Perturbation theory - fisica.ugto.mxmsabido/XII_taller/cursos/perturbation_1.pdfChapter 1....
Transcript of Perturbation theory - fisica.ugto.mxmsabido/XII_taller/cursos/perturbation_1.pdfChapter 1....
Chapter 1. Homogeneous Universe
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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,
:
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The unperturbed Universe
where
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,
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The expanding Universe by E.A. Poe
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 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
JUAN CARLOS HIDALGO (ICF-UNAM). XII TALLER DGFMCosmological Perturbations
• Comoving (orthogonal) observer,
• Kinematic quantities defined with projection tensor ,
• Expansion , vorticity , shear , and acceleration .
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Observers and kinematics
JUAN CARLOS HIDALGO (ICF-UNAM). XIV-EXFA la caza del Inflatón
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
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Geometrical quantities1
3dV
HV d
K K H
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• 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
JUAN CARLOS HIDALGO (ICF-UNAM). XII TALLER DGFMCosmological Perturbations
• 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
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• Stress Energy tensor with anisotropic stress.
• Proper frame of perfect fluid (with )
• Dark matter presents no velocity dispersion, no pressure, no shear .
• From distribution function
• Energy density, momentum density and pressure
• For ultra-relativistic particles , thus .
• also in equilibrium this is a perfect fluid
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Unperturbed ingredients
diag( , , , ).T p p p p
E P 3P
JUAN CARLOS HIDALGO (ICF-UNAM). XII TALLER DGFMCosmological Perturbations
• Stress Energy tensor with anisotropic stress.
• Proper frame of perfect fluid (with )
• Dark matter presents no velocity dispersion, no pressure, no shear .
• A scalar field has
• The homogeneous field can be described as a perfect fluid
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Unperturbed ingredients
diag( , , , ).T p p p p
JUAN CARLOS HIDALGO (ICF-UNAM). XII TALLER DGFMCosmological Perturbations
• Einstein equations (and Conservation of ) for a perfect fluid ( )
• Solutions
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Evolution of horizons in FLRW
JUAN CARLOS HIDALGO (ICF-UNAM). XIV-EXFA la caza del Inflatón
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
JUAN CARLOS HIDALGO (ICF-UNAM). XIV-EXFA la caza del Inflatón
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.
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Solution: Inflation• Homogeneous Universe beyond the rH at recombination.
• Require a shrinking rH for early times: inflation
Scalar field (inflaton)
Slow roll conditions.
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Chapter 2. Perturbations
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• Approximating scheme to solve problems from solutions to related simplifications. Example:
For tensors
And a Taylor expansion
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What are perturbations?
126 25 1 5 1 5(1 1/ 50) 5.1( 5.099)
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JUAN CARLOS HIDALGO (ICF-UNAM). XII TALLER DGFMCosmological Perturbations
• 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?
JUAN CARLOS HIDALGO (ICF-UNAM). XIV-EXFA la caza del Inflatón
( , ) ( ) ( , ) ( )(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
JUAN CARLOS HIDALGO (ICF-UNAM). XII TALLER DGFMCosmological Perturbations
• 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)
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• 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
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• The metric tensor perturbations
• Split from Helmholtz Theorem
• Scalar, vector and tensors decouple at first order.
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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 shiftLocal scale factor
k
kg 213( ) 'i j E B Shear scalar
JUAN CARLOS HIDALGO (ICF-UNAM). XII TALLER DGFMCosmological Perturbations
• The metric tensor perturbations
• Geometrical quantities:• Curvature of spatial hypersurfaces
• Acceleration
• Expansion
• Proper time
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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 shiftLocal scale factor
k
kg 213( )i j E B Shear scalar σ
(1 )d dt
JUAN CARLOS HIDALGO (ICF-UNAM). XII TALLER DGFMCosmological Perturbations
• 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
JUAN CARLOS HIDALGO (ICF-UNAM). XII TALLER DGFMCosmological Perturbations
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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
JUAN CARLOS HIDALGO (ICF-UNAM). XII TALLER DGFMCosmological Perturbations
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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
JUAN CARLOS HIDALGO (ICF-UNAM). XII TALLER DGFMCosmological Perturbations
• 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
JUAN CARLOS HIDALGO (ICF-UNAM). XII TALLER DGFMCosmological Perturbations
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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
JUAN CARLOS HIDALGO (ICF-UNAM). XII TALLER DGFMCosmological Perturbations
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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.
• 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
JUAN CARLOS HIDALGO (ICF-UNAM). XII TALLER DGFMCosmological Perturbations
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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
JUAN CARLOS HIDALGO (ICF-UNAM). XII TALLER DGFMCosmological Perturbations
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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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Gauge Invariants• Combine two scalars
• Obtain a Gauge-invariant quantityThe Uniform density curvature
11 1
'
H H H
F : flat
=0
=0 : uniform density
: uniform field =0
JUAN CARLOS HIDALGO (ICF-UNAM). XII TALLER DGFMCosmological Perturbations
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Gauge Invariants• Combine two scalars
• 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
=0 : uniform density
: 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
=0 : uniform density
: uniform field =0
1Q
1Q
,( , )i i
JUAN CARLOS HIDALGO (ICF-UNAM). XII TALLER DGFMCosmological Perturbations
Chapter 3. Perturbation Dynamics
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Conservation Equations• Energy conservation at first order (continuity equation)
• In terms of uniform density curvature (with ) :
JUAN CARLOS HIDALGO (ICF-UNAM). XII TALLER DGFMCosmological Perturbations
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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
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Conservation Equations• Momentum conservation (Euler equation)
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Conservation Equations• Momentum conservation (Euler equation)
• In a comoving gauge: 0V v B
Acceleration produced by pressure gradients
JUAN CARLOS HIDALGO (ICF-UNAM). XII TALLER DGFMCosmological Perturbations
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Conservation Equations• Momentum conservation (Euler equation)
• In a comoving gauge:
• For pressureless dust:
0V v B
Acceleration produced by pressure gradients
In synchronous dust velocity evolves as 1 1/V a
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Conservation Equations• Momentum conservation (Euler equation)
• In a comoving gauge:
• For pressureless dust:
• In Longitudinal gauge, Euler + continuity:
0V v B
Acceleration produced by pressure gradients
In synchronous dust velocity evolves as 1 1/V a
2 2
1 1 12 4 0sH G c ¡Tarea!
JUAN CARLOS HIDALGO (ICF-UNAM). XII TALLER DGFMCosmological Perturbations
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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.
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Chapter 3. Perturbation Solutions
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