Cosmological inflation with a Dirac-Born-Infeld fieldparticle-theory.group.shef.ac.uk/Joel/files/AppliedMathsShef.pdf · Dirac-Born-Infeld eld Joel M. Weller University of She eld,

Cosmological inflation with aDirac-Born-Infeld field

Joel M. WellerUniversity of Sheffield, 3rd November 2010

Astro-Particle Theory and Cosmology GroupDepartment of Applied Mathematics, University of Sheffield

Joel M. Weller (Sheffield) Inflation with a DBI field 1 / 44

Outline

1 Basics of the Big Bang ModelThe Friedmann equationsEvolution of the UniverseProblems with the Big Bang Model

2 Cosmic InflationAn accelerated expansionSingle field InflationDBI Inflation

3 Summary


Basics of the Big Bang Model

Basics of the Big Bang Model


Basics of the Big Bang Model The Friedmann equations

The Friedmann equations

To describe the Universe on large scales we turn to General Relativity.

The geometry of spacetime is specified by the metric,

ds2 = gµνdxµdxν ,

which gives the distance between two points in terms of thecoordinates.

Standard cosmology is based upon the maximally spatially symmetricFriedmann-Robertson-Walker (FRW) metric,

ds2 = −dt2 + a2(t)

[dr 2

1− κr 2+ r 2(dθ2 + sin2 θdφ2)

].

a(t) is the scale factor, relating coordinate distances to physicaldistances by dphys = a(t)dcoord.




The physical velocity and distance are related by Hubble’s law.

We have dphys = a(t)dcoord

so v = dphys = Hdphys.

H is the Hubble parameter,H(t) = a/a.

On the largest scales, thegreater the separationbetween two objects, thefaster they are movingapart.

v = Hd




The Einstein equations relate the matter content Tµν to the spacetimecurvature

Rµν − 12 Rgµν =

8πG

c4Tµν .

The matter content of the universe can often be assumed to be a perfectfluid with energy density ρ and pressure p so

Tµν = diag(−ρ, p, p, p)

The pressure is given in terms of the energy density by p = wρ, wherew = 0 for non-relativistic matter (dust) and w = 1/3 for relativisticmatter (radiation).




Putting this into the Einstein equations gives the Friedmann equations

H2 ≡(

a

a

)2

=8πG

3ρ− κ

a2

a

a= −4πG

3(1 + 3w)ρ

We can rewrite the first equation as

Ω− 1 =κ

a2H2,

where Ω = ρ/ρcrit andρcrit = 3H2/8πG ≈ 10−30gcm−3.

There is a one-to-onecorrespondence between Ω andthe spatial curvature of theUniverse.


Basics of the Big Bang Model Evolution of the Universe

Evolution of the UniverseThe continuity equation for matter is ρ+ 3Hρ(1 + w) = 0.

log a

log ρ

radiation: ρ ∝ a-4

matter: ρ ∝ a-3

matter domination: w = 0

ρ ∝ a−3

a ∝ t2/3

radiation domination: w = 1/3

ρ ∝ a−4

a ∝ t1/2

The energy density of radiation decays faster than non-relativisticmatter so the early Universe was radiation dominated.



Evolution of the Universe

Using a result from statistical mechanics, theenergy density of a gas with g internal degrees offreedom (and negligible chemical potential) is

ρ =g

(2π)3

∫ ∞m

(E 2 −m2)1/2

eE/T ± 1E 2dE ∝ T 4,

so the temperature of the radiation bath decreasesas the Universe expands: T ∝ a−1.

The big bang theory rests upon three observationalpillars

I The Hubble diagramI Abundances of light elementsI Cosmic microwave background



Photon decoupling and recombination

At high temperatures in the early universe, atoms were ionised and theUniverse was opaque to photons. As it cooled, neutral atoms were formedand the density of free electrons became too low for radiation and matterto remain in thermal contact. This gives the surface of last scattering.



