THE BIG BANG COSMOLOGICAL MODEL: THEORY AND OBSERVATIONS
MARTINA GERBINO INFN, sezione di Ferrara
ISAPP 2021 Valencia July 22st, 2021
Maps of CMB anisotropies show the Universe as it was at the time of recombination. The CMB field is isotropic and the rms fluctuations (in total intensity) are very small, < | | > ~10 (even smaller in polarization). Density perturbations � ≡ ��/� are proportional to CMB fluctuations. It is possible to show that, at recombination, perturbations could be from a few (for baryons) to at most 100 times (for CDM) larger than CMB fluctuations. We need a theory of structure formation that allows to link the tiny perturbations at z~1100 to the large scale structure of the Universe we observe today (from galaxies to clusters and beyond).
General picture: small density perturbations grow via gravitational instability (Jeans mechanism). The growth is suppressed during radiation-domination and eventually kicks-off after the time of equality (z~3000). When inside the horizon, perturbations grow proportional to the scale factor as long as they are in MD and remain in the linear regime (� ≪ 1).
M. GERBINO 2 ISAPP-VALENCIA, 22 JULY 2021 Preliminaries
⃗ Density contrast �(�⃗) ≡ and its Fourier expansion � = ∫ � � �(�⃗) exp(��. �⃗) Credits: Kolb&Turner
2� � � ≡ ; � = ; � = ��; � � � � � , �� � ≡ � �; � � = �(�) � ∝ 6 6 � / , ��
� �(�) ≈ � 2�
The amplitude of perturbations as they re-enter the horizon is given by the primordial power spectrum. Once perturbations re-enter the horizon, micro-physics processes modify the primordial spectrum
Scale factor
M. GERBINO 3 ISAPP-VALENCIA, 22 JULY 2021 Jeans mechanism (non-expanding)
The Newtonian motion of a perfect fluid is decribed via the Eulerian equations. Let’s linearize them!
We obtain a second-order differential equation Where � ≡ = is the sound speed A solution to this equation is
! Where the dispersion relation is given by � = � � − 4��� = 4��� ! − 1 "
M. GERBINO 4 ISAPP-VALENCIA, 22 JULY 2021 Jeans mechanism (non-expanding)
The critical Jeans wavenumber kJ sets the threshold for gravitational instability:
� � = � � − 4��� = 4��� − 1 �
k>kJ : � > 0, solution is oscillating sound wave (collapse counterbalanced by pressure) k
k/kJ ~ tpressure/tgrav
M. GERBINO 5 ISAPP-VALENCIA, 22 JULY 2021 Jeans mechanism (expanding)
To apply this formalism to the evolution of cosmological perturbations, we must take the expansion into account
Novelties: the Hubble friction term (I-order derivative) and a redefinition of the Jeans wavenumber 4��� � � ≡ � The Jeans wavenumber still represents the threshold for gravitational instabilites. However, the two solutions differ:
k>kJ : acoustic waves with slowly decreasing amplitude k
M. GERBINO 6 ISAPP-VALENCIA, 22 JULY 2021 Jeans mechanism (expanding) When we have multiple components in the Universe, we must take them into account in the Poisson term
In the case of CDM in a RD epoch, for k< M. GERBINO 7 ISAPP-VALENCIA, 22 JULY 2021 Jeans mechanism (expanding) Let’s summarize what happens to the growth of CDM and baryon perturbations. Consider their physical kJ,phys=kJ/a: CDM: kJ,phys~ → ∞ à kJ,phys is always much larger than any other scale and CDM perturbations always grow # as soon as they re-enter the horizon < �, � > � BARYON: : kJ,phys~ { à kJ,phys is ~ acoustic horizon (smaller than the causal horizon) as long as baryons are # → ∞, � < � tightly coupled to photons: baryons undergo acoustic oscillations as soon as they re-enter the horizon. After decoupling, the baryon sound speed drops and baryon perturbations can catch up with CDM perturbations M. GERBINO 8 ISAPP-VALENCIA, 22 JULY 2021 Free streaming Once decoupled, a given species evolves like free falling particles. If they have large velocity dispersion, they can stream out of overdense region and into underdense regions. This free-streaming effect efficiently erase perturbations on scales smaller than a characteristic free-streaming scale kfs. In analogy with the Jeans scale, we can define � � = , � Where v=c for massless particles (or massive particles in the UR regime) and v= /m for massive particle (when NR). Hence, massless and UR particles are free-streaming on all scales smaller than the horizon. The Jeans wavenumber allows to identify the scale below which (k>>kJ -> � ≪ � ) pressure counterbalances gravitational instability and prevents gravitational