Forecast: Planck + JDEM SN + DUNE WL

Thawing, freezing or non-monotonic?

• Thawing: w monotonically deviate from -1.• Freezing: w monotonically approaches -1.• Our parameterization with flat priors. Roughly 15 percent thawing, 8 percent

freezing, most are non-monotonic.

With freezing prior:

With thawing prior:

Conclusions

For a wide class of quintessence/phantom models, the functional form V(φ) in the near future is observationally immeasurable. Only a key trajectory parameter es = (1/16πG) <(V’/V)2> can be well measured.

The second parameter as can only be constrained to be less than ~0.3.

For current observational data, even with (physically motivated) dynamic w(a) parametrization, cosmological constant remains to be the best and simplest model.