Cosmic Microwave Background Radiation

The CMB radiation is an almost uniform blackbody spectrum correspondingto T = 2.725K giving us a snapshot of the extremely homogeneous state ofthe Universe at early times when T ≈ 104K.

Anisotropies are of the order 1 in 105 and analysis of the angular powerspectrum gives an enormous amount of information about the source ofthese primordial fluctuations, the geometry of the Universe and theexpansion rate since last scattering.













Nucleosynthesis

Looking even further back intime, the temperature was toohigh for nuclei to form andprotons and neutrons were notbound into nuclei.

When T . 1010K, stable nucleicould form via nuclear reactionswith rates that depend stronglyon temperature.

By measuring the abundances oflight elements, one can constrainthe expansion rate H in the firstfew minutes after the big bang.


Basics of the Big Bang Model Problems with the Big Bang Model

The flatness problemEarlier we found (where Ω = ρ/ρcrit)

Ω− 1 =κ

a2H2,

observations of the CMB (and other datasets) find the geometry of theUniverse to be extremely close to flat

Ω0 = 1.000+0.003−0.008

If κ = 0, the Universe is flat at all times. However, if there is even a smallcurvature term then this would become more important as the Universeexpands.

during radiation domination

H2 ≈ ρrad/3 ∝ a−4

∴ Ω− 1 ∝ κ

a2a−4∝ a2

during matter domination

H2 ≈ ρmat/3 ∝ a−3

∴ Ω− 1 ∝ κ

a2a−3∝ a



The flatness problem

If the curvature contribution is small now, it must have been eventinier in the early universe. How small?

The ratio of the curvature contribution now (T0 ≈ 1K) to that atnucleosynthesis (TN ≈ 1010K) is

|Ω− 1|T=TN

|Ω− 1|T=T0

≈(

a2N

a20

)≈(

T 20

T 2N

)≈ O(10−20),

as a ∝ T−1.

To get a small but non-zero curvature contribution today requires anextraordinary amount of fine-tuning! It only gets worse as we gofurther back in time.



The particle horizon

Since in the big bang model, time has a beginning, there is amaximum distance light could have travelled to reach us since t = 0.

In general relativity, light travels on null paths, for which ds2 (in themetric) is equal to 0. For a simplified FRW metric we have

ds2 = −dt2 + a2(t)dr 2 = 0 ⇒ rmax =

∫ t

0

dt ′

a(t ′).

The physical distance to the horizon scales with a(t) soRH = a(t)rmax.



The particle horizon

During the radiation and matter dominated eras the scale factorscales as a power law a(t) ∝ tn so H = a/a = nt−1. Therefore

RH = tn

∫ t

0

dt ′

t ′n=

t

1− n=

n

1− nH−1 ∼ H−1.

In the radiation and matter dominated eras the horizon isapproximately equal to the reciprocal of the Hubble parameter (thecurvature radius) : RH ∼ H−1.

This defines the maximum scale on which events are causallyconnected.



The horizon problem

log a

phys

ical

leng

th

λ∝a

H-1∝a2 (radiation)

H-1∝a3/2 (matter)

When a given scale λ is outside the horizon it cannot affect phenomenonoccuring at smaller scales.



The horizon problem

Looking back to last scattering (TLS ≈ 104K), the length scalecorresponding to the observable universe λH(tLS) was much smaller than thevalue today (RH)

λH(tLS) = RH

(aLS

a0

)= RH

(T0

TLS

).

In the matter dominated era H2 ∝ ρmat ∝ a−3 ∝ T 3 the horizon at lastscattering was

H−1LS =

H0

HLSH−1

0 = RH

(T0

TLS

)3/2

Thus, at last scattering there were

λH(tLS)3

H−3LS

=

(T0

TLS

)−3/2

≈ 106

casually disconnected regions within the volume that corresponded to thepresent size of our Universe!



The horizon problem

How can we reconcile the high degree of uniformity of the CMB radiationwith the fact that that regions on opposite sides of the sky were not incausal contact?