collapse. Similarly, the free-streaming wavenumber identifies the scale below which the high thermal velocity allows collisionless particles to escape gravitational wells. Free streaming inhibits structure formation on scales much smaller than the free-streaming scale. Hot dark matter cannot be the dominant dark matter component (top-down vs bottom-up). M. GERBINO 9 ISAPP-VALENCIA, 22 JULY 2021 Lambda-dominated epoch At recent times, the cosmological constant takes over CDM and becomes the dominating species. When this happens, the growth of perturbations is modified due to the change in the expansion history. Lambda is responsible for the accelerated expansion we observe today. Therefore, it enhances the effect of the Hubble friction term in the Boltzmann equations for the evolution of perturbations. As a result, the growth of perturbations is slowed down in a way that is independent of the wavenumber: �(�) ∝ ��(�, Ω , Ω ) where g(a) is a decreasing function of the scale factor that goes from 1 during MD to vanishingly small values at later times and depends on the amount of CDM and Lambda. Observations of the distribution of cosmic structures at late times are key to investigate dark energy properties! M. GERBINO 10 ISAPP-VALENCIA, 22 JULY 2021 Super-horizon perturbations To correctly describe the behaviour of super-horizon perturbations, general-relativistic analysis is needed. Here, we will follow an approximate scheme (on super-horizon scales, micro-physics can be neglected). Consider a flat Universe with average density � and a second Universe with the same expansion rate but a slighlty higher density � > � (hence, it must have curvature parameter k=1). If we equate the Friedmann equations in the two Universes, we can work out the density contrast as / ! � ≡ $ % = % %/ As long as the contrast remains small, we can easily see that � � , �� � ∝ ∝ � �, �� Super-horizon perturbations are untouched by micro-physics effects. When they re-enter the horizon, their amplitude at horizon-crossing is set by the inflationary spectrum. When inside the horizon, micro-physics kicks in and the initial spectrum is modified. M. GERBINO 11 ISAPP-VALENCIA, 22 JULY 2021 Matter power spectrum khc=ahcH~ keq -1 -1/2 (ahc , RD; ahc , MD) -3 -3 Phc~k Pin~k P(k,t)~k P(k,t)~k-3 1) thc 2) thc> teq: 2 4 P(k,0)~(1/ahc) Phc~k Phc~k Credits: NeutrinoCosmology M. GERBINO 12 ISAPP-VALENCIA, 22 JULY 2021 Matter power spectrum We have seen that the spectrum of perturbations at horizon crossing is given by the primordial spectrum of scalar perturbations. Different scales cross the horizon at different times. It is useful to have an expression for the spectrum at a given time as a function of the scale/wavenumber. To do so, we must find 1) when a given scale enters the horizon, 2) How the scale is modified between horizon crossing and the time at which we want the spectrum to be computed. In practice, we can express the spectrum as � �, � = � �, � � (�) If we want the CDM spectrum today: 1) Let’s call keq the scale that enters the horizon at matter-radiation equality; -1 -1/2 -3 -3 2) Let’s see that khc=ahcH~ (ahc , RD; ahc , MD) and Phc~k Pin~k ; 2 -3 3) Scales k>keq have entered the horizon at thc After teq, all scales evolve as a(t) and the shape of the (linear) spectrum remains the same. There is a time tnl when a given scale enters in the non-linear regime: clustering is enhanced and initial conditions are Rapidly forgot. The evolution in the (midly) non-linear regime must be studied with different dedicated tools. M. GERBINO 13 ISAPP-VALENCIA, 22 JULY 2021 Matter power spectrum We can add the effect of baryons. We know that: 1) On super-horizon scales, baryon perturbations evolve as the dominant component 2) Before recombination, baryon oscillates with photons (BAO) 3) After recombination, baryon perturbations catch-up with CDM We can also add the effect of neutrinos: 1) Neutrinos contribute to clustering only on scales larger than the free streaming scale (k On top of these direct effects, baryons and neutrinos backreact on CDM slowing down the overall growth of perturbations with respect to the case of a pure CDM Universe. M. GERBINO 14 ISAPP-VALENCIA, 22 JULY 2021 Cosmic Microwave Background Soon after matter-radiation decoupling, CMB photons propagate (almost) undisturbed. The equilibrium distribution of CMB photons is redshifted as the Universe