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Gennaro Miele » 10.A Brief Introduction to Cosmic Microwave Background Anisotropy Formation


Physical Basis: the relativistic kinetic theory

In order to describe the evolution of cosmic microwave background anisotropies we need to apply the relativistic kinetic and perturbation theories. Kinetic theory is necessary to describe the property of CMB photons in the metric perturbed by gravitationally unstable density fluctuations, where such fluctuations evolve according to the relativistic perturbation theory. These two phenomena combined are responsible for anisotropy formation in any model where structure formation proceeds by gravitational instability.

As already presented in the previous lessons, the relativistic kinetic theory is based on the phase space distributions conservation along geodesics unless collision terms due to scattering are at work

{d f \over dt} \equiv {\partial f \over \partial t}+{\partial f \over \partial x^i}{ d x^i \over dt}+ {\partial f \over \partial p}{ d p \over dt}+{\partial f \over \partial \gamma^i}{ d \gamma^i \over dt}= C[f]

where γi are the direction cosines of the photon momentum p (hereafter we will use for direction cosines both notations of \gamma_i \,\, {\rm and}\,\, \hat{p}_i) and C[f] stands for the collisional term which will be discussed in the following. For the moment we restrict our analysis to a vanishing collisional term. In order to simplify the previous equation let us recall that Pi ≡ dxi/dλ and P0 ≡ dt/dλ and hence dxi/dt = Pi/P0.

Physical Basis: the relativistic kinetic theory

As already discussed in previous lessons, in a smooth universe the metric is characterized by the single function a(t) only, whereas this is not the case if perturbation of the metric are considered. Hereafter we will take into account scalar perturbation only, since they are the only ones coupling with matter perturbations and the most relevant for photons. Under this hypothesis the perturbed metric takes the form

g_{00}(\vec{x},t) =-1 -2 \Psi(\vec{x},t)           g_{0i}(\vec{x},t)=0             g_{ij}(\vec{x},t)=a^2 \delta_{ij}( 1 + 2 \Phi(\vec{x},t))

Note that according to such notation, it corresponds to the longitudinal gauge (or conformal Newtonian) and that to recover the results presented in previous trasparencies one should simply substitute the present paramaters Ψ,Φ with -Φ,Ψ. This to account for the different signature and notation.

Physical Basis: the relativistic kinetic theory

By defining p2≡ gij Pi Pj we have up to the first order in the perturbation P0 = p ( 1 - Ψ ). According to this sign convention, overdense regions correspond to ψ < 0, hence photons loose energy if they move away from these regions characterized by the potential well. Such equation is the genereralization of the relativistic expression E = p c for photons. Analogously one can show that   Pi = p γi (1 - Φ)/a   which implies

\frac{d x^i}{dt} = \frac{\gamma^i}{a}\left(1 + \Psi - \Phi \right)   (*)

Since in overdense regions Ψ<0 and Φ>0, this means that photons slow down when travel across an overdense region. From these considerations one obtains that the term of the kinetic equation  {\partial f \over \partial \gamma^i}{ d \gamma^i \over dt} can be safely neglected since it is of the second order, moreover one can ignore the dependence on the scalar perturbation of eq. (*) since it would introduce second order corrections as well. From these we have the simplified kinetic equation for collisionless photons, namely

{d f \over dt} \equiv {\partial f \over \partial t}+{ \gamma^i \over a}{\partial f \over \partial x^i}+ {\partial f \over \partial p}{ d p \over dt}

By using the geodesic equation one can prove that

\frac{1}{p}\frac{dp}{dt} = -H - \frac{\partial \Phi}{\partial t} - \frac{\gamma^i}{a}\frac{\partial \Psi}{\partial x^i}

Physical Basis: the relativistic kinetic theory

and hence substituting in the kinetic equation one gets

{d f \over dt} \equiv {\partial f \over \partial t}+{ \gamma^i \over a}{\partial f \over \partial x^i}-p {\partial f \over \partial p} \left[ H + \frac{\partial \Phi}{\partial t} + \frac{\gamma^i}{a}\frac{\partial \Psi}{\partial x^i}\right]

Let us expand the photon distribution around the zero-order Bose-Einstein distribution, namely

f(\vec{x},p,\gamma^i,t) = \left[ \exp\left\{ \frac{p}{T(t)\left[1 + \Theta(\vec{x},\gamma^i,t) \right]}\right\} - 1\right]^{-1}

Note that the perturbation is characterized by the function \Theta(\vec{x},\gamma^i,t) = \frac{\delta T}{T}

The perturbed distribution can be expanded as

f \approx f^{(0)} - p \frac{\partial  f^{(0)}}{\partial p} \Theta  where f^{(0)} = \frac{1}{e^{p/T}-1 }

At zero order the kinetic equation for photons leads to the well known equation already proved that T \propto a^{-1}, whereas the first order kinetic equation for photons (l.h.s of the Boltzmann equation) for the collisionless case reads

\left. \frac{df}{dt}\right|_{\rm first}= - p \frac{\partial  f^{(0)}}{\partial p}\left[ \frac{\partial \Theta}{\partial t} + \frac{\gamma^i}{a}\frac{\partial \Theta}{\partial x^i} + \frac{\partial \Phi}{\partial t} + \frac{\gamma^i}{a}\frac{\partial \Psi}{\partial x^i} \right]

The first two terms account for the free streaming whereas the last two for the effect of gravity.

Collision terms

Let us introduce the collisional term by summarizing  the main features of Compton scattering

i) Scattering couples the photons to the baryons and forces perturbations in their number and hence energy density to evolve together.

ii) Scattering leads to isotropic photon distribution in the electron rest frame, hence coupling the local CMB dipole to the electron velocity.

iii) In the Thomson limit, there is no energy transfer in the scattering.

iv) Through Compton scattering the number of photons is preserved.

From previous considerations we can state that scattering rules the intrinsic temperature perturbations at last scattering, and the bulk velocity perturbation. Since through scattering there is no change in the net energy or photon number in the CMB to lowest order, the distortions to the black body spectrum cannot arise in linear theory.  Unless the electrons are heated sensibly above the CMB temperature or energy is dumped into the CMB from an external source. A further effect must be considered and concerns with the fact that photons propagate in a space-time distorted by density fluctuations, hence one must employ the geodesic equation in presence of perturbations.  This generates fluctuations on CMB spectrum via the gravitational redshift effects from the dp/dt term in the previous equation.  On the othe hand, The dΥi/dt term, related to the gravitational lensing, gives contribution up to the first order in a curved universe only.

Compton scattering

In order to quantify the influence of the Compton scattering on the photon distribution let us remind the process we are interested in e^-(\vec{q}) + \gamma(\vec{p}) \leftrightarrow e^-(\vec{q}\,') + \gamma(\vec{p}\,') Since we want to describe the variation in the distribution of photons with momentum \vec{p}f(\vec{p}) , we have to sum on the probability of all processes with any choice of momenta \vec{q}, \, \vec{q}\,', \, \vec{p}\,' , namely

C[f(\vec{p})] = \sum_{\vec{q},\vec{q}\,',\vec{p}\,' } \left| {\rm Amplitude}\right|^2 \, \left\{ f_e(\vec{q}\,') f(\vec{p}\,') - f_e(\vec{q}) f(\vec{p}) \right\}

where we have neglected the stimulated emission and Pauli blocking terms as can be safely done up to the first order in perturbation theory. The explicit form of the collisional term is the involved expression

C[f(\vec{p})] = \frac 1p \int \frac{d^3q}{(2 \pi)^3 2 E_e(q)} \int \frac{d^3q'}{(2\pi)^3 2 E_e(q')}\int \frac{d^3p'}{(2 \pi)^3 2 E(p')}\left| {\cal M}\right|^2 \delta^4(p+q-p'-q')(2 \pi)^4 \left\{ f_e(\vec{q}\,') f(\vec{p}\,') - f_e(\vec{q}) f(\vec{p}) \right\}

In the non relativistic limit, the Compton scattering is an almost elastic process, this allows for a relevant simplification of the result. In particular one gets

C[f(\vec{p})] = \frac {\pi}{4 m_e^2 p} \int d^3q\frac{f_e(\vec{q})}{(2 \pi)^3} \int \frac{d^3p'}{(2 \pi)^3 p'} \left| {\cal M}\right|^2 \left\{\delta(p-p')+\frac{(\vec{p}-\vec{p}\,')\cdot \vec{q}}{m_e} \frac{\partial \delta(p-p')}{\partial p'}\right\} \left\{ f(\vec{p}\, ') - f(\vec{p}) \right\}

