Condensed Matter, Particle Physics and Cosmologyfjeger/Krakow_6.pdf · F. Jegerlehner, IFJ-PAN, Krakow Lectures 2014, — nx Lect. 6 xo 379. The SM running Parameters at high scales

Lecture 6:Back to the SM: Higgs inflation in detail

Outline of Lecture 6:´ Analysis of SM Higgs Inflation

Topics:

o running couplings impact on cosmological solutionso Higgs field at the Planck scaleo how to get enough inflationo slow-roll inflation criteriao evolution of the various energy componentso metric perturbations, primordial fluctuationso polarization, BICEP2 and trans-Planckian physics

F. Jegerlehner, IFJ-PAN, Krakow Lectures 2014, —≪x Lect. 6 x≫ 379

http://www-com.physik.hu-berlin.de/~fjeger/Krakow_5.pdf


The SM running Parameters at high scalesMS parameters at various scales for MH = 126 GeV and µ0 ' 1.4 × 1016 GeV. C1 and C2 are the one- and

two-loop coefficients of the quadratic divergence. The last two columns show corresponding results fromDegrassi et al. [DEGea]

coupling \ scale MZ Mt µ0 MPl Mt [DEGea] MPl [DEGea]g3 1.2200 1.1644 0.5271 0.4886 1.1644 0.4873g2 0.6530 0.6496 0.5249 0.5068 0.6483 0.5057g1 0.3497 0.3509 0.4333 0.4589 0.3587 0.4777yt 0.9347 0.9002 0.3872 0.3510 0.9399 0.3823yb 0.0238 0.0227 0.0082 0.0074yτ 0.0104 0.0104 0.0097 0.0094√λ 0.8983 0.8586 0.3732 0.3749 0.8733 i 0.1131λ 0.8070 0.7373 0.1393 0.1405 0.7626 - 0.0128

C1 −6.768 −6.110 0 0.2741C2 −6.672 −6.217 0 0.2845

m[GeV] 89.096 89.889 97.278 96.498 97.278




The location of the zeros of Ci, C′i = Ci + λ and Xi = 18

(2 C′i − λ

)as a function of scale in GeV, at one (i = 1) and

two (i = 2) loops.

C1 C2 C′1 C′2 X1 X2

1.42 × 1016 1.82 × 1016 7.77 × 1014 9.94 × 1014 3.25 × 1015 4.15 × 1015

The SM dimensionless couplings in the MS scheme as a function of therenormalization scale.




Left: the coefficient of the quadratic divergence term at one and two loopsas a function of the renormalization scale. Right: the Higgs phase transition

in the SM: shown is X = sign(m2bare) × log10(|m2

bare|) which representsm2

bare = sign(m2bare) × 10X. The band represents the parameter uncertainties.




Effect of finite temperature on the phase transition




o SM properties near the Planck scale

l perturbation theory works up to MPlanck

l Higgs vacuum remains stable up to MPlanck

l The bare Higgs mass changes sign around 1.4 × 1016 GeV below MPlanck

l this triggers the Higgs transition into the symmetric phase

l the bare Higgs mass is enhanced by the “quadratic divergence”

l the SM Higgs triggers inflation and reheating

l all properties known: SM physics boosted to MPlanck (parameters known)

l only the value of the Higgs field φ(µ = MPlanck) in unknown




Higgs inflation, the Higgs talking to gravityScalar field φ now the Higgs!

The SM Higgs affects the evolution of the universe by its contribution toenergy density and pressure:

ρφ =12φ2 + V(φ) ; pφ =

12φ2 − V(φ) .

They enter the dynamics via the Friedmann equations and the field equation:

a/a = −`2

2 (ρ + 3p)

a2/a2 + k/a2 = `2 ρ

The slow-roll inflation condition 12φ

2 V(φ) implies inflation withcorresponding dark energy equation of state w = p/ρ ' −1.




The Hubble constant is given by H2 = `2[V(φ) + 1

2 φ2]

= `2 ρ.

The field equationφ + 3Hφ = −V ′(φ) ≡ −dV(φ)/dφ

tells us that φ(t) in general performs damped oscillations and predominantly

r is oscillating if |φ| 3H|φ|

r or exponentially decaying if |φ| 3H|φ|

l our LEESM: Higgs potential with quadratically enhanced mass term

Ü seems to satisfy inflation condition 12 φ

2 V(φ)!!!

Higgs a good candidate for the inflaton




l a dominant mass term also looks to imply the inflaton to representessentially a free field (Gaussian).

