Physical Cosmology 6/5/2016

Physical Cosmology 6/5/2016

Alessandro Melchiorri

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

oberon.roma1.infn.it/alessandro/cosmo2016

Today lecture is also slightly based on:

Structure formation

We are moving away from an homogeneous and isotropic universe.

We have introduced the density contrast:

“Mean”density.

Average is over Volume

And we assumed it as very small (linear perturbation theory):

Structure formation

If we now consider a pressure less fluid (w=0), with a very qualitative treatment, we have found that in an

expanding universe:

Remember: the density contrast is in the matter (w=0) component, but the expansion of the universe can be dominated by a different energy component !

Structure Formation in Expanding Universe

A key point is that the matter component that is collapsing could be different from the dominant energy component.

Introducing the time dependent density parameter:

We can write:

Structure Formation in Expanding Universe

Let us first consider the radiation here. During this epoch:

The equation can be therefore written as:

with solution:

during radiation dominated epoch the perturbations in matter grow only logarithmically !!!

Structure Formation in Expanding Universe

If we move to matter dominated we have:

Structure Formation in Expanding Universe

The equation is therefore:

The solution is a power law:

During matter dominated, matter fluctuations grow with a power of 2/3 (less than in a static universe).

We discard the 1/t solution since we expect a more homogeneous

Universe in the past

Structure Formation in Expanding Universe

Finally, in an epoch dominated by a cosmological constant:

The equation is:

And the solution:

If Lambda dominates the expansion, then the fluctuations in the matter component remain constant.

They are, in practice, frozen.

First Summary

If we consider perturbations in a pressureless matter component (Jeans length always zero) their growth depend on which kind of energy component is

dominating the expansion.

We have substantial growth only

here !

Moving to Fourier Space

In order to move to a more physical description let us consider an expansion in Fourier modes of the density contrast field:

Each Fourier mode is given by:

Each Fourier component is a complex number, which can be written in the form:

It is possible to show that assuming a linear perturbation theory, i.e. :

And following a more formal approach, one gets, in the Newtonian regime, for a pressure less fluid:

Moving to Fourier Space

Moving to Fourier Space

This is the same equation we got for the density contrast of a sphere of radius R, but now it applies to each

Fourier mode of a generic density contrast !

The fact that we assumed linear perturbation theory implies that the evolution of each Fourier mode is

independent from the other, i.e. we don’t have

density contrasts of different modes in the equation and their time evolution does not mix.

Pressure term

The previous equation holds for a pressure-less fluid (w=0). If we consider pressure, it is possible to show that the equation modifies to:

New term due to fluid pressure.

k is in comoving coordinates, so at each wavenumber k corresponds a physical scale at time t of:

From the above equation we identify the Jeans wavelength:

Each Fourier mode will therefore evolve with time in a different way if the corresponding k is larger or

smaller than the Jeans wavelength.

If

We can neglect this.

and we have the growth as discussed for a fluid with w=0.

Pressure term

If, on the contrary, we are below the Jeans lenght:

We neglect the first term and the solution is given by a more complicate oscillation term damped in time.

Pressure term

Horizon Scale

Given a time t, the quantity:

provides a causal horizon, i.e. particles that are a distances larger than it are not causally connected.

What happens if I consider a perturbation on scales larger than the horizon scale ?

We can treat them only using general relativity.

As we will see, the solution is also gauge-dependent.

Horizon Scale Horizon Scale

Assuming a synchronous gauge, it is possible to show that perturbations on scales larger than the horizon, i.e. such that at a given time t have:

they always grow, as:

Summary (single fluid)

We have therefore two important scales for structure formation: the horizon scale and the Jeans scale.

For cold dark matter (w=0) what is important is the

horizon scale at equivalence. Perturbations that enter the horizon before the epoch of equivalence are damped

respect to perturbations that enter the horizon later.

For CDM the Jeans scale is always zero.

For baryons, the Jeans length is approximately the

horizon scale until decoupling. The crucial scale is the horizon scale at decoupling. After decoupling baryons have w=0 (approximately).

Perturbations that enter the horizon before decoupling (z=1100) are strongly damped respect to perturbations that enter the horizon later.

For example, we can consider two modes, one entering the horizon before the matter-radiation equivalence and another one entering after it.

log δ

a log

Perturbation in Red: enters the horizon AFTER the equivalence.

Perturbation in Blue: enters the horizon BEFORE equivalence.

aEQ

a2

k1

k2

) (

/ ₂ ¹ ₂

2 a c H a

k = ⁻

1

2

k

k >

a1

Evolution for a w=0 (no pressure)

component.

