Dick Bond

Constraining Inflation Trajectories, now & then

Bad Timing: I arrived at Cambridge in summer 82 from Stanford for a year, sadly just after Nuffield VEU.

armed with Hot, warm, cold dark matter, classified by the degree of collisionless damping

Transparencies of the time: fluctuation spectra were breathed out of the mouth of the great dragon, quantum gravity

Linear then non-linear amplifier

Primordial spectrum as variable n, could be variable anything. Plots used

HZP argument that scale invariant avoids nonlinearities at large or small scales. Gaussian because of central limit theorem & simplicity.

Bad Timing: I arrived at Cambridge in summer 82 from Stanford for a year, sadly just after Nuffield VEU.

armed with Hot, warm, cold dark matter, classified by the degree of collisionless damping

Transparencies of the time: fluctuation spectra were breathed out of the mouth of the great dragon, quantum gravity

Linear then non-linear amplifier

Primordial spectrum as variable n, could be variable anything. Plots used

HZP argument that scale invariant avoids nonlinearities at large or small scales. Gaussian because of central limit theorem & simplicity.

PS, Chibisov & Mukhanov 81: ‘phonon’ appendix as exercise for Advanced GR class at Stanford then !

PS, Chibisov & Mukhanov 81: ‘phonon’ appendix as exercise for Advanced GR class at Stanford then !

INFLATION THEN INFLATION THEN

1987

2003

Radical BSI inflationSBB89: multi-field, the hybrid inflation prototype, with

curvature & isocurvature & Ps(k) with any shape

possible & Pt(k) almost any shape (mountains &

valleys of power), gorges, moguls, waterfalls, m2eff <

0, i.e., tachyonic, non-Gaussian, baroqueness,

radically broken by variable braking (k); SB90,91 Hamilton-Jacobi formalism to do non-G (&

Bardeen pix of non-G) (k) - H()

cf. gentle break by smooth brake in the slow roll limit.

f||

fperp

2003

Blind power

spectrum analysis cf. data, then

& now

measures matter

“theory prior”

informed priors?

Dick Bond

Inflation Then k=(1+q)(a) ~r(k)/16 0<

= multi-parameter expansion in (lnHa ~ lnk)

~ 10 good e-folds in a (k~10-4Mpc-1 to ~ 1 Mpc-1 LSS)

~ 10+ parameters? Bond, Contaldi, Kofman, Vaudrevange 05…08 H(), V() Cosmic Probes now & then CMBpol (T+E,B modes of polarization), LSS

Inflation Now1+w(a) goes to 2(1+q)/3

~1 good e-fold. only ~2params Zhiqi Huang, Bond & Kofman 07

Cosmic Probes Now CFHTLS SN(192),WL(Apr07),CMB,BAO,LSS,Ly

Cosmic Probes Then JDEM-SN + DUNE-WL + Planck +ACT/SPT…

V’()@ pivot pt

~0 to 2 to 3/2 to ~.4 now, on its way to 0?

Constraining Inflation Trajectories, now & then

Standard Parameters of Cosmic Structure Formation

Òk Òbh2 ÒË nsÒdmh2 ntlnAs r = A t=Asüc

òø `à1s

r < 0.6 or

< 0.28 95% CLø lnû2

8 nçs

ns = .958 +- .015 (+-.005 Planck1)

.93 +- .03 @0.05/Mpc run&tensor

r=At / As < 0.28 95% CL (+-.03 P1)

<.36 CMB+LSS run&tensor;

< .05 ln r prior!

dns /dln k=-.038 +- .024 (+-.005 P1)

CMB+LSS run&tensor prior change?

As = 22 +- 2 x 10-10 fNL = (+- 5-10 P1)

The Parameters of Cosmic Structure FormationThe Parameters of Cosmic Structure FormationCosmic Numerology: aph/0611198 – our Acbar paper on the basic 7+; bckv07

WMAP3modified+B03+CBIcombined+Acbar06+LSS (SDSS+2dF) + DASI (incl polarization and CMB weak lensing and tSZ)

bh2 = .0226 +- .0006

ch2 = .114 +- .005

= .73 +.02 - .03

h = .707 +- .021

m= .27 + .03 -.02

zreh = 11.4 +- 2.5

1+w = 0.02 +/- 0.05 ‘phantom DE’ allowed?!

New Parameters of Cosmic Structure FormationÒk

Òbh2lnH(kp)

ï (k); k ù HaÒdmh2

=1+q, the deceleration parameter history

order N Chebyshev expansion, N-1 parameters

(e.g. nodal point values)

P s(k) / H 2=ï ;P t(k) / H 2

òø `à1s ; cf :ÒË

Hubble parameter at inflation at a pivot pt

Fluctuations are from stochastic kicks ~ H/2 superposed on the downward drift at lnk=1.

