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# Cosmology cheatsheet 1

Cosmology cheatsheet for the first part (1/3) of the "Introduction to cosmology" undergraduate course, Prof. Rodrigo Nemmen, IAG USP.

Topics covered:
• Fundamental observations of cosmology
• General relativity concepts
• Curved spaces and the Robertson-Walker metric
• Distances in cosmology

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

## Rodrigo Nemmen

September 11, 2017

## Transcript

2. ### Rodrigo Nemmen© FUNDAMENTAL OBSERVATIONS 1 The night is dark 㱺

The universe has a ﬁnite age F = Z 1 0 nL dr = 1 Olber’s paradox Solution F = Z ct0 0 nL dr = nLct0 2 The universe is homogeneous and isotropic at large scales: The cosmological principle 50 Mpc 4 Gpc 3 Hubble’s law 㱺 The universe is expanding z = H0 c r 2.3. REDSHIFT PROPORTIONAL TO DISTANCE 17 Figure 2.5: A more modern version of Hubble’s plot, showing cz versus distance. In this case, the galaxy distances have been determined using Cepheid variable stars as standard candles, as described in Chapter 6. (from Freedman, et al. 2001, ApJ, 553, 47) velocity away from Earth. Since the values of z in Hubble’s analysis were all small (z < 0.04), he was able to use the classical, nonrelativistic relation for the Doppler shift, z = v/c, where v is the radial velocity of the light source (in this case, a galaxy). Interpreting the redshifts as Doppler shifts, Hubble’s law takes the form v = H0 r . (2.6) Since the Hubble constant H0 can be found by dividing velocity by distance, it is customarily written in the rather baroque units of km s−1 Mpc−1. When Hubble ﬁrst discovered Hubble’s Law, he thought that the numerical value of the Hubble constant was H0 = 500 km s−1 Mpc−1 (see Figure 2.4). However, it turned out that Hubble was severely underestimating the distances to galaxies. Figure 2.5 shows a more recent determination of the Hubble constant from nearby galaxies, using data obtained by (appropriately enough) the 4 The universe is made up of diﬀerent types of fundamental particles p e- n γ ν dark matter Hubble time: Hubble distance: c/H0 1/H0
3. ### Rodrigo Nemmen© FUNDAMENTAL OBSERVATIONS 5 Cosmic microwave background (CMB) radiation

㱺 Universe was initially hotter, denser Big bang model T(t) / 1 a(t)
4. ### Rodrigo Nemmen© GENERAL RELATIVITY CONCEPTS a Equivalence principle The gravitational

force near a massive object is equivalent to the pseudoforce in an accelerated (non-inertial) frame Implies that gravity is a property associated with spacetime curvature: starting point for general relativity Inertial frame with gravity Accelerated frame
5. ### Rodrigo Nemmen© GENERAL RELATIVITY CONCEPTS Newtonian gravity Mass M tells

gravity how to exert a force Force tells mass how to accelerate F = GMm r2 F = ma Einstein's gravity Mass-energy Tμν tells spacetime how to curve gμν Curved spacetime tells mass-energy how to move xμ or F = mr Why is it important to talk about GR and gravity in a cosmology course? At cosmological scales, gravity is the force that dominates over all other three fundamental interactions, even though it is the weakest force in particle physics Gravity shapes the cosmological evolution at large scales
6. ### Rodrigo Nemmen© Gravity curves spacetime. In order to properly perform

cosmological calculations, need to take into account spacetime curvature and learn the geometry of curved spaces CURVED SPACES Importance Distance between nearby points: metric or line element ds Hartle Metric for cosmological spacetimes. Obeys the cosmological principle, allows for curvature and expansion Robertson-Walker metric special relativity comoving coordinates cosmic dynamics space curvature ds2 = c2dt2 + a(t)2 ⇥ dr2 + Sk(r)2d⌦2 ⇤ Spacetime metrics are distances between nearby points in 4D (one of them is time)
7. ### Rodrigo Nemmen© COSMIC DYNAMICS: PRELUDE Distances in cosmology dp(t) =

a(t)r proper distance comoving distance cosmic dynamics Hubble law vp(t) = H(t)dp(t) H(t) ⌘ ˙ a a vp(t) = ˙ dp Cosmological redshift 1 + z = 1 a(tE) scale factor at emission epoch cosmological redshift The redshift depends only on the relative scale factor between observation and emission and nothing more
8. ### Github Twitter Web E-mail Bitbucket Facebook Group ﬁgshare rodrigo.nemmen@iag.usp.br rodrigonemmen.com

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