L. Alvarez-Gaume - Minimal Inflation

Minimal Inﬂation
Luis Alvarez-Gaume BSI August 29th, 2011

Work in collaboration with C. Gómez and R. Jiménez

1
Monday, 29 August, 2011

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...

2

Summary of FRLW
Matter is well represented by a perfect ﬂuid

Tab = (p + ρ)ua ub + pgab 2
˙
a k 8πG
+ 2 = ρ
a a 3
The Einstein equations are
¨
a 4πG
Gab = 8πG Tab = − (ρ + 3p)
a 3
These are the possible ρ + 3H(ρ + p) = 0
˙
k = ±1, 0 curvatures distinguishing the
space sections
p = wρ
ds2 = −dt2 + a(t)2 ds2 ˙
a(t)
3
ρ+3
˙ (1 + w)ρ = 0
a(t)
a 3(1+w)

0
˙
a(t) ρ k
ρ = ρ0
H≡ Ω≡ 1−Ω = 2 2 a
a(t) ρc a H

3H 2
ρc =
8πG 1 √ √ 1 ab
S = −g(R − 2Λ) + −g − g ∂a φ∂b φ − V (φ)
16πG 2

1 ˙2 1 ˙2
ρ = φ + V (φ) p = φ − V (φ)
2 2

3

Accelerating the Universe
We need to violate the dominanat energy
condition for a sufficiently long time. ¨
a 4πG
= − (ρ + 3p)
If we take during inflation H approx. constant, a 3
the number of e-foldings:
1
p − ρ
3 a(t) 0
¨
|1 − Ω| ∝ a−2
˙
Number of e-foldings could be
a(tf )
N = log = H(tf − ti ) 50-100, making a huge Universe.
a(ti )
We can use QFT to construct
|1 − Ω(tf )| = e −2N
|1 − Ω(ti )| some models
1 ˙2 1 ˙2 1

√

√

1 ab

ρ = φ + V (φ) p = φ − V (φ) S =
16πG
−g(R − 2Λ) + −g − g ∂a φ∂b φ − V (φ)
2
2 2

Slow roll paradigm ˙2 V V
Hybrid inflation
φ V (φ) 1, 1
V V

(Courtesy of Licia Verde)

4

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.

5

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.

6

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

7

SSB Scenarios
Observable Sector

Gauge mediation
MEDIATOR
Gravity mediation

Anomaly mediation
Hidden Sector

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

f µ2 µ → ∞
m3/2 = =
Mp Mp M → ∞ m3/2 ﬁxed

8

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 (φ + z F ) = V (φ)
V = ∂ W (φ) ∂ W (φ)

9

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

α
˙
Jµ = jµ + θα Sµα + θα S µ + (θσ ν θ) 2Tνµ + . . .
˙

√
X = x(y) + 2θψ(y) + θ2 F (y)

√
2 µ α˙ 2
ψα = σ ˙S , F = T + i∂µ j µ
3 αα µ 3

α
˙
D Jαα = Dα X
˙

10

General Lagrangian

4 2 2
i ¯¯ ¯ ¯ ¯¯
S= d θK(Φ , Φ ) + d θW (Φ ) + d θW (Φi )
i i

¯˙¯ ¯˙ ¯ ¯
Jαα = 2gi (Dα Φi )(Dα Φ) − 2 [Dα , Dα ]K + i∂α (Y (Φ) − Y (Φ))
˙ 3

1 2 1 2
X = 4 W − D K − D Y (Φ)
3 2

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:

2
= 0
X → XN L

U V → IR
XN L
SP oincare/P oincare

11

Some IR consequences

L = d θ XN L X N L +
4
d θ f XN L + c.c.
2

G2 √
XN L = + 2θG + θ2 F
2F

This is precisely the Akulov-Volkov Lagrangian

12

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...

msof t E Λ The goldstino superfield is the spurion

XN L 2 V
4
d θ m Qe Q + d2 θ XN L (B Q Q + AQ Q Q ) + c.c.
f f

Integrate out the massive superpartners

E msof t adding extra non-linear constraints

For light fermions, and similar conditions
XN L = 0,
2
XN L QN L = 0 for scalars, gauge fields,...

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

13

Some details

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...

¯
a(X + X) bX X¯ c X +X 2 ¯2 ¯
X +X
¯ ¯
K(X, X) = X X 1+ − − + ... − 2M 2 log +1
2M 6M 2 9M 2 M

W (X) = f0 + f X

K
¯ 3 1
V =e M2
−1
(KX,X DW DW
¯ − 2 |W |2 ) D W = ∂X W + ∂ KW
2 X
M M

14

The full lagrangian

15

Primordial density ﬂuctuations

2
2
Mpl V nS = 1 − 6 + 2η,
f ∼ 1011−13 GeV =
2 V
,
r = 16
nt = −2,

2 V V Mpl4
η= Mpl , ∆2 = .
V R
24π 2

16

Choosing useful variables

√
z = M (α + iβ)/ 2

ds2 = 2gzz dsd¯ = ∂z ∂z K(α, β)M 2 (dα2 + dβ 2 )
¯ z ¯

3 3 1 ˙
S=L dta g(α, β)M 2 (α2 + β 2 ) − f 2 V (α, β)
˙
2

1
t = τ M/f S=L f 3 2
m−1
3/2 dτ a 3
g(α, β)(α2 + β 2 ) − V (α, β)
2

17

Cosmological equations

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

a 1
α + 3 α + ∂α log g(α2 − β 2 ) + ∂β log gα β + g −1 Vα
= 0
a 2
a 1 Plus fermion
β + 3 β + ∂β log g(β 2 − α2 ) + ∂α log gα β + g −1 Vβ