Compton scattering

For a correct treatment one should obtain from Feynman rules the expression for \left| {\cal M \right|^2 which is depending both on the angle between inner and outer photons as well as on their polarization. One can imagine to neglect such small effect for a first order treatment but having in mind that a more precise computation requires the correct treatment for such tiny terms. Under this simplification ansatz, one obtains a constant expression for the squared amplitude, namely \left| {\cal M \right|^2 = 8 \pi \sigma_T m_e^2

where σT is the Thomson cross-section. This allows to integrate on the p’ direction encoded in the solid angle Ω’. Hence one gets

C[f(\vec{p})] = \frac {n_e \sigma_T}{p} \int_0^\infty dp'\, p' \left[\delta(p-p')\left( -p' \frac{\partial f^{(0)}}{\partial p'} \Theta_0 + p  \frac{\partial f^{(0)}}{\partial p} \Theta(\hat{p})\right) + \vec{p} \cdot \vec{v}_b \frac{\partial \delta(p-p')}{\partial p'}( f^{(0)}(p') -  f^{(0)}(p))\right]

where

\Theta_0(\vec{x},t) \equiv \frac{1}{4 \pi}\int d\Omega' \, \Theta(\hat{p}',\vec{x},t)

Since it is not depending on the photon direction it represents the monopole term of the temperature perturbation. Integrating on p’ one gets

C[f(\vec{p})] = - p \frac{\partial f^{(0)}}{\partial p} n_e \sigma_T \left[ \Theta_0 -\Theta(\hat{p}) + \vec{p} \cdot \vec{v}_b \right]

Note that Compton scattering tends to produce a very simple photon distribution, just characterized by a not vanishing monopole and dipole (if and only if electrons carry a bulk velocity) term.

The Boltzmann equation for photons

The Boltzmann equation for the temperature perturbations evolution reads

{\partial \Theta \over \partial \eta} + \gamma^i {\partial \Theta \over \partial x^i}+{\partial \Phi \over \partial \eta} + \gamma^i {\partial \Psi \over \partial x^i}=\dot \tau [ \Theta_0 -\Theta + \gamma_i v_e^i]

The quantity \dot \tau = x_e n_e \sigma_T a denoting the differential Compton optical depth where x_e is the ionization fraction, n_e the electron density.  Summarizing, the three main sources of anysotropies are

1) Gravitational redshifts due to metric fluctuations.

2) Hot and cold spots from the intrinsic temperature  at last scattering.

3) The Doppler effect due to the velocity of the last scatterers.  Due to the linearity of the expression one can Fourier transform it. In this case we have also an additional, since we are perturbating a smooth background the only dependence on the spatial coordinates comes through the pertubation functions only. By defining

\Theta(\vec{x}) = \int \frac{d^3k}{(2 \pi)^3} e^{i \vec{k} \cdot \vec{x}} \, \widetilde{\Theta}(\vec{k})

an useful quanities results to be

\mu = \frac{\vec{k} \cdot \vec{\gamma} }{k}

The Boltzmann equation for photons

Note that \vec{k} denotes the direction along which the temperature changes, whereas \vec{p} = p \,  \vec{\gamma} stands for the direction along which the photons propagate. One typically assumes that photons propagate along the same direction in which we measure the temperature variation, hence we get

\widetilde{\vec{v}}_b \cdot \vec{\gamma} = \widetilde{v}_b \, \mu

from which we have

\dot{\widetilde{\Theta}} + i k \mu \widetilde{\Theta} + \dot{\widetilde{\Phi}} + i k \mu \widetilde{\Psi} = - \dot{\tau} \left[ \widetilde{\Theta}_0 - \widetilde{\Theta} + \mu \,\widetilde{v}_b\right]

where the “dot” simply denotes the derivative with respect to the conformal time η. Since perturbations in CMB remain small in all cosmic epoch, the Fourier transform is very useful and implies the almost independence in the evolution of each single mode.

The Boltzmann equation for Cold Dark Matter

By following the approach outlined in the previous trasparencies we can deduce the form of the evolution equation for each specie present in the universe. In particular, in the case of cold dark matter it results much easier to obtain the equation from the energy-momentum conservation. Of couse what makes CDM different from photons is that by definition DM particles do not interact with any other component of the primordial fluid, and they are nonrelativistic.

From the kinematics we have

P^\mu = \left[E (1 - \Psi), p \gamma^i \frac{1 - \Phi}{a} \right]

The kinetic term of the Boltzmann equation for the CDM results

{d f_{\rm dm} \over dt} \equiv {\partial f_{\rm dm} \over \partial t}+{ d x^i \over dt} {\partial f_{\rm dm} \over \partial x^i}+ { d E \over dt} {\partial f_{\rm dm} \over \partial E}+{ d \gamma^i \over dt}{\partial f_{\rm dm} \over \partial \gamma^i}

again the last term can be neglected since of the second order. The previous assumes for CDM a different form with respect to the photon one this due to the non relativistic nature of DM particles. One can show with some algebra that the evolution equation can be cast in the following form

{\partial f_{\rm dm} \over \partial t}+{ \gamma^i \over a} {p \over E}{\partial f_{\rm dm} \over \partial x^i}- {\partial f_{\rm dm} \over \partial E}\left[ H \frac{p^2}{E} + \frac{p^2}{E} {\partial \Phi \over \partial t}+ \frac{\gamma^i p}{a} {\partial \Psi \over \partial x^i}\right]=0

The Boltzmann equation for Cold Dark Matter

Differently from the photon case we do not expand the CDM distributon function around an equilibrium one, but rather compute the evolution equation for its moments. Integrating the previous expression with respect to d3p/(2π)3 we get the zero-moment of the equation which reads  {\partial n_{\rm dm} \over \partial t}+{1\over a} {\partial (n_{\rm dm} v^i)\over \partial x^i}+ 3 \left[ H+ {\partial \Phi\over \partial t}\right] n_{\rm dm} =0

The same equation restricted to zero-order approximation leads to the very well known results n_{\rm dm}^{(0)} \propto a^{-3} wheras to get the first order term we need to expand the number density as n_{\rm dm} \approx  n_{\rm dm}^{(0)} \left[1 + \delta(\vec{x},t) \right]  which once substituted in the evoluton equation gives {\partial \delta \over \partial t}+{1\over a} {\partial v^i\over \partial x^i}+ 3  {\partial \Phi\over \partial t} =0 (*)

On the other hand the first moment of the Boltzmann equation gives

{\partial v^j \over \partial t}+H v^j+ \frac{1}{a}  {\partial \Psi\over \partial x^j} =0 (**)

The two equation are sufficient to describe the evolution of the density and velocity of CDM.

The Boltzmann equation for Cold Dark Matter

The equations (*) and (**) of the previous trasparency can be written in terms of the conformal time η and Fourier transforms. The density equation reads

\dot{\tilde{\delta}} + i k \tilde {v} + 3 \dot{\tilde{\Phi}} =0

where one has assumed an irrotation al velocity, namely

\tilde{v}^i = \frac{k^i}{k} \tilde{v}

The velocity equation reads

\dot{\tilde{v}} + \frac{\dot{a}}{a} \tilde {v} + i k {\tilde{\Psi}} =0

The Boltzmann equation for baryons

Apart of photons and CDM particles, the universe of CMB is also filled of electrons and protons, here simply denoted (improperly) as baryons. Due to the tight coupling forces acting on both electrons and protons, one can define a common overdensity δb , namely

\frac{\rho_e - \rho_e^{(0)}}{\rho_e^{(0)}} = \frac{\rho_p - \rho_p^{(0)}}{\rho_p^{(0)}} \equiv  \delta_b

and a common bulk velocity, namely \vec{v}_e = \vec{v}_p \equiv \vec{v}_b   .Electrons and protons interact via the Coulomb scattering e + p → e + p, hence one can simply write

\frac{df_e(\vec{x},\vec{q},t)}{dt}=\langle c_{ep}\rangle_{QQ'q'}+ \langle c_{e \gamma}\rangle_{pp'q'}\,\,\,\,\,\,\,\,\,\,\,\, \frac{df_p(\vec{x},\vec{Q},t)}{dt}=\langle c_{ep}\rangle_{qq'Q'}

The scattering of protons off photons can be neglected since inversely proportional to proton mass squared. In the previous expressions we used the notation

\langle (..)\rangle_{pp'q'} \equiv \int\frac{d^3 p}{(2 \pi)^3} \int \frac{d^3 p'}{(2 \pi)^3} \int \frac{d^3 q'}{(2 \pi)^3} (..)             and

c_{e \gamma} = (2 \pi)^4 \delta^4(p+q-p'-q') \frac{\left| {\cal M} \right|^2}{8 E(p)E(p')E_e(q)E_e(q')} \left\{ f_e(q') f_\gamma(p') -  f_e(q) f_\gamma(p)\right\}

The Boltzmann equation for baryons

As for the Dark Matter case one can integrate the previous Boltzmann equation for electron on the phase space term d3q/(2π)3. Interestingly the r.h.s. collisional term vanishes and one gets the simple expression

\frac{\partial n_e}{\partial t} + \frac 1a \frac{\partial (n_e v_b^i)}{\partial x^i}+ \left[ \frac 1a \frac{da}{dt} + \frac{\partial \Phi}{\partial t} \right] n_e =0

Hence the perturbed equation for baryon overdensity is then analogous to the DM case. In the Fourier space it reads  \dot{\tilde{\delta}}_b + i k \tilde {v}_b + 3 \dot{\tilde{\Phi}} =0

The equation for bulk velocity is then obtained by first momenta of both Boltzmann equations for protons and electrons. In particular adding them together one gets

\dot{\tilde{v}}_b + \frac{\dot{a}}{a}\tilde {v}_b + i k \tilde{\Psi} = \dot{\tau} \frac{4 \rho_\gamma}{3 \rho_b} \left[ 3 i \widetilde{\Theta}_1 + \tilde {v}_b \right] where \Theta_1 \equiv i \int_{-1}^1 \frac{d\mu}{2}\,  \mu \, \Theta(\mu)

Note that the above equation was obtained under the hypothesis of ne=np=nb. However it results valid also in presence of neutral hydrogen and helium due to their tight coupling with electrons.

