If we take the general relativity field equations, include homogeneity and isotropy (generally in this case expressed using the Robertson Walker metric, see https://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–Walker_metric), we derive the basis for the Friedmann equations.
(For non-math folks, I'm trying to write this in such a way that non-math folks will be able to understand most of it, so don't worry. Also, I'm skipping over difficult stuff and only including really simple stuff).
[GALLERY=media, 9493]Friedmann by Meow Mix posted Jun 23, 2021 at 11:15 PM[/GALLERY]
What I want you to take away from this:
That means that we can just do a little simple algebra to find a critical energy density at which we get a universe that looks like the universe that we see (where k = 0).
[GALLERY=media, 9494]Critical by Meow Mix posted Jun 23, 2021 at 11:28 PM[/GALLERY]
You can easily do this yourself: going back to the Friedmann equation, you can set k = 0 to get the tail right side of the equation to drop out, then algebraically isolate the energy density with some simple division/multiplying (and remember that the a(dot)/a = H).
So, in order to get a universe that looks like the universe we inhabit, we can know that if we total everything with energy up in the universe, we will have this energy density in total! (It's about 4870 MeV^-3, which is close to about one Milky Way galaxy per Mpc^-3).
Now, we can go further to being able to talk about dark matter and dark energy: we can define a density parameter against the critical density.
[GALLERY=media, 9495]Densparam by Meow Mix posted Jun 23, 2021 at 11:34 PM[/GALLERY]
This is defining an omega as the energy density of that particular thing over the critical density of the entire universe. That means everything should add up to 1 (if we add the density parameter for baryonic matter, and we add the density parameter for dark matter, and we add the density parameter for radiation, and so on and so forth, if we add everything, it will total up to 1 because we set it up that way). What this means is that if I say something like "the density parameter of baryonic matter is about 0.04," then what you should get out of this statement is that baryonic matter makes up a laughably tiny fraction of the energy density of the universe. (Baryonic matter being the "regular" matter you see and interact with around you: protons, neutrons and so forth).
So what parameters might we consider? There's a lot that can be said here, but again in the interest of brevity (ask questions in comments if you'd like), I'll tell you now that the parameters are radiation, matter, and dark energy. A very good model of the universe, called the Benchmark model, finds these parameters (and we'll get into how during posts specifically about these parameters):
[GALLERY=media, 9496]Benchmarkparam by Meow Mix posted Jun 23, 2021 at 11:44 PM[/GALLERY]
(The "r" is for radiation, the "m" is for matter, and the lambda is for dark energy here).
Notice that these parameters add up to 1, as they should (because each of them is defined against the critical density at which the universe is flat; and the universe is flat). The zeroes represent that these are their values at the present time: the parameters are different at different times because the scale factor (the size) of the universe is different at different times (and remember, we're ultimately talking about energy densities, so it makes sense that if the size of the container changes, the density in that container will change if you think about it).
Right now, the lambda term dominates the universe (and again, in a future post I will be going over how we actually get these values, so just take them on my word for right now). However, that may not have been the case in the past. What I want you to understand at this point is that these values change as the scale factor a changes; and that they change differently. Radiation scales per the scale factor as a^-4, matter scales per the scale factor as a^-3, and lambda does not scale as the universe changes size.
So as the scale factor a increases, matter's density parameter decreases proportionally to a^-3. If we move back in time, this means that matter's density parameter gets larger the further back we look, because the scale factor of the universe gets smaller (as we look back, the universe was smaller than it is today because it's expanding!)
So we can calculate questions for instance like "what size was the universe when the density parameter for matter was equal to the density parameter of lambda," and since there is a relationship between redshift and scale factor, that means we could calculate at what redshift out we have to look in order to see what the universe looked like under those circumstances! Remember, looking out is looking back because it takes time for light to reach us; so if we look at a certain redshift out, we are literally seeing the universe as it was at a given time, not as it would appear today. (For the curious, the scale factor at matter/dark energy equality was about 0.77, which corresponds to a redshift of about 0.31).
