2nd Vienna Central European Seminar – Vienna, 26th November 2005

INFLATIONARY MODELS

and a Running Spectral Index

Laura Covi CERN (& DESY)

based on work in collaboration with

David Lyth, Alessandro Melchiorri and Carolina Odman

PRD70 (2004) 123521 (astro-ph/0408129)

see also PRD67 (2003) 043507 (hep-ph/0210395)

OUTLINE

1. Introduction:– single field inflationary models

– the primordial spectrum and its running

2. The running mass model(s):– theoretical motivation

– characteristic predictions

3. Comparison with recent data:– CMBR & LSS & Ly-α

– reionization constraints

4. Conclusions and Outlook

a

INFLATION: period of quasi-exponential expansion, explaining the FLATNESS,

ISOTROPY, HOMOGENEITY of the Universe, the absence of UNWANTED RELICS

and producing the initial SMALL PERTURBATIONS.

How to sustain inflation ???

⇒ USE THE POTENTIAL ENERGY OF A SCALAR FIELD φ AS AN EFFECTIVE COSMOLOGICAL CONSTANT

OSCILLATION

FAST ROLL

SLO

W R

OLL

INFL

ATIO

N

Λ

Λ

eff

eff

= 0

α (φ)V

V (φ)

R = eH t

REHEATING

⇒ The scalar field has to

slow roll in an

ALMOST FLAT POTENTIAL

such that

φ ≪ 3Hφ ⇒ 3Hφ = −V ′

⇒ slow roll expansion

(Single field) inflationary model ⇔ Flat Potential V (φ)

- SLOW ROLL:

ǫ =M2

P

2

(

V ′

V

)2≪ 1

|η| = M2P

|V ′′|V

≪ 1

- Enough expansion: N =

∫ tf

ti

dtH =

∫ φi

φf

dφV (φ)

M2P V ′(φ)

> 50

- Sufficient reheating before Big Bang Nucleosynthesis: Trh > 1 MeV

Huge number of models in the literature:

old inflation, new inflation, chaotic inflation, hybrid inflation, inverted hybrid inflation, smooth

inflation, topological inflation, .....

Is that ALL ?

NOT REALLY: the Universe that we see is not perfectly uniform and homogeneous,

it presents a rich structure of galaxies, clusters of galaxies, etc.. etc... BUT such

inhomogeneities were small in the past and an initial fluctuation of the order of 10−5

is sufficient to produce the present Large Scale Structure. So the question is:

Was the universe during inflation perfectly homogeneous ???

NO !

Quantum Mechanics limits the homogeneity !

Quantum fluctuation of the inflaton

The inflaton rolls classically towards the minimum of the potential...

... but φ is a quantum field and in a de Sitter background its quantum fluctuations are given by

δφ ≃ H

2πGAUSSIAN

So the dynamics of the inflaton is slightly different in different parts of the universe and this

generates density fluctuations:δρ

ρ≃ Hδt ≃ H

δφ

φ≃ H2

2πφ

⇓The amplitude of the density fluctuations at a scale λ is given by

H2

2πφcomputed at the time when λ = H−1 during inflation.

λ )

R Rrad

log(R)

reheating

matter

log(

rh eq

H−1

Frozen

inflation

If H and φ were exactly constant dur-

ing inflation, all the wavelengths would

have exactly the same amplitude

⇒ SCALE INVARIANT SPECTRUM

(nearly so in slow roll !!!)

In this limit, the power spectrum is de-

termined by the inflaton potential:

H2

φ∝ V 3/2

V ′M3P

Initial condition for structure formation !

Testing inflation:Single field

inflation⇐⇒ Flat Potential

V (φ)

The scalar power spectrum is given by PR(k) =1

12π2M6P

V 3

V ′2

∣

∣

∣

∣

k=aH

∝ kn−1

and its spectral index is: n(k)−1 =d log(PR)

d log(k)

∣

∣

∣

∣

k=aH

= 2η−6ǫ+. . .

The tensor power spectrum is given by Pgrav(k) =1

6π2

V

M4P

∣

∣

∣

∣

k=aH

and its spectral index is ngrav(k) =d log(Pgrav)

d log(k)

∣

∣

∣

∣

k=aH

= −2ǫ+...

where the slow roll parameters are

ǫ =M2

P

16π

(V ′)2

V 2η =

M2P

8π

V ′′

V

(

and ξ =M4

P

64π2

V ′V ′′′

V 2

)

At first order in the slow roll expansion that is all, BUT n′ arises at 2nd order in SLOW ROLL:

n′(k) =2

3

(

(n − 1)2 − 4η2)

+ 2ξ

so naively for a safe perturbative expansion, we expect n′ ∝ (n − 1)2 < |n − 1|!!!Breakdown of slow-roll expansion for large |n′| ???

