## Physical Cosmology 27/5/2016

Alessandro Melchiorri

alessandro.melchiorri@roma1.infn.it slides can be found here:

oberon.roma1.infn.it/alessandro/cosmo2016

### The Cosmic Microwave Background

Discovered By Penzias and Wilson in 1965.

It is an image of the universe at the time of recombination (near

baryon-photons decoupling), when the universe was just a few thousand years old (z~1000).

The CMB frequency spectrum

is a perfect blackbody at T=2.73 K:

this is an outstanding confirmation of the hot big bang model.

Uniform...

Dipole...

Galaxy (z=0)

### The Microwave Sky

COBE

Imprint left by primordial

tiny density inhomogeneities (z~1000)..

### Fluctuation generator

### Fluctuation amplifier

### Dens Hot Sm oot e

### h

### Ra Cool

### refi ed

### Clum py

## CMB map from COBE

### 1992

## CMB map from WMAP

### 2003

## CMB map from Planck

### 2015

In the past 20 years we measured the CMB sky with improved sensitivity, frequency range and angular resolution.

7 degree 0.3 degree 0.08 degree

Planck maps for 9 frequencies

Main foreground:

Synchrotron

Main

foreground:

Dust

In practice: we see only dust for channels above 353 Ghz

### ( )

^{Δ}

### ( )

^{=}

### ∑

^{+}

### (

^{⋅}

### )

Δ

2 1

2

1 (2 1)

2

1 γ γ

γ π

γ *C* *P*

*T*
*T*
*T*

*T*

### The CMB Angular Power Spectrum

R.m.s. of has
power per decade in l:Δ*T /* *T* *l +*

### (

*l*

^{1}

### )

*C*

*l*

^{/}

^{2}

^{π}

( ) ( ) ( )

∑ ^{+} ^{≈} ∫ ^{+}

=

Δ *l* *l* *C* *d* *l*

*l* *C*
*T*

*T* _{l}

*l*

*rms* *l* ln

2 1 4

1
/ ^{2} 2

π π

Doroshkevich, A. G.; Zel'Dovich, Ya. B.; Syunyaev, R. A.

Soviet Astronomy, Vol. 22, p.523, 1978

A Brief History of the CMB Anisotropies Angular Spectrum (Theoretical predictions)

Wilson, M. L.; Silk, J., Astrophysical Journal, Part 1, vol. 243, Jan. 1, 1981, p. 14-25.

1981

A Brief History of the CMB Anisotropies Angular Spectrum (Theoretical predictions)

Bond, J. R.; Efstathiou, G.; Royal Astronomical Society, Monthly Notices (ISSN 0035-8711), vol. 226, June 1, 1987, p. 655-687, 1987

A Brief History of the CMB Anisotropies Angular Spectrum (Theoretical predictions)

Chung-Pei Ma, Edmund Bertschinger, Astrophys.J. 455 (1995) 7-25

A Brief History of the CMB Anisotropies Angular Spectrum (Theoretical predictions)

Hu, Wayne; Scott, Douglas; Sugiyama, Naoshi; White, Martin.

** Physical Review D, Volume 52, Issue 10, 15 November 1995, pp.5498-5515 **

A Brief History of the CMB Anisotropies Angular Spectrum (Theoretical predictions)

A Brief History of the CMB Anisotropies Angular Spectrum (Experimental Data)

In 1995 Big Bang Model was nearly dead…

Collection of CMB anisotropy data from C. Lineweaver et al., 1996

A Brief History of the CMB Anisotropies Angular Spectrum (Experimental Data)

Test flight: P. Mauskopf et al, ApJ letters 536, L59, (2000), astro-ph/9911444 A. Melchiorri et al, ApJ letters 536, L63, (2000),astro-ph/9911445 LDB-B00: P. de Bernardis et al, Nature 404 (2000) 955-969, asro-ph/0004404

A. E. Lange et al, PRD D63 (2001) 042001, astro-ph/0005004

Maxima: A. Jaffe et al, Phys. Rev. Lett, (2001) 86, 3475-3479, astro-ph/0007333 LDB-B01: B. Netterfield et al, ApJ in press, astro-ph/0104460

P. de Bernardis et al, ApJ in press, astro-ph/0105296.

### BOOMERanG Experiment

**Boomerang’s Track
**

(1 lap in 10.6 days...)