What I'm trying to relay during this portion is to reiterate that the universe has not been the same over time. There was initially an epoch where radiation energy density dominated the universe; then as the universe expanded (and cooled, because remember that temperature is proportional to a^-1), radiation's energy density dropped off faster than matter's energy density dropped off: so radiation went from being dominant over matter to being equal with matter. Then as time went on, radiation continued to drop off faster than matter did, so matter became the dominant energy density of the universe!
[GALLERY=media, 9483]Scale by Meow Mix posted Jun 22, 2021 at 9:43 PM[/GALLERY]
(Forgive the sudden change in image format, this is from a slide I made).
The above is a picture of the energy densities through the history of the universe, with today being the vertical line; and we have the Friedmann equation in terms of density parameters on the left side.
The thing to take away from this is again to note how the density parameters change with time (how they scale as the scale factor a changes). Right now, radiation is negligible in the big picture of the universe because it scales away so quickly (as a increases, it scales away as a^-4!). Matter's energy density diffuses nearly just as fast (as a^-3). We live in a time where matter is quickly becoming irrelevant and dark energy is beginning to dominate completely!
All of this should give us a good foundation to be able to talk about how we know what we know about dark matter and dark energy, now that I can use terms like "density parameter" and "scale factor" and so on.
Next post will be about different possible universes and how we know we live in a universe like the one I've described (with the density parameters of the Benchmark model), and in that same post I can most likely also cover dark matter. Dark energy will be its own post after that.
(For non-math folks, I'm trying to write this in such a way that non-math folks will be able to understand most of it, so don't worry. Also, I'm skipping over difficult stuff and only including really simple stuff).
[GALLERY=media, 9493]Friedmann by Meow Mix posted Jun 23, 2021 at 11:15 PM[/GALLERY]
What I want you to take away from this:
- On the left-hand side is simply the Hubble parameter squared. H = (da/dt) / a, and the dot over the a is shorthand for a time derivative (so, da/dt). For non-math people, this means it quantifies how a changes with time. As a reminder, a is the scale factor, or the size of the universe, as compared to the size of the universe today.
- The epsilon is the energy density of stuff in the universe: to understand what an energy density is, imagine some unit of space and how much energy is within that unit of space; that'd be an energy density. Remember that at above 100 Mpc (megaparsecs), the universe is homogeneous and isotropic, so finding the energy density at those scales is finding the energy density for the entire universe.
- The k term is for the curvature of the universe. Without getting into general relativity (GR) too much, the universe can have different kinds of curvature: it can be flat (k = 0), or it can have positive or negative curvature (k = 1 or k = -1). I would be here a while if I had to explain the geometry of space, so ask questions in the comments if you need to, but take my word for now that observations tell us that space is flat (k = 0).
That means that we can just do a little simple algebra to find a critical energy density at which we get a universe that looks like the universe that we see (where k = 0).
[GALLERY=media, 9494]Critical by Meow Mix posted Jun 23, 2021 at 11:28 PM[/GALLERY]
You can easily do this yourself: going back to the Friedmann equation, you can set k = 0 to get the tail right side of the equation to drop out, then algebraically isolate the energy density with some simple division/multiplying (and remember that the a(dot)/a = H).
So, in order to get a universe that looks like the universe we inhabit, we can know that if we total everything with energy up in the universe, we will have this energy density in total! (It's about 4870 MeV^-3, which is close to about one Milky Way galaxy per Mpc^-3).
Now, we can go further to being able to talk about dark matter and dark energy: we can define a density parameter against the critical density.
