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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 Introd... |
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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. ... |
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 appr... |
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 redshif... |
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 f... |
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 t... |
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
$$ \mathbf{v} ... |
1.1.4 Red-shift
We have seen that Hubble’s law tells us that
$$ 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 $$ (13)
The Doppler law tells us that the shift in the wavelength of radiation between emission at one poi... |
1.2 Newtonian cosmology
To study the expansion of the universe properly we need General Relativity (GR). However the key equations can be derived using Newton’s laws and a few assumptions. See App. A and B (appendices are non-examinable). |
1.2.1 Friedmann equation
The Friedmann equation
$$ \left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} $$ (17)
tells us how the evolution of the scale factor, $a(t)$, (i.e. the expansion of universe) depends on its density, $\rho(t)$. The constant $k$ arises from a full GR calculation. We’ll see ... |
1.2.2 Fluid equation
In order to use the Friedmann equation to calculate how the universe expands, i.e. $a(t)$, we first need to know the density $\rho(t)$ varies with time. The fluid equation
$$ \dot{\rho} = -3\frac{\dot{a}}{a}\left(\rho + \frac{p}{c^2}\right) $$ (18)
where $3\dot{a}/a = 3H$ tells us how the density o... |
1.2.3 Acceleration equation
Differentiating the Friedmann equation, Eq. (17) with respect to time we get
$$ 2\frac{\dot{a}}{a}\frac{a\ddot{a} - \dot{a}^2}{a^2} = \frac{8\pi G}{3}\dot{\rho} + 2kc^2 \frac{\dot{a}}{a^3} $$ (20)
using the fluid equation, Eq. (18), to eliminate $\dot{\rho}$ and rearranging we get
$$ \frac{\... |
1.2.4 Natural units
Cosmologists often use natural units, where the speed of light is set to unity $c = 1$. Mass density $\rho$ and energy density $\epsilon = \rho c^2$ then become interchangeable as $\rho = \epsilon$ if $c = 1$.
In most (but not all...) of Liddle’s textbook he sets $c = 1$, however we won’t follow thi... |
1.3 Simple cosmological models |
1.3.1 What is the universe made of?
Special relativity tells us that the total energy of a particle is
$$ E_{\mathrm{tot}}^2 = m^2 c^4 + p_m^2 c^2 $$ (23)
where $m$ is the rest mass and $p_m$ the momentum. For non-relativistic particles $v \ll c$
$$ E_{\mathrm{tot}} = mc^2 \left(1 + \frac{p_m^2}{m^2 c^2}\right)^{1/2} \... |
1.3.2 Equation of state
In cosmology the equation of state is written as $p = w\rho c^2$ (in some other fields it’s written as $p = (\gamma - 1)\rho c^2$, where $\gamma - 1 \equiv w$) i.e.
$$ w = \frac{p}{\rho c^2} $$ (25)
The equations of state for the constituents of the universe we discussed above fall into 3 catego... |
1.3.3 Matter domination
Matter has $p = 0$ and the fluid equation becomes
$$ \dot{\rho} + 3\frac{\dot{a}}{a}\rho = 0 $$ (27)
This can be solved by ‘brute force’:
$$ \frac{\dot{\rho}}{\rho} = -3\frac{\dot{a}}{a}, \quad \int \frac{d\rho}{\rho} = -3 \int \frac{da}{a}, \quad \ln \rho = -3 \ln a + \mathrm{const}, \quad \rho... |
Assume for now zero curvature $k = 0$ (which, as we'll see in Sec. 1.4, corresponds to the geometry of the universe being flat) and matter dominates the universe. In Sec. 1.5 we'll look at solutions for $k \neq 0$ and multiple components (matter, radiation and a cosmological constant).
