Dark matter was finished up in the last couple of posts, so now I'm going to move on to another topic people are frequently asking about: dark energy. This is a little harder to talk about at the very basic level these posts have tried to maintain, but I will continue trying to do this.
In the early posts, we discussed the Friedmann equations and deriving them from Einstein's equations. The base Friedmann equations model the expansion of space under the assumption of homogeneity and isotropy in the universe. We need to understand the evolution of the energy density in the universe, so we use another equation called the Fluid Equation. Lastly, we need to understand how the scale factor might accelerate as it changes, so we use another equation called the Acceleration Equation.
Only two of these are independent, and we have three unknowns (the scale factor, the energy density, and the pressure): we need to be able to relate the energy density and the pressure to be able to solve problems and answer questions with these equations.
So we are arriving to a concept called the equation of state, which is something that will connect a couple of these concepts: in this case, we need an equation of state that will relate cosmological energy density to pressure.
I will post the equations here, but don't expect to understand them just from looking at them (especially devoid of any derivation context); I mainly want you to see that they contain many of the same terms:
[GALLERY=media, 9516]Eqns by Meow Mix posted Jun 30, 2021 at 2:17 AM[/GALLERY]
We need something that connects epsilon (the energy density) and P (pressure): this would be called an equation of state.
Since no equation of state equation accurately predicts the properties of all things under all conditions, we need to introduce an additional term that we'll call the equation of state parameter. So, for instance, we'll have an equation of state for everything that looks like this:
[GALLERY=media, 9517]Eoseqn by Meow Mix posted Jun 30, 2021 at 2:21 AM[/GALLERY]
Where the lowercase omega (the w) is our equation of state parameter: this thing will be different depending on whether we're talking about matter (w = 0 for non-relativistic matter, this is an approximation), radiation (w = 1/3 for photons, calculated from statistical mechanics), and from the acceleration equation* we get that for an accelerating universe, dark energy would have to have an equation of state parameter that's less than -1/3.
(* -- you can do this yourself by taking the acceleration equation and plugging in the equation of state; you will see that in the parentheses, a critical value for w is -1/3 such that if w is more negative than that, the acceleration will be non-zero and non-negative!)
So, even if you didn't quite follow everything above, an important thing is that an equation of state parameter that's less than (more negative than) -1/3 describes a universe that is accelerating its expansion.
What are some examples of things that might have an equation of state parameter like this?
The first example of a negative pressure like this that causes the universe to "push outwards" is Einstein's cosmological constant, lambda. This would be a constant equation of state parameter w = -1 (for calculus folks, it's actually an integration constant set to lambda rather than zero, so it is a true constant).
Einstein's reasoning for this at the time was that the universe appeared to be static to him (he didn't know at the time it was expanding). But there's a problem: all the mass in the universe should have been causing it to be contracting in on itself (or with the caveat that the universe might be infinite with an infinite amount of balanced matter, any small perturbation would cause it to fall apart like a house of cards). So, Einstein introduced this lambda term to counterbalance the "damage" that he presumed gravity may have done to a static universe model.
He famously considered this one of his greatest blunders: it was a really ad hoc decision, and it of course turned out the universe isn't static, so it was introduced for the wrong reasons.
So, Einstein's cosmological constant was forgotten for a long time in physics. I've had instructors tell me that before 1998, they were always told in astrophysics and cosmology classes, "by the way, there might be this thing, but we're pretty sure lambda = 0," so they'd have to learn cosmological equations including lambda terms for about a week before forgetting them.
However, then a thing happened in 1998: the discovery of evidence that the universe was accelerating as it expanded.
Scientists in 1998, being the mindful folk that they are, were plotting very distant (high redshift) supernovae against different possible universe models, including some models that included lambda being nonzero, and to their utter astonishment, it looked like the data best fit models with nonzero lambda.
[GALLERY=media, 9518]1998supernovae by Meow Mix posted Jun 30, 2021 at 2:45 AM[/GALLERY]
(How to read this plot: the x axis is redshift, which cosmologists call "z," so remember higher numbers of redshift = further out in the universe, further back in time as we are looking at it because light takes time to reach us. The y axis is the magnitude [brightness] of the supernovae being plotted. Assume for now that we can align these axes if we know how far a special type of supernovae is by knowing how bright it is at its peak. The different dotted lines represent different models of the universe with different possible density parameters. The data falls along the lines of the model with a nonzero lambda term, not on the models that ignore lambda!)
Coming up in Post 7: the deceleration parameter (hint: they named it this before they suspected it would turn out to be negative, and thus become an acceleration parameter in all but name!), Type 1a supernovae, and the cosmic history of density parameters and their relation to the acceleration and fate of the universe. (All of this means: more dark energy post for Post 7).
