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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