We can substitute $\rho(a)$ into ... |
1.3.4 Radiation domination
Radiation has $p = \rho c^2/3$ and the fluid equation becomes
$$ \dot{\rho} + 3\frac{\dot{a}}{a}\left(\rho + \frac{\rho c^2}{3c^2}\right) = 0, \quad \dot{\rho} + 4\frac{\dot{a}}{a}\rho = 0 $$ (35)
This can be rewritten as
$$ \dot{\rho} + 4\frac{\dot{a}}{a}\rho = \frac{1}{a^4}\frac{d}{dt}(\rho... |
1.3.5 Cosmological constant
The cosmological constant is the energy of the vacuum, and is denoted by $\Lambda$. The cosmological constant was originally introduced by Einstein to produce a static non-expanding universe, ($H(t) = 0$), something he later regarded as his ‘biggest blunder’. It, or something similar called ... |
1.3.6 Deceleration parameter
The deceleration parameter is defined as
$$ q = -\frac{a\ddot{a}}{\dot{a}^2} $$ (47)
It is positive if the universe is deccelerating ($\ddot{a} < 0$) and negative if the universe is accelerating ($\ddot{a} > 0$). Matter and radiation dominated universes have $q > 0$ while a cosmological con... |
1.4 Geometry of the universe
In Sec. 1.2.1 we introduced the curvature $k$ in terms of the (constant) energy of a particular particle in a Newtonian derivation of the Friedmann equation
$$ \left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3} $$ (48)
for the expansion of the... |
1.4.1 Flat universe
A flat universe has $k = 0$ and the familiar rules of Euclidean geometry apply:
* angles of a triangle add up to $180^\circ$, $\sum \theta_i = 180^\circ$,
* circumference of a circle is $c = 2\pi r$,
* parallel lines remain parallel.
A flat universe is infinite; if it had edges the cosmologica... |
1.4.2 Spherical geometry (closed)
It is difficult (impossible?) to visualize 3d geometry. Therefore let’s consider the surface of a 2d sphere which has positive curvature $k > 0$. The surface looks the same from all points and while it is finite, it has no edge. Therefore it is isotropic and homogeneous, as required by... |
1.4.3 Hyperbolic geometry (open)
A 2d surface with negative curvature, $k < 0$, looks like a saddle. A universe with hyperbolic geometry is infinite and open; if you travel in a straight line you will never return to your starting point.
With hyperbolic geometry
* angles of a triangle add up to less than $180^\circ$,... |
The 3 geometries are illustrated in Fig. 2 and their properties are summarised in Table 1. Note that whatever the geometry, the *observable* Universe is finite due to the finite speed of light. We’ll study this in detail in the ‘Observing the Universe’ section. |
Table 1: The properties of the 3 possible geometries for the universe.
| curvature | geometry | $\sum \theta_i$ | Infinite? |
| :--- | :--- | :--- | :--- |
| $k > 0$ | spherical, closed | $> 180^\circ$ | No |
| $k = 0$ | flat | $180^\circ$ | Yes |
| $k < 0$ | hyperbolic, open | $< 180^\circ$ | Yes | |
1.4.4 Critical density
We can see from the Friedmann equation, Eq. (90), that the universe can only be flat, $k = 0$, if the universe has a particular density, known as the **critical density**, $\rho_c$:
$$ \rho_c = \frac{3H^2}{8\pi G} $$ (49)
Note that since the Hubble parameter varies with time the critical density ... |
1.4.5 Density parameter
The **density parameter** is defined as
$$ \Omega = \frac{\rho}{\rho_c} $$ (52)
Note that since $\rho$ and $\rho_c$ are functions of time the density parameter usually is too (unless the time dependences cancel). This definition holds for each of the individual components of the universe (‘m’=ma... |
1.4.6 Conventions for $k$ and $a_0$
Either the scale factor today, $a_0$, or $|k|$ for open and closed universes, but not both, can be set to unity i.e. there are two different conventions:
* Open and closed universes have $k < 0$ and $k > 0$ respectively and $a_0 = 1$.
* Open and closed universes have $k = -1$ and... |
1.5 Evolution including curvature
Here are some equations/definitions that we’ve met before which we’ll be using in this section:
Friedmann equation:
$$ \left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3} $$ (57)
critical density:
$$ \rho_c = \frac{3H^2}{8\pi G} $$ (58)
co... |
1.5.1 Late time behaviour
‘Late time’ behaviour means ‘once $a$ has become sufficiently large that only the slowest varying of the terms on the right hand side of the Friedmann equation is non-negligible’. |
* $\Lambda \neq 0$
The cosmological constant term on the RHS of the Friedmann equation is constant, while the density and geometry terms decrease with increasing $a$. Therefore, if we wait long enough, eventually the cosmological constant term dominates and the universe expands exponentially, $a \propto \exp (\sq... |
* $\Lambda = 0, k = 0$
In a flat universe with $k = 0$, $\rho \to 0$ as $t \to \infty$ and therefore $H \to 0$ i.e. the expansion rate tends to zero. However this is not the case if $k \neq 0$.
