Perturbation theory - fisica.ugto.mxmsabido/XII_taller/cursos/perturbation_1.pdfChapter 1. Homogeneous Universe Cosmological Perturbations JUAN CARLOS HIDALGO (ICF-UNAM). XII TALLER

Perturbation theoryAnd its observables

LECTURE 1

Chapter 1. Homogeneous Universe

2JUAN CARLOS HIDALGO (ICF-UNAM). XII TALLER DGFMCosmological Perturbations

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How do we describe the Universe?• Depends on what we see:

Galaxies and groups

A Cosmic background radiation

Distance tracers

• Depends on what we are looking for:

The origin of the Universe (as a whole)

Why the universe is in an accelerated expansion?

How structures assemble?

JUAN CARLOS HIDALGO (ICF-UNAM). XII TALLER DGFMCosmological Perturbations

• Cosmological principle estipulates that universe is invariant:Wherever you stand (homogeneous) and whichever direction you look at (isotropic)

• Comoving coordinates and uniform dynamics yield,

:

4

The unperturbed Universe

where



• Comoving coordinates and uniform dynamics yield,

5

The expanding Universe by E.A. Poe



• Comoving coordinates and uniform expansion yield,

• RW proved that the metric is the most general case of an homogeneous and isotropic expansion

Where spatial curvature k defines the geometry

:

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The unperturbed Universe

JUAN CARLOS HIDALGO (ICF-UNAM). XIV-EXFA la caza del Inflatón

where

22 2 2 2 2 2 2

2( ) ( sin( ) )

1

drds a d r d d

kr

• Comoving (orthogonal) observer,

• Kinematic quantities defined with projection tensor

Observers and kinematicsd

ndx

1n n

4-velocityproper time η

world-line

0 0( )[ ]P g n n a t

projector on

( )[ 1,0,0,0]n a t

P g n n


• Comoving (orthogonal) observer,

• Kinematic quantities defined with projection tensor ,

• Expansion , vorticity , shear , and acceleration .

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Observers and kinematics


dn

dx

1n n

P g n n

• The rate of change of an infinitesimally commoving volume V is given by

• The Lie derivative of the projection tensor along the velocity field is

the extrinsic curvature of spatial hypersurfaces

9

Geometrical quantities1

3dV

HV d

K K H


• The rate of change of an infinitesimally commoving volume V is given by

• The Lie derivative of the projection tensor along the velocity field is

the extrinsic curvature of spatial hypersurfaces

• In an FLRW universe, the comoving horizon is related to

the (comoving) Hubble radius• From

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Geometrical quantities1

3dV

HV d

K K H

0 0 0' log log

2

a aH

c H

H

rcr d d a r d a

raH

H

c cr

aH

H

Radiation dominationMatter domination

2 0ds d dr

thus Hubble HorizonH

c cr

aH

H


• Stress Energy tensor with anisotropic stress.

• Proper frame of perfect fluid (with )

• Dark matter presents no velocity dispersion, no pressure, no shear .

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Unperturbed ingredients

diag( , , , ).T p p p p





• From distribution function

• Energy density, momentum density and pressure

• For ultra-relativistic particles , thus .

• also in equilibrium this is a perfect fluid

12


diag( , , , ).T p p p p

E P 3P





• A scalar field has

• The homogeneous field can be described as a perfect fluid

13


diag( , , , ).T p p p p


• Einstein equations (and Conservation of ) for a perfect fluid ( )

• Solutions

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Evolution of horizons in FLRW


p T

3 ( )d

H Pdt

2

0

m

H

H

• Einstein equations (and Conservation of ) for a perfect fluid ( )

• Solutions

• Comoving horizon ever expanding

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Evolution of horizons in FLRW


p T

3 ( )d

H Pdt

2 2

4/ 3Hdr a G

a pdt a H

2

0

m

H

H

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(Problem for Big Bang)• Homogeneous Universe beyond the rH at recombination.


17

Solution: Inflation• Homogeneous Universe beyond the rH at recombination.

• Require a shrinking rH for early times: inflation

Scalar field (inflaton)

Slow roll conditions.


