# Dark Energy contribution to plasma temperature in galaxy clusters?

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• Suekdccia
Suekdccia
TL;DR Summary
Dark Energy contribution to isothermal temperature of plasma in clusters of galaxies?
I have a question about this work called "Dark energy and key physical parameters of clusters of galaxies"*There, towards the end, the authors talk about the isothermal velocities and tempreature parameters of the gas and particles circulating between galaxies in clusters. In particular they calculate the isothermal plasma temperature (equation 37)I can note there is a contribution from a dark energy parameter in that equation (the cube root of the dark energy density value is present in the equation, which should give a large number as the dark energy density in space is small). I tried to ask the authors themselves, but they only told me that "*Temperature is defined from virial relation. It contains Dark Energy due to its antigravity*"Does it mean that dark energy contributes to the value of that temperature? Does dark energy help to increase the value of that temperature?* https://arxiv.org/abs/1206.1433

Suekdccia said:
Does it mean that dark energy contributes to the value of that temperature?
Do you understand what the authors meant by "temperature is defined from virial relation"?

PeterDonis said:
Do you understand what the authors meant by "temperature is defined from virial relation"?
That the temperature is given by the virial theorem. If so, particles would reach a maximum turn-around radius and then fall into the overdensity and the potential energy would be transofrmed into kinetic energy. But how can dark energy contribute to this?

Suekdccia said:
That the temperature is given by the virial theorem.
Yes. And where does the virial theorem come from? How is it derived?

Suekdccia said:
If so, particles would reach a maximum turn-around radius and then fall into the overdensity and the potential energy would be transofrmed into kinetic energy.
No such thing has to happen for the virial theorem to apply. The virial theorem relates the time averages of potential energy and kinetic energy (and the latter is in turn directly related to temperature).

Suekdccia said:
how can dark energy contribute to this?
By modifying the potential energy. Looking at how the virial theorem is derived should make this evident.

PeterDonis said:
Yes. And where does the virial theorem come from? How is it derived?No such thing has to happen for the virial theorem to apply. The virial theorem relates the time averages of potential energy and kinetic energy (and the latter is in turn directly related to temperature).By modifying the potential energy. Looking at how the virial theorem is derived should make this evident.
Mmmh that is what I was thinking, if dark energy modifies the potential energy (presumably making it larger, as particles would be less bound and could reach a higher distance) then kinetic energy should be greater as well and thus temperature increases.But the authors themselves say in this paper (https://arxiv.org/abs/1109.1215, sections 3 & 4) that the potential energy (and therefore the kinetic energy) is decreased in the presence of dark energy (because when things fall into the overdensity, dark energy affects the gravitational pull, thus reducing the energy) so according to this, temperature should be lower (as kinetic energy is also reduced because there is less potential energy). This is what I don't really understand

Suekdccia said:
if dark energy modifies the potential energy (presumably making it larger, as particles would be less bound and could reach a higher distance)
"Larger" in the sense of "less negative", yes. But the virial theorem says that the time average of the kinetic energy is (half of) minus the time average of the potential energy. So a larger, i.e., less negative, potential energy means a smaller kinetic energy because of the minus sign.

PeterDonis said:
"Larger" in the sense of "less negative", yes. But the virial theorem says that the time average of the kinetic energy is (half of) minus the time average of the potential energy. So a larger, i.e., less negative, potential energy means a smaller kinetic energy because of the minus sign.
But then here what I understand is that dark energy ultimately reduces the kinetic energy, which should then reduce the temperature, not increase it (dark energy density seems to contribute to the temperature of the gas in the paper that I cited in my original question, so a larger dark energy density should mean that the isothermal teperature is larger as well)

If you look at eq. 37 (https://arxiv.org/abs/1206.1433) and you assume a larger value for dark energy density then the temperature increases not decreases

Last edited:
Suekdccia said:
what I understand is that dark energy ultimately reduces the kinetic energy, which should then reduce the temperature, not increase it
It reduces the temperature compared to a system that is otherwise the same, but without the dark energy. But that is not what is being done in the section of the paper where eq. 37 appears.

Suekdccia said:
If you look at eq. 37 (https://arxiv.org/abs/1206.1433) and you assume a larger value for dark energy density then the temperature increases not decreases
Yes, but this equation is not comparing a system with dark energy to an identical system without dark energy. So it is irrelevant to the comparison you have been asking about. To put this another way, eq. 37 is not giving the "contribution of dark energy" to the isothermal plasma temperature, which is what you have been asking about. It is just giving the overall temperature taking all contributions into account.

What eq. 37 is doing is assuming that the gravitating system is the maximum size that it can be based on the dark energy density; this size is called ##R_\Lambda## earlier in the paper. But that size decreases as the dark energy density increases; see eq. 6 in the paper. In fact, if we use eq. 6 to rewrite eq. 37 in terms of ##R_\Lambda##, we get

$$T_{iso} = \frac{m}{3k} \frac{M}{R_\Lambda}$$

In other words, as the dark energy density increases, the maximum possible size of the bound system gets smaller, which means it is more compact, and a more compact gravitating system will have a higher temperature.

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