Recombination

It is fundamental to summarize an out of equilibrium phenomenon just driving the decoupling of photons from matter and hence affecting the pattern of CMB anysotropies observed today.

When the temperature of primordial plasma was ≈ 1 eV  electron and protons were highly interacting via Coulomb scattering and at the same time phtotns and electrons strongly interacted via Compton scattering. This ensured that the process e- + p → H + γ was exactly balanced by the reverse one, thus keeping the primordial plasma in an almost ionized form. In other words, the neutral hydrogen H had no time to form that the large photon/baryon ratio would have dissociated it. This has a straightforward implication, namely

\frac{n_e n_p}{n_H} = \frac{n_e^{(0)} n_p^{(0)}}{n_H^{(0)}}

which ensures the mantaining of thermal equilibrium. Moreover, the neutrality of the universe guarantees that ne = np. Let us define the fraction of free electrons as

X_e \equiv \frac{n_e}{n_e+n_H} =  \frac{n_p}{n_p+n_H}

The previuously reported equilibrium condition then reads (Saha approximation)

\frac{X_e^2}{1 - X_e} \approx \frac{1}{n_e + n_H} \left[ \left( \frac{m_e T}{2 \pi}\right)^{3/2} e^{-[m_e+m_p-m_H]/T}\right]

Recombination

Defining Δ=me + mp – mH and neglecting the amount of helium atoms, ne + nH = nb, which determined by BBN gives nb ≈ηb nγ = 10-9 T3 . By using such value in the previous equation, one gets for a T ≈ Δ a r.h.s. of the equation of the order of 1015 that implies Xe almost equal to unity, namely no neutral hydrogen at all. To have a reasonable amount of hydrogen one has to wait that temperature drops below Δ, but in order to have a reliable prediction of the amount of hydrogen produced one has to solve a corresponding Boltzmann equation

a^{-3} \frac{d(n_e a^3)}{dt}= n_e^{(0)} n_p^{(0)} \langle \sigma v \rangle \left\{ \frac{n_H}{n_H^{(0)}} - \frac{n_e^2}{n_e^{(0)}n_p^{(0)}} \right\} =n_b \langle \sigma v \rangle \left\{ (1 - X_e) \left( \frac{m_e T}{2 \pi}\right)^{3/2} e^{-\Delta/T} - X_e^2 n_b \right\}

Defining the factors

\beta = \langle \sigma v \rangle \left( \frac{m_e T}{2 \pi}\right)^{3/2} e^{-\Delta/T} and the recombination rate as \alpha^{(2)} = \langle \sigma v \rangle = 9.78 \, \frac{\alpha^2}{m_e^e} \left(\frac{\Delta}{T} \right)^{1/2} \log\left(\frac{\Delta}{T} \right)

the previous Boltzmann equation becomes  \frac{dX_e}{dt}=\left\{ (1 - X_e) \beta - X_e^2 n_b \alpha^{(2)} \right\}

Recombination

The Saha approximation allows to get a good approximation of the redshift at recombination but fails as the electron fraction drops and the system goes out of equilibrium. For this reason one must use the Boltzmann equation in order to have a realiable picture of all the evolution. However the very important decoupling of photons occurs approximately when the rate of Compton scatters becomes equal to the expansion rate. The scattering rate is n_e \sigma_T = X_e n_b \sigma_T where \sigma_T = 0.665 \times 10^{-24}\, {\rm cm}^2 is the Thomson cross section. However since  \frac{m_p n_b}{\rho_{cr}} = \Omega_b a^{-3} we have  \frac{n _e \sigma_T}{H} = 0.0692 \, a^{-3} \, X_e \Omega_b \, h\, \frac{H_0}{H}

* Since at early times the main contribution to Hubble parameter came from radiation and matter one has  \frac{H}{H_0} = \Omega_m^{1/2} a^{-3/2} \left(1 + \frac{a_{eq}}{a}\right)^{1/2} which once substituted in the r.h.s. of equation (*) provides the way to get the decoupling redshift zdecouplig . Numerical solutions of the Boltzmann equation show that decoupling just occurs during recombination.

Perturbation Theory

General Relativity states that particles move in a space-time perturbed by the fluctuations of the matter density. In particular the stress-energy tensor of the total matter is covariantly conserved in the perturbed metric, T^{\mu\nu}_{\hphantom{\mu\nu};\mu}=0, whereas the matter fluctuations are the source of metric perturbations via a generalized Poisson equation.  Note that the equation T^{\mu\nu}_{\hphantom{\mu\nu};\mu}=0 for ν=0 provides the number or energy density conservation (continuity equation), whilst the spatial components give momentum conservation. Unfortunately, such simple principles must face a delicate subtlety due to gauge freedom that in general affects a relativistic perturbation theory.   In other words, in order to define a perturbation we must specify the relation between the physical spacetime and the hypothetical unperturbed background.  Hence we denote as a perturbation the difference between quantities at the same coordinate values. Since this choice is not unique, one has to fix a gauge before making any calculations. In practice, a gauge choice means fixing the constant-time hypersurfaces and the spatial grid on these surfaces.  There are two basic approaches to fixing a gauge or coordinate system. It is possible to fix a set of preferred observers and place the coordinate frame to be their rest frame.  For example, the common synchronous gauge is just one example of this approach. In this case one chooses freely falling observers to define the frame, namely the collisionless cold dark matter.  In such a rest frame the evolution equations result largerly simplified. The total matter gauge represents an extension of this idea. In this case, the coordinate system coincides with the rest frame of the combined relativistic and non-relativistic matter.  Such a choice is very useful in the early universe where radiation dominates and does not necessarily follow the perturbations induced by the non-relativistic matter. Since we have devoted a set of trasparencies to such issue, we will simply report here the form of Einstein equation for the potentials Φ, Ψ defining the metric perturbation in the longitudinal gauge.

Perturbation Theory

By following the analysis performed in the previous lesson concerning cosmological perturbations, and considering the different notation here adopted (signature and definition of newtonian potentials according to S. Dodelson book), we have that the 0-0 component of Einstein’s equation \delta G^0_0 = 8 \pi G \, \delta T^0_0 leads to the first order equation for the Fourier transform k^2 \tilde{\Phi} + 3 \frac{\dot{a}}{a} \left( \dot{\tilde{\Phi}} - \tilde{\Psi} \frac{\dot{a}}{a} \right) = 4 \pi G a^2 \, \left[\rho_{dm} \tilde{\delta} + \rho_b \tilde{\delta}_b + 4 \rho_\gamma \tilde{\Theta}_0 + 4 \rho_\nu \tilde{\cal N}_\nu \right]

The spatial components of Einstein’s equation provide the second equation involving Φ and Ψ, namely k^2 \left( \tilde{\Phi} + \tilde{\Psi}\right) = - 32 \pi G a^2 \, \left[\rho_\gamma \tilde{\Theta}_2 + \rho_\nu \tilde{{\cal N}}_2 \right]

Neglecting the small contributions coming from photons and neutrino quadrupoles one would get from the previous expression Φ = – Ψ . Such equations have to be added to the set of differential equations written for overdensities and bulk velocities of all particle species present at the time of recombination.

The Complete set of equations

We here collect all equations derived for photons, dark matter, baryons, massless neutrino and metric scalar perturbations. Remind that for seek of simplicity we have neglected the angular dependence of Compton scattering and polarization of photons that would have added a further equation.

