what is a Friedmann model?

  • Can you explain what is a Friedmann model to a layman?
    And also give some examples of Friedmann models, specially I would like to know if the lambda-CDM model is considered a Friedmann model.

  • pela

    pela Correct answer

    7 years ago

    The "Friedmann model" is a model of the Universe governed by the Friedmann equations, which describes how the Universe expands or contracts. These equations are a solution to Einstein's field equations, and with two very important assumptions they form 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. This is a layman's explanation, so I won't write the equations here, but farther down I've added the equation and go a bit more in detail what it means.

    The cosmological principle


    That the Universe is homogeneous means that its "the same" everywhere. Obviously, it's not, really. For instance, under your feet is a dense, rocky planet, whereas above your head there's thin air. We live in a galaxy full of stars and molecular clouds and what not, while 100,000 lightyears from the Milky Way, there is virtually nothing. But on very large scales, say above half a billion lightyears, the Universe actually looks the same all over.


    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, neither of these assumptions have to be true, but observations tell us that apparently, they are. See this image, where every dot is a galaxy (Maddox et al. 1990):


    You might think that a homogeneous Universe would also be isotropic and/or vice versa, but that's not the case.

    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. In the 2D analogy, the 2D surface of Earth seems flat locally, and for all practical purposes the parking lot outside is flat. But if you draw a triangle from Congo→Indonesia→the North Pole→Congo, you'd measure it's sum of angles to be roughly 270º. That's because the geometry of the surface of Earth is not flat, but is "closed".

    A closed universe

    If the Universe 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. So if you take you spaceship and fly straight away from Earth, you would end up back here (assuming the Universe doesn't collapse before you get back, or expand too fast for you).

    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, and this doesn't look "flat" by any means; it is simply your good old 3D ("Euclidian") space that you know from your everyday senses.

    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 first Friedmann equation is intuitively most understandable, I think, when 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} +
    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, 10 Mpc ($\simeq33$ lightyears), moves away from us at a current speed of 700 km/s. A galaxy 20 Mpc away recedes at 1400 km/s, and so on (and galaxies farther away than roughly 4.3 Gpc recede faster than the speed of light, but that's no problem and we can still see them).


    The size of the Universe is unknown, and quite possibly infinite (lest it wouldn't be homogeneous, but to be fair, we only know that the observable Universe is homogeneous). 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. Defining $a$ 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 $\Omega\equiv \rho_i / \rho_\mathrm{cr}$.


    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 $\Omega_\mathrm{M}=\Omega_\mathrm{M,0}/a^3$.


    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.


    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.


    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):


    Great answer, +1. Do you want to add in the equations, for convenience?

    Can a Friedmann model contain dark energy?

    @Mick: Yes, definitely. Dark energy, as well as everything else in the Universe contributing to the total energy density, is a part of the equations. When Friedmann formulate the equations, dark energy was unknown, so the expansion would always decelerate. Without dark energy, a closed universe will at some point decelerate enough to start collapsing. With, however, if the grows above a certain limit, dark energy will dominate and accelerate the expansion.

    You asked for a layman's explanation, so I didn't include the relevant equations, but since you ask, and by @HDE226868 's request, I'll add them.

    And yes, the ΛCDM Universe is a Friedmann model.

    Re $\Omega_k$, it must be taken into account as when space is curved (i.e. not flat), the proper volume of a sphere of fixed proper radius will change with expansion, which in turn will affect how the density of the Universe changes. For example in a Universe with spherical geometry the volume of the fixed proper radius sphere will decrease as the Universe expands, which in turn means its density of matter and radiation will decrease more slowly than in the flat case as the scale factor increases.

    Thanks, @JohnDavis, that makes sense. My comment about surface tension doesn't, but it makes sense that it goes as 1/a², since it affects "one less dimension than the 3D of space". So that in your example with the sphere analogy, it would go as 1/a.

    I recently watched a series of lectures given at MIT by Prof. Alan Guth, which may be of interest, https://www.youtube.com/playlist?list=PLUl4u3cNGP61Bf9I0WDDriuDqEnywoxra . I'm not a physicist, but I found them very useful.

    Very nice to have so many things tied together (even loosely) in one place. This kind of answer can be really helpful when noobs like me are trying to get "the big picture."

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Content dated before 7/24/2021 11:53 AM

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