This seems to be well supported by recent Planck mission constraints onnon-Gaussianity: Φ(~k) gravitational potential

〈Φ(~k1)Φ(~k2)Φ(~k3)〉︸︷︷︸three point correlation

= (2π)3 δ(3)(~k1 + ~k2 + ~k3)︸︷︷︸enforces triangular configuration

fNL F(k1, k2, k3)︸︷︷︸bispectrum

Three limiting cases

fNL

Local Equilateral Orthogonal2.7 ± 5.8 −42 ± 75 −25 ± 39

No evidence for non-Gaussianity




Non-Gaussianity: CMB angular bispectrum




The amount of inflation is quantified by the inflation exponent Ne given by

Ne = lna(tend)

a(tinitial)=

te∫ti

H(t) dt =

φe∫φi

Hφ

dφ

= −8πM2

Pl

φe∫φi

VV ′

dφ = H (te − ti) ,

follows from field equation Hdt = −H2/V ′ dφ and first Friedman equationH2 = `2V in the slow-roll approximation.

o times: ti beginning, te end of inflation; fields: φi = φ(ti), φe = φ(te)

o provided H = constant: Ne = H (te − ti), should be a good approximationwhen the total energy density ρtot ' ρΛ is dominated by thecosmological constant (CC).




o in symmetric phase V/V ′ > 0 and hence φi > φe .

o rescaling of the potential does not affect inflation, but the relative weightof the terms is crucial.

A precise analysis of the relative importance of the various possiblecomponents will presented below

r for the SM Higgs potential in the symmetric phase, denotingz ≡ λ

6 m2 , and V(φ) = V(0) + ∆V(φ)

we get two terms in the integrandV(0)2m2

1φ

11+zφ2 and ∆V

V′ =φ4

(1 + 1

1+zφ2

)and for the integral

I =

φi∫φe

VV ′

dφ =V(0)2m2

ln φ2i

φ2e− ln

φi2 z + 1

φe2 z + 1

+18

[φ2

i − φ2e +

1z

lnφi

2 z + 1φe

2 z + 1

]




such that

Ne =8πM2

Pl

I .

Below we will show that

V(0) =m2

2〈0|φ2|0〉 +

λ

24〈0|φ4|0〉

like m2 and z = λ6m2 all are known SM quantities! Ne large requires φi φe.

What is φi?

The Higgs field is the only quantity we cannot predict from low energy infor-mation! It is not an observable in the LEET!

r we adopt φi ' 4.51 MPl ,a value motivated by the amount of inflation required (see below)




Taking into account the running of parameters as given by the standard MSRG,

r we find φe ' 2.01 × 10−3MPl and Ne ≈ 64.68 at the end of inflation at aboutt ' 450 tPl , a value not far above the phenomenologically requiredminimum bound. Ne may be increased by increasing φi.




The inflation era I: Ne ≈ 66 is reached at time t ≈ 50 tPl . The Hubble constantH is satisfying H ≈ `

√V(φ) very well shortly after Planck time. The

evaluation of Ne agrees very well with the numerical Neff = ln a(t)/a(tPl)obtained by solving the coupled set of dynamical equations.




The inflation era II: dark energy contributions during inflation. Shown arethe separate terms of the Higgs potential together with the Higgs kineticterm. Slow-roll inflation stops at about t ≈ 450 tPl when Lkin ∼ Vmass Vint ,after which the Higgs behaves as a damped quasi-free oscillating field.




As we will see, SM Higgs inflation is far from working obviously. The reasonwhy SM inflation is quite tricky is the fact that the form of the potential isgiven and the parameters are known. What is at our disposal is essentiallyonly the value of the Higgs field at the Planck scale, since in the experimen-tally accessible low energy region the Higgs field is not an observable andwe only know its vacuum expectation value.

In the following we are dealing with physics near the Planck scale, where thebare theory resides, and by φ,V(φ), λ and m we denote the bare quantities(fields and parameters), if not specified otherwise.




The profile of Higgs inflation

Can the LEESM scenario explain inflation?

What we know:r for 126 GeV there is a Higgs transition at about µ0 ∼ 1.4 × 1016 GeV

Ü for µ > µ0 the SM is in the

symmetric phase

r we know λ(µ = MPl) and m2bare(µ = MPl)

r we do not know φ(µ = MPl) however!

However, as follows from the discussion above, inflation requires a largefield in order to get sufficient amount of inflation!

Bare mass term is

m2 ∼ δm2 'M2

Pl

32π2 C(µ = MPl) ' (0.0295 MPl)2 , or m2(MPl)/M2Pl ≈ 0.87 × 10−3 .




Ü four very heavy Higgses: H,H±, φ (very heavy Majorana singlet νM’s ?).

For large slowly varying fields, the field equation of motion simplifies to theslow-roll equation 3 Hφ ≈ −V ′ with H ≈ `

√V, which describes a decay of the

field.In the symmetric phase the key object of interest is the SM Higgs potential

V(Φ) = m2 Φ+Φ + λ3! (Φ+Φ)2

eventually dominated by the mass term m2 Φ+Φ . Here Φ is the complex SMS U(2) Higgs doublet field, which in the symmetric phase includes four heavyphysical scalars:

Φ =

(φ+

φ0

); φ+ = i φ1−i φ2√

2, φ0 =

H−i φ√

2, φ = φ3

in terms of the real fields H, φi , (i = 1, 2, 3). In the broken phase 〈0|H|0〉 = v,the φi’s transmute to gauge degrees of freedom and we get the Higgspotential




V(H) = m2

2 H2 + λ24 H4 ,

which is what we considered so far.