In the early posts, we discussed the Friedmann equations and deriving them from Einstein's equations. The base Friedmann equations model the expansion of space under the assumption of homogeneity and isotropy in the universe. We need to understand the evolution of the energy density in the universe, so we use another equation called the Fluid Equation. Lastly, we need to understand how the scale factor might accelerate as it changes, so we use another equation called the Acceleration Equation.
Only two of these are independent, and we have three unknowns (the scale factor, the energy density, and the pressure): we need to be able to relate the energy density and the pressure to be able to solve problems and answer questions with these equations.
So we are arriving to a concept called the equation of state, which is something that will connect a couple of these concepts: in this case, we need an equation of state that will relate cosmological energy density to pressure.
I will post the equations here, but don't expect to understand them just from looking at them (especially devoid of any derivation context); I mainly want you to see that they contain many of the same terms:
[GALLERY=media, 9516]Eqns by Meow Mix posted Jun 30, 2021 at 2:17 AM[/GALLERY]
We need something that connects epsilon (the energy density) and P (pressure): this would be called an equation of state.
Since no equation of state equation accurately predicts the properties of all things under all conditions, we need to introduce an additional term that we'll call the equation of state parameter. So, for instance, we'll have an equation of state for everything that looks like this:
[GALLERY=media, 9517]Eoseqn by Meow Mix posted Jun 30, 2021 at 2:21 AM[/GALLERY]
Where the lowercase omega (the w) is our equation of state parameter: this thing will be different depending on whether we're talking about matter (w = 0 for non-relativistic matter, this is an approximation), radiation (w = 1/3 for photons, calculated from statistical mechanics), and from the acceleration equation* we get that for an accelerating universe, dark energy would have to have an equation of state parameter that's less than -1/3.
(* -- you can do this yourself by taking the acceleration equation and plugging in the equation of state; you will see that in the parentheses, a critical value for w is -1/3 such that if w is more negative than that, the acceleration will be non-zero and non-negative!)
So, even if you didn't quite follow everything above, an important thing is that an equation of state parameter that's less than (more negative than) -1/3 describes a universe that is accelerating its expansion.
What are some examples of things that might have an equation of state parameter like this?
The first example of a negative pressure like this that causes the universe to "push outwards" is Einstein's cosmological constant, lambda. This would be a constant equation of state parameter w = -1 (for calculus folks, it's actually an integration constant set to lambda rather than zero, so it is a true constant).
Einstein's reasoning for this at the time was that the universe appeared to be static to him (he didn't know at the time it was expanding). But there's a problem: all the mass in the universe should have been causing it to be contracting in on itself (or with the caveat that the universe might be infinite with an infinite amount of balanced matter, any small perturbation would cause it to fall apart like a house of cards). So, Einstein introduced this lambda term to counterbalance the "damage" that he presumed gravity may have done to a static universe model.
He famously considered this one of his greatest blunders: it was a really ad hoc decision, and it of course turned out the universe isn't static, so it was introduced for the wrong reasons.
So, Einstein's cosmological constant was forgotten for a long time in physics. I've had instructors tell me that before 1998, they were always told in astrophysics and cosmology classes, "by the way, there might be this thing, but we're pretty sure lambda = 0," so they'd have to learn cosmological equations including lambda terms for about a week before forgetting them.
However, then a thing happened in 1998: the discovery of evidence that the universe was accelerating as it expanded.
Scientists in 1998, being the mindful folk that they are, were plotting very distant (high redshift) supernovae against different possible universe models, including some models that included lambda being nonzero, and to their utter astonishment, it looked like the data best fit models with nonzero lambda.
[GALLERY=media, 9518]1998supernovae by Meow Mix posted Jun 30, 2021 at 2:45 AM[/GALLERY]
(How to read this plot: the x axis is redshift, which cosmologists call "z," so remember higher numbers of redshift = further out in the universe, further back in time as we are looking at it because light takes time to reach us. The y axis is the magnitude [brightness] of the supernovae being plotted. Assume for now that we can align these axes if we know how far a special type of supernovae is by knowing how bright it is at its peak. The different dotted lines represent different models of the universe with different possible density parameters. The data falls along the lines of the model with a nonzero lambda term, not on the models that ignore lambda!)
Coming up in Post 7: the deceleration parameter (hint: they named it this before they suspected it would turn out to be negative, and thus become an acceleration parameter in all but name!), Type 1a supernovae, and the cosmic history of density parameters and their relation to the acceleration and fate of the universe. (All of this means: more dark energy post for Post 7).
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