PART II ASTROPHYSICS

LECTURE 3

Now move on to gravitation and cosmology— Newtonian gravity does quite well with the universe— But to understand properly, need aspects of general relativity (GR)

Aims — want to understand:

• need for General Relativity theory• Principle of Equivalence• metrics and geodesics• bending of light in a gravitational field• basics of orbits of planets

• Non-rotating black holes— what are they?— what happens when things fall

into them?

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As well will look at:

COSMOLOGY— the study of the universe on the largest scales— questions concern universe as a whole

Big Questions

1. How large is the universe?2. How old is the universe?3. What is the universe made of?4. How did the structure we see in the

universe get there?5. What is the future of the universe?

— will it continue expanding forever?— will it eventually recollapse?

The study of gravity we initiate here is fundamental to answering these questions.

Using the techniques from the first few lectures of this part of course, we will be able tofind the metric of the universe (Friedmann-Robertson-Walker) and get full dynamicalequations for the evolution of the universe.

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• In this first lecture, we will start by considering some simple aspects of cosmology,with the aim of showing that Newtonian gravity can get a surprisingly long way in thestudy of the universe as a whole.

• However, we run adrift when it comes to considering what the universe was like atearly times, or how light propagates in the universe at any time.

• This will show us why we need a new theory of gravitation in order to study theuniverse.

• Then, we will move on to consider why General relativity (GR) is required, and not forexample, just a simple extension to Special Relativity which incorporates gravity.

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MAIN OBSERVATIONAL FEATURES OF THE UNIVERSE

To start, what are the main features of the universe on the largest scales, and how doesit change with time? Begin with Hubble’s Law (Hubble, 1929 — predicted by Weyl,1923). The law is

v = H0d

i.e. galaxies are systematically moving away from us with speed proportional todistance.

Infer this from looking at spectra:

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Identifiable pattern of emission (or absorption) lines (H, Mg, Ca, Na, etc. all give opticallines).

Express this quantitatively via the redshift z defined by

z =λobs − λL

λL.

Note that other definitions of redshift would be possible (e.g. dividing by λobs) — theone given is the one used in cosmology.

If we choose to interpret this as a (non-relativistic) Doppler shift, then v ≈ zc (sinceνobs = νL(1− v/c)).

So the Hubble effect, is that for cases where we know the distance d to the object (e.g.as provided by Cepheids) then we find

v ≈ zc = H0d.

H0 is Hubble’s Constant. Exact value is controversial, but pretty sureH0 = 50− 80kms−1 Mpc−1.

Translated into SI units this is 1.6− 2.6× 10−18 s−1.

This has big implications for the time evolution of the universe.

Consider a galaxy a distance d from us, e.g. M101 which is ≈ 4Mpc away.

Projecting back in time, when wouldit have been on top of us?

v = 4×H0 kms−1 =⇒

t =d

v=

4Mpc

4Mpc×H0= H−1

0 s

I.e. tcollide is approximately:

15 billion years (byr) for H0 in the middle of the known range;

12.5 byr if H0 = 80kms−1 Mpc−1;

and 20 byr if H0 = 50kms−1 Mpc−1.

Clearly this time is independent of which galaxy we are considering, and thus the wholeof the universe must have been in a very small volume this time ago.

But the age of the Earth is known to be ∼ 4.5 byrand the oldest stars in the Galaxy are ∼ 11 byr old.

This poses great problems for higher values of H0, since the universe is then in dangerof being younger than some of its constituents!

(If include dynamics, find tcollide is modified somewhat, and is most likely even shorterthan we have just deduced!)

In any case, it is clear that the universe must have come from an earlier stage at whichit was much smaller and denser.

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Roughly, conservation of matter tells us that ρ ∝ R−3, where ρ is the matter densityand R some characteristic size — the scale factor — associated with the universe —we will see later how to define this physically.

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COSMIC MICROWAVE BACKGROUND

The CMB was first discovered 1965 by Penzias and Wilson.

(Actually the effects were first noticed 1939 in excitation of Cyanogen molecules in theinterstellar medium

— optical spectral absorption work towards stars byAdams and McKellar.)

Anywhere in empty space at the mo-ment there is radiation present corre-sponding to what a blackbody wouldemit at a temperature of ∼ 2.74K.

Best measurement of the spectrum of this radiation so far has come from the COBEsatellite (Cosmic Background Explorer satellite.)

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COBE shows the spectrum is accurately blackbody to approx. 1 part in 104.

Idea is that this radiation was emitted in the early universe (hot, dense conditions),when matter and radiation were in thermal equilibrium.

— hot means matter was ionized— therefore photons scattered frequently off the

free electrons

This is extremely important. Radiation must come from a region where the matter andradiation are in strict thermodynamic equilibrium (TE).

