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Parameterizing Dark Energy
Z. Huang, R. J. Bond, L. Kofman
Canadian Institute of Theoretical Astrophysics
w - Dark Energy Equation of State
Data: CMB + SN + LSS + WL + Lya
Constant w=w0
Code: modified cosmomc
w(a)=w0+wa(1-a)
Principal components
s1=0.12 s2=0.32 s3=0.63
w0 = -0.98 ± 0.05
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
simplicity of w(a).
we do need
assumptions
simplicity of V(φ).
Dynamics of Quintessence/Phantom
define
Popular story of quintessence: Fast rolling (large eV) in early universe (scaling regime);
Slow rolling (small eV) in late universe.
Field equation
+ Friedmann Equations
Simplicity assumptions
In the relevant redshift range (e.g. 0<z<4),
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 f
V(f) = V0 fn (n=0,±1, ±2, ±3,…)
1-parameter parametrization
• Additional assumption: slow-roll at 0<z<10
(initial velocity Hubble-damped at low redshift).
“average slope”
es=<ev>
where
es>0, quintessence
es=0, cosmological constant
es<0, phantom
CMB + SN + LSS + WL + Lya
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).
CMB + SN + LSS + WL +
Lya
Hubble damping
term
2-parameter parametrization
- residual velocity at low redshift.
3-parameter parametrization
In 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-parameter
1-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.