*Blue and Red
*

*are two separate
*

*analyses of same data*

*Beam*

*uncertainty
*
*(correlated)*

1.8% of the sky Boomerang Power Spectrum

20% calibration uncertainty

(correlated)

Error bars

10% correlated

11 days of observations 4 channels at 150GHz

CMB anisotropies pre-WMAP (January 2003)

**Wilkinson Microwave Anisotropy Probe**

Best fit model

cosmic variance

**Temperature**

**Temperature-** **polarization**

1 deg

85% of sky

Spergel et al, 2003

### BOOM03 Flight

Launched: January 6, 2003

! Polarization sensitive receivers 145/245/345 GHz

! Flight January 2003

! 195 hours (11.7 days) of data
f_{sky}= 1.8%

! First results published in July 2005

– Masi et al. astro-ph/0507509

– Jones et al. astro-ph/0507494

– Piacentini et al. astro-ph/0507507

– Montroy et al. astro-ph/0507514

– MacTavish et al. astro-ph/0507503

Feet

[MacTavish et al. 2005]

A Brief History of the CMB Anisotropies Angular Spectrum (2012)

### Planck 2015

CMB observations

@May 2016

### The shape of the CMB angular spectrum depends on

### cosmological parameters !

### How to get a bound on a cosmological parameter

DATA

Fiducial cosmological model:

(Ω_{b}h^{2 }, Ω_{m}h^{2 }, h , n_{s }, τ, Σm_{ν })

PARAMETER ESTIMATES

Cosmic Microwave Background Anisotropies: Theory

http://space.mit.edu/home/tegmark/varenna.html

### CMB Anisotropy: BASICS

### • Friedmann Flat Universe with 5 components:

### Baryons, Cold Dark Matter (w=0, always),

### Photons, Massless Neutrinos, Cosmological Constant.

### • Linear Perturbation theory. Conformal Newtonian Gauge. Scalar modes only.

40

### • Perturbation Variables:

### CMB Anisotropy: BASICS

### Key point: we work in Fourier space :

41

### CMB Anisotropy: BASICS

CDM:

Baryons:

Photons:

Neutrinos:

Their evolution is governed by a nasty set of coupled partial differential equations:

42

### Numerical Integration

- Early Codes (1995) integrate the full set of equations (about 2000 for each k mode, approx, 2 hours CPU time for obtaining one single spectrum).

COSMICS first public Boltzmann code http://arxiv.org/abs/astro-ph/9506070.

- Major breakthrough with line of sight integration method with CMBFAST

(Seljak&Zaldarriaga, 1996, http://arxiv.org/abs/astro-ph/9603033). (5 minutes of CPU time)

- Most supported and updated code at the moment CAMB (Challinor, Lasenby, Lewis), http://arxiv.org/abs/astro-ph/9911177 (Faster than CMBFAST).

Another recent code (CLASS) is available at http://class-code.net.

- Both on-line versions of CAMB and CMBFAST available on LAMBDA website.

43

### CMB Anisotropy: BASICS

CDM:

Baryons:

Photons:

Neutrinos:

Their evolution is governed by a nasty set of coupled partial differential equations:

44

### First Pilar of the standard model of structure formation:

Standard model: Evolution of perturbations is passive and coherent.

### Active and decoherent models of structure formation

### (i.e. topological defects see Albrecht et al, http://arxiv.org/

### abs/astro-ph/9505030):

Linear differential operator

Perturbation Variables

45

Oscillations supporting evidence for passive and coherent scheme.

46

### CMB anisotropies

The temperature fluctuation in a direction can be expressed as a line-of-sight integral:

We have 3 terms:

Gravity: photons coming out from CDM potential wells will suffer a redshift.

Intrinsic: where we have more baryons we have more photons (tight coupling) and a blue shift.

Doppler: baryons move and they leave a doppler effect when scattering off photons.

## CMB anisotropies @LSS

We have that:

n^{e} is the free electron fraction.

In the limit of infinitely thin LSS the function

approaches to a Dirac

delta-function centered at recombination.

## CMB anisotropies @LSS

As a first approximation we have just to compute the value of the first three terms at the time of

recombination !

For simplicity we also neglect the last term.

This term is called Integrated Sachs-Wolfe effect and is zero when the universe is dominated by matter.

## Equations for CDM

Continuity Euler

Poisson

Here we have the

total density perturbation.

We will assume to be dominated by CDM.

## Equation for photons

The relativistic species (photons and neutrinos) are not

characterized by a simple velocity field, but by a distribution function f whose evolution is governed by the Boltzmann

equation,

Here p is the magnitude of the momentum, γ is a direction cosine of the momentum, and C is a collision term having to do with scattering.

## Primary anisotropies

We will begin by assuming that, after the end of the radiation epoch, most of the mass in the Universe is in the form of cold dark matter:

Then the gravitational potential is completely determined by the CDM.

**We also make another approximation: we assume tight **
**coupling between photons and baryons. Specifically, we **

assume that the mean free time τ between photon collisions is small compared to the other important time scales:

This is an excellent approximation right up until around the time of last scattering.