Potential trajectory from HJ (SB 90,91):

üc

à ï = d lnH =d lna

1à ïà ï = d lnk

d lnH

d lnkd inf = 1à ï

æ ïp

V / H 2(1à 3ï );

ï = (d lnH =d inf)2or dual ln P s(k);P t(k)

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

criterion [e.g., a Chebyshev-filter: data cannot constrain high ln k frequencies]

P(trajectory|data, th) ~ P(lnHp,k|data, th)

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

Likelihood theory prior / evidence

“path integral” over probability landscape of theory and data, with mode-function expansions of the paths truncated by an imposed smoothness

criterion [e.g., a Chebyshev-filter: data cannot constrain high ln k frequencies]

P(trajectory|data, th) ~ P(lnHp,k|data, th)

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

Likelihood theory prior / evidenceData:

CMBall

(WMAP3,B03,CBI, ACBAR,

DASI,VSA,MAXIMA)

+

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

Data:

CMBall

(WMAP3,B03,CBI, ACBAR,

DASI,VSA,MAXIMA)

+

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

Theory prior

The theory prior matters a lot for current

data. Not so much for a Bpol future.

We have tried many theory priors

e.g. uniform/log/ monotonic in k

(philosophy of equal a-prior probability hypothesis,

but in what variables)

linear combinations of grouped Chebyshev nodal

points & adaptive lnk-space cf. straight Chebyshev

coefficients (running of running …)

Theory prior

The theory prior matters a lot for current

data. Not so much for a Bpol future.

We have tried many theory priors

e.g. uniform/log/ monotonic in k

(philosophy of equal a-prior probability hypothesis,

but in what variables)

linear combinations of grouped Chebyshev nodal

points & adaptive lnk-space cf. straight Chebyshev

coefficients (running of running …)

1980

2000

1990

-inflation Old Inflation

New InflationChaotic inflation

Double Inflation

Extended 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 pNGB inflation

Old view: Theory prior = delta function of THE correct one and only theoryOld view: Theory prior = delta function of THE correct one and only theory

Radical BSI inflation variable MP inflation

Old view: Theory prior = delta function of THE correct one and only theoryOld view: Theory prior = delta function of THE correct one and only theory

New view: Theory prior = probability distribution on an energy landscape

whose features areare 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 coordinates in

the low energy landscape:

moduli fields, sizes and shapes of geometrical structures such as holes in

a dynamical extra-dimensional (6D) manifold approaching stabilization

moving brane & antibrane separations (D3,D7)

New view: Theory prior = probability distribution on an energy landscape

whose features areare 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 coordinates in

the low energy landscape:

moduli fields, sizes and shapes of geometrical structures such as holes in

a dynamical extra-dimensional (6D) manifold approaching stabilization

moving brane & antibrane separations (D3,D7)

Theory prior ~ probability of trajectories given potential parameters of

the collective coordinates X probability of the potential parameters X

probability of initial conditions

lns (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

lns (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

s self consistency: order 5

log prior r <0.34; .<03 at 1!

s 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 ï

pP s(k) / H 2=ï ;P t(k) / H 2

ï = (d lnH =d inf)2

lnPs lnPt no consistency:

order 5 uniform prior r<0.42

lnPs lnPt no consistency:

order 5 log prior r<0.08

lns (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

lns (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-sigma

uniform prior

r < 0.64

CL BB for lns (nodal 5) + 4 params inflation trajectories reconstructed from

CMB+LSS data using Chebyshev nodal point expansion & MCMC

CL BB for lns (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

Spider balloon 2009.9

uniform prior

log prior

Spider+Planck broad-band

error

2004

2005

2006

2007

2008

2009

Polarbear(300 bolometers)@Cal

SZA(Interferometer) @Cal

APEX(~400 bolometers) @Chile

SPT(1000 bolometers) @South Pole

ACT(3000 bolometers) @Chile

Planck08.8

(84 bolometers)

HEMTs @L2

Bpol@L2

ALMA(Interferometer) @Chile

(12000 bolometers)SCUBA2

Quiet1

Quiet2Bicep @SP

QUaD @SP

CBI pol to Apr’05 @Chile

Acbar to Jan’06, 07f @SP

WMAP @L2 to 2009-2013?

