1. Inflation Then k=(1+q)(a) ~r/16 0<
Dick Bond
Dynamical Trajectories for Inflation now & then
Inflation Now1+w(a)= εsf(a/aeq; as/aeq) goes to (a)x3/2 = 3(1+q)/2 ~1 good e-fold. only ~2params
Zhiqi Huang, Bond & Kofman es= , weak as
Cosmic Probes Now CFHTLS SNe (192), WL (Apr07), CMB, BAO, LSS
Inflation Then k=(1+q)(a) ~r/ <
= multi-parameter expansion in (lnHa ~ lnk)
Dynamics ~ Resolution ~ 10 good e-folds (~10-4Mpc-1 to ~ 1 Mpc-1 LSS) ~10+ parameters?
r(kp) i.e. k is very prior dependent. Large (uniform ), Small (uniform ). Tiny (roulette inflation of moduli; almost all string-inspired models).

2. w-trajectories for V(f)
For a given quintessence potential V(f), we set the “initial conditions” at z=0 and evolve backward in time.
w-trajectories for Ωm (z=0) = 0.27 and (V’/V)2/(16πG) (z=0) = 0.25, the 1-sigma limit, varying the initial kinetic energy w0 = w(z=0)
Dashed lines are our 2-param approximation using an a-averaged es= (V’/V)2/(16πG) and χ2 -fitted as.
Wild rise solutions
S-M-roll solutions
Complicated scenarios: roll-up then roll-down

3. Approximating Quintessence for Phenomenology
Zhiqi Huang, Bond & Kofman 07
Friedmann Equations + DM+B
1+w=2sin2 

4. slow-to-moderate roll conditions
1+w< 0.3 (for 0<z<10) gives a 2- parameter model (as and es):
Early-Exit Scenario: scaling regime info is lost by Hubble damping, i.e.small as
1+w< 0.2 (for 0<z<10) and gives a 1- parameter model (as<<1):

5. w-trajectories for V(f): varying V

6. w-trajectories for V(f): varying es

7. w-trajectories for pNGB potentials
pNGB potentials (Sorbo’s talk Jun7/07) are also covered by our 2-param approximation
Dashed lines are using a-averaged
es= (V’/V)2/(16πG)
and χ2 fitted as.
Wild rise solutions
S-M-roll solutions
Complicated scenarios: roll-up then roll-down

8. w-trajectories for pNGB V(f): varying f,f(0)

9. effective constraint eq.
Some Models
Cosmological Constant (w=-1)
Quintessence
(-1≤w≤1)
Phantom field (w≤-1)
Tachyon fields (-1 ≤ w ≤ 0)
K-essence
(no prior on w)
w(a)=w0+wa(1-a)
Uses latest April’07
SNe, BAO, WL, LSS, CMB, Lya data
Higher Chebyshev expansion is not useful: data cannot determine >2 EOS parameters. e.g., Crittenden etal.06 Parameter eigenmodes
effective constraint eq.

10. w(a)=w0+wa(1-a) models cf. SNLS+HST+ESSENCE = 192 "Gold" SN
illustrates the near-degeneracies of the contour plot

11. Measuring constant w (SNe+CMB+WL+LSS)

12. Include a w<-1 phantom field, via a negative kinetic energy term
φ -> iφ  es= - (V’/V)2/(16πG) < 0
es>0  quintessence
es=0  cosmological constant
es<0  phantom field

13. 45 low-z SN + ESSENCE SN + SNLS 1st year SN + Riess high-z SN, all fit with MLCS
SNLS+HST
= 182 "Gold" SN
SNLS+HST+ESSENCE
= 192 "Gold" SN
SNLS1 = 117 SN
(~50  are low-z)

14. Measuring es and as(SNe+CMB+WL+LSS+Lya)
Well determined es
ill-determined as

15. estrajectories cf. the 1-parameter model
Dynamical es= (1+w)(a)/f(a) cf. shape es= (V’/V)2 (a) /(16πG)