\dot{\widetilde{\Theta}} + i k \mu \widetilde{\Theta} + \dot{\widetilde{\Phi}} + i k \mu \widetilde{\Psi} = - \dot{\tau} \left[ \widetilde{\Theta}_0 - \widetilde{\Theta} + \mu \,\widetilde{v}_b\right]  (photons)

\dot{\tilde{\delta}} + i k \tilde {v} = - 3 \dot{\tilde{\Phi}} \,\,\,\,\,\,\,\, \dot{\tilde{v}} + \frac{\dot{a}}{a}\tilde {v} =- i k \tilde{\Psi} (DM)

\dot{\tilde{\delta}}_b + i k \tilde {v}_b + 3 \dot{\tilde{\Phi}} =0 \,\,\,\,\,\,\,\,\, \dot{\tilde{v}}_b + \frac{\dot{a}}{a}\tilde {v}_b + i k \tilde{\Psi} = \dot{\tau} \frac{4 \rho_\gamma}{3 \rho_b} \left[ 3 i \widetilde{\Theta}_1 + \tilde {v}_b \right] (Baryons = electrons, protons)

\dot{\cal N}+ i k \mu {\cal N} = - \dot{\tilde{\Phi}}-i k \mu \tilde{\Psi}   (massless neutrinos)

{\cal N} plays the same role for neutrinos as \Theta for photons.

The Complete set of equations

k^2 \tilde{\Phi} + 3 \frac{\dot{a}}{a} \left( \dot{\tilde{\Phi}} - \tilde{\Psi} \frac{\dot{a}}{a} \right) = 4 \pi G a^2 \, \left[\rho_{dm} \tilde{\delta} + \rho_b \tilde{\delta}_b + 4 \rho_\gamma \tilde{\Theta}_0 + 4 \rho_\nu \tilde{\cal N}_0\right]   (metric scalar perturbation)

k^2 \left( \tilde{\Phi} + \tilde{\Psi}\right) = - 32 \pi G a^2 \, \left[\rho_\gamma \tilde{\Theta}_2 + \rho_\nu \tilde{{\cal N}}_2 \right]

where the quantity

\Theta_l \equiv \frac{1}{(-1)^l} \int_{-1}^{1} \frac{d\mu}{2} {\cal P}_l(\mu) \Theta(\mu)

and analogously for {\cal N}, with {\cal P}_l the Legendre polynomial of order l. In the following we will abandon the notation of using a “tilde” to denote the Fourier transform since all quantities will be evaluated in the momentum space.

Initial Conditions and Scalar perturbations

As we have already discussed, an inflationary stage is necessary to overcome several problems of standard cosmology. Moreover, inflation provides for free a spectrum of perturbation which represents the initial condition to solve the set of differential equations governing the primordial plama at recombination. To find the spectrum of perturbations of Ψ emerging from inflation (we assume Ψ and Φ of the same order of magnitude) results to be quite complicated, so in principle one can neglect such approach and rather focus the attention on the perturbations concerning the scalar field driving inflation δφ. Such perturbations will be eventually transferred on Ψ. The reason of this is very simple, until a mode is inside the horizon one can safely neglect Ψ, but after the horizon such consideration is not more valid. However since a linear combination of Ψ and δφ is constant, the information on δφ is straightforwardly converted on Ψ. An alternative way is to use the spatially flat slicing which ius a gauge in which the spatial part of the metric is unperturbed. In such a gauge one does not need to consider the perturbation of  Ψ , but then one has the problem to convert the results in the longitudinal gauge. This can be solved by working with a gauge invariant combination of both fields δφ and Ψ. We have already analyzed the dynamics of a scalar field in a perturbed space-time, which in case of slow roll conditions leads to the approximate EoM for the field perturbations \ddot{\delta \tilde{\phi}} + 2 a H \dot{\delta  \tilde{\phi}} + k^2 \delta \tilde{ \phi}=0 where the “tilde” reminds that we are in Fourier transform. This gives the following expression for the power spectrum of fluctuations, namely

P_{\delta \phi}= \frac{H^2}{2 k^3}

We recall the definition of a power spectrum as  \langle \delta \tilde{\phi}(\vec{k}) \delta \tilde{\phi}^*(\vec{k}') \rangle = (2 \pi)^3 P_{\delta \phi}(k)  \, \delta^3(\vec{k}- \vec{k}')

Scalar perturbations

As already stated the metric perturbations can be neglected as long as the wavelength of perturbations are equal or smaller than the horizon. Nevertheless, at the end of the inflation the metric perturbation have achieved a substantial role, so a starting δφ ends in a linear combination of itself and Ψ as well, or more precisely and a perturbation of the energy-momentum tensor. The trick is to find a linear combination of quantities which is constant at the crossing of the horizon, so such quantity evaluated in terms of the δφ only at the horizon crossing, provides the expression of Ψ after inflation. The quantity to consider is  \zeta \equiv - \frac{i k_i \, \delta T^0_i \, H}{k^2 \, (\rho+p)} - \Psi where ρ and p are zero-order energy density and pressure. For sub-horizon modes or those that have just left horizon Ψ is negligible. This gives that around the horizon crossing (the time when is satisfied the condition a(t) H = k for a given k) \zeta = - \left. \frac{a H \delta\phi}{\dot{\phi}^{(0)}} \right|_{aH = k} where we have used the fact that during inflation  p + \rho = \left( \frac{\dot{\phi}^{(0)}}{a}\right)^2

Scalar perturbations

After inflation radiation is dominating the universe, and i k_i \delta T^0_i = 4 a k \rho_r \Theta_1 hence proportional to the dipole of radiation. Since in RD the pressure is 1/3 of energy density we have

\zeta = - \frac{3 a H \Theta_1}{k} - \Psi = -\left.  \frac 32 \Psi \right|_{\rm post \,\,\,inflation}

where the last equation comes from the initial conditions that to satisfy the Boltzmann equation relate the dipole to the potential. Equating the two determinations for \zeta we get

\Psi \Big|_{\rm post \,\,\,inflation} = \frac 23 \left. \frac{a H \delta\phi}{\dot{\phi}^{(0)}} \right|_{aH = k}

This leads to a relation between the power spectra

P_\Psi = \left. \frac 49 \left(  \frac{a H}{\dot{\phi}^{(0)}}  \right)^2 \, P_{\delta \phi} \right|_{a H = k} = \frac{8 \pi G}{9 k^3} \left. \frac{H^2}{\epsilon} \right|_{aH = k}

where ε is a slow roll parameter already defined.

Inhomogeneities: three stages of evolution

According to the standard cosmology, gravitational instabilities are resposible for the structures observed in our universe. The idea is that as the time goes on matter accumulates in initially overdense regions, and this is almost independent on the value of the initial overdensity, even if it was of the order of 10-5. In very simple words, a generic equation ruling the overdensity evolution takes the form

\ddot{\delta} + [{\rm Pressure} - {\rm Gravity}] \delta = 0

Gravity forces matter to fall in the overdense region, increasing the amount of matter there contained, whereas the increase in temperature due to the matter infall tries to balance such attracting force.  The evolution of cosmological perturbations can be split in three stages:

1) Early times – all modes are outside the horizon ( k η <<1 ) and the potential is constant

2) Intermediate times – walengths fall inside horizon and the universe passes from RD to MD (at a=aeq). Large scale modes that enters the horizon after aeq evolve very differently from small scale modes entering before aeq. 3) Finally all modes evolve identically again.

Inhomogeneities: three stages of evolution

Since we observe the universe in the late time, we have to connect the almost present matter distribution with the primordial potential (the one produced at inflation), this is well described in Figure 7.2 of S. Dodelson’s book, and can be easily summarized by the following expression

\Phi(\vec{k},a) = \Phi_p(\vec{k}) \times \left\{ {\rm Transfer \, Function(k)}\right\}  \times \left\{ {\rm Growth \, Function(a)}\right\}

where Φp stands for the primordial potential, The transfer function, T(k), accounts for the perturbation evolution duering the horizon crossing and radiation/matter transition, whereas the growth function is responsible for the wave-length independent evolution at late times. As shown in Fig. 7.2 of S. Dodelson’s book (shaded region) the decline of perturbation during the radiation/matter transition reduces with the increasing of the wave-length considered. Hence almost no declie at all is present for large scale. For this reason one could simply assume T(k) equal to unity for large scale. However to account for a small decline still present one defines T(t) as T(k) = \frac{\Phi(k,a_{late})}{\Phi_{LS}(k,a_{late})} whera alate indicates a time well after the transfer regime, whereas ΦLS stands for the primordial potential just slightly reduced as it occurs for large wave-length. One can see that neglecting anisotropic stresses  such reduction factor is equal to 9/10. Moreover, concerning the growth function, it consider a phase which is wave-length independent, hence one can define a growth function D1(a) as the ratio \frac{\Phi(a)}{\Phi(a_{late})} = \frac{D_1(a)}{a}\,\,\,\,\,\,{\rm for}\,\,\,\,\, a> a_{late}

Inhomogeneities: three stages of evolution

For a constant potential D1(a) = a. Under these conventions one gets \Phi(\vec{k},a)= \frac{9}{10} \Phi_p(\vec{k}) T(k) \frac{D_1(a)}{a}\,\,\,\,\,\,{\rm for}\,\,\,\,\, a> a_{late}

From the observational point of view, one can measure the power spectrum of matter distribution which is determined by the potential at late time. For thise reason one needs to find a way to relate the above two quantities. This can be done by using the Poisson equation that in absence of radiation reads \Phi = \frac{4 \pi  G \rho_m a^2 \delta}{k^2}\,\,\,\,\,\,{\rm for}\,\,\,\,\, a >> a_{late}

where the background density matter ρmm ρcr /a3 and (4 π G) ρcr = (3/2) H02.

