Different forms of energy density in inflation

In summary, the conversation discussed the second Friedmann equation and its use in warm inflation. It was mentioned that in warm inflation, radiation is present throughout and does not require a proper reheating process. The equation for ##\rho## was also discussed, with the kinetic and potential energy terms for ##\rho_r## being compared to those of ##\rho_\phi##, the inflaton. Additional resources for further study were also provided.
  • #1
shinobi20
271
20
From the second Friedmann equation,

$$H^2 = \frac{1}{3M_p^2} \rho \quad (k=0, flat)$$

In warm inflation, radiation is present all the way therefore not requiring proper reheating process, so

$$\rho = \rho_\phi + \rho_r \, ; \quad \rho_\phi = inflaton, \, \rho_r = radiation$$

But, $$\rho = \Big(T + V \Big)$$

What should be the kinetic and potential energy of ##\rho_r## as opposed to ##\rho_\phi = \Big(\frac{1}{2} \dot \phi^2 + V \Big)## (i.e. ##V = \frac{1}{2}m^2\phi^2##)?
 
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  • #2
I'm not quite sure what Your trying to do but if I'm not mistaken
[tex]V_(\phi)[/tex] is your potential

[tex]\rho=\frac{1}{2}\dot{\phi}^2+V(\phi)[/tex]
[tex]p=\frac{1}{2}\dot{\phi}^2-V(\phi)[/tex]

Looks like you dropped the pressure term of [tex]M_p [/tex]

[tex]H^2=\frac{8\pi G_n}{3}\rho\rightarrow \frac{\dot{a}}{a}=\sqrt{\frac{8\pi G_nV(\phi)}{3}}[/tex]
 
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  • #3
Mordred said:
I'm not quite sure what Your trying to do but if I'm not mistaken
[tex]V_(\phi)[/tex] is your potential

[tex]\rho=\frac{1}{2}\dot{\phi}^2+V(\phi)[/tex]
[tex]p=\frac{1}{2}\dot{\phi}^2-V(\phi)[/tex]

Looks like you dropped the pressure term of [tex]M_p [/tex]

[tex]H^2=\frac{8\pi G_n}{3}\rho\rightarrow \frac{\dot{a}}{a}=\sqrt{\frac{8\pi G_nV(\phi)}{3}}[/tex]
How could ##ρ## suddenly become ##V(φ)## in the last equation? Also, there is no pressure term in the second friedmann equation.
 
  • #4
Are you sure about that? I left you a handy link showing the pressure term in one of your other threads.

https://redirect.viglink.com/?format=go&jsonp=vglnk_148505730666812&key=6afc78eea2339e9c047ab6748b0d37e7&libId=iy84rrwr010009we000MA49vip5hy&loc=https://www.physicsforums.com/threads/rescaling-the-equation-of-motion-of-inflation.900793/&v=1&out=http://www.google.ca/url?sa=t&source=web&cd=2&ved=0ahUKEwi67MqH0dPRAhVH4mMKHf9vBhgQFgggMAE&url=http%3A%2F%2Fwww3.imperial.ac.uk%2Fpls%2Fportallive%2Fdocs%2F1%2F56439.PDF&usg=AFQjCNFCbq4LLlR6366LhUvr8T_y6_f0eA&sig2=n5C7FRMAGfPcq5gYfq4hMw&ref=https://www.physicsforums.com/forums/cosmology.69/&title=Rescaling the equation of motion of inflation | Physics Forums - The Fusion of Science and Community&txt=http://www.google.ca/url?sa=t&source=web&cd=2&ved=0ahUKEwi67MqH0dPRAhVH4mMKHf9vBhgQFgggMAE&url=http://www3.im...see equation 1.2 and 1.3 in the article above

Start with the equation of state [tex] w=p/\rho [/tex]
Then look at the scalar field modelling equation

https://en.m.wikipedia.org/wiki/Equation_of_state_(cosmology)

Though the scalar field modelling equation is better explained in the first link. (I was trying to avoid posting the same two links in two different threads as you have 3 that are related.

If you don't want to use the first link the Friedmann fluid equation is the second one on this link.
https://en.m.wikipedia.org/wiki/Friedmann_equations

Note both pressure and energy density in the second equation. If you include the pressure term all three of your threads can be answered.