expands and cools down. The currently observed CMB frequency spectrum is a perfect black body spectrum (vanishingly small spectral distortions) with a very precisely measured temperature -4 T0=(2.725+/-0.001) K=(2.348+/-0.001) 10 eV On top of this average temperature, the standard model predicts (and we observe!) fluctuations in intensity and polarization. Let’s see how they evolved. In absence of primordial perturbations, if gravitation and pressure equilibrated perfectly, there would be no sound waves. The tiny shift from equilibrium due to primordial perturbations determines the propagation of acoustic waves in the tightly coupled baryon-photon fluid. The (comoving) sound horizon is the distance travelled by an acoustic wave � � ��′ � � = �(�′) with a given sound speed cs. M. GERBINO 15 ISAPP-VALENCIA, 22 JULY 2021 Cosmic Microwave Background (CMB) To fully characterize the evolution of CMB fluctuations, we need to follow an infinite hierarchy of Boltzmann equations. However, we can understand the basic picture if we begin with restricting to the tightly coupled regime well before Recombination. At that time, photons and baryons can be described as a tighlty coupled perfect fluid (we only need To follow the density and velocity). The two components are coupled via Thomson scattering (free electrons and photons) and, indirectly, via Coulomb scattering (electrons and protons). This is the equation of a damped, driven oscillator. We have 1) a damping term due to R ≡ &; 2) a time-varying mass term ' due to the variation of the sound speed � = [3 1 + � ] ; 3) a gravitational driving term on the RHS M. GERBINO 16 ISAPP-VALENCIA, 22 JULY 2021 Cosmic Microwave Background 2 Deep in the RD epoch, inside the sound horizon, the equation can be simplified noting that R->0, cs ->1/3 and the RHS is negligible (gravitational effects can be neglected with respect to pressure effects). We are left with the equation of a simple harmonic oscillator Outside the sound horizon, CMB fluctuations (as well as all other perturbations) are frozen to the initial values and independent of k. Moving to MD, R grows and cs decreases. The amplitude of oscillations is damped and the zero point of the oscillations move to a different value (baryon loading). As we approach decoupling, tight-coupling is compromised. Fluctuations are exponentially damped on scales smaller than the photon mean free path (diffusion length) : � = �� ; � = ∫ . This is the so-called Silk damping effect. ( ) M. GERBINO 17 ISAPP-VALENCIA, 22 JULY 2021 Cosmic Microwave Background Credits: NeutrinoCosmology M. GERBINO 18 ISAPP-VALENCIA, 22 JULY 2021 Cosmic Microwave Background After decoupling, CMB photons free stream in the Universe. We can define a last-scattering surface at the time of last scattering between photons and residual free electrons. We can define the optical depth in terms of the number of scatterings between us and the LSS % � � = �� �� � *)+ After decoupling, the Universe becomes transparent. Due to the very large baryon-to-photon ratio, the time of CMB decoupling is different from the time of baryon decoupling (baryon drag). The CMB field carries a picture of the Universe as it was at the time of decoupling. On top of this, secondary effects further enrich the information contained in the CMB. M. GERBINO 19 ISAPP-VALENCIA, 22 JULY 2021 CMB lensing At late times, CMB photons interact with forming cosmological structures and are gravitationally lensed by their potential. The observed CMB fields becomes a distorted image of the field as it was at decoupling: Θ � ~Θ � + ∇� Where � is the lensing potential (convolution of the matter power spectrum along the line of sight) and its gradient is the Deflection angle. Lensing is most visible at small scales. The typical deflection is ~2.5 arcmin. Credits: ESA/Planck M. GERBINO 20 ISAPP-VALENCIA, 22 JULY 2021 Polarization Thomson scattering creates a small level of linear polarization in the CMB field. As long as tight coupling is efficient, Polarization is isotropic due to the very large cross section. Around recombination, the CMB quadrupole begins to be Populated. This quadrupolar pattern of CMB fluctuations is responsible for sourcing a small level of CMB fluctuations In polarization. Credits: W.Hu tutorial http://background.uchicago.edu/~whu/intermediate/Polarization/polar1.html M. GERBINO 21 ISAPP-VALENCIA, 