L. Alvarez-Gaume - Minimal Inflation

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Minimal Inflation

Work in collaboration with C. Gómez and R. Jiménez

Monday, 29 August, 2011

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Inflation is 30 years old

Originally Inflation was related to the horizon, flatness and relic problems

Nowadays, its major claim to fame is seeds of structure. There is more and more evidence that the general philosophy has some elements of truth, and it is remarkably robust...


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Summary of FRLWMatter is well represented by a perfect fluid

Tab = (p + ρ)uaub + pgab

Gab = 8πGTab

a

a

2

+k

a2=

8πG

3ρ

a

a= −4πG

3(ρ + 3p)

ρ + 3H(ρ + p) = 0

The Einstein equations are

S =1

16πG

√−g(R− 2Λ) +

√−g

−1

2gab∂aφ∂bφ − V (φ)

ρ =12φ2 + V (φ) p =

12φ2 − V (φ)

p = w ρ

ρ + 3a(t)a(t)

(1 + w)ρ = 0

ρ = ρ0

a0

a

3(1+w)

k = ±1, 0These are the possible curvatures distinguishing the space sections

ds2 = −dt2 + a(t)2 ds23

H ≡ a(t)a(t)

ρc =3H

2

8πG

Ω ≡ ρ

ρc1− Ω =

k

a2H2


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Accelerating the Universe

a

a= −4πG

3(ρ + 3p)

We need to violate the dominanat energy condition for a sufficiently long time.

If we take during inflation H approx. constant, the number of e-foldings:

(Courtesy of Licia Verde)

p < −13

ρ a(t) > 0|1− Ω| ∝ a−2

N = log

a(tf )a(ti)

= H(tf − ti)

|1− Ω(tf )| = e−2N |1− Ω(ti)|

Number of e-foldings could be 50-100, making a huge Universe.

We can use QFT to construct some models

S =1

16πG

√−g(R− 2Λ) +

√−g

−1

2gab∂aφ∂bφ − V (φ)

ρ =

12φ2 + V (φ) p =

12φ2 − V (φ)

V

V<< 1,

V

V<< 1Slow roll paradigm

Hybrid inflation φ2 << V (φ)


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Origins of Inflation

The number of models trying generating inflation is enormous. Frequently they are not very compelling and with large fine tunings.

Different UV completions of the SM provide alternative scenarios for cosmology, and it makes sense to explore their cosmic consequences.


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Basic properties

Enough slow-roll to generate the necessary number of e-foldings and the necessary seeds for structure.

A (not so-) graceful exit from inflation, otherwise we are left with nothing.

A way of converting “CC” into useful energy: reheating.

Everyone tries to find “natural” mechanisms within its favourite theory.


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Supersymmetry is our choice

In the standard treatment of global supersymmetry the order parameter of supersymmetry breaking is associated with the vacuum energy density. More precisely, in local Susy, the gravitino mass is the true order parameter.

Having a vacuum energy density will also break scale and conformal invariance.

When supergravity is included the breaking mechanism is more subtle, and the scalar potential far more complicated.

Needless to say, all this assumes that supersymmetry exists in Nature

Claim:It provides naturally an inflaton and a graceful exit


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SSB Scenarios

Observable Sector

Hidden Sector

MEDIATOR

Gauge mediation

Gravity mediation

Anomaly mediation

It is normally assumed that SSB takes places at scales well below the Planck scale. The universal prediction is then the existence of a massless goldstino that is eaten by the gravitino. However in the scenario considered, the low-energy gravitino couplings are dominated by its goldstino component and can be analyzed also in the global limit.

This often goes under the name of the Akulov-Volkov lagrangian, or the non-linear realization of SUSY

m3/2 =f

Mp=

µ2

Mp

µ → ∞M → ∞ m3/2 fixed


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Flat directions

One reason to use SUSY in inflationary theories is the abundance of flat directions. Once SUSY breaks most flat directions are lifted, sometime by non-perturbative effects. However, the slopes in the potential can be maintained reasonably gentle without excessive fine-tuning.

For flat Kahler potentials, and F-term breaking, there is always a complex flat direction in the potential. A general way of getting PSGB, the key to most susy models. The property below holds for any W breaking SUSY.

Most models of supersymmetric inflation are hybrid models (multi-field models, chaotic, waterfall...)

F = −∂W (φ).

V = ∂W (φ) ∂W (φ)V (φ + z F ) = V (φ)


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Important properties of SSB

The Akulov-Volkov-type actions provide the correct framework to analyse the general properties of SSB. We can use the recent Komargodski-Seiberg presentation.