As far as the last equation which is under the Newtonian limit there are numerous solutions most involving GR. However chapter 4 here will give you the basics

https://www.google.ca/url?sa=t&sour...S8lQSyX0BFY9Rq0WQ&sig2=xUX7mahq-ORWjMmET6Q7Qg

Andrew Liddle's "Introductory to Cosmology" though has a decent chapter though done a little too Heuristic. Hope that helps I've usually only enough time for one forum and I usually spend that avaliable time on another as they can use my help more so than this excellent forum.

Another excellent textbook for introduction level is Ryder Lewis "Introductory to Cosmology " he has a decent chapter on three different inflation models. Lambda based ie ( the equations of state in the links above where w=-1), k inflation and quintessenece
 
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  • #5
Here is some additional resources you may find handy in your studies

http://arxiv.org/pdf/hep-ph/0004188v1.pdf :"ASTROPHYSICS AND COSMOLOGY"- A compilation of cosmology by Juan Garcıa-Bellido
http://arxiv.org/abs/astro-ph/0409426An overview of Cosmology Julien Lesgourgues
http://arxiv.org/pdf/hep-th/0503203.pdf"Particle Physics and Inflationary Cosmology" by Andrei Linde
http://www.wiese.itp.unibe.ch/lectures/universe.pdf:" Particle Physics of the Early universe" by Uwe-Jens Wiese Thermodynamics, Big bang Nucleosynthesis
 
  • #6
Mordred said:
Are you sure about that? I left you a handy link showing the pressure term in one of your other threads.

https://redirect.viglink.com/?format=go&jsonp=vglnk_148505730666812&key=6afc78eea2339e9c047ab6748b0d37e7&libId=iy84rrwr010009we000MA49vip5hy&loc=https://www.physicsforums.com/threads/rescaling-the-equation-of-motion-of-inflation.900793/&v=1&out=http://www.google.ca/url?sa=t&source=web&cd=2&ved=0ahUKEwi67MqH0dPRAhVH4mMKHf9vBhgQFgggMAE&url=http%3A%2F%2Fwww3.imperial.ac.uk%2Fpls%2Fportallive%2Fdocs%2F1%2F56439.PDF&usg=AFQjCNFCbq4LLlR6366LhUvr8T_y6_f0eA&sig2=n5C7FRMAGfPcq5gYfq4hMw&ref=https://www.physicsforums.com/forums/cosmology.69/&title=Rescaling the equation of motion of inflation | Physics Forums - The Fusion of Science and Community&txt=http://www.google.ca/url?sa=t&source=web&cd=2&ved=0ahUKEwi67MqH0dPRAhVH4mMKHf9vBhgQFgggMAE&url=http://www3.im...see equation 1.2 and 1.3 in the article above

Start with the equation of state [tex] w=p/\rho [/tex]
Then look at the scalar field modelling equation

https://en.m.wikipedia.org/wiki/Equation_of_state_(cosmology)

Though the scalar field modelling equation is better explained in the first link. (I was trying to avoid posting the same two links in two different threads as you have 3 that are related.

If you don't want to use the first link the Friedmann fluid equation is the second one on this link.
https://en.m.wikipedia.org/wiki/Friedmann_equations

Note both pressure and energy density in the second equation. If you include the pressure term all three of your threads can be answered.

As far as the last equation which is under the Newtonian limit there are numerous solutions most involving GR. However chapter 4 here will give you the basics

https://www.google.ca/url?sa=t&sour...S8lQSyX0BFY9Rq0WQ&sig2=xUX7mahq-ORWjMmET6Q7Qg

Andrew Liddle's "Introductory to Cosmology" though has a decent chapter though done a little too Heuristic. Hope that helps I've usually only enough time for one forum and I usually spend that avaliable time on another as they can use my help more so than this excellent forum.

Another excellent textbook for introduction level is Ryder Lewis "Introductory to Cosmology " he has a decent chapter on three different inflation models. Lambda based ie ( the equations of state in the links above where w=-1), k inflation and quintessenece
Thank you for your effort to put together some great resources, but what I don't know what you are talking about is how can
$$H^2 = \frac{8\pi G}{3} \rho$$
have any pressure term. I want to solve for ##H## so I think the second equation is not so useful.
 
  • #7
Would it help to know the definition of pressure is force per unit volume?

So [tex]\frac {8\pi G_n}{3}[/tex] gives you your pressure term for your volume. G is your Newtons gravitational constant. We have set the curvature term k=0 which is why I didn't include k.