= 0 terms on the RHS
a 2

a H 1 1
= =√ g(α2 + β 2 ) + V (α, β)
a M3/2 3 2

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

1
˙
DΦi /dt ∼ 0 H= 3V + 6V + 9V 2
18

18

The scalar potential

1.0

Β

0.5

0.0
0.10

0.05

0.00
0.0

0.5

Α
1.0

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

Ε
1.0

Β 0.5

0.0
1.0

0.5

0.0
0.0

0.5

Α 1.0

19

Attractor and inﬂationary trajectories
Β

1.0

Β
0.5

0.8
0.4

0.3

0.2

0.1

0.6
Α
0.2 0.4 0.6 0.8

V
0.10

0.08
0.4

0.06

0.04

0.02

0.2
t
0.05 0.10 0.50 1.00 5.00

Nearly a textbook example of inﬂationary potential

Α
0.05 0.10 0.15 0.20 0.25 0.30

20

Decoupling and the Fermi sphere

mINF 2
η = ( )
m3/2

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


f
qF =
η

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

21

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

22


Thank you

23


Back up slides

24

Graceful exit

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)

K
¯ 3 1
V =e M2
−1
(KX,X DW DW
¯ − 2 |W |2 ) D W = ∂X W + ∂ KW
2 X
M M

2 V ¯ ¯
K(X, X) = X X + . . . η = 1 + ...
η = Mpl ,
V

25

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.

XN L = 0,
2

G2 √
XN L = + 2 θ G + θ2 F.
2F

26

Vanishing cosmological constant

√ √ 2
2
(2h− 2α)(−72+α( 2+2α)(18+5α ))
1− √
3 √ 2 72(1+ 2α)
− −2h + 2α + α2 2
=0
4 1+ 3 + √ 2
(1+ 2α)

0.9980

0.9975

0.9970
h

0.9965

0.9960
0.10 0.15 0.20 0.25 0.30
Α

27

...continued

X = M (α + i β)
V = f 2 (1 + A1 (α2 + β 2 ) + B1 (α2 − β 2 ) + . . .)
2 f2 2 f2
m2
α = (A1 + B1 ), m2
β = (A1 − B1 ).
M2 M2
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
α, β ∼ f /M
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)

X N L 2 2 V XN L 1
L = − d4 θ ¯
f m Qe Q + d2 θ
f 2
Bij Qi Qj + . . . + 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

28

Slow roll conditions
2
2
Mpl V nS = 1 − 6 + 2η,
= ,
2 V r = 16
nt = −2,

2 V V Mpl4
η= Mpl , ∆2 = .
V R
24π 2
2
= 2 ( (A1 − B1 )β ) + . . .
V = f 2 (1 + A1 (α2 + β 2 ) + B1 (α2 − 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

1 dx βf dβ 1
N = √ = √

N = √ log βf . m3/2 =
f
M 2 βi 2 2|A1 − B1 | βi

M
2
1 m2 1 m2 √ m3/2
|A1 − B1 | =
β
2 , |A1 + B1 | = α
2 , N = 2 log βf
2 m3/2 2 m3/2 mβ βi

1/4
V f 1/2
= 1/2
= .027M, WMAP data
21/4 (|A1 − B1 |β)

29

Some numbers

√ 1/2 2
√ m3/2 2 βf m3/2 f mβ
N = 2 log

, 21/4 = 0.027, η =
mβ βi mβ M m3/2

µ √
≈ 5.2 10−4 η µ= f mβ = m3/2 η,
M
η ∼ .1 µ ∼ 1013 GeV

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 sqrt{eta}. Reheating
depends very much on the details of the theory, but some estimates can be made
3/2
TRH = 10−10 f /GeV GeV 107 TRH 109

nχ ∼ 10 70−90
Sf /Si = 10 ( f /GeV )−1/2
7

30

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 susy
breaking paradigm has some universal features, and this is all we wanted to explore. Better and
more 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!

31

Slow roll, more details
8πG ˙
a
H2 = ρ H= ρ ≈ V (φ)
3 a
2
¨ ˙ ˙
2
MP l V 2 V

φ + 3H(t) φ + V (φ) = 0

→ 3H(t) φ ≈ −V (φ) =
2 V
η = MP l
V

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

H2
φ(x) =
2

2π
Lphy = a(t) Lcom

ds2 = −dt2 + e2Ht (dx2 + dy 2 + dz 2 )

32

Power spectrum
In 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:

3 ∞
k dk
Pφ (k) = V 2 φ(k) φ(k)† φ(x)2 = Pφ (k)
2π 0 k
2
H
Precisely: Pφ (k) =
2π aH=a0 k

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)

2 2
H H 1 H2
PR (k) = =
˙ ˙

φ 2π aH=a0 k
4π 2 φ t=tk

Using the slow-roll equations and defining the spectral index: d log PR (k)
≡ n(k) − 1
d log k
1 1 V3 1 1 V k n−1
PR (k) = = PR (k) ∼ ( )
12π 2 MP l V 2
6 24π 2 MP l
4 kp
n=1 is Harrison-Zeldovich

33

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