Inflation & its Cosmic Probes, now & then

Dick Bond

Inflation Now1+w(a)= εsf(a/aeq;as/aeq;s) goes to εax3/2 = 3(1+q)/2 ~1 good e-fold. only ~2params

Inflation Then εk=(1+q)(a) ~r/16 0<ε

= multi-parameter expansion in (lnHa ~ lnk)

Dynamics ~ Resolution ~ 10 good e-folds (~10-4Mpc-1 to ~ 1 Mpc-1 LSS) ~10+ parameters? Bond, Contaldi, Kofman, Vaudrevange 07

r(kp) i.e. εk is prior dependent now, not then. Large (uniform ε), Small (uniform lnε). Tiny (roulette inflation of moduli; almost all string-inspired models)

Cosmic Probes NowCFHTLS SN(192),WL(Apr07),CMB,BAO,LSS,Ly Zhiqi Huang, Bond & Kofman 07 εs=0.0+-0.25 now, to +-0.08 Planck+JDEM SN, weak as

w-trajectories for V(): pNGB example e.g.sorbo et07

For a given quintessence potential V(), 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 first 2-param approximation using an a-averaged εs= (V’/V)2/(16πG) and χ2 -fitted as.

Wild rise solutions

Slow-to-medium-roll solutions

Complicated scenarios: roll-up then roll-down

Approximating Quintessence for Phenomenology

+ Friedmann Equations + DM+B

1+w=2sin2

Zhiqi Huang, Bond & Kofman 07

slow-to-moderate roll conditions

1+w< 0.2 (for 0<z<10) and gives a 1-parameter model (as<<1):

Early-Exit Scenario: scaling regime info is lost by Hubble damping, i.e.small as

1+w< 0.3 (for 0<z<10) gives a 2-parameter model (as and εs):

CMB+SN+LSS+WL+Lya

Higher Chebyshev expansion is not useful: data cannot determine >2 EOS parameters.

e.g., Crittenden etal.06 Parameter eigenmodes

w(a)=w0+wa(1-a)

effective constraint eq.

Some ModelsCosmological Constant (w=-1)

Quintessence

(-1≤w≤1)

Phantom field (w≤-1)

Tachyon fields (-1 ≤ w ≤ 0)

K-essence

(no prior on w)

Uses latest April’07 SNe, BAO, WL, LSS, CMB, Lya data

cf. SNLS+HST+ESSENCE = 192 "Gold" SN

illustrates the near-degeneracies of the contour plot

w(a)=w0+wa(1-a) models

w(z) probes:

redshift w DE fraction Present day probes

0<z<0.4 w0 dominant CMB,SN,WL,Lya,LSS,BAO

0.4<z<2 w1 subdominant CMB,SN,WL,Lya

2<z<4 w2 tiny ? CMB,WL,Lya

z>4 w3 negligible ? CMB

• Piecewise w parameterization (4)

• Rich information at z<2, weak information at 2<z<4, almost no information at z>4

piecewise parameterization

with prior –3<w0,w1,w2,w3<1

CMB+SN+WL+LSS+Lya

What we know about w

Measuring constant w (SNe+CMB+WL+LSS)1+w = 0.02 +/- 0.05

Include a w<-1 phantom field, via a negative kinetic energy

termφ -> iφ εs= - (V’/V)2/(16πG)< 0

εs>0 quintessenceεs=0 cosmological constantεs<0 phantom field

Measuring εs

(SNe+CMB+WL+LSS+Lya)

Modified CosmoMC with Weak Lensing and time-varying w models

45 low-z SN + ESSENCE SN + SNLS 1st year SN+ Riess high-z SN, all fit with MLCS

SNLS1 = 117 SN(~50  are low-z)

SNLS+HST

= 182 "Gold" SN

SNLS+HST+ESSENCE

= 192 "Gold" SN

εstrajectories are slowly varying: why the fits are good

Dynamicalεw= (1+w)(a)/f(a) cf. shape εV= (V’/V)2 (a) /(16πG)

εs= εvuniformly averaged over 0<z<2 in a.

3-parameter parameterizationnext order corrections:

m (a) (depends on εs redefines aeq)

εvεs (a) (adds new s parameter)

enforce asymptotic kinetic-dominance w=1 (add as power suppression)

refine the fit to encompass even baroque trajectories.

this choice is analytic. The correction on w is only ~ 0.01

3-parameter parameterization

3-parameter fittingεs and ζs are all calculated from εtrajectory (linear

least square). Only as is χ2 fitted.• Perfectly fits slow-roll.

fits wild rising trajectories

Measuring the 3 parameters with current data• Use 3-parameter formula over 0<z<4 &

w(z>4)=wh (irrelevant parameter unless large).

Comparing 1-2-3-parameter results1-parameter: use 1-param formula for all redshifts.

2-parameter: use 2-param formula for 0<z<2 & w(z>2)=w(z=2).

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

Conclusion: all the complications are irrelevant:

we cannot reconstruct the quintessence potential

we can only measure the slope εs

CMB + SN + WL + LSS +Lya

Planck + JDEM-SN forecast

Planck + JDEM-SN forecast

Planck + JDEM

In progress: adding JDEM-WL

• The data cannot determine more than 2 w-parameters (+ csound?). general higher order Chebyshev expansion in 1+w as for “inflation-then” ε=(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 (εs >0) and the phantom equivalent (εs <0); use a 3-parameter model to well-approximate even rather baroque w-trajectories. Here we ignore constraints on Q-density effect on photon-decoupling and BBN because further trajectory extrapolation is needed.