Chapter 2. Perturbations


• Approximating scheme to solve problems from solutions to related simplifications. Example:

For tensors

And a Taylor expansion

19

What are perturbations?

126 25 1 5 1 5(1 1/ 50) 5.1( 5.099)

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• Approximating scheme to solve problems from solutions to related simplifications.

For tensors

And a Taylor expansion

In Cosmological Perturbation Theory we deal with deviations from FLRW Universe:

• Inhomogeneities or anisotropies.

• Why Perturbations? Observations from CMB:

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What are perturbations?


( , ) ( ) ( , ) ( )(1 )x t t x t t

510T

T

• Unambiguous definition of perturbations requires a map.

• The map must account for a small deviation from background.• What if ? Then is independent of mapping.

• Then is Gauge-Independent (Stewart-Walker Lemma)

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What are perturbations? ( , ) ( ) ( , ) ( )(1 )x t t x t t

0Q QQ


• Unambiguous definition of perturbations requires a map.

• The map must account for a small deviation from background.

• No unique way of defining this map (re: coordinate choice).

• No unique way of defining perturbations (i.e. gradient expansion).

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What are perturbations? ( , ) ( ) ( , ) ( )(1 )x t t x t t

2 22 2 1Q aH k Q aH (at super-horizon scales)


• The metric tensor perturbations

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Metric Perturbations

2

, ,2 ( 2 )ij i ij jF hEa

( ) ( , )g g t g x t

22a 2

,( )i ia B S

Gravitational potential

Potential shift



• Split from Helmholtz Theorem

• Scalar, vector and tensors decouple at first order.

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2


( ) ( , )g g t g x t

22a 2

,( )i ia B S


Potential shiftLocal scale factor

k

kg 213( ) 'i j E B Shear scalar



• Geometrical quantities:• Curvature of spatial hypersurfaces

• Acceleration

• Expansion

• Proper time

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2


( ) ( , )g g t g x t

22a 2

,( )i ia B S


Potential shiftLocal scale factor

k

kg 213( )i j E B Shear scalar σ

(1 )d dt


• The Stress-Energy tensor split

• Split not explicitly shown but

• Scalar field:

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Perturbations( ) ( , )T T t T x t

Matter density perturbation

Velocity potentialLocal pressure

k

kT Anisotropic stress

T

0 2

0 ,

0 2

,

2

,

'( ' ')

'( )

'( ' ')

i i

i i

j j

T a U

T a

T a U

1

,1 , iia v vu


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Gauges• depends on our choice of equal-time hypersurface at each point (x,t) :• So will also depend on this choiceof time-slicing or gauge choice

Gauge Transformation:Coordinate transformation which map points of one slicing to another1) Must be small change2) Helmholtz split 3) Imposed to specific characteristics of

Q(t)

( , )Q r t

x x

( , )Q r t

,( , )i i


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Gauges• depends on our choice of equal-time hypersurface at each point (x,t) :• So will also depend on this choiceof time-slicing or gauge choice

Gauge Transformation:Coordinate transformation which map points of one slicing to another1) Must be small change2) Helmholtz split 3) Imposed to specific characteristics of4) OJO: spurious quantities may appear

Q(t)

( , )Q r t

x x

( , )Q r t

,( , )i i


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Gauges• Active approach: Map transforming perturbed quantities

• Vector field generating transformation

• Expansion of exponential map

• Split of tensor transformation

1, 1 22

( , )i i

1 1 1

2

2 2 2 1 1 1

£

£ (£ ) 2£

T T

T T T

T T T T T


• Passive approach: Provide relation between coordinates and

• Require total quantities invariant

• Expansion of both sides in perturbations

• Result: Transformation rule at first order:

• Same applies for any other 4-scalar

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Gauges ,

211 22

( , )i i

( )x q ( )x q

1 1 1'Q Q Q


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Gauges• Vector and tensor transformations computed through exponential map.• Relevant results from vectors

• Velocity transformation

• Scalar off-diagonal metric

• Results from tensor transformations• Scalar metric potentials

• Gravitational Waves

1 1 1 'v v

1 1 1 1'B B


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Gauges…what Gauges?• Observers may measure different observables • Observers that see uniform field:

• Require:

• Result: Curvature perturbation in uniform density gauge.