2017

(1000 HEMTs) @Chile

Spider

Clover @Chile

Boom03@LDB

DASI @SP

CAPMAP

AMI

GBT

2312 bolometer @LDB

JCMT @Hawaii

CBI2 to early’08

EBEX@LDB

LMT@Mexico

LHC

WMAP3 sees 3rd pk, B03 sees 4th ‘‘Shallow’ scan, 75 hours, fShallow’ scan, 75 hours, fskysky=3.0%, large scale TT=3.0%, large scale TT

‘‘deep’ scan, 125 hours, fsky=0.28% 115sq deg, ~ Planck2yrdeep’ scan, 125 hours, fsky=0.28% 115sq deg, ~ Planck2yr

B03+B98 final soon

Current state

October 06

You are seeing this before people in the

field

Current state

October 06

You are seeing this before people in the

field

Current state

October 06

Polarization

a Frontier

Current state

October 06

Polarization

a Frontier

WMAP3 V band

CBI E CBI B

CBI 2,5 yr EE, ~ best so far, ~QuaD

Does TT Predict EE (& TE)? (YES, incl wmap3 TT) Inflation OK: EE (& TE) excellent agreement with prediction from TT

pattern shift parameter 0.998 +- 0.003 WMAP3+CBIt+DASI+B03+ TT/TE/EE pattern shift parameter 1.002 +- 0.0043 WMAP1+CBI+DASI+B03 TT/TE/EE Evolution: Jan00 11% Jan02 1.2% Jan03 0.9% Mar03 0.4%

EE: 0.973 +- 0.033, phase check of CBI EE cf. TT pk/dip locales & amp EE+TE

0.997 +- 0.018 CBI+B03+DASI (amp=0.93+-0.09)

Current high L frontier state Nov 07

Current high L frontier state Nov 07

CBI5yr sees 4th 5th pkCBI5yr excess 07,

marginalization critical to get

ns & dns /dlnk

WMAP3 sees 3rd pk, B03 sees 4th

Jan08@AAS: CBI5yr+ full ACBAR data ~ 4X includes 2005 observations

Planck1yr simulation: input LCDM (Acbar)+run+uniform tensor

r (.002 /Mpc) reconstructed cf. rin

s order 5 uniform prior

s order 5 log prior

GW/scalar curvature: current from CMB+LSS: r < 0.6 or < 0.25 (.28) 95%; good shot at 0.02 95% CL with BB polarization (+- .02 PL2.5+Spider), .01 target

BUT foregrounds/systematics?? But r-spectrum. But low energy inflation

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

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

s order 5 uniform prior

s order 5 log prior

lnPs lnPt (nodal 5 and 5)

http://www.astro.caltech.edu/~lgg/spider_front.htmhttp://www.astro.caltech.edu/~lgg/spider_front.htm

No Tensor

SPIDER Tensor Signal

Tensor

• Simulation of large scale polarization signal

forecast Planck2.5

100&143

Spider10d

95&150

Synchrotron pol’n

Dust pol’n

are higher in B

Foreground Template removals

from multi-frequency data

is crucial

B-pol simulation: ~10K detectors > 100x Planck

input LCDM (Acbar)+run+uniform tensor

r (.002 /Mpc) reconstructed cf. rin

s order 5 uniform prior s order 5 log prior

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

also input trajectory is recovered

Chaotic inflation

Power-law inflation

Roulette inflation Kahler moduli/axion

Natural pNGB inflation

Radical BSI inflation

Uniform acceleration, exp V: rnsrns

Uniform acceleration, exp V: rnsrns

V/MP4 ~2rns

4 rns

V/MP4 ~2rns

4 rnsV (f|| , fperp

(k) but isoc feed, r(k), ns(k)

V/MP4 ~red

4sin2fred -1/2nsfred-2

to match ns.96, fred~ 5, r~0.032to match ns.97, fred~ 5.8, r ~0.048 ,

V/MP4 ~red

4sin2fred -1/2nsfred-2

to match ns.96, fred~ 5, r~0.032to match ns.97, fred~ 5.8, r ~0.048 ,

D3-D7 brane inflation, a la KKLMMT03 typical r < 10-10typical r < 10-10

. 2/nbrane1/2 << 1 BM06

General argument (Lyth96 bound): if the inflaton < the Planck mass, then over

since = (dd ln a)2 & r hence r < .007 …N-flation?

r <~ 10-10 As & ns~0.97 OK but by statistical selection!

running dns /dlnk exists, but small via small observable window

r <~ 10-10 As & ns~0.97 OK but by statistical selection!

running dns /dlnk exists, but small via small observable window

Roulette inflation Kahler moduli/axion

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.

focus on “4-cycle Kahler moduli in large volume limit

of IIB flux compactifications” Balasubramanian,

Berglund 2004, + Conlon, Quevedo 2005, + Suruliz 2005

Real & imaginary parts are both important BKPV06

V~MV~MPP44 PPs s r r (1-(1-3)3) 3/2 3/2

~~ (10(101616 GevGev))44 r/0.1 r/0.1 (1-(1-3)3)