Physics in symmetric phase:

S U(3)c ⊗ S U(2)L ⊗ U(1)YÜ

Ü S U(3)c ⊗ U(1)em

unbroken gauge group assignments QED: γ’s, charges!

r We adopt the “would be” charge assignments, as ifin the broken phase.

The SM RG alone would tell us:

l m2 is small relative to δm2, As the Higgs field in the LEESM only dependslogarithmically on the cutoff,

l m2 〈0|Φ+Φ|0〉 ?

l Higgs field equation helps: predicts under slow-roll conditiondecays exponentially




Note: large Higgs fields work against Gaussianity, which requires thedominance of the Higgs mass term, and hence

φ2 <12 m2

λ=

3 C8 π2 λ

M2Pl =

3 (2λ + 32g′2 + 9

2g2 − 12y2

t )8π2λ

M2Pl

during inflation. With our input parameters, mass term dominance holdswhen

|φ| √

12/λ m (MPl) ≈ 0.2726 MPl .

o Since φ is decreasing rapidly during inflation, the condition of Gaussianitygets dynamically established at some point, at which however thedark energy density V(φ) has to be large enough to keep inflation going.

o We also note that RG evolution may yield a substantially smaller value forλ(MPl) in case yt(Mt) would be slightly larger than our estimate.




o If λ(MPl) = 0, the minimum value allowed for our scenario to work, we haveC(µ) ≤ 0 for yt(MPl) > 0.353, and there is no Higgs transition below MPl.

o Thus, for our scenario to work we need yt(MPl) < 0.353, given the gaugecouplings at MPl.

In any case, it is very interesting that the whole scenario based on theexistence of the Higgs phase transition sufficiently below the Planck scale,

requires a

window in parameter space which is very close to whatever

SM parameters estimates yield.

l Can this be an accident?

As our SM inflation scenario is supported by CMB data, we expect that atthe end something close to our scenario should turn out to describe reality.

At the Big Bang radiation in any case wins!




The Hubble constant in our scenario, in the symmetric phase, during theradiation dominated era is given by

H = `√ρ ' 1.66 (kBT )2

√102.75 M−1

Pl

such that at Planck timeHi ' 16.83 MPl .

One expects that V(φ) does not exceed too much a possible vacuum energyof size M4

Pl . The condition V(φ0) = M4Pl yields an initial value φ0 as follows:

with a = 6m2/λ, b = 24M4Pl/λ and r =

√b + a2 we find

φ0 =√

r − a ' 4.40 × 1019 GeV = 3.61MPl .

It is well known that the CMB horizon problem requires an inflation indexNe > 60. This index may be considered as a direct measure of the unknowninitial value φ0, and hence observational inflation data actually provides alower bound on this input. With the plausible estimate we actually obtainNe ∼ 57 but we easily can reach an index above Ne ∼ 60 by slightlyincreasing. We will adopt an initial field enhanced by 25% i.e. as our




standard input we choose

φ0 = 4.51MPl .

l in view of the fact that at Planck scale the Planck medium is known to beextremely hot, actually even higher values of φ0 make sense.We sill have ρV ρrad at tPl

Note that ρrad at tPl is an unambiguous SM prediction, just given by allrelativistic DOFs and Stefan Boltzmann’s law.

l the need for post Planckian fields makes inflation a rather involved issueas the dominance of terms changes rapidly with time(see last Figure presented above).

l keep in mind that the low energy effective theory expansion stops to makeany sense at tPl! Which does not mean that an effective Higgs field cannotbe present in the Planck medium, already.




An analytic solution for a mixed universe with radiation anddark energy for arbitrary curvature type

Here and early universe solution, assuming matter comes in latter.

We may write Friedmann’s equations in the form(aa

)2= `2

(ργ + ρΛ − ρk

)and a

a = `2(−ργ + ρΛ

),

where

ργ = ργi

(ai

a

)4; ρΛ =

Λ

3`2 ; ρk =kc2

`2a2 ,

represent the radiation, dark energy and curvature contribution to theenergy densities, respectively.

o indexed by i initial quantities at the Planck scale.




Adding the two Friedmann equations one obtains

12

d2

dt2 a2(t) = 2`2 ρΛ a2 − kc2 ,

which may be written as

X = E2Λ X ; X ≡ a2 − 2kc2t2

Λ ; EΛ = 2`√ρΛ = 1/tΛ ,

with solution

a(t) =

(c1 e−tEΛ +

12

c2 etEΛ + 2kc2 t2Λ

)1/2

.

o integration constants c1 and c2 may be fixed by assuming special initialvalues for ai = a(ti) and Hi = ai/ai at the initial time ti, which we choose to bethe Planck time.




