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PHYS4003 Introduction to Cosmology |
Prof. Anne Green |
CAPT B110 |
anne.green@nottingham.ac.uk |
Contents |
1 The expanding universe |
1.1 Introduction |
1.2 Newtonian cosmology |
1.3 Simple cosmological models |
1.4 Geometry of the universe |
1.5 Evolution including curvature |
2 Observing the Universe |
2.1 Expansion rate and age |
2.2 Light travel and horizons |
2.3 Distances |
3 Thermal history |
3.1 Introduction |
3.2 Nucleosynthesis |
3.3 Cosmic Microwave Background radiation and Summary |
4 Dark side |
4.1 Dark matter |
4.2 Dark energy |
5 Early universe |
5.1 Problems with the Big Bang |
5.2 Inflation |
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1 The expanding universe |
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1.1 Introduction |
1.1.1 Cosmological principle |
The cosmological principle states that the universe is, on average, isotropic (same in all directions) and homogeneous (same at all points). In other words, our location is not special. It was introduced by Einstein to simplify his calculations and is now confirmed observationally (by e.g. galaxy redshift surveys) on l... |
1.1.2 Hubble parameter and scale factor |
The velocities of galaxies can be measured via the Doppler effect using the red-shift, $z$, of spectral lines: |
$$ z = \frac{\lambda_r - \lambda_e}{\lambda_e} $$ (1) |
where $\lambda_e$ and $\lambda_r$ are the wavelengths of the emitted and received radiation respectively. This approach was pioneered by Slipher in the 1910s with Hubble using it to measure the expansion rate of the Universe in 1929. |
For nearby objects (with speed, $v$, much less than the speed of light, $c$) |
$$ z = \frac{v}{c} $$ (2) |
while for more distant objects special relativity gives |
$$ 1 + z = \sqrt{\frac{1 + v/c}{1 - v/c}} $$ (3) |
A galaxy which is receding from us has $v > 0$ and hence $z > 0$ and $\lambda_r > \lambda_e$, i.e. it is redshifted, while the opposite is true for a galaxy which is approaching us. Hubble showed that not only are the majority of galaxies moving away from us, their recession speed is proportional to their distance: |
$$ \mathbf{v} = H\mathbf{r} $$ (4) |
This is Hubble’s law. $H$ is the Hubble parameter. As we will see later on it is time dependent. Its present day value (often referred to as the Hubble constant) is denoted by $H_0$ and is written as |
$$ H_0 = 100h \, \mathrm{km \, s^{-1} \, Mpc^{-1}} $$ (5) |
Little $h$ parameterises the uncertainty in the measurement of the Hubble constant. Measurements of $H_0$ (or equivalently $h$), which we will discuss in Section 2.1 ‘Observing the Universe: expansion rate and age’, find $H_0 \approx 70 \, \mathrm{km \, s^{-1} \, Mpc^{-1}}$, i.e. $h \approx 0.7$. Note that while the Hu... |
It’s important to emphasize that we are not at the centre of the expansion; an observer at any point in the universe sees $\mathbf{v} \propto \mathbf{r}$. Imaging baking a cake with raisins in or blowing up a balloon with dots on it. The expansion looks the same from the point of view of any of the raisins or dots. Not... |
Consider a distribution of galaxies expanding uniformly. Position vectors at time $t$, $\mathbf{r}(t)$, are just scaled versions of their values today at time $t_0$: |
$$ \mathbf{r}(t) = a(t)\mathbf{r}(t_0) $$ (6) |
and the scaling quantity $a(t)$ is known as the scale factor. Because the universe is homogenous the scale factor is a function of time only. Differentiating this relationship with respect to time, $\dot{} \equiv (d/dt)$, gives |
$$ \dot{\mathbf{r}}(t) = \dot{a}(t)\mathbf{r}(t_0) $$ (7) |
and substituting in the original relationship we get |
$$ \dot{\mathbf{r}}(t) = \frac{\dot{a}(t)}{a(t)} \mathbf{r}(t) $$ (8) |
which is the Hubble law, Eq. (4), (since $\mathbf{v} \equiv \dot{\mathbf{r}}$) and the Hubble parameter is related to the scale factor by |
$$ H(t) = \frac{\dot{a}(t)}{a(t)} $$ (9) |