Thus one gets \delta(\vec{k},a)= \frac{k^2 \Phi(\vec{k},a) a}{(3/2) \Omega_m H_0^2}\,\,\,\,\,\,{\rm for}\,\,\,\,\, a >> a_{late}

By using the expression for Φ one gets \delta(\vec{k},a)= \frac 35 \frac{k^2 }{\Omega_m H_0^2} \Phi_p(\vec{k}) T(k) D_1(a)\,\,\,\,\,\,{\rm for}\,\,\,\,\, a >> a_{late}

In the inlationary scenario P(k,a)= 2 \pi^2 \delta_H^2 \frac{k^n}{H_0^{n+3}} T^2(k) \left( \frac{D_1(a)}{D_1(a=1)}\right)^2\,\,\,\,\,\,{\rm for}\,\,\,\,\, a >> a_{late}

with δH denoting the scalar amplitude at horizon crossing.

Inhomogeneities: three stages of evolution

Generally one prefers to use a dimensionless variable after the integration of the different directions \Delta^2(k) \equiv \frac{k^3 P(k)}{2 \pi^2} The difference between cosmological models can be easily appreciated by comparing for example sCDM (flat universe + CDM) with ΛCDM (flat universe, with CDM and a comological constant). In the first case the aeq results to be anticipated by the larger amount of matter present. This means that only smaller scales enter the horizon during RD phase that is the phase depressing the inhomogeneities. For this reason the turning point of sCDM is moved on smaller scales with respect to ΛCDM (Figure 7.4 of  S. Dodelson’s book). The evolution of matter overdensities does not need the solution of the complete set of equations, but just a simplified subset  of them. In first approximation one can neglect baryons with respect to DM (their abundance are smaller) and quadrupole terms, hence handling with a single gravitational potential only (Φ = - Ψ). Concerning the photons at early times (before recombination a < a*) they are tightly connected with electrons/protons, hence only Θ0 and Θ1 are involved in the equations, being the higher momenta suppressed. After a* this is not true anymore, but then the behaviour of photons becomes irrelevant for the evolution of matter overdensity.

Inhomogeneities: three stages of evolution

Under these hypotheses we get a set of equations to be solved

\dot{\Theta}_{r,0} + k {\Theta}_{r,1} = - \dot{\Phi}

\dot{\Theta}_{r,1} - \frac k3 {\Theta}_{r,0} = - \frac k3 {\Phi}

\dot{\delta} + i k v = - 3 \dot{\Phi}

\dot{v}+\frac{\dot{a}}{a}v = i k \Phi

with “r” denoting both photons and neutrinos. One has to add the equation for the gravitational potential which reads

k^2 \Phi + 3 \frac{\dot{a}}{a}\left( \dot{\Phi}+ \frac{\dot{a}}{a} \Phi \right) = 4 \pi G  a^2 \left[ \rho_{dm} \delta + 4 \rho_r \Theta_{r,0}\right]

or alternatively the algebraic equation

k^2 \Phi = 4 \pi G a^2 \left[ \rho_{dm} \delta + 4 \rho_r \Theta_{r,0} + \frac{3 a H}{k}\left( i \rho_{dm} v + 4 \rho_r \Theta_{r,1}\right)\right]

Unfortunately an analytic solution of the complete set of equations is generally not available, whereas one can get fair approximation of the solution in very particular conditions. On the other hand, it is always possible to solve numerically the set of equation. In the following we will try to sketch the general behaviour of the solutions by considering two limiting conditions only (large and small scales), where analytical considerations are possible.

Large scales

On very large scales, one can obtain analytic solutions for the potential through the matter-radiation transition and horizon crossing sequently.

Super horizon solution

Let us consider modes well outside the horizon, for them kη<< 1 and one can neglect in all the set of equations the terms dependent on k. Under this condition v and Θr,1 decouple and the remaining set of equation to solve reads \dot{\Theta}_{r,0} = - \dot{\Phi}        \dot{\delta} = - 3 \dot{\Phi}   (*)3 \frac{\dot{a}}{a}\left( \dot{\Phi}+ \frac{\dot{a}}{a} \Phi \right) = 4 \pi G  a^2 \left[ \rho_{dm} \delta + 4 \rho_r \Theta_{r,0}\right]

From the first two equations one gets that \delta - 3 \Theta_{r,0} = const.

Such constant is however fixed to be vanishing due to the initial conditions, hence \Theta_{r,0} = \frac{\delta}{3}.

From this we get 3 \frac{\dot{a}}{a}\left( \dot{\Phi}+ \frac{\dot{a}}{a} \Phi \right) = 4 \pi G  a^2 \rho_{dm} \delta \left[ 1 + \frac{4}{3y} \right]

where y \equiv \frac{a}{a_{eq}} = \frac{\rho_{dm}}{\rho_r}

that together with Eq. (*) is the system to solve.

Large scales

We transform such system in a single second order differential equation with respect to the evolution quantity η, which results

\Phi^{''} + \frac{21 y^2 + 54 y + 32}{2y (y+1)(3y +4)} \Phi^{'} + \frac{\Phi}{y(y+1)(3y+4)} = 0

that admits the following solution

\Phi = \frac{\Phi(0)}{10} \frac{1}{y^3}\left[16 \sqrt{1+y} + 9 y^3 + 2 y^2 - 8y -16\right]

Note that for small y (radiation dominated universe) \Phi= \Phi(0), whereas for large y (a matter dominated universe)\Phi= \frac{9}{10}\Phi(0).

This can be summarized stating that at large scales the potential drops of a factor 9/10 as the universe passes the equality epoch. Similar considerations on the whole set of differential equations lead to prove that potential remains constant as the mode crosses the horizon.  Potentials remain constant as long as the universe is MD. If at later times the universe becomes dominated by other forms of energy, like dark energy for a > 1/10 for example, in this case the potential decays but this affects the growth function only and not the transfer function.

Small scales

In the previous slides we have solved the equations in the case of large scales where the modes cross the horizon well after the equality epoch. Thus the problem splitted in two phases:

i) super-horizon modes evolving during equality epoch;

ii) modes in MD era crossing the horizon. Exactly the converse occurs for small scales taht cross the horizon when the universe is deep in the radiation epoch. Thus in this case the problem splits in two phases as well:

i) modes crossing the horizon in RD epoch;

ii) sub-horizon modes evolving during equality epoch.

Horizon crossing

In a RD universe the dark matter perturbation is determined by the gravitational potential, which is essentially fixed by radiation only. For this reason one must solve the set of equation for Θr,0 , Θr,1 and Φ, and consequently one can obtain the matter distribution by treating  the potential as a driving force. The Einstein equation in its algebraic expression reads \Phi = \frac{6 a^2 H^2}{k^2} \left[\Theta_{r,0}+ \frac{3 a H}{k} \Theta_{r,1} \right] that allows to eliminate Θr,0 from the radiation equations. As done for Large scales, we can transform the set of two first order differential equations for Θr,1 and Φ in a single second order differential equation which results \ddot{\Phi} + \frac4\eta \dot{\Phi} + \frac{k^2}{3} \Phi = 0

Small scales

The previous equation admits a single solution not diverging for η→0 that is \Phi = 3 \Phi_p \left(\frac{\sin(k\eta/\sqrt{3}) - (k\eta/\sqrt{3}) \cos(k\eta/\sqrt{3})}{(k\eta/\sqrt{3})^3} \right) with Φp denoting the primordial value of Φ. From this expression one can see that as a mode enters the horizon in a RD epoch its potential start decaying. After such decay it oscillates. Since we have obtained the expression for the gravitational potential, it is possible to derive the evolution of matter perturbation. Again we turn a set of two first order differential equation for δ and v, in a single second order differetial equation for δ only, namely \ddot{\delta} + \frac 1\eta \dot{\delta} = - 3 \ddot{\Phi} + k^2 \Phi - \frac 3\eta \dot{\Phi} One can see (S. Dodelson’s book) that after the mode has entered the horizon \delta(k,\eta) = A \Phi_p \log(B \eta) with A and B two integration constants. One can summarize the results stating that the matter perturbations grow even during RD epoch even though in a less prominent way than during MD epoch.