[GALLERY=media, 9495]Densparam by Meow Mix posted Jun 23, 2021 at 11:34 PM[/GALLERY]
This is defining an omega as the energy density of that particular thing over the critical density of the entire universe. That means everything should add up to 1 (if we add the density parameter for baryonic matter, and we add the density parameter for dark matter, and we add the density parameter for radiation, and so on and so forth, if we add everything, it will total up to 1 because we set it up that way). What this means is that if I say something like "the density parameter of baryonic matter is about 0.04," then what you should get out of this statement is that baryonic matter makes up a laughably tiny fraction of the energy density of the universe. (Baryonic matter being the "regular" matter you see and interact with around you: protons, neutrons and so forth).
So what parameters might we consider? There's a lot that can be said here, but again in the interest of brevity (ask questions in comments if you'd like), I'll tell you now that the parameters are radiation, matter, and dark energy. A very good model of the universe, called the Benchmark model, finds these parameters (and we'll get into how during posts specifically about these parameters):
[GALLERY=media, 9496]Benchmarkparam by Meow Mix posted Jun 23, 2021 at 11:44 PM[/GALLERY]
(The "r" is for radiation, the "m" is for matter, and the lambda is for dark energy here).
Notice that these parameters add up to 1, as they should (because each of them is defined against the critical density at which the universe is flat; and the universe is flat). The zeroes represent that these are their values at the present time: the parameters are different at different times because the scale factor (the size) of the universe is different at different times (and remember, we're ultimately talking about energy densities, so it makes sense that if the size of the container changes, the density in that container will change if you think about it).
Right now, the lambda term dominates the universe (and again, in a future post I will be going over how we actually get these values, so just take them on my word for right now). However, that may not have been the case in the past. What I want you to understand at this point is that these values change as the scale factor a changes; and that they change differently. Radiation scales per the scale factor as a^-4, matter scales per the scale factor as a^-3, and lambda does not scale as the universe changes size.
So as the scale factor a increases, matter's density parameter decreases proportionally to a^-3. If we move back in time, this means that matter's density parameter gets larger the further back we look, because the scale factor of the universe gets smaller (as we look back, the universe was smaller than it is today because it's expanding!)
So we can calculate questions for instance like "what size was the universe when the density parameter for matter was equal to the density parameter of lambda," and since there is a relationship between redshift and scale factor, that means we could calculate at what redshift out we have to look in order to see what the universe looked like under those circumstances! Remember, looking out is looking back because it takes time for light to reach us; so if we look at a certain redshift out, we are literally seeing the universe as it was at a given time, not as it would appear today. (For the curious, the scale factor at matter/dark energy equality was about 0.77, which corresponds to a redshift of about 0.31).
What I'm trying to relay during this portion is to reiterate that the universe has not been the same over time. There was initially an epoch where radiation energy density dominated the universe; then as the universe expanded (and cooled, because remember that temperature is proportional to a^-1), radiation's energy density dropped off faster than matter's energy density dropped off: so radiation went from being dominant over matter to being equal with matter. Then as time went on, radiation continued to drop off faster than matter did, so matter became the dominant energy density of the universe!
[GALLERY=media, 9483]Scale by Meow Mix posted Jun 22, 2021 at 9:43 PM[/GALLERY]
(Forgive the sudden change in image format, this is from a slide I made).
The above is a picture of the energy densities through the history of the universe, with today being the vertical line; and we have the Friedmann equation in terms of density parameters on the left side.
The thing to take away from this is again to note how the density parameters change with time (how they scale as the scale factor a changes). Right now, radiation is negligible in the big picture of the universe because it scales away so quickly (as a increases, it scales away as a^-4!). Matter's energy density diffuses nearly just as fast (as a^-3). We live in a time where matter is quickly becoming irrelevant and dark energy is beginning to dominate completely!
All of this should give us a good foundation to be able to talk about how we know what we know about dark matter and dark energy, now that I can use terms like "density parameter" and "scale factor" and so on.
Next post will be about different possible universes and how we know we live in a universe like the one I've described (with the density parameters of the Benchmark model), and in that same post I can most likely also cover dark matter. Dark energy will be its own post after that.