i.e. the Hubble parameter is the (relative) expansion rate of the Universe. |
1.1.3 Comoving coordinates |
In cosmology it’s common, and useful, to use comoving coordinates, $\mathbf{x}$, which are carried along with the expansion of the universe. They are related to physical coordinates, $\mathbf{r}$, by |
$$ \mathbf{r}(t) = a(t)\mathbf{x} $$ (10) |
It is conventional to define $a(t_0) = 1$ so that today physical and comoving coordinates coincide (and in the past $a < 1$). As comoving coordinates move with the expansion by definition, $\dot{\mathbf{x}} = 0$. |
Objects that are moving apart only due to the expansion of the universe remain at fixed locations in $\mathbf{x}$. Note that Hubble’s law isn’t exact (as the cosmological principle doesn’t hold perfectly) galaxies also have random motions, known as peculiar velocities, $\mathbf{v}_{\mathrm{pec}}$ so that |
$$ \mathbf{v} = H\mathbf{r} + \mathbf{v}_{\mathrm{pec}} $$ (11) |
1.1.4 Red-shift |
We have seen that Hubble’s law tells us that |
$$ v = Hr = \frac{\dot{a}}{a} r $$ (12) |
Consider 2 nearby points separated by $dr$. Their relative speed, $dv$, is given by |
$$ dv = \frac{\dot{a}}{a} dr $$ (13) |
The Doppler law tells us that the shift in the wavelength of radiation between emission at one point and observation at the other, $d\lambda = \lambda_r - \lambda_e$ is given by |
$$ \frac{d\lambda}{\lambda} = \frac{dv}{c} $$ (14) |
Combining this equation with Eq. (13), and writing the separation in terms of the light travel time $dt = dr/c$ we get |
$$ \frac{d\lambda}{\lambda} = \frac{\dot{a}}{a} \frac{dr}{c} = \frac{\dot{a}}{a} dt = \frac{da}{a} $$ (15) |
Integrating this gives $\ln \lambda = \ln a + \mathrm{const}$ and hence $\lambda \propto a$. |
This is telling us that as the universe expands the wavelength of radiation is effectively stretched. For radiation that is observed today $a_r = a(t_0) = 1$ and |
$$ 1 + z = \frac{\lambda_r}{\lambda_e} = \frac{1}{a_e} $$ (16) |
This derivation only holds for objects that are close together, however (as we’ll see in Section 2.2.2 ‘Observing the Universe’) the result is true in general. |
1.2 Newtonian cosmology |
To study the expansion of the universe properly we need General Relativity (GR). However the key equations can be derived using Newton’s laws and a few assumptions. See App. A and B (appendices are non-examinable). |
1.2.1 Friedmann equation |
The Friedmann equation |
$$ \left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} $$ (17) |
tells us how the evolution of the scale factor, $a(t)$, (i.e. the expansion of universe) depends on its density, $\rho(t)$. The constant $k$ arises from a full GR calculation. We’ll see in Sec. 1.4 that it tells us about the geometry of the universe, and is usually referred to as the curvature. Since the universe is ho... |
1.2.2 Fluid equation |
In order to use the Friedmann equation to calculate how the universe expands, i.e. $a(t)$, we first need to know the density $\rho(t)$ varies with time. The fluid equation |
$$ \dot{\rho} = -3\frac{\dot{a}}{a}\left(\rho + \frac{p}{c^2}\right) $$ (18) |
where $3\dot{a}/a = 3H$ tells us how the density of any fluid changes as the universe expands. The first term on the RHS ($3H\rho$) comes from the dilution due to the expansion of the universe. Since the universe has 3 spatial dimensions the volume, $V$, is proportional to $a^3$, $\dot{V}/V = 3\dot{a}/a$ and the change... |
$$ \frac{1}{V}\frac{d(\rho V)}{dt} = \frac{1}{V}(\dot{\rho}V + \rho \dot{V}) = \dot{\rho} + \rho \frac{\dot{V}}{V} = \dot{\rho} + 3\rho \frac{\dot{a}}{a} $$ (19) |
The 2nd term on the RHS ($3Hp/c^2$) comes from the loss of energy due to the pressure doing work as the volume increases. Note that energy is conserved; the energy that is lost from the fluid due to the work done goes into gravitational potential energy. |
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