The starting point of their analysis is the Ferrara-Zumino (FZ) multiplet of currents that contains the energy-momentum tensor, the supercurrent and the R-symmetry current

Jµ = jµ + θαSµα + θαSαµ + (θσνθ) 2Tνµ + . . .

DαJαα = DαX

X = x(y) +√

2θψ(y) + θ2F (y)

ψα =√

23

σµααS

αµ, F =

23T + i∂µjµ


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S=

d4θK(Φi, Φi) +

d2θW (Φi) +

d2θW (Φi)

Jαα = 2gi(DαΦi)(DαΦ)− 23 [Dα, Dα]K + i∂α(Y (Φ)− Y (Φ))

General Lagrangian

X = 4 W − 13D

2K − 1

2D

2Y (Φ)

X is a chiral superfield, microscopically it contains the conformal anomaly (the anomaly multiplet), hence it contains the order parameter for SUSY breaking as well as the goldstino field. It may be elementary in the UV, but composite in the IR. Generically its scalar component is a PSGB in the UV. This is our inflaton. The difficulty with this approach is that WE WANT TO BREAK SUSY ONLY ONCE! unlike other scenarios in the literature

The key observation is: X is essentially unique, and:

X → XNL

UV → IRX2

NL = 0SPoincare/Poincare


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Some IR consequences

L =

d4θ XNLXNL +

d2θ f XNL + c.c.

XNL =G2

2F+√

2θG + θ2F

This is precisely the Akulov-Volkov Lagrangian


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Coupling goldstinos to other fields: reheating

We can have two regimes of interest. Recall that a useful way to express SUSY breaking effects in Lagrangians is the use of spurion fields. The gluino mass can also be included...

msoft << E << Λ

E << msoft

The goldstino superfield is the spurion

Integrate out the massive superpartners adding extra non-linear constraints

d4θ

XNL

f

m2 QeV Q +

d2θXNL

f(B QQ + AQ QQ ) + c.c.

X2NL = 0, XNL QNL = 0 For light fermions, and similar conditions

for scalars, gauge fields,...

Reheating depends very much on the details of the model, as does CP violation, baryogenesis...


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An important part of our analysis is the fact that the graceful exit is provided by the Fermi pressure in the Landau liquid in which the state of the X-field converts once we reach the NL-regime. This is a little crazy, but very minimal however...

Some details

W (X) = f0 + f X

V = eK

M2 (K−1X,X

DWDW − 3M2

|W |2) D W = ∂X W +1

M2∂XK W

K(X, X) = XX

1 +

a(X + X)

2M− bXX

6M2−

cX2 + X2

9M2+ . . .

− 2M2 log

X + X

M+ 1


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The full lagrangian


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Primordial density fluctuations

f ∼ 1011−13 GeV =

M2pl

2

V

V

2

,

η = M2pl

V

V,

nS = 1− 6 + 2η,

r = 16

nt = −2,

∆2R =

V M4pl

24π2.


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ds2 = 2gzzdsdz = ∂z∂zK(α,β)M2(dα2 + dβ2)

S = L3

dta3

1

2g(α,β)M2(α2 + β2)− f2V (α,β)

t = τM/f S = L3f2m−13/2

dτa3

1

2g(α,β)(α2 + β2)− V (α,β)

z = M(α+ iβ)/√2

Choosing useful variables


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α + 3

a

aα +

1

2∂α log g(α2 − β

2) + ∂β log gαβ + g

−1V

α = 0

β + 3

a

aβ +

1

2∂β log g(β

2 − α2) + ∂α log gα

β + g

−1V

β = 0

a

a=

H

M3/2=

1√3

1

2g(α2 + β

2) + V (α,β)

H =

1

18

3V +

6V + 9V 2

DΦi/dt ∼ 0

Cosmological equations

The full equations of motion, neglecting fermions for the moment are:

Looking for the attractor and slow roll implies that the geodesic equation on the target manifold is satisfied for a particular set of initial conditions. This determines the attractor trajectories in general for any model of hybrid inflation. Numerical integration shows how it works. We have not tried to prove “theorems’ but there should be general ways of showing how the attractor is obtained this way