NO, |n′| can still be larger than expected if ξ dominates.

Surprisingly WMAP seemed to require a large running...; possible for large ξ, but are

there “natural” models giving it ? ⇒ RUNNING MASS model !

Before discussing a particular model, a couple of general remarks:

• is |n′| < |n − 1| a requirement of slow roll ???

Not really, slow roll can be a perfectly good approximation, even for ”large” n′.

E.g. a pathological potential

V (φ) ∝ φm → n − 1 = 2η − 6ǫ =(

2m(m − 1) − 3m2) M2

P

φ2

⇒ for m = −2 the first order vanishes exactly, but slow roll still holds for φ ≫ MP !

In this case both n − 1 and n′ are 2nd order and are expected to be similar.

• at which scale k should the inequality hold ???

If the potential changes curvature, V ′′′ 6= 0, the spectral index can naturally cross 1 and

there we must have |n′| > |n − 1| ≃ 0 !!! The only way to keep a scale dependent

spectral index very close to 1, as required by the data, is to have n − 1 change sign !

Only the simple polynomial potentials give a fixed sign for n − 1...

Running mass model:

theoretical motivation

SUSY broken

in inflation=⇒ SUGRA !

A model is defined by superpotential W (Φ) & Kahler potential K(Φ, Φ)

L = Kn∗m∂µΦn∂µΦm − V (Φ, Φ)

V (Φ, Φ) = eK(Φ,Φ)(

FmKmn∗F∗n − 3|W |2

)

+ VD

where Fm = ∂W∂Φm

+ ∂K∂Φm

W (φ), Kn∗m = ∂2K∂Φm∂Φ

n∗

Kmn∗ =(

K−1)mn∗

Take a canonical Kahler K = ΦnΦn and we have

V = e|Φn|2

(

∣

∣

∣

∣

∂W

∂Φn+ ΦnW

∣

∣

∣

∣

2

− 3|W |2)

+ VD

so that from the exponential one obtains

V ′′ = V + .... → η ≃ 1 η problem

NO SLOW ROLL POSSIBLE IN SUGRA ?!

There are a couple of ways out... ... one of them: the running mass !

[Stewart ’96, ’97]

The running mass model: φ → flat direction of the V ′SUSY (φ) = 0

SUSY potential

SUSY breaking generates a soft mass for φ: V (φ) = V0+1

2m2φ2+. . . for φ < MP .

At tree level, for a generic scalar field one has naturally |m2| ≃ V0/M2P η problem !

→ V (φ) is NOT flat at high scale

But if the inflaton field interacts not so weakly, the one loop corrections to the potential give

m2 → m2(Q = φ) running mass

The running of the mass can flatten the potential somewhere in the region φ ≪ MP .

⇓Slow roll inflation

In general any type of coupling can be responsible for the inflaton’s mass running:

dm2

d log(Q)= −2C

παm2+

D

16π2|λ|2m2

s

gauge Yukawa

The inflaton has to couple sufficiently strongly, but still in the perturbative regime...

Different models exist depending on the sign of the running and the initial conditions:

φ*

Model (ii)

φ

φV( )

Model (i)

φ*

Model (iii) Model (iv)

φ

φV( )

What are the observable consequences of non-weakly coupled inflaton ???

n(k) − 1 ≪ 1 on

cosmological scales⇒ linear expansion

around pivot φ0 (↔ k0)

So expand the running mass around φ0 as m2(φ) ≃ m2(φ0) + c ∗ log

(

φ

φ0

)

Then we can

write the potential as a function of two parameters s and c as

V

3H2I M2

P

≃ 1 +1

2

(

s +1

2c − c log

(

φ

φ0

))

φ2

M2P

Note that s, c are related to physical parameters of the lagrangian of specific models rescaled

by the inflationary Hubble scale H2I :

c ≡ −βm(φ0)

3H2I

s +1

2c ≡ m2(φ0)

3H2I

NOTE: this is equivalent to stopping the perturbative expansion to one loop and neglecting the

change of βm... , but a higher order can be used to run from MP down to the scale φ0.

Connect to simple supersymmetric examples:

→ gauge coupling α dominance for φ in the adjoint representation of SU(N)

c = 2Nα(MP )π

m2(MP )3H2

I

α3(φ0)α3(MP )

m gaugino mass

s = − c2 + m2(MP )−2m2(MP )

H2

I

+ 2m2(MP )3H2

I

α2(φ0)α2(MP )

m inflaton mass

→ Yukawa coupling λ dominance

c = −λ2(MP )12π2

1

1− 3

8π2λ2(MP ) log

(

φ0

MP

)

2

for mscalars ≃ HI

s = − c2 + 2

3

1

1− 3

8π2λ2(MP ) log

(

φ0

MP

) − 12

Result for the spectrum: a ”strongly” scale-dependent n(k) and PR(k) !