Sub-horizon modes

As previously stated, radiation pressure leads to a decay of gravitational potential as modes eneter the horizon. On the other hand, the growth of radiation perturbation itself is suppressed during this phase. This is in contrast with matter perturbations that have a logarithmic increasing. This leads to the condition that ρdm δ > ρr Θr,0 even though ρdm > ρr. Also in this case we would like to reduce the set of three equations (two of them are first order differential equations) in a second order differential equation \delta' + \frac{ikv}{a H y} = - 3 \Phi' \,\,\,\,\,\,\,\,\, v' + \frac vy = \frac{i k \Phi}{a H y}\,\,\,\,\,\,\,\,\,\, k^2 \Phi = \frac{3y}{2(y+1)} a^2 H^2 \delta

Small scales

Note that the gravitational potental dependes on δ only in view of the dominance condition ρdm δ > ρr Θr,0 . Moreover since we are in early times where dark energy and curvature are negligible we have 4 \pi G \rho_{dm} a^2 \rightarrow (3/2) a^2 H^2 y /(y+1) with y the ratio of the scale factor to its value at equality.  By a quite simple algebraic manipulation well decsribed in the S. Dodelson book one gets the Meszaros equation governing the evolution of sub-horizon modes of CDM perturbations once radiation perturbations can be neglected \delta'' + \frac{2 + 3y}{2y(1+y)} \delta' + - \frac{3}{2y(y+1)} \delta =0

Once the equation is solved one has to match it with the logarithmic behaviour at the horizon crossing epicted in the previous trasparencies. It is possible to show that the general solution of Meszaros equation takes the form \delta(k,y) = C_1 D_1(y) + C_2 D_2(y)\,\,\,\,\,\, y >> y_H where D_1(y) = y + \frac 23 and D_2(y) = D_1(y) \log\left[ \frac{\sqrt{1+y}+1}{\sqrt{1+y}-1}\right] - 2 \sqrt{1+y}

Small scales

The previous equation is valid within the horizon but before equality, namely yH << y << 1.

In this case one can fix the constants C1 and C2 by requiring a match condition for the function δ and its first derivative

A \Phi_p \log\left( \frac{B y_m}{y_H}\right) = C_1 D_1(y_m) + C_2 D_2(y_m)\,\,\,\,\,\,\,\,\,A \frac{\Phi_p}{y_m} = C_1 D_1'(y_m) + C_2 D_2'(y_m)

where ym has to satisfy the condition

yH << ym << 1.

Reminding the way in which δ is related to the transfer function T, namely   \delta(\vec{k},a)= \frac 35 \frac{k^2 }{\Omega_m H_0^2} \Phi_p(\vec{k}) T(k) D_1(a)\,\,\,\,\,\,{\rm for}\,\,\,\,\, a >> a_{late}

and comparing it with the previously determined expression one get explicitely the form of T.

In the figure is reported the form of the power spectrum for a standard CDM model

In the figure is reported the form of the power spectrum for a standard CDM model


Photon Anisotropies

The primordial perturbations, determined during inflation, show themselves through phenomena affecting both radiation and matter. We have already analyzed the pertubation in the matter distribution measured via the matter power spectrum. In the following we will study a similar quantity but related to radiation. In particular we will see the evolution of photon perturbations that leads to predictions concerning the anisotropy spectrum measured today.

The photon perturbations have an evolution that can be split in two phases. Before recombination the photons were tightly coupled to electrons and protons, hence all together they formed a sigle fluid, “baryon-photon fluid”. After ricombination photons decoupled from baryons and hence free-streamed from the surface of last-scattering to us today.

As clearly shown in Figure 8.1 of S. Dodelson’s book, different modes of the combination Θ0+Ψ evolve in different way. Note that the presence in the previous combination of the gravitational potential Ψ is related to the fact that photons, that we see today, had to travel out of the potential in which they were at the time of recombination.   From the Figure one sees that super-horizon modes  exibit a very little evolution since no casual processes can be involved. This means that large scale anisotropies, sensitive to modes whose wavelengths were larger than horizon at recombination,  give information about perturbation at very early times.

A different behaviour is shown by modes whose wavelength is corresponding to the so-called “first peak”. By definition they have an increasing amplitude till recombination, hence they will give the maximum contribution to the anisotropy spectrum. Then one can consider the other modes that peaking earlier reach at the recombination different values. As a function of k, their sum produces an oscillating behaviour of the total spectrum.

Large-scale anisotropies

From the super-horizon equation  \dot{\Theta}_0 = - \dot{\Phi} one gets \theta_0 = - \Phi + {\rm const}  , since the initial condition fix \Theta_ 0 = \frac{\Phi}{2} one gets \theta_0 = - \Phi + \frac 32 \Phi_p

As previously discussed the evolution of Φ is given by \Phi = \frac{\Phi(0)}{10} \frac{1}{y^3}\left[16 \sqrt{1+y} + 9 y^3 + 2 y^2 - 8y -16\right]

If the recombination occurs long after the equality epoch one can take the previous expression for large y, namely Φ→9 Φp/10.

This implies that at recombination \Theta_0(k,\eta_*) = \frac 23 \Phi(k,\eta_*).

Since the observed anisotropy is Θ0+Ψ ≈Θ0-Φ  we get \left(\Theta_0 + \Psi \right)(k,\eta_*) = \frac 13  \Psi(k,\eta_*)  (*) By using the equation for the matter density, namely \dot{\delta} = - 3 \dot{\Phi}, and the initial condition δ = 3/2 Φ we get \delta(\eta_*) = 2 \Phi(\eta_*).

Comparing such relation with  Eq. (*) we have \left(\Theta_0 + \Psi \right)(k,\eta_*) =-  \frac 16  \delta(\eta_*)

Note that the l.h.s. of previous equation provides δT/T, whereas the r.h.s. δρ/ρ.

This means that at a δT/T ≈ 10-5 corresponds a δρ/ρ ≈ 6 10-5

Acoustic Oscillations

Strong coupling limit for the Boltzmann Equation

At the epoch in which all electrons were free travelling and hence matter results ionized, hence before η*, the interaction length was much smaller that the horizon so Compton scattering made the electron-baryon fluid to be tightly coupled with photons. In this scenario is possible to show that the only not negligible momenta of photon perturbations are the monopole (l=0), Θ0, and the dipole (l=1), Θ1.  Such simplification allows to write the two first order differential equations for photon perturbation momenta \dot{\Theta}_0 + k \Theta_1 = - \dot{\Phi}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \dot{\Theta}_1 - \frac{k \Theta_0}{3} = \frac{k \Psi}{3} + \dot{\tau} \left[ \Theta_1 - \frac{i v_b}{3}\right] These equations have to be supplemented by the ones related to the electron-baryon fluid. The equation for the velocity reads v_b = - 3 i \Theta_1 + \frac{R}{\dot{\tau}}\left[ \dot{v}_b + \frac{\dot{a}}{a}v_b + i k \Psi\right] with \frac{1}{R} \equiv \frac{4 \rho_\gamma^{(0)}}{3 \rho_b^{(0)}} Observing the on the l.h.s of the previous equation the main term is -3 i Θ1 , one can apply an iterative method that leads up to the second order to the expression v_b \approx  - 3 i \Theta_1 + \frac{R}{\dot{\tau}}\left[ - 3 i \dot{\Theta}_1  - 3 i  \frac{\dot{a}}{a} \Theta_1  + i k \Psi\right] This expression can be inserted in the previous equation for \dot{\Theta}_1 yielding \dot{\Theta}_1 + \frac{\dot{a}}{a} \frac{R}{1+R} \Theta_1 - \frac{k}{3\left[1+R\right]} \Theta_0 = \frac{k \Psi}{3}

Acoustic Oscillations

Differentiating the equation for \dot{\Theta}_0 and using the last expression we get after some algebraic manipulation

\ddot{\Theta}_0 + \frac{\ddot{a}}{a} \frac{R}{1+R} \dot{\Theta}_0 + k^2 c_s^2 \Theta_0 = - \frac{k^2}{3} \Psi - \frac{\dot{a}}{a} \frac{R}{1+R} \dot{\Phi} - \ddot{\Phi}

where the r.h.s. represents a driving force and c_s= \sqrt{\frac{1}{3(1+R)}}

is the sound speed that depends on the abundance of baryons. Note that the second order differential equation can be cast in a very elegant form, namely \left\{ \frac{d^2}{d\eta^2} + \frac{\dot{R}}{1+R} \frac{d}{d\eta} + k^2 c_s^2 \right\} \left[\Theta_0 + \Phi \right] = \frac{k^2}{3}\left[ \frac{1}{1+R} \Phi - \Psi\right]

The proper solution of such equation is obtained in the S.Dodelson’s book in the form

\Theta_0(\eta) + \Phi(\eta) = \left[\Theta_0(0) + \Phi(0)\right] \cos(k r_s) + \frac{k}{\sqrt{3}} \int_0^\eta d\eta' \left[ \Phi(\eta') - \Psi(\eta')\right] \sin\left[ k (r_s(\eta) - r_s(\eta'))\right]

Such equation is obtained dropping  all occurrences of R except in the arguments of trigonometric functions. This expression provides the anysotropy in the tightly coupled limit. In the limit in which the first term of the above equation dominates the peaks in the anysotropies should appear at the extrema of cosine, namely for k_p = n \pi/r_s \,\,\,\,\,\, n=1,2,..

