Physical Cosmology 12/5/2017

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

Structure FormationUntil now we have assumed an homogeneous and isotropic universe.

The scale factor, the Friedmann equations, the energy densities they depend just on time.

But this is just an approximation, the observed universe is never perfectly homogeneous and isotropic !

Structure FormationGalaxy distribution from the Sloan Digital Sky Survey:

We see clusters, filaments, structure.

How can we reconcile this with isotropy and homogeneity ?

How did this structure form ?

What we can learn from it ?

Structure FormationLet’s start from the most simple case: a static universe !

We consider a sphere a radius R. Outside the sphere we have everywhere a density , constant with time. Inside the sphere we have a density

with defined as the density constrast and that varies with time. How does this overdensity evolves with time ?

Structure FormationIf the density excess δ is uniform within the sphere, then the gravitational acceleration at the sphere’s surface, due to the excess mass, will be

Thus, a mass excess (δ > 0) will cause the sphere to collapse inward ( < 0).

Structure FormationConservation of mass tells us that the mass of the sphere,

remains constant during the collapse.Thus, we can write another relation between R(t) and δ(t) which must hold true during the collapse:

where

Structure FormationLet us now assume that the density contrast is small:

we can retain values at linear order for δ(t) and neglect higher powers:

Taking the second time derivative yields :

Thus, mass conservation tells us that

Structure Formation

The most general solution of this equation has the form

where the dynamical time for collapse is

Structure formationIf we consider just the growing solution (the initial universe is homogeneous):

In a static universe fluctuations grow exponentially with a dynamic timescale:

i.e. denser the medium faster is the collapse.

PressureThe density of the air around you is ρ≈ 1kgm−3, yielding a dynamical time for collapse of tdyn ≈ 9 hours. What keeps small density perturbations in the air from undergoing a runaway collapse over the course of a few days? The answer, of course, is pressure.A non-relativistic gas has an equation-of-state parameter

where T is the temperature of the gas and μ is the mean mass per gas particle. Thus, the pressure of a ideal gas will never totally vanish, but will only approach zero in the limit that the temperature approaches absolute zero.

PressureIf the pressure is nonzero, the attempted collapse will be countered by a steepening of the pressure gradient within the perturbation. The steepening of the pressure gradient, however, doesn’t occur instantaneously. Any change in pressure travels at the sound speed.Thus, the time it takes for the pressure gradient to build up in a region of radius R will be

where cs is the local sound speed. In a medium with equation-of-state parameter w > 0, the sound speed is

Pressure

Time scale for collapse Time scale for pressure

We have collapse only if:

Or, if the size of the perturbation is larger than the Jeans lenght:

Structure FormationCold dark matter, always non relativistic:

=0

practically any over density will collapse. Pressure is negligible.

Photons, or relativistic particles:

=1/3

photons are free streaming. In an expanding universe the Jeans length is close to the horizon size ! We don’t have structure.

Structure FormationSome energy component has w that depends on time. For example: baryons.For z>1100 baryons are tightly coupled to CMB photons. They make a relativistic plasma with w=1/3. For z<1100 baryons are decoupled. They have masses >0.5 MeV and are non relativistic. They have w=0. Before z=1100 thus we have:

Structure FormationBut before and after decoupling the baryon sound speed changes a lot:Before:

After:

The Jeans length collapses after decoupling, we have:

Perturbations in the baryon density, from supercluster scales down the the size of the smallest dwarf galaxies, couldn’t grow in amplitude until the time of photon decoupling.

Structure Formation in Expanding Universe

Suppose you are in a universe filled with pressureless matter which has mass density ρ (̄t). As the universe expands, the density decreases at the rate ρ (̄t) ∝ a(t). Within a spherical region of radius R, a small amount of matter is added, or removed, so that the density within the sphere is

In performing a Newtonian analysis of this problem, we are implicitly assuming that the radius R is small compared to the Hubble distance and large compared to the Jeans length.

Structure Formation in Expanding Universe

The total gravitational acceleration at the surface of the sphere will be

The equation of motion for a point at the surface of the sphere can then be written in the form

Structure Formation in Expanding Universe

Mass conservation tells us that the mass inside the sphere,remains constant as the sphere expands.

and, since

we have

Structure Formation in Expanding Universe

Taking two time derivatives of

We get:

Combining with:

Structure Formation in Expanding Universe

We get:

This is the second Friedmann equation.

These two terms cancel

Structure Formation in Expanding Universe

We have:

or, finally:

The expansion of the Universe enters here !

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.

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.

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.