In fact the slow-roll parameters for this potential become:

ǫ ≃ s2φ2

M2P

e2c∆N η ≃ sec∆N − c ξ ≃ −csec∆N

where ∆N = log

(

k

k0

)

=1

clog

[

1 − c

slog

[

φ

φ0

]]

.

c suppressed by a coupling, s also to have slow roll... Then we have for the spectral index

n(k) − 1

2= s

(

k

k0

)c

− c and n′(k) = 2sc

(

k

k0

)c

→ ξ!

“Strong (exponential !)” scale dependence !!

Look for such strong scale dependence in the data, trying to extend the lever arm as far as

possible:

[Figure by M. Tegmark]

• CMB: first year WMAP data

astro-ph/0302207

• LSS: Sloan Digital Sky Survey re-

sults for the galaxy power spectrum

astro-ph/0310723

• LSS: Sloan Digital Sky Survey re-

sults on Lyman-α

astro-ph/0405013 & 0407372

What are the constraint from the new data for s, c in such models ?

[LC, Lyth, Melchiorri & Odman astro-ph/0408129]

WMAP+SLOAN+Ly-α

WMAP+SLOAN

WMAPWMAP strongly constrains

along the direction s = c, i.e.

n(k0) − 1 = 0

Theoretically expected

region

Ly–α data tighten the bound

on scale dependence and

require

|c| ≤ 0.12

Look at the constraints in the n′0 vs n0 plane instead

WMAP+SLOAN+Ly-α

WMAP+SLOAN

WMAP

NOTE: negative n′0 is allowed by

the model only for

n′0 ≤ − (n0 − 1)2

4

due to the dependence on s, c.

The rest of the parameter space is

unphysical !

Fitting for arbitrary n0, n′0 is not

equivalent as fitting for the running

mass model !

Again the most stringent bound on

n′0 comes from Ly–α data giving

n′0 ≤ 0.2

Compare the result with the fit for a general Taylor expansion: n(k) = ns+αs log

(

k

k0

)

.

Using the same data Seljak et al. (astro-ph/0407372) find, contrary to WMAP,

ns = n0 = 0.977+0.025−0.021

αs = n′0 = −0.003 ± 0.010

NO RUNNING !

ss

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.8 0.85 0.9 0.95 1 1.05 1.1

WMAP+lyaWMAP+gal+bias

nα

That is fully compatible with our result; the data cannot yet distinguish between the different

parameterizations ! In fact αs ≤ 0.2 is similar to our result...

NOTE: the blue data ”like” a spectral index crossing unity at a scale log(

kk0

)

≃ −n−1αs

.

What are the bounds on the “physical parameters” ?

m2(φ0)3H2

I

WMAP+SLOAN+Ly-α

c ∝ m′2(φ0)3H2

I

Strong running, i.e. large inflationary scale, is dis-

favoured... From the WMAP normalization

HI = 2πP1/2R |φ0||s| ∼ 3 × 10−4|φ0||s|

and assuming linear running from MP

m2(φ0) ≃ 0 → φ0

MP∼ exp

(

− 1

|c||m2(MP )|

3H2I

)

i.e. |c| ≤ 0.1 and |m2| = 3H2I

→ φ0 ≤ 10−5MP .

Small |c| implies φ0 ≪ MP and therefore also

HI ≪ φ0 ≪ MP ... HI highly sensitive to c !

Another hint for a running index: REIONIZATION..... ???

Estimate the reionization epoch zR using the Press-Schechter formula as the epoch of collapse of a

fraction f of matter into objects of mass 106M⊙:

1 + zR ≃√

2σ(106M⊙)

1.7g(ΩM )erfc−1(f)

where σ is the present linear rms density contrast

computed from the primordial spectrum and the

CDM transfer function and g(ΩM ) accounts for the

suppression of the growth when ΩM < 1.

There is a strong correlation between c and zR:

zR grows cery quickly for large c

s has been fixed to c and c − 0.05.

The red line corresponds to zR = 6.0

10

20

30

40

50

60

−0.2 −0.1 0 0.1 0.2

z

c

R

Conclusions and Outlook

• The simple (single field) inflationary paradigm is very successful in describingpresent observations. Unfortunately it is still not clear which model of the manyproposed is favoured..., therefore let us try to look for specific observationalsignatures, e.g. investigate the scale-dependence of the spectral index !

• The running mass model is very well-motivated within particle physics and has avery characteristic expression for the spectral index.

• Present data allow still a relatively strong scale dependence and cannot yet

exclude this type of models. MORE DATA are expected soon (WMAP...?!?) and

then we will know more !