16. Measuring es (SNe+CMB+WL+LSS+Lya)
Modified CosmoMC with Weak Lensing and time- varying w models

17. Measuring es (SNe+CMB+WL+LSS+Lya)
Marginalizing over the ill- determined as (2-param model)
or setting as =0 (1-param model)
has little effect on
prob(es , Wm)

18. as is not well-determined by the current data
Inflation now summary
The data cannot determine more than 2 w-parameters (+ csound?). general higher order Chebyshev expansion in 1+w as for “inflation-then” =(1+q) is not that useful
The w(a)=w0+wa(1-a) phenomenology requires baroque potentials
Philosophy of HBK07: backtrack from now (z=0) all w-trajectories arising from quintessence (es >0) and the phantom equivalent (es <0); use a few-parameter model to well-approximate the not-too-baroque w-trajectories
For general slow-to-moderate rolling one needs 2 “dynamical parameters” (as, es) & WQ to describe w to a few % for the not-too-baroque w-trajectories
(cf. for a given Q-potential, velocity, amp, shape parameters are needed to define a w-trajectory)
as is not well-determined by the current data
In the early-exit scenario, the information stored in as is erased by Hubble friction over the observable range & w can be described by a single parameter es.
phantom (es <0), cosmological constant (es =0), and quintessence (es >0) are all allowed with current observations which are well-centered around the cosmological constant es=
To use: given V, compute trajectories, do a-averaged es & test (or simpler es -estimate)
Aside: detailed results depend upon the SN data set used. Best available used here (192 SN), but this summer CFHT SNLS will deliver ~300 SN to add to the ~100 non-CFHTLS and will put all on the same analysis/calibration footing – very important.
Newest CFHTLS Lensing data is important to narrow the range over just CMB and SN

19. Inflation Then Trajectories & Primordial Power Spectrum Constraints
Constraining Inflaton Acceleration Trajectories Bond, Contaldi, Kofman & Vaudrevange 07
Ensemble of Kahler Moduli/Axion Inflations
Bond, Kofman, Prokushkin & Vaudrevange 06

20. Inflation in the context of ever changing fundamental theory
1980
-inflation
Old Inflation
New Inflation
Chaotic inflation
SUGRA inflation
Power-law inflation
Double Inflation
Extended inflation
1990
Hybrid inflation
Natural inflation
Assisted inflation
SUSY F-term inflation
SUSY D-term inflation
Brane inflation
Super-natural
Inflation
2000
SUSY P-term inflation
K-flation
N-flation
DBI inflation
inflation
Warped Brane inflation
Tachyon inflation
Racetrack inflation
Roulette inflation Kahler moduli/axion

21. (equal a-prior probability hypothesis)
Constraining Inflaton Acceleration Trajectories Bond, Contaldi, Kofman & Vaudrevange 07
“path integral” over probability landscape of theory and data, with mode-function expansions of the paths truncated by an imposed smoothness (Chebyshev-filter) criterion [data cannot constrain high ln k frequencies]
P(trajectory|data, th) ~ P(lnHp,ek|data, th)
~ P(data| lnHp,ek ) P(lnHp,ek | th) / P(data|th)
Likelihood theory prior / evidence
Data:
CMBall
(WMAP3,B03,CBI, ACBAR,
DASI,VSA,MAXIMA) +
LSS (2dF, SDSS, s8[lens])
Theory prior
uniform in lnHp,ek
(equal a-prior probability hypothesis)
Nodal points cf. Chebyshev coefficients (linear combinations)
monotonic in ek
The theory prior matters alot
We have tried many theory priors

22. Old view: Theory prior = delta function of THE correct one and only theory
New view: Theory prior = probability distribution on an energy landscape whose features are at best only glimpsed, huge number of potential minima, inflation the late stage flow in the low energy structure toward these minima. Critical role of collective geometrical coordinates (moduli fields) and of brane and antibrane “moduli” (D3,D7).