Since the matter density decreases less rapidly than the radiation density ($\rho_m \propto a^{-3}$ while $\rho_r \p... |
* $\Lambda = 0, k < 0$
In an open universe with $k < 0$, $-k = |k|$ so that the Friedmann equation, Eq. (90), can be written as
$$ H^2 = \frac{8\pi G}{3}\rho + \frac{|k|c^2}{a^2} $$ (60)
Both terms on the RHS are positive, $H$ is never zero and the universe expands for ever. At late times the RHS will be ... |
* $\Lambda = 0, k > 0$
In a closed universe $k > 0$ and hence the curvature term on the RHS of the Friedmann equation is negative. Since the magnitude of the curvature term decreases less rapidly than the magnitude of the energy density term there is a time, when
$$ \frac{8\pi G}{3}\frac{\rho_{m,0}}{a^3} = \f... |
In the absence of a cosmological constant or dark energy, ‘geometry is destiny’ i.e. the geometry of the universe determines its long term fate. However if the cosmological constant is non-zero this relationship no longer holds and at late times the universe expands exponentially independent of its geometry (as we saw ... |
1.5.2 Matter domination with non-zero curvature
In this case, with $\rho = \rho_0 a^{-3}$, the Friedmann equation has a parametric solution (c.f. the equation of a circle with radius $r$, $x^2 + y^2 = r^2$ can be written in parametric form as $x = r \cos \theta, y = r \sin \theta$ with $0 < \theta < 2\pi$). |
$$ a(\theta) = \frac{4\pi G}{3kc^2}\rho_0(1 - \cos \theta) $$
$$ t(\theta) = \frac{4\pi G}{3k^{3/2}c^3}\rho_0(\theta - \sin \theta) $$ (63)
We can verify that this is a solution of the Friedmann equation, by substituting Eqs. (63) into the Friedmann equation. We find $\dot{a}$ by using
$$ \frac{da}{dt} = \frac{da}{d\th... |
For an open universe ($k < 0$) the parametric solutions are
$$ a(\psi) = \frac{4\pi G}{3|k|c^2}\rho_0(\cosh \psi - 1) $$
$$ t(\psi) = \frac{4\pi G}{3|k|^{3/2}c^3}\rho_0(\sinh \psi - \psi) $$ (72)
(see Problem Sheet 1). By considering the behaviour of these solutions for small and large $\psi$ respectively we can show t... |
1.5.3 General solutions
For a universe with non-zero curvature, composed of matter, radiation and a cosmological constant the Friedmann equation is
$$ H^2 = \frac{8\pi G}{3}(\rho_m + \rho_r) - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3} $$ (73)
We’ve already seen that solving this equation for a non-flat universe dominate... |
Fig. 6 shows the evolution of the scale factor with time for closed, flat and open universes with $\Lambda = 0$ and a $\Lambda \neq 0$ universe, which have the same present day expansion rate. If $\Lambda \neq 0$ at late times the cosmological constant dominates and the scale factor grows exponentially (as we saw in Se... |
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2 Observing the Universe
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2.1 Expansion rate and age |
2.1.1 Present day expansion rate |
The Hubble parameter is the (time dependent) constant of proportionality in the Hubble law:
$$ \mathbf{v} = H\mathbf{r} \, , $$ (1)
and its present day value (often known as the Hubble constant) is written as
$$ H_0 = 100h \, \mathrm{km \, s^{-1} \, Mpc^{-1}} \, , $$ (2)
with $h$ parametrizing the uncertainty in its va... |
In principle to measure $H_0$ we just need to measure the speeds of galaxies at different distances. In practise there are complications in measuring both the speeds and distances. As recently as the 1990s there was a factor of 2 uncertainty in the value of the Hubble constant (and the value measured by Hubble himself ... |
The velocity of a galaxy is actually the sum of its recession velocity and its peculiar velocity:
$$ \mathbf{v} = H_0\mathbf{r} + \mathbf{v}_{\mathrm{pec}} \, . $$ (3)
However if we observe galaxies at sufficiently large distances ($\gtrsim 10 \mathrm{Mpc}$) then we can ensure that $v_{\mathrm{pec}} \ll H_0 r$ (see pro... |
Distances are hard to measure. In astronomy the distances of nearby objects can be measured using parallax (the apparent movement of an object due to the Earth’s orbit), however for a galaxies at distances larger than 1 Mpc the parallax is smaller than a micro arcsecond, and hence far too small to be measured. Instead ... |
To measure absolute distances, and hence $H_0$, we need to know not just that the objects are standard candles, but what their absolute brightness is. This is done by using a chain of different standard candles to measure the distance of distant objects (this is known as the cosmic distance ladder). For instance parall... |
2.1.2 Rough estimate of age |
A very rough estimate of the age of the Universe, $t_0$, can be obtained from the Hubble law
$$ t_0 \sim \frac{r}{v} = \frac{1}{H_0} \, . $$ (4)
Converting the units into time gives
$$ t_0 \sim \frac{1}{100h \, \mathrm{km \, s^{-1} \, Mpc^{-1}}} = \frac{3.1 \times 10^{22}}{100h \times 10^3} \, \mathrm{s} = 3.1 \times 1... |
For most of its evolution the Universe is matter dominated, therefore we can obtain a better estimate for the age of a flat, $k = 0$, Universe by using the expression for the Hubble parameter which we derived in the ‘Expanding Universe’ section: $H = 2/(3t)$. This gives
$$ t_0 = \frac{2}{3}H_0^{-1} \approx 7 \, \mathrm... |
2.1.3 Observational limits on age |
The Universe must be older than the objects in it. This allows us to use measurements of the age of various objects to place lower limits on the age of the Universe: |
* From geological data the age of the Earth is of order 5 Gyr.