One obtains

c1 =12

(a2

i − 2aiaitΛ − 2kc2t2Λ

)etiEΛ ; c2 =

(a2

i + 2aiaitΛ − 2kc2t2Λ

)e−tiEΛ ,

where ai can be calculated in terms of the initial densities ργi, ρki and ρΛ,which yields

(2tΛaa)i = a2i

(ργi

ρΛ

−ρki

ρΛ

+ 1)1/2

; 2kc2t2Λ =

12

a2iρki

ρΛ

.

The solution then reads

a(t) = ai

(cosh τ + 1

2ρkiρΛ

(1 − cosh τ) +(ργiρΛ−

ρkiρΛ

+ 1)1/2

sinh τ)1/2

where τ = (t − ti) EΛ is the reduced time . The Hubble function is then

H(t) =EΛ

2

(ai

a

)2(1 − 1

2ρki

ρΛ

)sinh τ +

(ργi

ρΛ

−ρki

ρΛ

+ 1)1/2

cosh τ

,F. Jegerlehner, IFJ-PAN, Krakow Lectures 2014, —≪x Lect. 6 x≫ 406



while the acceleration reads

aa

=E2

Λ

2

(1 −

12ρki

ρΛ

(ai

a

)2)− H2 .

We can then calculate the evolution of the various components, which weshow in the figure together with the top quark density and the heavy Higgsdensity to be introduced later.




The logs of the densities as a function of time, taking into account the Higgsdecay into tt [1] up to before the Higgs transition (φ0 ' 4.5 MPl). ρt [2] (in theplot little below ργ) is the top component in ργ, which denote the totalradiation density without the reheating part.




This graph shows that the dark energy term is dominating the density quitesoon after the big bang, and thus indeed is driving inflation. For laterreference we also show the top quark density ρt obtained from reheating [1]in comparison with the fraction of top quark radiation which is part of ργ.Included is also the non-relativistic heavy Higgs component ρH).




As an initial value for the Higgs field at MPl we adopt. In general, if the initialvalue of φ is exceeding about 1

5 MPl one can neglect φ in the field equation aswell as the kinetic term 1

2 φ2 in the Friedman equation. We expect inflation to

start at Planck time ti ≡ tinitial = tPl ' 5.4 × 10−44 sec and to stop definitely attCC ' 2.1 × 10−40 sec the at drop of the CC to be discussed later. As mentionedearlier the efficient era of slow-roll inflation ends at about t ' 450 tPl. Weadopt, somewhat arbitrary, a period including the bare Higgs transition pointat te ≡ tend = tHiggs ≈ 4.7 × 10−41 sec . As m is substantially lower than MPlactually for strong fields the interaction term is dominating. Then φ decaysexponentially like

φ(t) = φ0 e−E0 (t−t0) ; E0 =

√2λ

3√

3`≈ 4.3 × 1017 GeV; Vint Vmass ,

while a dominant mass term leads to a decay linear in time

φ(t) = φ0 − X0 (t − t0) ; X0 =

√2m

3`≈ 7.2 × 1035 GeV2; Vmass Vint .




However, this is not quite what corresponds to the true SM prediction. Aswe will argue below we have to account for a cosmological constant termV(0) ≡ 〈V(φ)〉, which in the field equation contributes the Hubble constant asH ' `

√V(0) + ∆V(φ). At the begin of inflation V(0) is of size comparable to

Vint while later on the mass term starts to dominate over the interaction term,but still V(0) Vmass, such that actually also during this era we have anexponential decay

φ(t) ≈ φ0 e−E0 (t−t0) ; E0 ≈m2

3`√

V(0)≈ 6.6 × 1017 GeV; Vmass Vint

of the field. Thus in any case, during slow-roll inflation, the decay of thedynamical part of the Higgs field is exponential at dramatic rate. Actually, aswe will see, the cosmological constant proportional to ρΛ = V(φ) ≈ V(0) anda corresponding Hubble constant H ≈ `

√V(0) long after slow-roll inflation

has ended, will decrease dramatically when V(0) drops essentially to zero ata scale µCC ' 5.0 × 1015 GeV . Without the contribution V(0), the fast decay ofthe Higgs field could be in contradiction with the observationally favored




slow-roll scenario. An important point here is that the Higgs potential has acalculable non-vanishing vacuum expectation value in the bare system.Vacuum contractions are ruled by Wick ordering: φ2 = 〈φ2〉 + φ′2 andφ4 = 〈φ4〉 + 3! 〈φ2〉φ′2 + φ′4 which yields a constant V(0) = 〈V(φ)〉 and a massshift m′2 = m2 + λ

2〈φ2〉 plus the potential in terms of the fluctuation field ∆V(φ) .

Note that in SM notation 〈0|Φ+Φ|0〉 = 12〈0|H

2|0〉 ≡ 12 Ξ is a singlet contribution.