Diffusion damping

Comparing the analytical expression with the numerical predictions of CMB anisotropies it appears clear that one has to consider the effect of diffusion. It is characterized by the presence of a small but non negligicle quadrupole momentum, Θ2 . For this reason we have to supplement the equations previously analyzed with an equation for the quadrupole. Since we are analyzing small scales, the potentails Φ and Ψ are very small, so one can neglect them everywhere. Also in this case since higher momenta are involved by increasing power of the 1/\dot{\tau} we can neglect momenta higher than the second. Under these approximations the set of equations to be solved reads

\dot{\Theta}_0 + k \Theta_1 = 0        \dot{\Theta}_1 + k \left( \frac 25 \Theta_ 2 - \frac 13 \Theta_0 \right) = \dot{\tau} \left( \Theta_1 - \frac{i v_b}{3}\right)             \dot{\Theta}_2 - \frac{2k}{5} \Theta_1 = \frac{9}{10} \dot{\tau} \Theta_2 3 i \Theta_1 + v_b = \frac{R}{\dot{\tau}} \left[ \dot{v}_b + \frac{\dot{a}}{a} v_b\right]

The last equation can be simplified by neglecting the second term in its r.h.s.. In fact since the damping occurs at small scales , or high frequencies \dot{v}_b >> (\dot{a}/a) v_b. In this case one can apply an iterative approach up to the second order to solve the equation for the velocity. After some algebraic manipulation, as shon in Dolson’s book one gets

\Theta_0,\,\,\Theta_1 \sim \exp\left\{ i k \int d\eta c_s \right\} \exp\left\{ - \frac{k^2}{k_D^2} \right\}

where the damping wave number is defined as \frac{1}{k_D^2(\eta)} \equiv \int_0^\eta \frac{d\eta'}{6(1+R) n_e \sigma_T a(\eta')} \left[ \frac{R^2}{1+R}+ \frac 89\right]

Diffusion damping

Neglecting terms order 1, we see that    1/k_D \sim \left[ \eta/n_e \sigma_T a\right]^{1/2}.

In prerecombination limit all electrons except the ones forming helium atoms are free. The total number of Helium 4 atoms are Yp/4, with Yp denoting the Helium 4 mass fraction. For this reason the free electrons density results

n_e^{free} = n_p^H = n_p - 2 n_{He4} = n_p - (n_p + n_n)\frac{Y_p}{2} = (n_p + n_n)\left( 1 - \frac{n_n}{n_n + n_p} - \frac{Y_p}{2} \right) = (n_p + n_n)\left( 1 -  Y_p \right)

Thus one gets

n_e^{free} \sigma_T a = 2.3 \times 10^{-5} {\rm Mpc}^{-1} \Omega_b h^2 a^{-2} (1 - Y_p)

Note that in absence of recombination kD scales as Ωb1/2. Of course such description of damping is a good approximation far from recombination. As η approaches η* one has to consider the amount of electrons swept up into neutral hydrogen.

Inhomogeneities to Anisotropies

Free streaming

In the previous trasparencies we have derived the expression of the momenta of photon perturbation at recombination epoch, η*. To evaluate the form of the anisotropies we need to evolve the momenta at recombination till present time, η0. To address such issue we have to go back to the equation previously reported, but now in the complete form \dot{{\Theta}} + i k \mu {\Theta} = - \dot{{\Phi}} - i k \mu {\Psi} - \dot{\tau} \left[ {\Theta}_0 - {\Theta} + \mu \, {v}_b - \frac 12 {\cal P}_2(\mu) \Pi\right]

where Π is a combination of polarization field as well. This expression can be cast in a more convenient form

\dot{{\Theta}} + (i k \mu - \dot{\tau}) {\Theta} = e^{-ik\mu \eta + \tau} \frac{d}{d\eta} \left[ \Theta  e^{ik\mu \eta - \tau} \right] = \widetilde{S}  with \widetilde{S} = - \dot{{\Phi}} - i k \mu {\Psi} - \dot{\tau} \left[ {\Theta}_0 + \mu \, {v}_b - \frac 12 {\cal P}_2(\mu) \Pi\right]

Multiplying both sides of Eq. (*) by the exponential and integrating on η we get

\Theta(\eta_0) = \Theta(\eta_{init}) e^{i k \mu (\eta_{init}- \eta_0)} e^{-\tau(\eta_{init})} + \int_{\eta_{init}}^{\eta_0} d\eta \, \widetilde{S}(\eta) e^{i k \mu (\eta - \eta_0)-\tau(\eta)}

By definition \tau(\eta_0)=0, and moreover since \tau(\eta_{init}) is large enough for early \eta_{init}, the first term on the r.h.s. of the previous equation is negligible. Thus we get \Theta(k,\mu,\eta_0) = \int_0^{\eta_0} d\eta \widetilde{S}(k,\mu,\eta) e^{i k \mu (\eta - \eta_0)-\tau(\eta)}

The dependence of \widetilde{S} on μ does not allow to write immediately the expression for Θl.

Inhomogeneities to Anisotropies

The dependence on μ contained in \widetilde{S} can be taken into account by replacing the μn factors of its Taylor expansion by (1/ik)n dn/dηn eikμ(η-η0) in the integral that allows for an integration by parts. In this way one gets for the single momenta the expression

\Theta_l(k,\eta_0) = \int_0^{\eta_0} d\eta S(k,\eta) \, j_l[k(\eta_0 - \eta)] with S(k,\eta) \equiv e^{-\tau} \left[ - \dot{\Phi} - \dot{\tau} \left( \Theta_0 + \frac 14 \Pi\right)\right] + \frac{d}{d\eta} \left[ e^{-\tau} \left( \Psi - \frac{i v_b \dot{\tau}}{k}\right) \right] - \frac{3}{4 k^2}  \frac{d^2}{d\eta^2} \left[ e^{-\tau} \dot{\tau} \Pi\right]

It is useful to define the visibility function g(\eta) \equiv - \dot{\tau} e^{-\tau}. It has the useful property to be already normalized, namely,

\int_{0}^{\eta_0} d\eta \, g(\eta) = 1

that allows to interprete it as a probability density. In particular it is the probability that a photon last scattered at η and results to be peaked around the recombination epoch η*.

The function S(k,η) can be expressend in terms of g(η) as

S(k,\eta) \approx g(\eta) [\Theta_0(k,\eta)+\Psi(k,\eta)]+ \frac{d}{d\eta} \left(\frac{i v_b(k,\eta) g(\eta)}{k} \right) + e^{-\tau} \left[ \dot{\Psi}(k,\eta)-\dot{\Phi}(k,\eta) \right]

Inhomogeneities to Anisotropies

Substituting such expression in the formula for Θl we get

\Theta_l(k,\eta_0) = \int_{0}^{\eta_0} d\eta \, g(\eta) [\Theta_0(k,\eta)+\Psi(k,\eta)] j_l[k(\eta_0 - \eta)]- \int_{0}^{\eta_0} d\eta \, g(\eta) \left(\frac{i v_b(k,\eta)}{k} \right) \frac{d}{d\eta} j_l[k(\eta_0 - \eta)]+ \int_{0}^{\eta_0} d\eta \, e^{-\tau} \left[ \dot{\Psi}(k,\eta)-\dot{\Phi}(k,\eta) \right] j_l[k(\eta_0 - \eta)]The term proportional to e gives conribution after recombination but are different from zero only in case of variable gravitational potentials. In many theories such potentials become constant after recombination so this term tends to vanish and in any case do not modify the qualitative behaviour of the solution. Hence, the dominant terms result the ones proportional to g(η).

