Inflation its Cosmic Probes, now then Dick Bond Inflation Now 1+ w (a)= s f(a/a eq ;a s /a eq ;...
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Transcript of Inflation its Cosmic Probes, now then Dick Bond Inflation Now 1+ w (a)= s f(a/a eq ;a s /a eq ;...
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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