Perturbations with k larger

than

are damped respect to perturbations with k smaller Cold dark

matter (only)

For example, we can consider two modes, one entering the horizon before the decoupling and another one entering after it.

log δ

a log

Perturbation in Red: enters the horizon AFTER the decoupling.

Perturbation in Blue: enters the horizon BEFORE decoupling.

a2

k1

k2

) (

/ ₂ ¹ ₂

2 a c H a

k = ⁻

1

2

k

k >

a1

Evolution

for the baryon component.

Perturbations with k larger

than

are strongly damped respect to perturbations with k smaller Baryons

(only)

We ar

e below the Jens lenght.

Damping+Oscillations

log δ

a log

Baryon/CDM Perturbation in Red: enters the horizon AFTER the decoupling.

Baryon Perturbation in Blue: enters the horizon BEFORE equivalence.

CDM Perturbation in Green: enters the horizon BEFORE equivalence.

a2

k1

) (

/ ₂ ¹ ₂

2 a c H a

k = ⁻

a1

CDM

+Baryons

aEQ

The situation is different if we consider a CDM+Baryon case.

Baryons “feel" the CDM gravitational potential.

Baryons,

after decoupling fall in the CDM potential wells.

k2

k2

Cosmological «Circuit»

Generator of Perturbations (Inflation)

Amplifier (Gravity)

Low band pass filter.

Cosmological and

Astrophysical effects

Tend to erase small scale (large k) perturbations

Power Spectrum

Each Fourier component is a complex number, which can be written in the form

The mean square amplitude of the Fourier components defines the power spectrum :

where the average is taken over all possible orientations of the wavenumber. If δ(⃗r) is isotropic, then no information is lost,

statistically speaking, if we average the power spectrum over all angles and we get an isotropic power spectrum:

Correlation function

Let us consider the autocorrelation function of the density field (usually called the correlation function):

Where the brackets indicates an average over a volume V.

We can write:

and, performing the integral we have:

Correlation function

Since the correlation function is a real number, assuming an isotropic power spectrum we have:

If the density field is gaussian, we have that the value of δ at a randomly selected point is drawn from the Gaussian

probability distribution:

where the standard deviation σ can be computed from the power spectrum:

Summary

In practice, our theory cannot predict the exact value of

in a region of the sky. But if we assume that the initial perturbations are gaussian we can predict the correlation function, the variance of the fluctuations and its the power spectrum P(k).

These are things that we can measure using, for example, galaxy surveys and assuming that galaxies trace the

CDM distribution.

Cosmological «Circuit»

Generator of Perturbations (Inflation)

Amplifier (Gravity)

Low band pass filter.

Cosmological and

Astrophysical effects

Tend to erase small scale (large k) perturbations

Power spectrum for CDM

The analysis we have presented up to now is very qualitative and approximated (just to have an idea…)

The true power spectrum for CDM density fluctuations can be computed by integrating a system of differential equations.

We will see this in better detail in the next lectures.

In any case, we assume a power law as initial power spectrum as:

The motivation of using this type of primordial spectrum comes from inflation (we will discuss this also in a future lecture). We have two free parameters: the amplitude A and the spectral index n^s.

Power spectrum for CDM

The first numerical results on the CDM power spectrum appeared around 1980.

Bardeen, Bond, Kaiser and Szalay (BBKS) in 1986 proposed the following fitting function:

where: This is correct fit for a pure CDM model (no baryons or massive neutrinos).

Note the dependence on

that defines the epoch of equivalence.

P(k) for LCDM (from numerical computations).

Spectral index is assumed

n=0.96

Note these oscillations in the

CDM P(k).

Gravitational feedback from

baryons.

The position of the peak is related to size of the

horizon at equivalence, i.e.

to the matter density since radiation is fixed.

Primor dial r

egime

Damping (scales that enter

ed horizon befor

e quality)

Effect of the Cosmological parameters

Cold dark matter

Effect of the Cosmological parameters

Baryon density

Effect of the Cosmological parameters

Spectral index

CDM vs data (1996)

If we assume a flat universe made just of matter with

we get too much power on small scales

to match observations.

Already in 1996 the best fit was for

i.e. suggesting a low matter density universe.

CDM vs data (2003)

The 2dfGRS provided

the following constraints:

(assuming just CDM, no massive neutrinos)

2 sigma detection of baryons:

Measurements from SDSS

Mass fluctuations

Given a theoretical model a quantity can be often easily compared with observations is the

variance of fluctuations on a sphere of R Mpc:

where:

Usually one assumes R=8 Mpc hˆ-1, where the linear approximation is valid.

Examples from CAMB

http://lambda.gsfc.nasa.gov/toolbox/tb_camb_form.cfm