The "Friedmann model" is a model of the Universe governed by the Friedmann equation, which describes how the Universe expands or contracts. Sometimes written as two independent equations, it's based on Einstein's general theory of relativity for gravity, and with two very important assumptions it forms the basis for our understanding of the evolution and structure of our Universe. These assumptions, together called "the cosmological principle", are that the Universe is homogeneous, and that it's isotropic. I'm not going to write the equation here, but they can be found e.g. here.
EDIT: In the bottom, I've added the equation and go a bit more in detail what they mean.
The cosmological principle
Homogeneity
That the Universe is homogeneous means that its "the same" everywhere. Obviously, it's not, really. For instance, we live on a dense, rocky planet, whereas just 100 km upward, the air is very thin. We also live in a galaxy, the Milky Way, while 10,000 lightyears from the Milky Way, space is extremely dilute. But on very large scales, say above half a billion lightyears, the Universe actually looks the same all over.
Isotropy
That it's isotropic, means that it looks the same in all directions. Again, obviously it doesn't on small scales, but on large scales, it does. If it didn't, it would mean that we occupied a special place in the Universe, and we don't think we do.
So, both of these assumption don't have to be true, but observations tell us that apparently, they are.
Three possible solutions
It turns out that, for these assumptions, there are three possible solutions to the Friedmann equation. We call the three possible universes flat, positively curved (or "closed"), and negatively curved (or "open"). Which of these possible universes we live in, turns out to depend upon the average density in the Universe, so by measuring this, we can determine the "geometry" of our own Universe. And it seems that it's "flat".
A flat universe
The reason it's called flat is that the geometry is like that of a flat 2D table, only in 3D. That is, a triangle has 180º, parallel lines never meet, etc. And it's infinitely large. Intuitively, we'd think that this is the way the Universe is, and definitely on small scales (say within our own Galaxy), it's an adequate approximation. Just like the Earth seems flat locally, and for all practical purposes the parking lot outside is flat.
A closed universe
If it were "closed", it's geometry would, in the 2D analogy, correspond to that of the surface of a ball, i.e. a triangle has more than 180º, lines that start out being parallel will at some point meet, etc. But like the surface area of a ball is finite (but doesn't have a border), so is the Universe.
An open universe
If it were "open", it's geometry would, in the 2D analogy, correspond to that of the surface of a saddle, i.e. a triangle has less than 180º, lines that start out being parallel will diverge, etc. And it's infinitely large.
This picture from here visualizes the 2D analogies. In 3D, only a flat geometry can be pictured.
Expansion of the Universe
The Friedmann equation, together with the densities of the constituents of the Universe (radiation, normal matter, dark matter, and dark energy) tell us how the Universe expands. So, again by measuring these densities, we can prediect the evolution of the Universe. And it seems that the Universe not only expands, but actually expands faster and faster.
Beyond the layman's explanation
Here, I'll expand a bit on how to understand the equation:
The Friedmann equations are intuitively most understandable, I think, when combined and written like this:
$$frac{H^2}{H_0^2} =
frac{Omega_mathrm{r,0}}{a^4} +
frac{Omega_mathrm{M,0}}{a^3} +
frac{Omega_k}{a^2} +
Omega_mathrm{Lambda}.$$
This equation tells us the connection between the expansion rate of the Universe (the left hand side), and the density of its components and its size (the right hand side). Below, I'll go through the components of the equation.
The Hubble parameter
In the equation, $H$ is the Hubble parameter describing how fast a galaxy at a given distance recedes (or approaches, for a collapsing universe), at a given time in the history of the Universe. $H_0$ is its value today, and is measured to be roughly $70,mathrm{km},mathrm{s}^{-1},mathrm{Mpc}^{-1}$. This means that a galaxy at a distance of, say, 70 Mpc (=230 lightyears), moves away from us at a current speed of 70 km/s.
Size
The size of the Universe is unknown, and probably infinite. Thus, we cannot talk about its absolute size. But we can talk about how much a certain volume of space expands in a given time. We us the parameter $a$, called the expansion factor. Defing $a$ to be 1 today, that means that at the time when the Universe was so small that all distances between the galaxies was, say, half of today's values, $a$ was equal to 0.5 (this happens to be 8 billion years ago).
Density parameters
Whether the geometry of the Universe as described above is flat, closed, or open, depends on whether the total density $rho_mathrm{tot}$ is exactly equal to, above, or below a certain critical threshold $rho_mathrm{cr} sim 10^{-29},mathrm{g},mathrm{cm}^{-3}$. It is customary to parametrize the density of the $i$'th component as $Omegaequiv rho_i / rho_mathrm{cr}$.
Matter
The term "matter" includes "normal" matter (gas, stars, planets, bicycles, etc.), and the mysterious dark matter. As the universe expands, the volume grows as $a^3$. That means that the density falls as $Omega_mathrm{M}=Omega_mathrm{M,0}/a^3$.
Radiation
Photons redshift as space expands, and this redshift goes as $1/a$. This is in addition to having their number density decrease, so the total energy density of radiation decreases faster than matter, namely as $1/a^4$. Today, the energy density of radiation is dominated by the CMB, and can be neglected, but in the early times, they would dominate.
Curvature
If space is not flat, its curvature contributes to $Omega_mathrm{tot}$. The reason is that the curvature affects the volume in which we measure densities (thanks to John Davis for this explanation). This scales as $1/a^2$.
Dark energy
Finally, there's the magical dark energy, of which even less is known than the dark matter. If it exists, it's thought to be a property of space itself, i.e. its energy density grows proportionally to the volume of the Universe, and thus there's no $a$-dependency.
Interpretation
From the equation, it is readily seen that if we're able to measure all the Omegas, then we know how fast the Universe has been expanding at all times. That means that we can integrate backwards in time and calculate when $a$ was 0, i.e. we can calculate the age of the Universe. Also, from the $a$-dependencies we can see when the Universe went from being radiation-dominated to being matter-dominated. We can also see that not only is it now dominated by dark energy (since $Omega_mathrm{Lambda}simeq0.7$, but $Omega_mathrm{M}simeq0.3$), but due to the $a$ factor, it will only get "worse". That is, all other densities will keep decreasing, but $rho_Lambda$ stays the same, and since dark energy has a repulsive effect rather than attracting, the expansion of the Universe accelerates.
Observationally, it is found (in several independent ways) that all the $Omega$s add up to one, i.e. that the total energy density of the Universe happens to be exactly equal to the critical density. This is pretty amazing. This figure, taken from here, shows the contribution from the different components now (top), and at the time of the CMB emission (380,000 yr after the Big Bang; bottom):
No comments:
Post a Comment