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Lecture 14: Olbers’ Paradox

• Named for Wilhelm Olbers, but • Olbers’ paradox known to Kepler and Halley – Consider spherical shell of radius r • and the expansion of the and thickness dr • The – Number of in this shell is 4πr2n dr, where n is number density • Ω0 and the curvature of space of stars 2 • The model – Flux from each is L/4πr , therefore flux from shell is – Primordial nucleosynthesis F= 4πr2n dr L/4πr2= nL dr, – The Cosmic Microwave Background independent of r – therefore, in an infinite universe, • The age and future of the Universe night sky should be as bright as a typical stellar surface – since any line of sight will hit a star, and stars block from behind them • So - why is the sky dark at night?

Resolutions of Olbers’ paradox Cosmic redshift

• Light is absorbed by • Universe has finite age • When a is receding, light waves travelling to intervening dust – light from stars more us are redshifted – suggested by Olbers than ctUni distant has not – doesn’t work: dust would had time to reach us heat up over time until it • Universe is expanding reached the same – effective temperature of temperature as the stars distant is that illuminate it redshifted down • Hubble measured the spectrum of nearby and found the • Universe has finite – this effect not known – suggested by Kepler until 19th century spectral lines to be redshifted – this works (integral is • These last two provide truncated at finite r) the resolution to Olbers’ • The more distant the galaxy, – but static finite universe paradox the greater its redshift would collapse under its own gravity - i.e. unstable

1 Cosmic redshift Cosmic redshift

Redshift z is given by: It is more useful to think of the cosmic redshift as arising from the expansion of space, rather Where λ is the observed wavelength, λ is the rest-frame 0 than being a wavelength and Δλ = λ – λ 0 Doppler shift For small z (i.e. v<

Where v is the recession velocity and c is the .

The Cosmological Principle The

The Cosmological Principle: Cosmological theories are based on the idea that on sufficiently large scales, the Universe looks the same • at all locations (homogeneity), and • in every direction (). One implication of this is that the Universe cannot have an “edge”.

Note that this is an assumption. We cannot see objects beyond a light travel time of ~14 Gyr, because light from them has not had enough time to reach us since the beginning of the Universe.

2 The curvature of space The Density Parameter - Ω0 is defined as: • According to - space is warped by mass Ω0 • Cosmological space is also being stretched by the expansion • The net curvature of space in the Universe depends upon the balance between these two effects • 3 simple types of curvature - consider 2-D analogies Where ρ0 is the current density of the universe and ρc is the critical density – the density which makes space flat.

So:

. Ω0 < 1 gives negative curvature – Open Universe Flat - no curvature Negative curvature Positive curvature . Ω0 = 1 gives a Flat Universe - e.g. - e.g. saddle . Ω0 > 1 gives positive curvature - a finite, Closed Universe

The Hot Big Bang model Primordial Nucleosynthesis

If the universe is expanding, then it follows that by going For the first 10-15 minutes after the Big Bang, the temperature back in time, we move towards a moment when the universe of the Universe was very high – high enough to produce light was infinitely dense. elements from by nuclear fusion. This singularity, from which the universe is expanding is the The theory of primordial nucleosynthesis combines our so-called Big Bang. understanding of nuclear physics with the Big Bang model, to predict the relative abundances of the light elements: The Big Bang model is a theory for the formation, 2 3 4 development and future history of the universe. It is now Deuterium ( 1H), Helium-3 ( 2He), Helium-4 ( 2He), 7 widely accepted because it explains three very important Lithium-7 ( 3Li) observational facts: These predictions can be tested by studying metal-poor 1. The expansion of the Universe galaxies (i.e. those where stellar-processing has had very little effect) 2. The relative abundances of light elements Predictions match observations very well. 3. The cosmic microwave background

3 Cosmic Microwave Background Cosmic Microwave Background In the early stages of the Universe, the temperature was So, at the Universe was high enough to keep all fully ionized. filled with radiation with scattered on all the free electrons, keeping blackbody temperature T~3000K. matter and radiation in equilibrium. Since then expansion has caused the radiation to “cool”. However, once the temperature of the Universe dropped Big Bang theory, therefore, to ~3000K, neutral hydrogen could form – this was predicts a “sea” of photons filling about ½ million years after the Big Bang. the Universe now with a blackbody temperature of ~3K. At this point, the number of free electrons dropped In 1965 Penzias and Wilson dramatically, and the photons became free to move The spectrum of the CMB is independently of matter: accidentally discovered the radiation using a microwave extremely close to that of a blackbody with T=2.726 +/- 0.010 K Era of decoupling (or recombination) detector (a Nobel prize was to follow).

The of the Universe The

For a massless Universe, the expansion rate da/dt is a a unbound constant, so that the time since the Big Bang is

1 1 1s.1Mpc 106 pc 3×1018 km 1yr marginally t = = = =14 Billion years 0 H 70km / s / Mpc 70km 1Mpc 1pc 3 107 s bound 0 ×

1/H0 is known as the Hubble bound time.

a=1 So the age of the Universe is From the cosmological principle, the expansion of the Universe must be equal to the Hubble time if it the same everywhere. It can therefore be represented by the changing has expanded at a constant value of a scale factor a, which is normally defined to be 1 at the present rate. Otherwise the Hubble time. For an empty Universe, da/dt=const., whilst matter decelerates a. time gives just a rough The Hubble parameter H is then just H=(da/dt)/a measure of the age of the

H0 (“Hubble constant”) is the current value of the Hubble parameter. Universe.

4 The Age of the Universe The future of the Universe

In general the expansion of the Universe could be either • There is good evidence (e.g. from the size scales of fluctuations in the decelerating or accelerating. CMB) that Ω0=1. The rate of deceleration of the • However, observations of high redshift supernovae now show that expansion can be the expansion of the Universe seems represented by a deceleration to be accelerating (i.e. q0 is -ve). parameter, q0 . • This is believed to be due to dark A massive Universe with energy (also known as Einstein’s Ω =1 has q =1/2 , and an age ), which now ρ 0 0 Ω = dominates the energy density of the ρcrit 2 1 2 1 Universe. t = = = 9.3 Billion years 0 • If this is the case, then the future of 3 H 0 3 70km / s / Mpc the Universe will be one of ever- For comparison, the oldest globular clusters in the accelerating expansion. Galaxy are about 13 Gyr old…

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