Thus universe was not just small and dense at early times, but in TE. This is verydifferent from today.

Stars shine out at ∼ 6000K into empty space at 2.7K.

Early universe was not only in TE, but very hot, since the radiation cools as theuniverse expands.

Think of this either as stretching of the photons’ wavelength:

or as the work done by the photon gas in expansion.

Either way find:

Temperature of radiation cools as T ∝ 1/R where (R is the scalefactor of the universe).

Thus at times ∼ tcollide before the present, universe was filled with black body radiationat a very high temperature:

Running the film backwards, see that universe cools away from infinite temperature atsome sort of singularity — the Big Bang.

As the universe expanded, it cooled down and matter became neutral:

— This happens at temp. T ∼ 4000K

— photons no longer scattered— time about 0.3− 0.5 million years after the big bang— redshift ≈ 1400

Known as the epcoh of recombination — corresponds to last scattering for the cosmicmicrowave background (CMB) photons we see today.

So the CMB provides us with a picture of the universe as it was at these early times:

FLUCTUATIONS IN THE CMB

The radiation decouples from the matter

— propagates unhindered towards us today givingus a picture of the universe at a time approx.300,000 years after the big bang.

First evidence for structure in this picture (fluctuations) came from the COBE satellite.

— level found was at just 1 part in 105

at an angular scale of 10degrees

This uniformity is astonishing on Standard Hot Big Bang picture.

— can show that regions on sky more than approx. 1.5degrees apart, cannot havebeen in causal contact in their past at the time the radiation was emitted.

How did they know to synchronize their temperatures to the level of 1 part in 105?

Leave this question for a moment. Instead concentrate on the idea of CMB structure asgiving a picture of the seeds needed to form galaxies etc.

The 1 part in 105 uniformity of the CMB can be contrasted with the non-uniformity ofthe present day matter distribution:

• treating the stars as a continuous fluid the mean density of a galaxy is∼ 6× 10−21 kgm−3

• treating the galaxies themselves as a continuous fluid, the mean density ofuniverse as a whole is ∼ 5× 10−28 kgm−3

Discuss how to find these numbers later.

Assuming that the anisotropy of the CMB provides an estimate of the fractionalperturbations in the matter distribution at recombination, we see that the galaxies, likestars, represent strong condensations (δρ/ρ ∼ 107) out of an originally veryhomogeneous fluid (δρ/ρ ∼ 10−5). How did this come about?

STRUCTURE FORMATION

Basic cosmological question is the origin of structure

— how did galaxies and clusters of galaxies get there?

Basic answer to this is gravitational instability (e.g. Bentley/Newton correspondence).

— uniform gravitating medium with slight perturbation δρ/ρ

— will collapse further

=⇒ more δρ/ρ =⇒ more collapse.

Jeans analysis =⇒ get exponential growth, even in a medium with pressure, if mass ofperturbation is above a certain size. Find

δρ

ρ∝ exp(A× t)

(A is a positive constant)

Therefore might expect only need very small seeds of structure to be present in theearly universe.

BUT: Universe is expanding

— counteracts collapse tendency

In fact get δρρ ∝ t2/3 only (Lifshitz, 1946)

Now discuss how much cosmology we can understand using Newtonian physics only

NEWTONIAN GRAVITY AND THE UNIVERSE

We now address the question of the dynamics of the universe, and how far Newtoniangravity can get in describing this.

Consider a galaxy at the edge of a sphere of radius R. Gauss’s theorem is what weneed to tell us that all the material outside the sphere is irrelevant to our considerations— we need only consider the material inside the sphere, assumed uniformly distributedwith density ρ, and the ‘particle’ at its edge.

We can write the following series of equations:

E =1

2mv2 −

GMm

R, v = HR, M =

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3πρR3.

Thus defining

Ω = 8πGρ/3H2 = ρ/ρc say, where ρc =3H2

8πG,

we get

E = (1−Ω)1

2mH2R2,

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which since H = R/R can be rearranged to(R

R

)2

−8πGρ

3=

2E

mR2.

Now, using the full theory of general relativity, the equivalent equation for the scalefactor R is (

R

R

)2

−8πGρ

3=−kc2

R2.

Here k is a constant which has value

k =

1 “positive curvature, closed universe”0 “spatially flat universe”

−1 “negative curvature, open universe”

Simple Newtonian treatment recovers exact GR equation if identify the energy per unitmass of the particle (E/m) as −k′c2/2, and rescale the radial R coordinate to make k′

come out normalised to unity, as for k above.

(Rescaling corresponds to the distinction between the Newtonian radial coordinate andthe general relativistic ‘scale factor’.)

Can go further with the Newtonian analogy. Consider the eventual fate of the universe.Usual Newtonian classification for orbits (which applies also to purely radial motion)