Inflation

Cosmic Inflation


Inflation An accelerated expansion

An accelerated expansion

H2 =ρ

3− κ

a2,

a

a= −ρ

6(1 + 3w), ρ+ 3Hρ(1 + w) = 0

Both of these problems can be solved with an accelerated expansion. Thisis only possible if the equation of state parameter w = p/ρ satisfies

w < −1/3.

In the extreme case with the Universe dominated by vacuum energy withw = −1, we have a de Sitter stage with constant energy density. Solvinggives

H2 ≡(

a

a

)2

= constant ⇒ a(t) ∝ eHI t

and the scale factor increases exponentially.



Resolving the horizon problem

log a

phys

ical

leng

th

λ∝a


H-1∝a3/2 (matter)

During inflation the curvature radius is constant so physical scales largerthan the present horizon were in causal contact in the past.



Resolving the horizon problem

log a

phys

ical

leng

th

λ∝a


H-1∝a3/2 (matter)

H=constant

During inflation the curvature radius is constant so physical scales largerthan the present horizon were in causal contact in the past.



How much inflation is required?

The amount of inflation is given in terms of the number of efolds N

N = ln (af /ai ) = ln[HI (tf − ti )].

At the beginning of inflation the length scale corresponding to the horizontoday was equal to

λH(ti ) = H−10

(ati

at0

)= H−1

0

(atf

at0

)(ati

atf

)= H−1

0

(T0

Tf

)e−N

To solve the horizon problem we need this to be less than the curvatureradius during inflation λH(ti ) . H−1

I

N & ln

[(T0

H0

)(HI

Tf

)]≈ ln

(1

10−30

)− ln

(Tf

HI

)& 60



Resolving the flatness problem

During inflation, as H = HI =const, the curvature quickly becomesnegligible

Ω− 1 =k

a2H2∝ a−2.

We can compare the curvature contribution at the beginning and end ofinflation to find

|Ω− 1|t=tf

|Ω− 1|t=ti

=

(ai

af

)= e−2N ,

So even if the Universe has non-negligible curvature initially, after inflationit will be flat to a high degree of precision!



Resolving the flatness problem


Inflation Single field Inflation

Inflation with a scalar field

The action for a scalar field φ with potential V (φ) in curved space is

Sφ = −∫

d4x√−g[

12 gµν(∂µφ)(∂νφ) + V (φ)

]To find the energy density and pressure one can vary the action wrt themetric to get the stress energy tensor

Tµν = −δµν

[1

2gαβφ,αφ,β + V (φ)

]+ gµαφ,αφ,ν ,

giving, for a homogeneous field,

ρ = −T 00 = 1

2 φ2 + V (φ), p = T 1

1 = 12 φ

2 − V (φ)



Inflation with a scalar field

In order to get a > 0, we require w < −1/3. The equation of stateparameter for the scalar field is

w = p/ρ =12 φ

2 − V12 φ

2 + V.

Conditions for an accelerated expansion.

The scalar field must be the dominant species of matter:

3H2 ≈ ρ.

The field must be slowly rolling, i.e. the potential term dominatesover the kinetic term. This means that the potential must be very flat.

φ2 V



The slow-roll approximation

The equation of motion of the field is

φ+ 3Hφ+ V ′(φ) = 0

where V ′ = dVdφ .

As we are considering a slowly rolling field (φ2 V ), the φ term isnegligible (φ 3Hφ) as well.

These conditions can be written in terms of the slow-roll parameters,which satisfy ε, η 1 during inflation.

Slow-roll Parameters

ε ≡ − H

H2=

1

2

(V ′

V

)2

, η ≡ V ′′

V

ε quantifies the deviation from an exponential expansion and η is ameasure of the steepness of the potential.





ε ≡ − H

H2=

1

2

(V ′

V

)2

, η ≡ V ′′

V





The equation of motion of the field is

φ+ 3Hφ+ V ′(φ) = 0

where V ′ = dVdφ .

As we are considering a slowly rolling field (φ2 V ), the φ term isnegligible (φ 3Hφ) as well.