We thus obtain a quasi-constant vacuum density

V(0) =m2

2Ξ +

λ

8Ξ2 ; Ξ =

M2Pl

16π2 ,

the VEV of the potential, which does contribute to the cosmologicalconstant. The field equation, which only involves the time dependent part, isaffected via a modified Hubble constant and the shifted effective mass (seebelow). The Z2 symmetry Φ→ −Φ and the SM gauge symmetry remainuntouched. The Φ+Φ VEV 1

2 Ξ in principle should be calculable in a latticeSM. One has to be aware of course that we do not know the true underlingPlanck ether system. Such estimates in any case would be instructive in




understanding underlying mechanisms. Since, as a result of the SM RGevolution, effective SM parameters are well within the perturbative regimeone can actually calculate Ξ . In leading order we just have Higgs self-loops

〈H2〉 =: ; 〈H4〉 = 3 (〈H2〉)2 =:

given by Ξ = 〈H2〉 = Λ2

16π2 . Again, for the fluctuation field, which decaysexponentially, in the early phase of inflation we adopt the initial valueφ0 ≈ 4.51MPl estimated above. In the potential of the fluctuation field and thecorresponding field equation the mass square now is given by

m′2 = m2 +λ

2Ξ .

This actually is a very interesting shift as it modifies the Higgs transitionpoint to lower values with new effective coefficient

C′1 = C1 + λ = 3 λ +32g′2 +

92g2 − 12 y2

t .




For our values of the MS input parameters, we obtains

µ0 ≈ 1.4 × 1016 GeV→ µ′0 ≈ 7.7 × 1014 GeV ,

as a relocation of the Higgs transition point.

We can now solve the coupled system of equations numerically e.g. by theRunge-Kutta method. It adds to the above analytic solution for a constant“cosmological constant” the Higgs field dynamics. Results for the FRWradius a(t) and the field φ(t) together with the derivatives are displayed inFigs.




Expansion start: the FRW radius and its derivatives for k = 1 as a function oftime all in units of the Planck mass, i.e. for MPl = 1.




Expansion before the Higgs transition: the FRW radius and its derivativesfor k = 1 as a function of time all in units of the Planck mass, i.e. for MPl = 1.




The Higgs field and its derivatives for k = 1 as a function of time all in unitsof the Planck mass at start.




The Higgs field and its derivatives for k = 1 as a function of time all in unitsof the Planck mass. The Higgs field decay before the Higgs transition. Thefield starts oscillating strongly like a free field once Lkin ∼ Vmass whileVint Vmass.




The mass-, interaction- and kinetic-term of the bare Lagrangian in units ofM4

Pl as a function of time. The mass term is dominating in the ranget ' 100 to 450 tPl , where the slow-roll era ends and damped quasi-free fieldoscillations start. Note the behavior of V(φ).




As previous. Left: correct solution in terms of effective couplings. Right:fake solution based on constant couplings.

Figure shows how the different terms of the bare Lagrangian evolve. At laterinflation times the mass term is dominating as originally expected, but thedominance is not very pronounced. The temporary mass term dominance isimportant for the observed Gaussianity by the Planck mission. At aboutt ' 450 tPl slow-roll inflation ends and free field oscillations begin. The twopanels illustrate the difference obtained between working with running




couplings vs. keeping couplings fixed as given at the Planck scale. It turnsout to be crucial to take into account the scale dependence of the coupling,throughout the calculation. How can it be that the minor changes in SMcouplings between MPl and µ0 make up such dramatic difference? Thereason is quite simple, the effects are enhanced by the quadratic

“divergence” enhancement factorM2

Pl32π2 , which actually makes the whole thing

work. One of the key criteria during the inflation era is the validity of thedark energy equation of state w = p/ρ = −1, which we display in Fig as afunction of time before the bare Higgs transition point µ0. In fact w = −1 isperfectly satisfied quite early after Planck time. This shows how dark energyis supplied by the Higgs system.




Slow-roll inflation criteria

In text books often slow-roll inflation criteria are discussed, whichessentially test properties of the scalar potential of an inflation model. Asdiscussed earlier slow-roll means |φ| |3 Hφ|, |V ′|

Ü

3 Hφ = −V ′

slow roll equation of motion

i) given p ' −ρ Ü 1 φ2

V=

V′2

9VH2 , H2 '8πV3 M2

Pl

=M2

Pl

24π

(V ′

V

)2

≡23ε .

ε =M2

Pl8π

12

(V′V

)2




1st slow-roll condition:ε 1 ensures p ' −ρ

ii) ensure slow-roll for a long time: Ü maintain φ 3 Hφ using slow-rollequation of motion

φ2 =V′2

9H2 =M2

Pl

24πV′2

V; φφ =

M2Pl

48π

(V′2

V

)′· φ

φ =M2

Pl

24πV ′ ·

(V′′

V−

12

V′2

V2

)Define:

η ≡M2

Pl8π

V′′

V

thenφV′ =

φ

3 Hφ = 13 (ε − η) 1

2nd slow-roll condition:η 1 if ε 1




then:ε ' εH = − H

H2 = 1 − 1H2

aa ; η ' ηH =

εHεHH

means ε 1 implies εH 1 Ü a > 0 consistent.