Such terms can be simplified by reminding that the visibility function is peaked around η*.Thus we get \Theta_l(k,\eta_0) \approx \left[ \Theta_0(k,\eta_*)+\Psi(k,\eta_*)\right]j_l[k(\eta_0-\eta_*)]

+ 3 \Theta_1(k,\eta_*) \left( j_{l-1}[k(\eta_0-\eta_*)] - \frac{(l+1) j_{l}[k(\eta_0-\eta_*)]}{k(\eta_0-\eta_*)}\right) + \int_0^{\eta_0} d\eta \, e^{-\tau} \left[ \dot{\Psi}(k,\eta) - \dot{\Phi}(k,\eta)\right]  j_{l}[k(\eta_0-\eta)]

 

Inhomogeneities to Anisotropies

In the previous equation one has used the fact that v_b \approx - 3 i \Theta_1  at η*.Such equation essentially states that the anisotropies today are determined by the monopole (Θ0), the dipole (Θ1), and the potential (Ψ) at recombination. The additional effects related to the time dependence of potentials, also known as “integrated Sachs-Wolfe effect“, are contained in the integral term of previous expression.  It is interesting to note that since \lim_{l \rightarrow \infty}j_l(x) = \frac 1l \left( \frac xl \right)^{l - 1/2} , the Bessel function is very small for x < l.

This implies that Θl(k, η0) is almost vanishing for l > k η0. We can summarize the previous results  by stating that a perturbation with wavenumber k mainly contributes on angular scales of the order of l ≈ k η0.

The Cl coefficients

In the previous trasparencies we have defined the field Θ as a deformation of Bose-Einsetein distribution function, namely T(\vec{x},\hat{p},\eta) = T(\eta) [1+ \Theta(\vec{x},\hat{p},\eta)] where \hat{p} = \vec{\gamma} represents the arrival direction of photon, or equivalenty the line of sight of our telescope. The coordinates (\eta, \vec{x})  stand for the point of space-time where the measurement is performed. In the following since we refer to a present measurement we implicitely assume (\eta_0, \vec{x}_0) . The map of temperature in the sky is then determined by the angles θ, and φ fixing the direction of \hat{p}. Let us expand the field Θ in spherical harmonics, namely \Theta(\vec{x}, \hat{p},\eta) = \sum_{l=1}^{\infty} \sum_{m=-l}^{l} a_{lm} (\vec{x},\eta) \1, Y_{lm}(\hat{p}) Looking at the Fourier transform of Θ with respect to \vec{x}, we can invert the previous expression as a_{lm}(\vec{x},\eta) = \int \frac{d^3 k}{(2 \pi)^3} e^{i \vec{k} \dot \vec{x}} \int d\Omega\,  Y^*_{lm}(\hat{p}) \, \Theta(\vec{k},\hat{p},\eta) since alm come from primordial fluctuations one can only determine the distribution from which their measurement derives. The mean value of  alm is vanishing, but the variance is generally different from zero. In particular we have \langle a_{lm} \rangle = 0 \quad \langle a_{lm} a_{l'm'}^* \rangle = \delta_{ll'} \delta_{mm'} C_l This relation states that alm come from the same distribution for each m given the value of l. This means that one has a good sample of the distribution if one considers large l, whereas for small l, like 2 for example, the measurements used to define the distribution are 5 only, hence introducing a fundamental uncertainty in the information one can extract from C2. Such uncertainty is called “cosmic variance” and its amplitude can be easily obtained by the relation \left( \frac{\Delta C_l}{C_l}\right)_{\rm cosmic \, \, variance} = \sqrt{\frac{2}{2l+1}}

The Cl coefficients

From the previous expectation values we saw that C_l are obtained squaring the coefficients a_{lm} and then taking the average on the distribution. For this reason we end with computing the quantity \langle \Theta(\vec{k},\hat{p}) \Theta^*(\vec{k}',\hat{p}') \rangle. Such quantity depends on two phenomena:

1) the initial amplitude and phase of the perturbations produced at the inflationary epoch taken from a Gaussian distribution

2) the evolution of the perturbations till the present day as presented in the previous trasparencies

To separate these two contributes it is convenient to write Θ = δ × (Θ/δ), with the DM overdensity δ not depending on \hat{p}. Note that the ratio does not depend on the initial distribution so it can be extracted from the average

\langle \Theta(\vec{k},\hat{p}) \Theta^*(\vec{k}',\hat{p}') \rangle = \langle \delta(\vec{k}) \delta^*(\vec{k}') \rangle \frac{ \Theta(\vec{k},\hat{p})}{\delta(\vec{k})} \frac{ \Theta^*(\vec{k}',\hat{p}')}{\delta^*(\vec{k}')} = (2 \pi)^3 \delta^3(\vec{k}-\vec{k}') P(k) \frac{ \Theta(k, \hat{k} \cdot \hat{p})}{\delta(k)} \frac{ \Theta^*(k', \hat{k}' \cdot \hat{p}')}{\delta^*(k')}

From this expression one gets

C_l = \frac{2}{\pi} \int_0^\infty dk \, k^2\, P(k) \left| \frac{\Theta_l(k)}{\delta(k)} \right|^2

The anisotropy spectrum today (Sachs-Wolfe effect)

Large scales anisotropies are not affected by microphysics since at the time of recombination the perturbations were on scales too large to be connected via causal processes. For this reason the anisotropies are determined by monopole contribution only   \Theta_l(k,\eta_0) \approx \left[ \Theta_0(k,\eta_*)+\Psi(k,\eta_*)\right]j_l[k(\eta_0-\eta_*)] (*)

For the large scale structure we have shown that \left[ \Theta_0(k,\eta_*)+\Psi(k,\eta_*)\right] = \frac 13 \Psi(k,\eta_*)

If the recombination occurs far after the matter/radiation equality we can approximate the potential at recombination with its present value modulo the growth factor, so we have \left[ \Theta_0(k,\eta_*)+\Psi(k,\eta_*)\right] = \frac {1}{3 D_1(a=1)} \Psi(k,\eta_0) = - \frac {1}{3 D_1(a=1)} \Phi(k,\eta_0)

In this case we may use the expression Φ(η0) as a function of DM distribution and hence we get \left[ \Theta_0(k,\eta_*)+\Psi(k,\eta_*)\right] \approx -\frac{\Omega_m H_0^2}{2 k^2 D_1(a=1)} \delta(\eta_0)

This expression provides the source term in Eq. (*) and thus yields C_l^{SW} \approx  \frac{\Omega_m^2 H_0^4}{2 \pi D_1^2(a=1)} \int_0^\infty \frac{dk}{k^2}j_l^2[k(\eta_0 - \eta_*)] P(k)

The anisotropy spectrum today (Sachs-Wolfe effect)

By using the expression of the power spectrum

C_l^{SW} \approx  \pi H_0^{1-n} \left(\frac{\Omega_m}{D_1(a=1)}\right)^2 \delta_H^2 \int_0^\infty \frac{dk}{k^{2-n}}j_l^2[k(\eta_0 - \eta_*)]

The expression of Cl has the superscript SW to remind Sachs and Wolfe, the first people to compute the large-angle anisotropy. The integral over the spherical Bessel function once performed gives

C_l^{SW} \approx  2^{n-4}\pi^2 (\eta_0 H_0)^{1-n} \left(\frac{\Omega_m}{D_1(a=1)}\right)^2 \delta_H^2 \frac{\Gamma(l+\frac n2 - \frac 12)}{\Gamma(l+\frac 52 - \frac n2)} \frac{\Gamma(3-n)}{\Gamma^2(2 - \frac n2)}

for a Harrison-Zel’dovich-Peebles spectrum (n=1) we simply have

l(l+1) C_l^{SW} \approx  \frac \pi2  \left(\frac{\Omega_m}{D_1(a=1)}\right)^2 \delta_H^2

The small scale anisotropy spectrum is more involved since it depends not only on the monopole, but on the dipole and the integrated Sachs-Wolfe effect as well.

Cosmological parameters

The anisotropy spectrum depends on many cosmological parameters, in particular one can chose

  • Curvature density, Ωk ≡ 1- Ωm- ΩΛ
  • Normalization, C10
  • Primordial tilt, n
  • Tensor modes, r
  • Reionization, parameterized by τ back to recombination
  • Baryon density, Ωb h2
  • Matter density, Ωm h2
  • Cosmological constant energy density, ΩΛ

Such parameters show an high level of degeneracy, which can be removed by considering a wide range of cosmological observables. In figure we show the anisotropy spectrum as presently measured.

CMB data after 5 years of WMAP, improved ground-based and balloon-borne experiments. ΛCDM fairly agrees with the data.

CMB data after 5 years of WMAP, improved ground-based and balloon-borne experiments. ΛCDM fairly agrees with the data.


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Progetto "Campus Virtuale" dell'Università degli Studi di Napoli Federico II, realizzato con il cofinanziamento dell'Unione europea. Asse V - Società dell'informazione - Obiettivo Operativo 5.1 e-Government ed e-Inclusion