These conditions can be written in terms of the slow-roll parameters,which satisfy ε, η 1 during inflation.


ε ≡ − H

H2=

1

2

(V ′

V

)2

, η ≡ V ′′

V




ReheatingIn the simplest models the decay of the inflation is modeled by adding afriction term to the equation of motion.

Log[a]

Log[ρ]

Inflaton

Radiation

φ+ (3H + Γφ)φ+ U(φ) = 0

ρrad + 4Hρrad = Γφφ2



Perturbation theory

We can split the field into a ’classical’ homogeneous part and a smallperturbation

φ = φ(t) + δφ(t, x)

which can be expanded in Fourier modes

δφ(t, x) =

∫d3k

(2π)3/2e ik·xδφk(t)

that evolve according to

δφk + 3H δφk +k2

a2δφk + m2

φδφk = 0

where m2 ≡ d2V /dφ2 is the mass of the field.



Perturbation theory

As a first approximation consider the case where the field is massless.

δφk + 3H δφk +k2

a2δφk = 0

To solve this, we can redefine our time variable dτ = dt/a so the flatFRW metric becomes

ds2 = a2(τ)[−dτ2 + δijdx idx j ]

When a is growing exponentially, we have

a(τ) = − 1

HI ττ ∈ (−∞, 0)



Perturbation theory

Defining a new perturbation variable νk = a(τ)δφk gives

d2νkdτ2

+

(k2 − 1

a

d2a

dτ2

)νk = 0.

which has the solution

νk =e−ikτ

√2k

(1 +

i

kτ

).

On small scales (k aH) the perturbations oscillate but on large scales(k aH) we have νk ∝ a(τ) so δφk is constant



Perturbation theory

τ

log δϕ

k=aH

νk =e−ikτ

√2k

(1 +

i

kτ

).

On small scales (k aH) the perturbations oscillate but on large scales(k aH) we have νk ∝ a(τ) so δφk is constant



Power spectrum of fluctuations

The power spectrum is defined in terms of the two point function,

< 0|(δφ(t, x))2|0 >=

∫d3k

(2π)3|δφk|2 =

∫dk

kPδφ(k)

where

Pδφ(k) ≡ k3

2π2|δφk|2

The modes freeze in at a value |δφk| ≈ H/√

2k3 so the power spectrum isscale independent

Pδφ(k) =

(H

2π

)2



Power spectrum of fluctuations

In the following we have ignored the deviations from an exponential expansion.Including this effects gives a small scale dependence in terms of the slow-rollparameters ε and η

|δφk| ≈H√2k3

(k

aH

)3/2−√

9/4+9ε−3η

In General Relativity fluctuations in energy source perturbations in the space-timecurvature R. These primordial perturbations are responsible for the anisotropiesin the CMB. The power spectrum of curvature perturbations in this case is

PR =1

ε

k3

2π2|δφk|2 ≡ A2

R

(k

aH

)ns−1

with ns = 1 + 2η − 6ε,

which is is excellent agreement with the measured value ns = 0.963± 0.012.


Inflation DBI Inflation

Inflation from fundamental theory

The inflationary paradigm is an extremely successful, albeitphenomenological, description of the very early universe.

The general predictions are quite robust, however, for the details weneed to understand the mechanism from the perspective of particlephysics and/or a fundamental theory.

Since inflation takes place at high energies, there has been muchinterest in model building in the context of string theory.



String Theory and extra dimensions

In string theory, fundamental particles are not point-like, but arisefrom the vibrational modes of 1 dimension strings with tension andkinetic energy.

Theories are often formulated in 9 spatial dimensions. As we clearlydo not observe more than 3 dimensions in everyday life, the 6 extradimensions must be very small and ’folded up’ or compactified.

Although there are a large number of scalar fields in string theory, it isvery difficult to use them to build a good inflationary model as thepotentials are steep.

As well as strings, there are other objects known as D-branes thatmove in the compactified dimensions. Open strings are attached toD-branes but they can also interact with closed strings (that describegravity).