## Primary anisotropies

In a tight coupling approximation, the photon and baryon densities are coupled adiabatically:

The behavior of the photon-baryon fluid is therefore

characterized by a single variable. We will find it convenient to take as our variable the fractional temperature fluctuation Θ,

given by:

## Primary anisotropies

With these approximations, the dynamics of the photon-baryon fluid is described by the single equation:

Where:

Conformal time

Baryon/photon ratio

Gravitational driving

## Primary anisotropies

It is possible to write:

With the approximations that we’re making, there are no

anisotropic stresses, so the two gravitational potentials are simply related to each other:

## Primary anisotropies

The Poisson equation:

In Fourier space since k is co-moving we have:

So if the expansion is matter dominated we have that

δ≈a and by using the Poisson equation in Fourier space we see that Ψ is constant with time.

The gravitational driving term therefore simplifies to

## Primary anisotropies

If we assume that R is also slowly varying with time, we have:

This is the equation for a simple harmonic oscillator, with solution:

where:

## Primary anisotropies

In this approximation, then, each Fourier mode represents an
acoustic plane wave propagating at sound speed c^{s}.

There is a simple physical picture underlying this result. The baryon-photon fluid wants to fall into the potential wells, but it is supported by radiation pressure. The balance between

**pressure and gravity sets up acoustic oscillations. **

Let’s make things even simpler and set R = 0. Then

**In many theories, the initial perturbation is adiabatic, meaning **
that the matter and radiation fluctuations are the same at any
particular point. In this case we have the initial conditions:

## Primary anisotropies

In this very simple case we therefore have:

Since we have to compute:

Ignoring the Doppler term for the moment, note that the other two terms give a pure cosine oscillation in k:

LS

with frequency given by the sound horizon at LS (≈10 Mpc).

## Primary anisotropies

Let us consider now the Doppler term.Using the continuity equation and the relation δ = 3Θ, we find that

Since for R=0 we have and since we assume ψ time independent, we have:

Since the r.m.s. value of is , the r.m.s. Doppler contribution to equation is:

This has the same amplitude as the (Θ + Ψ) contribution, but is 90 degrees out of phase in both time (it goes like a sine

instead of a cosine) and space (it has an extra factor of i). This has the rather disastrous consequence of completely erasing the Doppler peaks: the total ∆T/T is the quadrature sum (we are considering the power spectrum) is:

## Primary anisotropies

The k dependence, which led to the peaks, is gone.

## Primary anisotropies

Let us remove the assumption R=0, but keep the

approximation that R is time-independent. Then the solution for Θ(η) changes in two ways.

1) The sound speed gets smaller by a factor:

2) The driving term F (η) gets bigger by a factor The adiabatic solution to Θ is now:

By allowing R to be nonzero, we have increased the amplitude of the cosine oscillations by a factor (1 + 3R). Furthermore,

there is now an offset in the combined Sachs-Wolfe and

adiabatic contributions to ∆T/T: in the limit R → 0, we found that Θ + Ψ oscillated symmetrically about zero; now it

oscillates about −RΨ.

## Primary anisotropies

We also have a smaller amplitude of the Doppler contribution,
since v is proportional to c^{s} and c^{s} has gotten smaller. Since
the cosine oscillations are now larger in amplitude than the
**sine oscillations, we do indeed expect to see a series of **
**peaks at**

Why does including the dynamical effect of the baryons effect these changes in the solution? The essential reason is that

**baryons contribute to the effective mass of the photon-baryon **
fluid, but not to the pressure. The effect of the baryons,

therefore, is to slow down the oscillations, and also to make the fluid fall deeper into the potential wells.

## Primary anisotropies

Hu, Sugiyama, Silk, Nature 1997, astro-ph/9604166

65

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Baryonic abundance and CMB peaks

CMB spectrum is therefore strongly dependent from R, i.e. the baryon

abundance.

Increasing the baryon density

increases the odd peaks and erases the even peaks.

### Projection

A mode with wavelength λ will show up on an angular scale θ ∼ λ/R, where R is the distance to the last-scattering surface, or in other words, a mode with wavenumber k shows up at multipoles l∼k.

The spherical Bessel function j^{l}(x) peaks at x ∼ l, so
a single Fourier mode k does indeed contribute

most of its power around multipole l^{k} = kR, as
expected. However, as the figure shows, jl does
have significant power beyond the first peak,

meaning that the power contributed by a Fourier
mode “bleeds” to l-values different from l^{k}.

Moreover for an open universe (K is the curvature) :

l=30

l=60

l=90

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### Projection

In Fourier space we have oscillations With frequency (or physical scale):

In Legendre space oscillations are smeared and have a frequency that Depends on the angular diameter distance at recombination.

68