• For general slow-to-moderate rolling one needs 2 “dynamical parameters” (as, εs) & Q to describe w to a few % for the not-too-baroque w-trajectories. a 3rd param s, (~dεs /dlna) is ill-determined now and even with Planck1yr+JDEM

• (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: to +-0.3 with Planck1yr+JDEM 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 εs.

• phantom (εs <0), cosmological constant (εs=0), and quintessence (εs >0) are all allowed with current observations which are well-centered around the cosmological constant εs=0.0+-0.25 to +-0.08 with Planck1yr+JDEM

• To use: given V, compute trajectories, do a-averaged εs & test (or simpler εs -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

Inflation now summary

Inflation Then Trajectories & Primordial Power

Spectrum ConstraintsConstraining Inflaton Acceleration Trajectories

Bond, Contaldi, Kofman & Vaudrevange 07

Roulette Inflation of Kahler Moduli and their AxionsBond, Kofman, Prokushkin & Vaudrevange 06

Inflation in the context of ever changing fundamental theory

1980

2000

1990

-inflation Old Inflation

New Inflation Chaotic inflation

Double InflationExtended inflation

DBI inflation

Super-natural Inflation

Hybrid inflation

SUGRA inflation

SUSY F-term inflation SUSY D-term

inflation

SUSY P-term inflation

Brane inflation

K-flationN-flation

Warped Brane inflation

inflation

Power-law inflation

Tachyon inflationRacetrack inflation

Assisted inflation

Roulette inflation Kahler moduli/axion

Natural inflation

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,εk|data, th)

~ P(data| lnHp,εk ) P(lnHp,εk | th) / P(data|th)

Likelihood theory prior / evidenceData:

CMBall

(WMAP3,B03,CBI, ACBAR,

DASI,VSA,MAXIMA)

+

LSS (2dF, SDSS, 8[lens])

Theory prior

uniform in lnHp,εk

(equal a-prior probability hypothesis)

Nodal points cf. Chebyshev coefficients (linear

combinations)

uniform in / log in / monotonic in εk

The theory prior matters a lot for current data

We have tried many theory priors

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).

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

no self consistency: order 5 in scalar and tensor power

r = .21+- .17 (<.53)

Power law scalar and constant tensor + 4 params

effective r-prior makes the limit stringent

r = .082+- .08 (<.22)

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

no self consistency: order 5 in scalar and tensor power

r = .21+- .17 (<.53)

Power law scalar and constant tensor + 4 params

effective r-prior makes the limit stringent

r = .082+- .08 (<.22)

run+tensor r = .13+- .10 (<.33)

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

log prior

r <0.075

lnPs lnPt no self consistency: order 5 in scalar and tensor

power

uniform prior

r = .21+- .11 (<.42)

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 < 0.075uniform prior

r < 0.42

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 2008.6

Spider balloon 2009.9

Spider+Planck broad-band

error

uniform prior

log prior

lnεs (nodal 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

log prior

r (<0.34; .<03 at 1-sigma!)

εs self consistency: order 5 uniform prior

r = (<0.64)

1à ïà ï = d lnk

d lnH

V / H 2(1à 3ï ); d lnk

d inf = 1à ïæ ïpP s(k) / H 2=ï ;P t(k) / H 2

ï = (d lnH =d inf)2

lnεs (nodal 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 < 0.33, but .03 1-sigmauniform prior

r < 0.64

CL BB for lnεs (nodal 5) + 4 params inflation trajectories reconstructed from CMB+LSS data using Chebyshev nodal point expansion & MCMC

Planck satellite 2008.6

Spider balloon 2009.9

uniform prior

log prior

Spider+Planck broad-band

error

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 2008.6

Spider balloon 2009.9

Spider+Planck broad-band

error

uniform prior

log prior

B-pol simulation: input LCDM (Acbar)+run+uniform tensor

r (.002 /Mpc) reconstructed cf. rin

εs order 5 uniform prior εs order 5 log prior

a very stringent test of the ε-trajectory methods: A+

Planck1 simulation: input LCDM (Acbar)+run+uniform tensor

r (.002 /Mpc) reconstructed cf. rin

εs order 5 uniform prior εs order 5 log prior

Planck1 simulation: input LCDM (Acbar)+run+uniform tensor

Ps Pt reconstructed cf. input of LCDM with scalar running & r=0.1

εs order 5 uniform prior εs order 5 log prior

r=0.144 +- 0.032 r=0.096 +- 0.030

Planck1 simulation: input LCDM (Acbar)+run+uniform tensor

Ps Pt reconstructed cf. input of LCDM with scalar running & r=0.1

εs order 5 uniform priorεs order 5 log prior

lnPs lnPt (nodal 5 and 5)

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

Inflation then summarythe 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 ε(k) ~ r(k): with current data, the scalar power downturns (ε(k) goes up) at low k 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 ε drives to low r. a B-pol could break this prior dependence, maybe Planck+Spider.

Complexity of trajectories arises in many-moduli string models. Roulette example: 4-cycle complex Kahler moduli in Type IIB string theory TINY r ~ 10-10 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

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