11 1 1 1' 0 '

'

1

'

H


33

Gauges…what Gauges?• Observers may measure different observables • Observers that see uniform field:

• Require:

• Result: Curvature perturbation in uniform density gauge.

• Observers with unperturbed spatial hypersurfaces:• Require

• Result: Field perturbation in flat gauge.

11 1 1 1' 0 '

'

1

'

H

0E 1 1 1 1, E H

11 1 ' MS

Q

H


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Gauges…what Gauges?• Observers may measure different observables • Observers that see uniform scalar field:

• Require:

• Result: Curvature perturbation in Uniform field gauge.

11 '

'

11 1

'

H


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Gauges…what Gauges?• Observers may measure different observables • Observers that see uniform scalar field:

• Require:

• Result: Curvature perturbation in Uniform field gauge.

• Observers that experience no shear:• Require

• Result: Metric potentials in Longitudinal or Newtonian Gauge.

11 '

'

11 1

'

H

1 1 1 1 'B E

1 1 1 1 1 1

1 1 1 1

' ' '

'

B E B E

B E

H

- H


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Gauge Invariants• Combine two scalars

• Obtain a Gauge-invariant quantity from their difference

1 1

'

H

F : flat

=0

=0 : uniform density

: uniform field =0


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• Obtain a Gauge-invariant quantityThe Uniform density curvature

11 1

'

H H H

F : flat

=0


: uniform field =0


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• Obtain a Gauge-invariant quantityThe Uniform density curvature

• Bardeen Potentials are the first but not only gauge-invariants.

• Curvature Perturbation in Uniform field (comoving) gauge

11 1

'

H H H

1 1 1 1 1 1

1 1 1 1

' ' '

'

B E B E

B E

H

- H

11 1

'

H R

F : flat

=0


: uniform field =0


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Gauge lessons• Physically meaningful are found by fixing a gauge completely.

• Gauge-invariant is any fixed-gauge quantity.

• Gauge transformations show only two degrees of freedom

• Different problems do with specific gauges.

F : flat

=0


: uniform field =0

1Q

1Q

,( , )i i


Chapter 3. Perturbation Dynamics


41

Conservation Equations• Energy conservation at first order (continuity equation)

• In terms of uniform density curvature (with ) :


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Conservation Equations• Energy conservation at first order (continuity equation)

• In terms of uniform density curvature (with ) :

• Result valid at all orders, is conserved if no entropy perturbations appear.

Constant for adiabatic and large scales 1P


43

Conservation Equations• Momentum conservation (Euler equation)


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• In a comoving gauge: 0V v B

Acceleration produced by pressure gradients


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• In a comoving gauge:

• For pressureless dust:

0V v B


In synchronous dust velocity evolves as 1 1/V a


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• In a comoving gauge:

• For pressureless dust:

• In Longitudinal gauge, Euler + continuity:

0V v B


In synchronous dust velocity evolves as 1 1/V a

2 2

1 1 12 4 0sH G c ¡Tarea!


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Einstein Equations• Energy and momentum constraints

• In longitudinal gauge:

2 2

1com4 Ga

Poisson Equation at all scales!


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Einstein Equations• Evolution equations:

• In longitudinal gauge: Equivalent potentialsif no anisotropic stress


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References• Julien Lesgourges. Neutrino Cosmology. CUP, 2013• David Lyth, Andrew Liddle. The primordial density perturbation. CUP, 2009• Karim Malik, David Wands. Cosmological perturbations arXiv:0809.4944v2• David Langlois & Filippo Vernizzi.


Chapter 3. Perturbation Solutions


Chapter 4. Inflation Perturbations

51JUAN CARLOS HIDALGO (ICF-UNAM). XIV-EXFA la caza del Inflatón

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Perturbation theory - fisica.ugto.mxmsabido/XII_taller/cursos/perturbation_1.pdfChapter 1. Homogeneous Universe Cosmological Perturbations JUAN CARLOS HIDALGO (ICF-UNAM). XII TALLER