1. Parameterizing Dark Energy Z. Huang, R. J. Bond, L. Kofman Canadian Institute of Theoretical Astrophysics

2. w - Dark Energy Equation of State Constant w=w 0 w(a)=w 0 +w a (1-a)Principal components Data: CMB + SN + LSS + WL + Lya Code: modified cosmomc w 0 = -0.98 ± 0.05  1 =0.12  2 =0.32  3 =0.63

3. What we know about w Phenomenologically w is close to –1 at low redshift Rich information at z 4 Results depend on parametrization. Need theoretical priors. What if we start from physics?

4. Use physics to solve problems When Canadian plug does not fit UK socket…

5. Dynamics of Quintessence/Phantom we do need assumptions simplicity of w(a). simplicity of V(φ).

6. Dynamics of Quintessence/Phantom define Field equation+ Friedmann Equations Popular story of quintessence: Fast rolling (large  V ) in early universe (scaling regime); Slow rolling (small  V ) in late universe.

7. Simplicity assumptions 1.V(  ) is monotonic. 2.Quintessence rolls down (dV/dt 0). 3.w is close to -1 at low redshift. Quantitatively, |1+w|<0.4 at 0<z<1. 4.V(  ) is a “simple” function. Quantitatively, |  V | is less than or of the same order of either Planck scale or  V. examples, V(  ) = V 0 exp(-  ) V(  ) = V 0 + V 1  V(  ) = V 0  n (n=0,±1, ±2, ±3,…) In the relevant redshift range (e.g. 0<z<4),

8. 1-parameter parametrization Additional assumption: slow-roll at 0<z<10 (initial velocity Hubble-damped at low redshift). where  s >0, quintessence  s =0, cosmological constant  s <0, phantom CMB + SN + LSS + WL + Lya “average slope”  s =

9. Initial velocity is damped by Hubble friction

10. Time variation of  V is not important Given solution w(a) and  V (a), define trajectory variables:  s =  V  uniformly averaged at 1/3<a<1.  w = (1+w)/f(a/a eq ). (remind: 1-param formula is w fit =-1+  s f(a/a eq ))

11. Constraint Equation Define w 0 =w| a=1, w a =-dw/da| a=1 w 0 and w a are functions of (  s, Ω m ). Numerical fitting yields

12. binned SNe samples (192 samples) Some w 0 -w a mimic cosmological constant

13. 2-parameter parametrization Assumption: slow-roll at 0<z<2 (with possible non-damped velocity). Hubble damping term CMB + SN + LSS + WL + Lya

14. 2-parameter parametrization - residual velocity at low redshift.

15. 3-parameter parametrization In general  V varies. Assuming no oscillation, we model Other corrections can only be numerically fitted: redefine a eq. O(θ 3 ) term numerical fitting. a s -  s power suppression (if  s and a s are both large, the power of Hubble damping term would be suppressed). When all smoke clears 

16. 3-parameter parametrization

17. 3-parameter fitting Perfectly fits slow-to-moderate roll.

18. Fit wild rising trajectories

19. Measuring 3 parameters Use 3-parameter for 0 4)=w h (free parameter).

20. Comparing 1-2-3-parameter 1-parameter: use 1-param formula for all redshift. 2-parameter: use 2-param formula for 0 2)=w h (free constant). 3-parameter: use 3-param formula for 0 4)=w h (free constant). Conclusion: all the complications are irrelevant, now only can measure  s CMB + SN + WL + LSS +Lya

21. Forecast: Planck + JDEM SN + DUNE WL

22. 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:

23. 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  s = (1/16πG) can be well measured.  The second parameter a s 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.