In string theory, fundamental particles are not point-like, but arisefrom the vibrational modes of 1 dimension strings with tension andkinetic energy.

Theories are often formulated in 9 spatial dimensions. As we clearlydo not observe more than 3 dimensions in everyday life, the 6 extradimensions must be very small and ’folded up’ or compactified.

Although there are a large number of scalar fields in string theory, it isvery difficult to use them to build a good inflationary model as thepotentials are steep.

As well as strings, there are other objects known as D-branes thatmove in the compactified dimensions. Open strings are attached toD-branes but they can also interact with closed strings (that describegravity).



DBI inflation

Dirac-Born-Infeld (DBI) inflation is a specific example of ’stringy’inflation in which a D3 brane moves in a simplified compactifiedspace, falling into a throat similar to a potential well.

A speed limit is imposed upon the motion of the brane, which isdependent upon the throat geometry, which allows inflationarysolutions with steep potentials.



DBI inflation

The action for the DBI field χ is

S =

∫d4x√−g[f −1(χ)(1− γ−1)− V (χ)

]f (χ) is the warp factor, determined by the background geometry ofthe space. As a first approximation one can write

f (χ) = λχ−4

γ is the boost factor, which plays a role similar to the Lorentz factorin special relativity,

γ =1√

1− f χ2



DBI Inflation

χ+ 3Hγ−2χ+ 12

fχf 2

(1− 3γ−2 + 2γ−3) + γ−3Vχ = 0

As the DBI field starts to roll down its potential, the boost factorbecomes large.

Using f = λχ−4, the late-time solution is

χ− 2

λχ3 ≈ 0 ⇒ χ→

√λ/t

This gives power law inflation a = ai t1/ε with

1

ε≈√

3

λ

1

m, γ →

√4

3λmt2



DBI inflation



DBI inflation

To understand the evolution of the curvature perturbation R one can define a

new variable z = aγ3/2χH , νk = zR.

d2vk

dτ 2+

(k2

γ2− 1

z

d2z

dτ 2

)vk = 0.

The sound speed of the perturbations is c2s = γ−2



DBI Inflation

PR =1

8π2

(H2

csε

)|csk=aH

≈ 1

36π2m4λ

One can use the power spectrum amplitude to fix the parameters, butthis can lead to a relatively small number of efolds of inflation.

Modes freeze-in at smaller scales as inflation progresses, cancellingthe red-tilt due to the evolution of H.

The spectral index is dependent on the warped geometry and thebackground dynamics. When γ 1, the spectral is very close toscale invariant.



DBI Inflation

In standard single field inflation, the perturbations are Gaussian in the sensethat all higher order correlation functions are given in terms of the two-pointfunction.

In DBI inflation, there can be non-Gaussian corrections to the powerspectrum Fluctuations can be correlated as the modes freeze-in at differentlength scales. The non-linearity parameter is a typical measure of the levelof non-Gaussianity in the perturbation.

ζ = ζL −3

5fNLζ

2L

Perturbations in DBI inflation are characterised by high levels ofnon-Gaussianities

fNL ≈ 0.32γ2 ⇒ γ . 20,

which could be used to distinguish these types of models from single fieldinflation.


Summary

Summary

Despite its successes, the big bang model suffers from the horizon andflatness problems.

In simple models of inflation the field slowly rolls down the potentialand afterwards oscillates around the minimum, whereupon it decaysinto radiation

Fluctuations in the scalar field freeze-in as they cross the horizon,giving rise to an almost scale invariant spectrum of curvatureperturbations.

DBI provides an interesting example of inflation realised in thecontext of string theory.

The field perturbations behave differently to the standard case, whichcan give rise to observable signatures in the cosmic microwavebackground radiation.


Documents

Cosmological inflation with a Dirac-Born-Infeld fieldparticle-theory.group.shef.ac.uk/Joel/files/AppliedMathsShef.pdf · Dirac-Born-Infeld eld Joel M. Weller University of She eld,