23. r = .082+- .08 (<.22) r = .21+- .17 (<.53)
lnPs Pt (nodal 2 and 1) + 4 params cf Ps Pt (nodal 5 and 5) + 4 params reconstructed from CMB+LSS data using Chebyshev nodal point expansion & MCMC
Power law scalar and constant tensor + 4 params
effective r-prior makes the limit stringent
r = (<.22)
no self consistency: order 5 in scalar and tensor power
r = (<.53)

24. lnPs lnPt no self consistency: order 5 in scalar and tensor power
lnPs lnPt (nodal 5 and 5) + 4 params. Uniform in exp(nodal bandpowers) cf. uniform in nodal bandpowers reconstructed from April07 CMB+LSS data using Chebyshev nodal point expansion & MCMC: shows prior dependence with current data
lnPs lnPt no self consistency: order 5 in scalar and tensor power
uniform prior
r = (<.42)
log prior
r <0.075)

25. r < 0.075 r < 0.42 logarithmic prior uniform prior
lnPs lnPt (nodal 5 and 5) + 4 params. Uniform in exp(nodal bandpowers) cf. uniform in nodal bandpowers reconstructed from April07 CMB+LSS data using Chebyshev nodal point expansion & MCMC: shows prior dependence with current data
logarithmic prior
r < 0.075
uniform prior
r < 0.42

26. CL BB for lnPs lnPt (nodal 5 and 5) + 4 params inflation trajectories reconstructed from CMB+LSS data using Chebyshev nodal point expansion & MCMC
Planck satellite
Spider balloon

27. CL TT BB for e (ln Ha) inflation trajectories reconstructed from a CMBpol expt (noise power = Planck/180) using Chebyshev nodal point expansion in lne (order 5, adjusting to make a uniform prior in e) & MCMC: output = input even with r= (sytematics & foregrounds aside)

28. Ps Ps & e (ln Ha) inflation trajectories reconstructed from a CMBpol expt (CLnoise = 1.7(-6) mK2 (x2 for polarization)) using Chebyshev nodal point expansion in lne (order 5, adjusting to make a uniform prior in e) & MCMC: output = input even with r= (systematics & foregrounds aside)
Input model = tilted LCDM using Acbar06 params + variable r, here tiny but reconstructable

29. The ‘house’ does not just play dice with the world.
Roulette: which minimum for the rolling ball depends upon the throw; but which roulette wheel we play is chance too.
The ‘house’ does not just play dice with the world.
even so, even smaller r’s may arise from string-inspired inflation models e.g., KKLMMT inflation, large compactified volume Kahler moduli inflation

30. Inflation then summary
the basic 6 parameter model with no GW allowed fits all of the data OK
Usual GW limits come from adding r with a fixed GW spectrum and no consistency criterion (7 params). Adding minimal consistency does not make that much difference (7 params)
r (<.28 95%) limit comes from relating high k region of 8 to low k region of GW CL
Uniform priors in (k) ~ r(k): with current data, the scalar power downturns ((k) goes up) at low L if there is freedom in the mode expansion to do this. Adds GW to compensate, breaks old r limit. T/S (k) can cross unity. But log prior in  drives to low r. a CMBpol could break this prior dependence.
Complexity of trajectories arises in many-moduli string models. Roulette example: 4-cycle complex Kahler moduli in Type IIB string theory TINY r ~ a general argument that the normalized inflaton cannot change by more than unity over ~50 e-folds gives r < 10-3
Prior probabilities on the inflation trajectories are crucial and cannot be decided at this time. Philosophy: be as wide open and least prejudiced as possible
Even with low energy inflation, the prospects are good with Spider and even Planck to either detect the GW-induced B-mode of polarization or set a powerful upper limit against nearly uniform acceleration. Both have strong Cdn roles. CMBpol

31. end

32. w-trajectories cf. the 2-parameter model
es= (V’/V)2/(16πG) a-averaged at z<2
the field starts to slow-roll at a~as

33. w-trajectories cf. the 1-parameter model
Require S-M roll conditions at 0<z<10, es= (V’/V)2/(16πG) a-averaged at z<2