εH, ηH ’physical’ Hubble slow-roll parameters

What do we need more? what stops inflation?

v need V(φ) that satisfies ε, η 1 at some φi

v ε, η < 1 for Ne ' 60 e-folds at least, then ε > 1 must be reached at someφe Ü slow-roll ends, φ oscillates rapidly about φ = 0 when |φ| < MPl/5 Üoscillations lead to abundant particle production reheating the universe

Note: when φ is large the friction term in φ + 3Hφ = −m2φ is large and thefield changes slowly towards lower values, once the field is sufficientlysmall the friction term is small and oscillation sets in (see later).




Perturbations of the metric

l In General Relativity, need to take into account perturbations of the wholemetric, not just the inflaton field

l Decompose metric perturbations into scalar, vector and tensor perturba-tions

l Inflation generates scalar ( curvature ) and tensor perturbations

(

gravitational waves ), but no vector perturbations

l Properties of the perturbations depend on the inflaton potential




Primordial Perturbations Inflated

0

H−1

a

inflation

reheating

radiationdominated

︸︷︷︸︸︷︷︸

physicalscale

λ > 1/Hfluctuationsfrozen in

λ < 1/Hfluctuations

evolve

λ < 1/Hfluctuations

evolve

horizon scaleλ ∼ 1/H

wavelenght λ of adensity fluctuation

Quantum fluctuations of φ are stretched beyond the horizon and freeze in




Perturbations and their Power Spectral Scalar (curvature) perturbations

Ps(k) ∝V3

(V ′)2

∣∣∣∣∣∣k=aH

≈ A∗s

(kk∗

)ns−1+...

l Tensor (gravitational waves) perturbations

Pt(k) ∝ V |k=aH ≈ A∗t

(kk∗

)nt+...

l Tensor-to-Scalar ratio

r ≡Pt(k)Ps(k)

∣∣∣∣∣k=0.002 Mpc−1




A∗s, A∗t scalar, tensor amplitudes, ns, nt scalar, tensor spectral indicesk wave vector evaluated at the epoch of horizon exit k = aHk∗ reference wave vector k∗ = 0.05 Mpc−1

r no significant non-trivial higher-order correlations (non-Gaussianity)r if single field: adiabatic perturbations (i.e., no isocurvature modes)

o Predictions of the simplest inflaton modelssingle-field canonical slow-roll inflation

ε ≡M2

Pl

8π12

(V ′

V

)2

, η ≡M2

Pl

8πV ′′

V

1st slow-roll condition: need dark energy φ2/V 1ε 1 ensures p ' −ρ

2nd slow-roll condition: persistence of inflation φ 3Hφη 1 in addition to ε 1




3 Spatial flatness ΩK ∼ 10−5

3 Almost (but not exactly) scale-invariant curvature perturbationsns = 1 − 6ε + 2η , nt = −2ε

3 Adiabatic initial conditions3 Nearly Gaussian initial fluctuations fNL < 13 Background of gravitational waves (tensor perturbations) r = 16ε

Note: only if the Higgs is the inflaton , the form and the parameters of thepotential are known except for φ(MPl), which must be of trans-Planckian mag-nitude in order to get N >

∼ 60

all other inflaton models have more flexibility to adjust the shape of thepotential and its parameter values!




o Gravitational waves and the Lyth bound

à gravitational waves expected to be significant only for ` <∼ 100

à first e-folds (Nin ∼ 4) after waves leave horizondNdφ = −H

φ= V

M2PlV′

= 1√

2εMPl

with dN → Nin ∼ 4, dφ→ ∆φ and ε = r/16 one obtains the

Lyth bound

(Lyth 1997 )

∆φ ' 0.5 MPl

√r

0.1

“Primordial gravitational waves will not be detectable by Planck unlessr >∼ 0.1, and are unlikely to be detected in the foreseeable future unlessr >∼ 0.01. Most models of inflation give a much smaller value. To see why,note first that the waves are significant only up to ` <∼ 100, corresponding tothe first 4 or so e-folds of inflation after our Universe leaves the horizon.”At the end we need more than about Ne ≈ 50 e-folds:




∆φ ' 2.0 MPlNe50

√r

0.01

à detectable gravitational waves require ∆φ >∼ 2 to 6 MPl

l BICEP2 r = 0.15 to 0.27 Ü ∆φ >∼ 7.7 to 10.4 MPl

Requires trans-Planckian fields “placing the inflation model out oftheoretical control”!?