~(few x1013 Gev)4

ns ~ - dln ~ - dln dlnk /(1-dlnk /(1-i.e., a finely-tuned potential shape

V~MV~MPP44 PPs s r r (1-(1-3)3) 3/2 3/2

~~ (10(101616 GevGev))44 r/0.1 r/0.1 (1-(1-3)3)

~(few x1013 Gev)4

ns ~ - dln ~ - dln dlnk /(1-dlnk /(1-i.e., a finely-tuned potential shape

INFLATION NOW

Inflation Now1+w(a)= sf(a/aeq;as/aeq;s) to ax3/2 = 3(1+q)/2 ~1

good e-fold. only ~2 eigenparams Zhiqi Huang, Bond & Kofman07: 3-param formula accurately fits slow-to-moderate roll & even wild rising baroque late-inflaton trajectories, as well as thawing & freezing trajectoriesCosmic Probes Now CFHTLS SN(192),WL(Apr07),CMB,BAO,LSS,Ly

s= (dlnV/d)2/4 = late-inflaton (potential

gradient)2

=0.0+-0.25 now;

weak as < 0.3 (zs >2.3) now

s to +-0.07 then Planck1+JDEM SN+DUNE WL, weak as <0.21 then, (zs >3.7)

3rd param s (~ds /dlna) ill-determined now & then

cannot reconstruct the quintessence potential, just the slope s & hubble drag info

(late-inflaton mass is < Planck mass, but not by a lot)

Cosmic Probes Then JDEM-SN + DUNE-WL + Planck1

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

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

as <0.3 data (zs >2.3)

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

45 low-z SN + ESSENCE SN + SNLS 1st year SN+ Riess high-z SN, 192 “gold”SN all fit with MLCS

illustrates the near-degeneracies of the contour plotillustrates the near-degeneracies of the contour plot

Forecast: JDEM-SN (2500 hi-z + 500 low-z)

+ DUNE-WL (50% sky, gals @z = 0.1-1.1, 35/min2 ) +

Planck1yr

s=0.02+0.07-0.06

as<0.21 (95%CL)

(zs >3.7)

Beyond Einstein panel: LISA+JDEM

ESA (+NASA/CSA)

s (~ds /dlna) ill-determined

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 (k) ~ r(k): with current data, the scalar power downturns ((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 drives to low r. a B-pol r~.001? breaks this prior dependence, maybe

Planck+Spider r~.02

Complexity of trajectories arises in many-moduli string models. Roulette example: 4-cycle complex Kahler moduli in large compact volume Type IIB string theory

TINY r ~ 10-10 if the normalized inflaton < 1 over ~50 e-folds then r < .007

~10 for power law & PNGB inflaton potentials. Is this deadly???

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 vs. nearly uniform acceleration. Both have strong Cdn roles. Bpol2050

PRIMARY END @ 2012? PRIMARY END @ 2012?

CMB ~2009+ Planck1+WMAP8+SPT/ACT/Quiet+Bicep/QuAD/Quiet +Spider+Clover

• 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. Parameter eigenmodes show what is probed

• 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 (s >0) and the phantom equivalent (s <0); use a 3-parameter model to well-approximate even rather baroque w-trajectories.

• We ignore constraints on Q-density from photon-decoupling and BBN because further trajectory extrapolation is needed. Can include via a prior on Q (a) at z_dec and z_bbn

• For general slow-to-moderate rolling one needs 2 “dynamical parameters” (as, s) & Q to describe w to a few % for the not-too-baroque w-trajectories.

• as is < 0.3 current data (zs >2.3) to <0.21 (zs >3.7) in Planck1yr-CMB+JDEM-SN+DUNE-WL future

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 s.

• a 3rd param s, (~ds /dlna) is ill-determined now & in a Planck1yr-CMB+JDEM-SN+DUNE-WL future

• To use: given V, compute trajectories, do a-averaged s & test (or simpler s -estimate)• for each given Q-potential, velocity, amp, shape parameters are needed to define a w-trajectory

• current observations are well-centered around the cosmological constant s=0.0+-0.25 • in Planck1yr-CMB+JDEM-SN+DUNE-WL future s to +-0.07• but cannot reconstruct the quintessence potential, just the slope s & hubble drag info• late-inflaton mass is < Planck mass, but not by a lot

• Aside: detailed results depend upon the SN data set used. Best available used here (192 SN), soon CFHT SNLS ~300 SN + ~100 non-CFHTLS. 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

End End