Rodrigo Nemmen
October 09, 2017
340

# Cosmology cheatsheet 2

Cosmology cheatsheet for the second part (2/3) of the "Introduction to cosmology" undergraduate course, Prof. Rodrigo Nemmen, IAG USP.

Topics covered:
• fundamentals of cosmological models (dynamics)
• main equations
• main components and equations of state
• critical density
• concordance cosmological model
• solving the Friedmann equation

https://rodrigonemmen.com/teaching/introducao-a-cosmologia/

October 09, 2017

## Transcript

2. ### Rodrigo Nemmen© COSMIC DYNAMICS ✓ ˙ a a ◆2 =

8⇡G 3 ⇢ kc2 R0a2 ˙ " + 3 ˙ a a (" + P) = 0 ¨ a a = 4⇡G 3c2 (" + 3P) Friedmann Fluid Acceleration 3 unknowns, 2 independent equations above Equation of state P = P(⇢) = w⇢c2 a(t) k scale factor geometry: curvature constant a is dimensionless and normalized such that a(t0) = 1 Goal of cosmic dynamics: relate a(t) and k to the energy content of the universe and its pressure P " = ⇢c2 Main variables Main equations
3. ### Rodrigo Nemmen© ⇢m / a 3 ⇢r / a 4

⇢⇤ / constant Type of component w Evolution Ωi in ΛCDM model Matter (non- relativistic) 0 0,3 Radiation 1/3 8×10-5 Dark energy < -1/3 Cosmologica l constant Λ -1 0,7 COMPONENTS ⇢i = ⇢i0a 3(1+wi) From ﬂuid equation: 10-7 10-5 0.001 0.100 10-5 105 1015 1025 a Matter Radiation Λ (dark energy) matter- radiation equality arm ≈ 3×10-4 matter-Λ equality amΛ ≈ 0.75 ρi
4. ### Rodrigo Nemmen© DENSITY PARAMETER k = 0 ) ⇢c ⌘

3H2 8⇡G ρ0 = 10-26 kg/m3 Critical density Density parameter ⌦i = ⇢i ⇢c Ω0 = 1 ± 0.005 Latest measurements: Flat universe, <1% uncertainty (Planck 2015) Average cosmological density
5. ### Ωb0 = 0.04 26% 70% DARK MATTER DARK ENERGY NORMAL

MATTER Modiﬁed by R. Nemmen, original version here RADIATION γ γ γ γ γ γ FÓTONS 0.005% 0.003% NEUTRINOS Ωm0 = 0.3 ΩΛ0 = 0.7 Ωr0 = 8×10-5 Standard cosmological model H0 = 70 km/s/Mpc k = 0 (plane) Mostly CMB photons Ω0 = 1 4% (ΛCDM)
6. ### Rodrigo Nemmen© SOLVING THE FRIEDMANN EQUATION 1 First deﬁne the

components and curvature of your model universe 2 Assemble the Friedmann equation in the most convenient form for your problem 3 Find a(t) by solving the equation in the appropriate way. If your model has one or two components, an analytical solution should be possible. For three components, you have to do it numerically radiation matter cosmological constant ⇢ = ⇢m + ⇢r + ⇢⇤ ⌦ = ⌦m + ⌦r + ⌦⇤ or If k=0, then ρ = ρcrit or Ω = 1 Which components will you keep? and equivalently for Ωi Remember: ⇢m = ⇢m0a 3 ⇢r = ⇢r0a 4 ⇢⇤ = ⇢⇤0 ⇢m = ⇢m0a 3 ⇢r = ⇢r0a 4 ⇢⇤ = ⇢⇤0 ⇢m = ⇢m0a 3 ⇢r = ⇢r0a 4 ⇢⇤ = ⇢⇤0 ✓ ˙ a a ◆2 = 8⇡G 3 ⇢ kc2 R0a2 or ✓ H H0 ◆2 = ⌦ + 1 ⌦0 a2
7. ### IMPORTANT SOLUTIONS OF FRIEDMANN EQUATION Density is destiny One component

Matter only, curved ΛCDM
8. ### Github Twitter Web E-mail Bitbucket Facebook Group ﬁgshare [email protected] rodrigonemmen.com

@nemmen rsnemmen facebook.com/rodrigonemmen nemmen blackholegroup.org bit.ly/2fax2cT