Well: who beliefs that we ever would be able to theoretically control thePlanck medium? Alone, the fact that we are dealing with a plenitude ofinflation models and scenarios tells us that although we have some basicconcepts we have little chance to pin down a specific theory. In the LEESMscenario the expansion in E/ΛPl looses its meaning at E = ΛPl, which doesnot mean that we are not able to learn about some collective properties ofthe system. The Lyth bound is no physical bound (at best a technical one).




CMB data confront inflation model predictionsCMB temperature power spectrum

CMB bispectrum fNL < 1 CMB B-polarization r = 0.20+0.07−0.05?

3 Spatial flatness ΩK ∼ 10−5

3 Adiabatic initial conditions3 Almost scale-invariant

curvature perturbations ns = 0.9603 ± 0.0073

Nearly Gaussian

à

initial fluctuations

Background ofgravitational waves à(tensor perturbations)




PolarizationLinks: see Wayne Hu R≫ , Scott and Smoot R≫ , Jan Hamann R≫

v results from the anisotropy ofThomson scattering

v originates from last scattering duringrecombination and from laterscattering off re-ionized material

v it implies a quadrupole anisotropyin the photon distribution

l Polarization in CMB largest in range 100 ≤ ` ≤ 1000 andfor much smaller `, if caused by re-ionization

l Polarized part amounts to about 1/10 of unpolarized signals i.e. ∼ 10−6 K


http://background.uchicago.edu/~whu/Presentations/polar.pdf

http://hepdata.cedar.ac.uk/lbl/2009/reviews/rpp2009-rev-cosmic-microwave-background.pdf

http://theorieseminar.desy.de/sites2009/site_theorieseminar/content/e41199/e100259/e172224/Hamann_2014.pdf



l Polarization is characterized by Stokes-parameters:intensity I, linear polarization Q, phase U, circular polarization V(easier to measure)

l Parity-even, curl-free E-mode and parity-odd, grad-free B-mode(easier to handle theoretically). Temperature anisotropy = T-mode.

l Power spectrum of polarization exhibits complementary information to C`.

CMB signals form primordial perturbations:

scalar vector tensor

T E B

B-polarization is the best probe of tensor perturbations




illustration by W. Hu




CMB polarization spectra

3 observables T,E,B à TT, EE, BB, TE, TB, EB illustration by W. Hu




T, E: scalarB: pseudoscalar à TB,EB=0




Cross power spectrum of the temperature anisotropies and E-modepolarization (TE) signal from WMAP, together with estimates fromBOOMERANG, DASI, QUAD, CBI and BICEP. Note that the y-axis here is notmultiplied by the additional C, which helps to show both the large and small




angular scale features.

Power spectrum of E-mode polarization (EE) from several differentexperiments, plotted along with a theoretical model which fits WMAP plus




other CMB data.

Computed Planck TE (left) and EE spectra (right). The red lines show thepolarization spectra from the base CCDM Planck+WP+highL model, which isfitted to the TT data only.




Spectral index

The Planck TT power spectrum plotted as ‘2D‘ against multipole, comparedto the best-fit base CCMD model with ns = 0.96 (red dashed line). The best-fitbase CCDM model with ns constrained to unity is shown by the blue line.




BICEP2 and tensor vs scalar fluctuationsRecently BICEP2 has obtained a first measurement of the amplitudes oftensor fluctuations relative to the scalar ones:




BICEP2 apodized E-mode and B-mode maps filtered to 50 < ` < 120. Right: Theequivalent maps for the first of the lensed- CDM+noise simulations. The colorscale displays the E-mode scalar and B-mode pseudoscalar patterns while thelines display the equivalent magnitude and orientation of linear polarization. Notethat excess B mode is detected over lensing+noise with high signal-to-noise ratioin the map (s/n > 2 per map mode at ` ≈ 70). (Also note that the E-mode andB-mode maps use different color and length scales.)




BB angular power spectrummeasured by BICEP2

[BICEP2 2014]

Consistent with expected lensing

from E-polarisation

BB angular power spectrummeasured by BICEP2

[BICEP2 2014]

Excess signalDue to tensor modes (?!)

Is the signal real?

Experimental systematics?– Pointing error

– Beam uncertainty

Passed consistency checks:– jackknife tests

– no EB- and TB-signal

→ very unlikely to accountfor excess signal

Is the signal of cosmological origin?

Astrophysical foregrounds– Polarised point sources

– Synchrotron emission

– Polarised dust emission

taken from Jan Hamann




The result

r = 0.20+0.07−0.05

o exceeds typical predictions of inflation models by a factor about 20

o assume this signal is real and that it is caused by primordial tensor pertur-bations from inflation

l is it compatible with LEESM Higgs inflation?