22 JULY 2021 Polarization It is useful to decompose the polarization field introducing two scalar quantities: - A curl-free component, the E-modes (no handedness) - A divergence-free component, the B-modes (handedness) Credits: W.Hu tutorial Scalar perturbations do not have any handedness. Therefore, they can only give rise to E-mode fluctuations. We need A different source for (primordial) B-mode fluctuations in CMB polarization. M. GERBINO 22 ISAPP-VALENCIA, 22 JULY 2021 Polarization While it is not possible to generate primordial B-modes from scalar perturbations, a secondary B-mode like pattern Arises due to gravitational lensing. The same mechanism that deflects CMB photons is responsible for a convertion From E-modes to lensing B-modes Credits: APS/A.Stonebraker M. GERBINO 23 ISAPP-VALENCIA, 22 JULY 2021 Polarization At late times (z~a few), the ignition of the first stars re-ionizes the intergalactic medium. A fraction of free electrons Was therefore available for scattering off CMB photons. At the time of reionization, a second source of CMB polarization Is available. Since this source is localized at very late times, the effect on the CMB field is peaked at very large scales. M. GERBINO 24 ISAPP-VALENCIA, 22 JULY 2021 Tensor modes A prediction of many early Universe models is the generation of tensor perturbations to the metric. These are responsible For the generation of tensor modes in CMB fluctuations, both in total intensity and polarization (E and B modes). By working out the Boltzmann equations for tensor modes, one can show that tensor modes are peaked at the scale that Enters the horizon at recombination (and reionization), then quickly decay inside the horizon. The most crucial effect is that tensor perturbations can generate primordial B-modes. This is the reason why The observation of primordial B-modes in the CMB field is such a holy grail in observational cosmology! M. GERBINO 25 ISAPP-VALENCIA, 22 JULY 2021 CMB spectrum Similarly to the matter field, we are interested in the statistical properties of the CMB field. Let’s expand the CMB field In spherical harmonics Θ � = ∑ � � (� ): ∗ 1) Since CMB fluctuations are gaussian, � are also gaussian distributed and we are only interested in < � � > ∗ 2) Since different Fourier modes are uncorrelated, so are different multipoles: < � � >∝ � ,� , 3) Since the Universe is isotropic, there should be no dependence on m We are left with the power spectrum of CMB fluctuations in harmonic space: ∗ � ≡< � � > The multipole l is the inverse of an angular scale on the sky and can be roughly related to the wavenumber k by �~��, where r is the distance to the last-scattering surface. In Fourier space, the correlation function of the CMB field is given by From which one can easily understand that the power spectrum Cl in terms of the transfer function and the primordial spectrum is M. GERBINO 26 ISAPP-VALENCIA, 22 JULY 2021 CMB spectrum A similar expression can be worked out for the power spectra of polarization and cross-correlation of CMB: In the standard cosmological model, there is no TB, EB power spectra, due to the different behaviour of E and B modes Under parity. It may seem that we need to know the solution to the Boltzmann equations for each multipole to compute the spectra. However, it can be seen (with some algebra) that the main contributions arise from a very few terms. For scalar T, The line of sight integral gives: At each multipole (selected via the Bessel function jl), the observed T today has contribution from: 1) Sachs-Wolfe: The fluctuations at the time of decoupling, corrected for a gravitational shift, only where g is non-zero 2) Doppler: photons are leaving a tightly coupled fluid with given peculiar velocity (where g is non-zero) 3) Integrated SW: photons are red/blue shifted when travelling through time-varying potentials M. GERBINO 27 ISAPP-VALENCIA, 22 JULY 2021 CMB spectrum Credits: NeutrinoCosmology In scalar polarization, the source function is simpler, as it only contains the contribution from the quadrupole terms. For tensor modes, the source function contains both the contribution from the tensor perturbations to the metric (in T) and the contribution from quadrupoles M. GERBINO 28 ISAPP-VALENCIA, 22 JULY 2021 CMB spectra M. GERBINO 30 ISAPP-VALENCIA, 22 JULY 2021 CMB with lensing effect M. GERBINO 31 ISAPP-VALENCIA, 22 JULY 2021