Plus fermionterms on the RHS


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0.0

0.5

1.0Α

0.0

0.5

1.0

Β

0.00

0.05

0.10

Ε

0.0

0.5

1.0Α

0.0

0.5

1.0

Β

0.0

0.5

1.0

The scalar potential

a=0, b=1, c=0a=0, b=1, c=-1.7


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0.2 0.4 0.6 0.8Α

0.1

0.2

0.3

0.4

0.5

Β

0.05 0.10 0.50 1.00 5.00t

0.02

0.04

0.06

0.08

0.10V

0.05 0.10 0.15 0.20 0.25 0.30Α

0.2

0.4

0.6

0.8

1.0

Β

Attractor and inflationary trajectories

Nearly a textbook example of inflationary potential


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Decoupling and the Fermi sphere

η = (mINF

m3/2)2

The energy density in the universe (f^2) contained in the coherent X-field quickly transforms into a Fermi sea whose level is not difficult to compute, we match the high energy theory dominated by the X-field and the Goldstino Fock vacuum into a theory where effectively the scalar has disappeared and we get a Fermi sea, whose Fermi momentum is

To produce the observed number of particles in the universe leads to gravitino masses in the 10 TeV region.

qF =

f

η


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Summary of our scenario

We take as the basic object the X field containing the Goldstino. Its scalar component above SSB behaves like a PSGB and drives inflation

Its non-linear conversion into a Landau liquid in the NL regime provides an original graceful exit

Reheating can be obtained through the usual Goldstino coupling to low energy matter

In the simplest of all possible such scenarios, the Susy breaking scale is fitted to be of the order of 10^13-14 GeV


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Thank you


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Back up slides


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Graceful exit

V = eK

M2 (K−1X,X

DWDW − 3M2

|W |2) D W = ∂X W +1

M2∂XK W

η = M2pl

V

V, K(X, X) = XX + . . . η = 1 + . . .

The theory is constructed with a Kahler potential violating explicitly the R-symmetry which phenomenologically is a disaster.

We also input our conservative prejudice that the values of the inflaton field should be well below the Planck scale.

The general supergravity lagrangian up to two derivatives and four fermions is given by a very simple set of function:

The Jordan frame function

The Kahler potential

The superpotential (and the gauge kinetic function and moment maps)


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Summary

The chiral superfield X is essentially defined uniquely in the ultraviolet. It has the following properties:

In the UV description of the theory, it appears in the right hand side the supercurrent equation (FZ) where it represents a measure of the violation of conformal invariance.

The expectation value of its F component is the order parameter of supersymmetry breaking.

When supersymmetry is spontaneously broken, we can follow the flow of X to the infrared (IR). In the IR this field satisfies a non-linear constraint and becomes the “goldstino" superfield. The correct normalization is 3X / 8 f. At low energies this field replaces the standard spurion coupling.

X2NL = 0,

XNL =G2

2 F+√

2 θ G + θ2 F.


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−3

4

−2h+

√2α

2+

1− (2h−

√2α)(−72+α(

√2+2α)(18+5α2))

72(1+√2α)

2

1 + α2

3 + 2

(1+√2α)2

= 0

0.10 0.15 0.20 0.25 0.300.9960

0.9965

0.9970

0.9975

0.9980

Α

h

Vanishing cosmological constant


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...continued

One could be more explicit, and choose some supersymmetry breaking superpotential. leading to an effective action description of X for scales well below M. It is however better to explicit examples of UV-completions of the theory. The potential is taken to be stable A - B >0.

We consider the beginning of inflation well below M, hence the initial conditions are such that a,b are much smaller than 1. In fact, since b is the lighter field, we take this one to be the inflaton. The inflationary period goes from this scale until the value of the field is close to the typical soft breaking scale of the problem, where we certainly enter the non-linear, or strong coupling regime, the field X become X_NL and behaves like a spurion. Its couplings to low-E fields is (not including gauge couplings)

m2α =

2 f2

M2(A1 + B1), m2

β =2 f2

M2(A1 −B1).

L = −

d4θ

XNL

f

2

m2QeV Q +

d2θXNL

f

12BijQ

iQj + . . . + h.c.

Once we reach the end of inflation, the field X becomes nonlinear, its scalar component is a goldstino bilinear and the period of reheating begins. The details of reheating depend very much on the microscopic model. At this stage one should provide details of the ``waterfall" that turns the huge amount of energy f^2 into low energy particles. Part of this energywill be depleted and converted into low energy particles through the soft couplings, we need to compute bounds on T-reheating

X = M(α + i β)V = f2(1 + A1(α2 + β2) + B1(α2 − β2) + . . .)

α,β ∼

f/M


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Slow roll conditions

=M2

pl

2

V

V

2

,

η = M2pl

V

V,

nS = 1− 6 + 2η,

r = 16

nt = −2,

∆2R =

V M4pl

24π2.