à the SM is what we see at low energy E/ΛPl 1

à only the renormalizable tail is seen of a unknown cutoff system sittingat the Planck scale




à the fields we see are all some collective excitations, with effectivemasses tamed by symmetries as the chiral symmetry and the localgauge symmetries

à except from the Higgs field itself the form of the effective Lagrangian, inparticular of the Higgs potential, and the values of the scale dependentparameters are given by extrapolation of known low energy physics,within uncertainties

à one thus can calculate tensor and scalar fluctuation amplitudes

At(k) =

128 V3 M4

Pl

k=aH

, As(k) =

128 πV3

3 M6Pl (V ′)2

k=aH

and the observed tensor to scalar ratio can be estimated:




r = (At/As)k=0.002 Mpc−1 ; r = PtPs

= 16ε ≤ 0.003(

50Ne

)2 (∆φMPl

)2

In terms of the observable Hubble constant and its derivative we may write

k3 Ps = −2π H4

M2PlH≈

(8πV)3

6 V ′2=

16π2

3V

M4Pl

1ε

= 2π2 As

k3 Pt = 32π H2

M2Pl≈

256π2

3V

M4Pl

= 2π2 At .

r ≡ PtPs

= 1π

(V ′MPl/V)2 = 16 ε .

F the Lyth bound ∆φMPl

= O(1)√

r0.01, says that for inflation to last sufficiently

long, ∆φ = φi − φe has to take on super-Planckian values, and as φi φe

this implies

φi = φ(MPl) ' ∆φ must be trans-Planckian




F the BICEP2 result r = 0.15 to 0.27 yields ∆φ ' 4.5 to 9.0 MPl, which is in therange of what LEESM inflation requires anyway (as many others)

F in any case statements like

“In effective field theory, Planck-mass suppressedhigher order operators would mess up things...”

are not conclusive, since the expansion in E/MPl at the Planck scale getsmeaningless in any case and we cannot any longer argue withfield monomials and the operator hierarchy appearing in thelow energy expansion

F what is important: the inflaton field is decaying very fast!




F once the sub-Planckian regime is reached, formally, given a truncatedseries of operators in the potential, the highest power is dominatingin the close-to Planckian regime

F once φ4 is dominating the decay is exponential, for higher dimensionaloperators it is faster than exponential, such that the fieldvery rapidly reaches the Planck- and sub-Planck regime

F soon the mass term is dominating after a very short period and beforethe kinetic term becomes relevant and slow-roll inflation ends

r so fears that in low energy effective scenarios with trans-Planckian fieldshigher order operators would mess up things are not in any sense justi-fied6.

6The constructive understanding of LEETs we have learned from Ken Wilson’s renormalization group based onintegrating out short distance fluctuations. This produces all kinds, mostly of irrelevant higher order interactions. Atypical example is the Ising model, which by itself seen as the basic microscopic system has simple nearest neighborinteractions only and by the low energy expansion develops a tower of higher order operators, which at the shortdistance scale are simply absent altogether. Such operators don’t do any harm at the intrinsic short distance scale.




r obviously, without the precise knowledge of the Planck physics, very closeto the Planck scale we never will be able to make a precise prediction ofwhat is happening. This however seems not to be a serious obstacle toquantitatively describe inflation and its properties as far as they can beaccessed by observation.

r the LEESM scenario in principle predicts not only the form of the effectivepotential not far below the Planck scale but also its parameters and theonly quantity not fixed by low energy physics is the magnitude of the fieldat the Planck scale

r we also have shown that taking into account the running of the parametersis mandatory for understanding inflation and reheating and all that.




à Trans-Planckian temperatures and fields

r trans-Planckian fields are not unnatural in a low energy effective scenariobecause the Planck medium exhibits a high temperature and temperaturefluctuations make higher excitations quite probable

r while the Planck medium will never be accessible to direct experimental tests, aphenomenological approach to constrain its effective properties is obviouslypossible, especially by data on the Cosmic Microwave Background (CMB) as wellas today’s form, composition and distribution of matter

In the extremely hot Planckian medium, the Hubble constant in the radiationdominated state with effective number

g∗(T ) = gB(T ) + 78 g f (T ) = 102.75

of relativistic degrees of freedom is given byH = `

√ρ ' 1.66 (kBT )2

√102.75 M−1

Pl ,at Planck time

Hi ' 16.83 MPl ,




such that a Higgs field of size

φi ' 4.51 MPl ,is not unexpected and could be as well somewhat higher.

Often it is argued that trans-Planckian field are unnatural in particular in aLEET scenario. I cannot see any argument against strong fields and LEETarguments (ordering operators with respect to a polynomial expansion andtheir dimension) completely loose their sense when E/ΛPl >∼ 1.

Provided the Higgs field decays fast enough towards the end of inflation weexpect the mass term to be dominant such that a Gaussian fluctuationspectrum prevails. The quasi-cosmological constant V(0) at these timesmainly enters the Hubble constant H and does not affect the fluctuationspectrum.

FFFFFFFFFF




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Condensed Matter, Particle Physics and Cosmologyfjeger/Krakow_6.pdf · F. Jegerlehner, IFJ-PAN, Krakow Lectures 2014, — nx Lect. 6 xo 379. The SM running Parameters at high scales