* Uranium isotopes produced in SNe decay at different rates. Their ratios can be used to measure the age of the Milky Way, $t_{\mathrm{MW}} \sim 7 \, \mathrm{Gyr}$.
* Studies of the cooling of old white dwarfs find $t_{\mathrm{WD}} \sim 10 \, \mathrm{... |
In each case we need to add the time after the Big Bang at which these objects form (roughly 1 Gyr for white dwarfs and globular clusters). |
The fact that the observational limits on the age of the Universe are roughly comparable with the estimates from the Big Bang model is reassuring (and to some extent a vindication of the Big Bang). However the theoretical estimate for a flat matter dominated Universe was somewhat smaller that the observational limits. ... |
2.1.4 Accurate calculation of age |
The exact value of the age of the Universe can be expressed as
$$ t_0 = \int_0^{t_0} dt = \int_0^{a_0=1} \frac{da}{\dot{a}} = \int_0^{1} \frac{da}{aH} \, . $$ (7)
Since redshift is defined as $1 + z = 1/a$ this can be rewritten, using
$$ \frac{da}{a} = -\frac{dz}{1 + z} \, , $$ (8)
as
$$ t_0 = -\int_{\infty}^{0} \frac{... |
2.2 Light travel and horizons |
In this part we’ll answer the question ‘How big is the Universe?’ (or, more accurately, ‘How far has light travelled since the Big Bang?’). To do this we’ll use some results from general relativity, and in the process derive the red-shifting of wavelength properly. |
2.2.1 Robertson-Walker metric |
In general relativity the metric is a fundamental quantity which describes the geometry of spacetime, giving the distance between neighbouring points. A proper derivation of this requires general relativity, so instead we’re just going to motivate its form in a more qualitative way. If you find this confusing, don’t wo... |
On a flat 2d surface the distance, $\Delta s$ between 2 points with coordinates $(X_{a1}, X_{b1})$ and $(X_{a2}, X_{b2})$ is given by
$$ \Delta s^2 = \Delta X_a^2 + \Delta X_b^2 \, , $$ (11)
where $\Delta X_a = X_{a2} - X_{a1}$ and $\Delta X_b = X_{b2} - X_{b1}$. This is just Pythagoras’ rule. In comoving coordinates, ... |
The cosmological principle states that the Universe has no preferred locations, which tells us that the spatial part of the metric, $ds_3^2$, has constant curvature. The most general spatial metric for which this is true is, in spherical polar coordinates,
$$ ds_3^2 = \frac{dr^2}{1 - kr^2} + r^2(d\theta^2 + \sin^2 \the... |
2.2.2 Redshift revisited |
We can use the Robertson-Walker metric to derive the redshift of radiation properly. Light obeys $ds = 0$ so therefore for a light ray which travels radially ($d\theta = d\phi = 0$)
$$ \frac{c \, dt}{a} = \frac{dr}{\sqrt{1 - kr^2}} \, . $$ (16)
Consider a light ray which travels radially from $r = 0$ at $t = t_e$ to $r... |
2.2.3 Cosmological horizon distance |
The **cosmological horizon distance** is the maximum distance light has travelled since the Big Bang (at $t = 0$). In comoving units it is given by, $r_H$ where
$$ \int_{0}^{r_H} \frac{dr}{\sqrt{1 - kr^2}} = \int_{0}^{t} \frac{c \, d\tilde{t}}{a(\tilde{t})} \, . $$ (24)
In a flat Universe, $k = 0$, this simplifies to
$... |
2.2.4 Cosmological event horizon |
The **cosmological event horizon**, $r_{\mathrm{ev}}$, is the radius within which signals emitted at time $t$ can be observed by time $t_{\mathrm{max}}$. The comoving cosmological event horizon is given by
$$ \int_{0}^{r_{\mathrm{ev}}} \frac{dr}{\sqrt{1 - kr^2}} = \int_{t}^{t_{\mathrm{max}}} \frac{c \, d\tilde{t}}{a(\t... |
2.3 Distances |