V = f2(1 + A1(α2 + β2) + B1(α2 − β2) + . . .), = 2 ( (A1 −B1)β )2 + . . .

η = 2 (A1 −B1) + . . . ,

Delta^2 is the amplitude of the initial perturbations. Since a,b are very small, the value of epsilon is also very small. For eta we need to make a slight fine-tuning. In fact it is related to the ratio between the inflaton and the gravitino masses. Now we can compute some cosmological consequences. For instance, the number of e-folding to start

N =1M

dx√2

=

βf

βi

dβ

2√

N =1√

2|A1 −B1|

logβf

βi

. m3/2 =f

M

|A1 −B1| =12

m2β

m23/2

, |A1 + B1| =12

m2α

m23/2

, N =√

2

m3/2

mβ

2 logβf

βi

V

1/4

=f1/2

21/4 (|A1 −B1|β)1/2= .027M, WMAP data


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Some numbers

N =√

2

m3/2

mβ

2 logβf

βi

, 21/4 m3/2

mβ

√f

M

1/2

= 0.027, η =

mβ

m3/2

2

µ

M≈ 5.2 10−4 η µ =

f

η ∼ .1 µ ∼ 1013 GeV

mβ = m3/2√

η,

βi ∼ 1013/M, βf ∼ 103/M N ∼ 110

With moderate values of eta, we can get susy breaking scales in the range 10^11-13 without major fine tunings. We get enough e-foldings and the inflaton is lighter than the gravitino by sqrteta. Reheating depends very much on the details of the theory, but some estimates can be made

TRH = 10−10

f/GeV3/2

GeV 107 < TRH < 109

nχ ∼ 1070−90 Sf/Si = 107(

f/GeV )−1/2


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Comments

Supersymmetry breaking can be the driving force of inflation. We have used the unique chiral superfield X which represents the breaking of conformal invariance in the UV, and whose fermionic component becomes the goldstino at low energies. Its auxiliary field is the F-term which gets the vacuum expectation value breaking supersymmetry.

To avoid the eta problem it is crucial to have R-symmetry explicitly broken.

The imaginary part of X plays the role of the inflaton. It represents a pseudo-goldstone boson, or rather a pseudo-moduli. There are many model in the market with such constructions (Ross, Sarkar, German...).

We have been avoiding UV completions on purpose. Similar to the inflation paradigm, the susybreaking paradigm has some universal features, and this is all we wanted to explore. Better andmore detail numbers require microscopic details.

An interested consequence (work in progress) is the detail theory of density perturbations in terms of X_NL correlators. This is related to current algebra arguments and hence it is quite general.

If these comments are taken seriously, we may have plenty to learn about Susy from the sky!


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Slow roll, more details

H2 =

8πG

3ρ H =

a

aρ ≈ V (φ)

φ + 3H(t) φ + V(φ) = 0 → 3H(t) φ ≈ −V

(φ)

During inflation, it is a good approximation to treat the inflaton field as moving in backgroun De-Sitter space, with H constant. The quantum fluctuations of the inflaton are the source of primordial inhomogeneities. The existence of this quantum fluctuations are similar to the Hawking radiation in dS-space. Schematically the picture shows what happens to the fluctuations. Remember that in dS there is a particle horizon 1/H

Lphy = a(t) Lcom

ds2 = −dt2 + e2Ht(dx2 + dy2 + dz2)

φ(x)2 =H

2

2π

=M2

Pl

2

V

V

2

η = M2Pl

V

V


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Power spectrumIn flat De Sitter coordinates we can Fourier transform F.T. To compute the power spectrum, we quantize the field in dS space. Then, expanding the field in oscillators:

Pφ(k) = Vk3

2π2φ(k) φ(k)† φ(x)2 =

∞

0

dk

kPφ(k)

Precisely: Pφ(k) =

H

2π

2

aH=a0k

Finally, the curvature perturbations can be computed using the perturbed Einstein equations, and a number of laborious and subtle steps, we get the power spectrum for the induced density perturbations (at horizon exit)

PR(k) =

H

φ

H

2π

2

aH=a0k

=1

4π2

H

2

φ

2

t=tk

Using the slow-roll equations and defining the spectral index:

PR(k) =1

12π2

1M6

Pl

V 3

V 2 =1

24π2

1M4

Pl

V

d logPR(k)d log k

≡ n(k)− 1

PR(k) ∼ (k

kp)n−1

n=1 is Harrison-Zeldovich


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L. Alvarez-Gaume - Minimal Inflation