THE BIG BANG COSMOLOGICAL MODEL: THEORY AND OBSERVATIONS MARTINA GERBINO INFN, sezione di Ferrara ISAPP 2021 Valencia July 22st, 2021 1 Structure formation 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 ,-./ -01 6 �+/#, �� �+ �(�) ≈ �3 -01 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 4, , Where �# ≡ = is the sound speed / 4& & A solution to this equation is ! # # # * Where the dispersion relation is given by � = �/ � − 4���5 = 4���5 ! − 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���5 = 4���5 # − 1 �6 # k>kJ : � > 0, solution is oscillating sound wave (collapse counterbalanced by pressure) # k<kJ: � < 0, solution is an exponentially growing mode (gravitational collapse) 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���5� �6 ≡ # �/ The Jeans wavenumber still represents the threshold for gravitational instabilites. However, the two solutions differ: k>kJ : acoustic waves with slowly decreasing amplitude k<kJ: power-law (non-exponential!) growing mode à the expansion slows down the exponential growth 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<<kJ, the equation above has the solution � � ∝ ln � : The higher expansion rate in a RD epoch suppresses the growth of perturbations (stagnation) 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: 7& CDM: kJ,phys~ → ∞ à kJ,phys is always much larger than any other scale and CDM perturbations always grow 8# as soon as they re-enter the horizon 7& < �, � > �9:; BARYON: : kJ,phys~ { à kJ,phys is ~ acoustic horizon (smaller than the causal horizon) as long as baryons are 8# → ∞, � < �9:; 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=<p>/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 -> � ≪ �6) 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 �5 and a second Universe with the same expansion rate but a slighlty higher density �@ > �5 (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 & $& */A! � ≡ $ % = &% BC7&%/+ As long as the contrast remains small, we can easily see that $# � �#, �� � ∝ ∝ U � �, �� 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 <teq: P(k,0)~[(1/aeq) X 2 ln(1+aeq/ahc)] Phc -3 # ~k ln(1+k) � �, � = � �, � �-;(�) 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 <teq: P(k,0)~[(1/aeq)ln(1+aeq/ahc)] Phc~k ln(1+k) 2 4 4) Scales k<keq will enter the horizon at thc> teq: P(k,0)~(1/ahc) Phc~k Phc~k. 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<kfs) 2) On smaller scales, neutrino free streaming erases perturbations in the neutrino fluid 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.