In this section we’ll see how redshift affects how the properties of objects, such as luminosity and diameter, appear to us. |
2.3.1 Proper distance |
The **proper distance** is the length of the spatial geodesic at some time $t$ i.e $d_p = \int ds$. The proper distance to a galaxy with comoving coordinates $(r_0, 0, 0)$ is given by
$$ d_p(t) = a(t) \int ds = a(t) \int_{0}^{r_0} \frac{dr}{\sqrt{1 - kr^2}} \, . $$ (31)
We’re usually interested in the proper distance t... |
2.3.2 Luminosity distance |
The **luminosity distance**, $d_{\mathrm{lum}}$, is defined as the distance an object appears to have, assuming the inverse square law holds. This is not the actual distance as the Universe is expanding and the geometry is not necessarily flat.
If a flux of photons, $f$, is measured from an object with luminosity, $L$,... |
2.3.3 Angular diameter distance |
The **angular diameter distance**, $d_{\mathrm{diam}}$, is defined as the distance an object of known physical extent (i.e. a ‘standard ruler’) appears to be at assuming Euclidean geometry. In other words it is a measure of how large objects appear.
If an object with physical extent $l$ is perpendicular of the line of ... |
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3 Thermal history
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3.1 Introduction |
3.1.1 Black-body spectrum |
If particles interact frequently then their energy distribution is given by equilibrium thermodynamics. In a thermal distribution the interaction rates in the forward and backwards directions are the same, so that $N(E)$, the number of particles with a given energy $E$, depends only on the temperature $T$. Photons are ... |
The **total photon energy density**, $\epsilon_\gamma$, can be found by integrating $\epsilon(f)$:
$$ \epsilon_\gamma = \int_{0}^{\infty} \epsilon(f) df = \frac{8\pi k_B^4 T^4}{h^3 c^3} \int_{0}^{\infty} \frac{y^3 dy}{\exp y - 1} \, , $$ (2)
where $y = hf/(k_B T)$. It can be shown that
$$ \int_{0}^{\infty} \frac{y^3 dy... |
3.1.2 Photon to baryon ratio |
Today the photon distribution has $T_0 = 2.725 \, \mathrm{K}$ (see the next section on the Cosmic Microwave Background radiation), so that its energy density is
$$ \epsilon_{\gamma,0} = \alpha T_0^4 = 7.57 \times 10^{-16} \times (2.725)^4 = 4.2 \times 10^{-14} \, \mathrm{J \, m^{-3}} \, . $$ (9)
The mass density, $\rho... |
3.1.3 Matter-radiation equality |
In this section we’ll calculate the time of matter-radiation equality, when the densities of matter and radiation are equal. Radiation is composed of neutrinos as well as photons, $\rho_r = \rho_\gamma + \rho_\nu$, so we need to take into account the neutrino density. If we assume that there are 3 light species of neut... |
3.1.4 Temperature-time relation |
Later in this section we’ll need to know how time and temperature are related to each other in a radiation-dominated universe. We can work this out by using the evolution of the scale factor. During radiation domination $a \propto t^{1/2}$ so that $T \propto t^{-1/2}$ and hence
$$ \left(\frac{t_{\mathrm{eq}}}{t}\right)... |
3.2 Nucleosynthesis |
3.2.1 Introduction |
Nucleosynthesis is the synthesis of the nuclei of the light elements (Deuterium, D, Helium-3, ${}^3\mathrm{He}$, Helium-4, ${}^4\mathrm{He}$, and Lithium-7, ${}^7\mathrm{Li}$) when the Universe was seconds to minutes old. (Heavier elements are created in stars.)
After the quark-hadron phase transition (when quarks form... |
3.2.2 Simplified calculation |
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