Physical Cosmology 18/5/2017

Alessandro Melchiorri alessandro.melchiorri@roma1.infn.it slides can be found here: oberon.roma1.infn.it/alessandro/cosmo2017

SummaryIf 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 SpaceIn 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 termThe 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 ScaleGiven 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 ScaleHorizon ScaleAssuming 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.

Horizon at equivalenceThe redshift of equivalence is given by:

The Hubble parameter at equivalence is then

The physical size is then:

And, in comoving coordinates:

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

δlog

alog

Perturbation in Red: enters the horizon AFTER the equivalence. Perturbation in Blue: enters the horizon BEFORE equivalence.

EQa2a

1k

2k

)(/ 21

22 aHcak −=

12 kk >

1a

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.

δlog

alog

Perturbation in Red: enters the horizon AFTER the decoupling. Perturbation in Blue: enters the horizon BEFORE decoupling.

2a

1k

2k

)(/ 21

22 aHcak −=

12 kk >

1a

Evolution for the baryon component.

Perturbations with k larger

than

are strongly damped

respect to perturbations with k smaller

Baryons (only)

We are below the

Jens lenght.

Damping+Oscillations

δlog

alog

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.

2a

1k

)(/ 21

22 aHcak −=1a

CDM +Baryons

EQa

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.

2k

2k

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 SpectrumEach 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 functionLet 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 functionSince 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:

SummaryIn 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 CDMThe 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 ns.

Power spectrum for CDMThe 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.

Primordial re

gime

Damping

(scales that entered

horizon before quality)

If we plot the P(k) in function of h/Mpc, then the dependence is just on

Effect of the Cosmological parameters

Cold dark matter

Effect of the Cosmological parameters

Cold dark matter

Effect of the Cosmological parameters

Baryon density

Effect of the Cosmological parameters

Baryon density

Effect of the Cosmological parameters

Spectral index

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

Massive NeutrinosWe saw that massive neutrinos contribute to the matter density when they are non-relativistic as:

However, until they are relativistic, neutrino have w=1/3, i.e. perturbations in the neutrino component dissipates when they enter the horizon. The wavelength of the horizon when they start to be non relativistic is given by:

Evolution of the density fields in the synchronous gauge (top panels) and the conformal Newtonian gauge (bottom panels) in the Ων = 0.2 CDM+HDM model for 3 wavenumbers k= 0.01, 0.1 and 1.0 Mpc−1. In each figure, the five lines represent δc , δb , δγ , δν and δh for the CDM (solid), baryon (dash-dotted), photon (long-dashed), massless neutrino (dotted), and massive neutrino (short-dashed) components, respectively.

Numerical evolution of density fluctuationsModel with CDM, neutrinos, massive neutrinos, photons, baryons

Focus just on the Top panel

(Syncronous Gauge)

These fluctuations enter the horizon

just after decoupling

CDM, Baryons and MN grow as a^2 and a

Photons and massless neutrinos oscillates and damp

after entering the horizon

Focus just on the Top panel

(Syncronous Gauge)

CDM grow as a^2, ln aˆ2 and a

Until decoupling baryons oscillates and are damped, they quickly follow CDM afterwards.

Massive neutrinos (nearly 1.5 eV in this case) are semi relativistic at equality and damped. They slowly catch CDM afterwards.

These fluctuations enter the horizon just after equality

Photons and massless neutrinos oscillates and damp

after entering the horizon

Focus just on the Top panel

(Syncronous Gauge)

Until decoupling baryons oscillates and are damped, they quickly follow CDM afterwards.

Massive neutrinos are first relativistic and strongly damped because of free streaming. They try to catch CDM afterwards.

These fluctuations enter the horizon well before equality

Photons and massless neutrinos oscillates and damp

after entering the horizon

CDM grow as a^2, ln aˆ2 and a small feedback from b and mnu

Baryons vs Massive Neutrinos

If a perturbation in baryons enters the horizon before decoupling oscillates with a decreasing amplitude because of diffusion damping (photons can go from hotter to colder regions after several scattering). After decoupling, baryons fall in the CDM potential well.

If a perturbation in massive neutrinos enter the horizon before the non relativistic regime is strongly damped because of free streaming (neutrinos are collision-less). Neutrino are very light and also afterwards they still suffer from free streaming and practically don’t cluster.

Massive NeutrinosThe growth of the fluctuations is therefore suppressed on all scales below the horizon when the neutrinos become nonrelativistic

the small scale suppression is given by:

Larger is the neutrino mass, large is the suppression.Hu et al, arXiv:astro-ph/9712057

eff

Massive Neutrinos

Massive Neutrinos

...but we have degeneracies...

• Lowering the matter density suppresses the power spectrum

• This is virtually degenerate with non-zero neutrino mass

Inclusion of CMB data is important to break degeneracies

Conservative limit from CMB+P(k) measurements:

Mass fluctuationsGiven 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.

P(k) for LCDM (from numerical computations). Spectral index is assumed n=0.96

The gives the P(k) amplitude around these scales

Examples from CAMB

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

Output from CAMB

Output from CAMB

Output from CAMB

Output from CAMB