Also as we are working with a homogeneous and isotropic fluid expansion can be described as a perfect fluid following all the gas law rules of an adiabatic fluid ( adiabatic meaning no net inflow/outflow of energy). We are also applying Newtons shell theorem in the above. As we are working with a homogeneous and isotropic fluid.

As you are working with two time derivitaves you must describe each volume seperately the pressure at R and the pressure at [tex]\dot {R}[/tex] in order to derive H^2 or alternately the two time derivitaves of [tex]\frac {\dot {a}}{a}[/tex] as according to the ideal gas laws an increase in volume decreases pressure/density and temperature. So for H you cannot use 1 value for pressure and energy/mass density but the time derivitaves of both at each time derivitave of your commoving volume

So using R for volume my two time derivitaves between two commoving volumes becomes
[tex]R =\frac {8\pi G_n}{3}\rho [/tex]
[tex]\dot {R }=\dot {\frac{{8\pi G_n}}{3}}\dot {\rho} [/tex] where [tex]\frac {8\pi G_n}{3}[/tex] is in essence the force per unit volume of time derivitave R

[tex]H^2=\frac {\dot {R}}{R}[/tex] you can literally plug the two time derivitaves of pressure and energy/density into the last formula to get H

The same time derivitave rules apply to your scalar field equations in my first reply

[tex]\dot {\phi}[/tex] denotes the potential is a time derivitave and needs to be treated as a time derivitave. The curvature term can be safety assumed as constant between the time derivitaves
Does that help?

Edit ;important side note Do not think of expansion as due to pressure/(force per unit volume) as were dealing with a homogeneous and isotropic fluid there is no net flow of force as there is no pressure gradient at a particular time
 
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  • #8
shinobi20 said:
Thank you for your effort to put together some great resources, but what I don't know what you are talking about is how can
$$H^2 = \frac{8\pi G}{3} \rho$$
have any pressure term. I want to solve for ##H## so I think the second equation is not so useful.
There is no pressure term in this equation.
 
  • #9
Mordred said:
Would it help to know the definition of pressure is force per unit volume

Pressure is force per unit area, not volume.
 
  • #10
Doh right lol my bad working too late in am
 
  • #11
bapowell said:
There is no pressure term in this equation.
That is what I'm wondering about, as Mordred posted.
 
  • #12
I should have said derive not give. Did you look at equations 1.1 to 1.3 of the first link yet?
1.1 your Friedmann metric
1.2
[tex]H^2=(\frac{\dot{a}}{a})^2=\frac{8\pi G}{3}\rho-\frac{k}{a^2}[/tex]
1.3
[tex]\frac{\ddot{a}}{a}=-\frac{-4\pi G}{3}(\rho+3p)[/tex]
 
  • #13
Mordred said:
I should have said derive not give. Did you look at equations 1.1 to 1.3 of the first link yet?
1.1 your Friedmann metric
1.2
[tex]H^2=(\frac{\dot{a}}{a})^2=\frac{8\pi G}{3}\rho-\frac{k}{a^2}[/tex]
1.3
[tex]\frac{\ddot{a}}{a}=-\frac{-4\pi G}{3}(\rho+3p)[/tex]
Yes, I'm using (1.2) and there is no pressure term there.
 
  • #14
I take it you missed the sentence above relating those two equations then?

"These are related to the energy density and pressure terms through the Friedmann equations 1.2 and 1.3."

key word AND not one or the other but both
 
  • #15
Mordred said:
I take it you missed the sentence above relating those two equations then?

"These are related to the energy density and pressure terms through the Friedmann equations 1.2 and 1.3."

key word AND not one or the other but both
Yes but what is your point? I'm only interested getting ##H##, so I can obtain it through $$H^2 = \frac{1}{6M_p^2}(\dot \phi^2 + m^2\phi^2)$$
 
  • #16
what does the subscript p stand for on M?
 
  • #17
Mordred said:
what does the subscript p stand for on M?
##M_p## stands for the reduced Planck mass.
 
  • #18
shinobi20 said:
##M_p## stands for the reduced Planck mass.
shinobi20 said:
##M_p## stands for the reduced Planck mass.

right and your reduced Planck mass formula is?
[tex]m_p=\sqrt{\frac{\hbar c}{8\pi G}}[/tex]
 
  • #19
Mordred said:
right and your reduced Planck mass formula is?
[tex]m_p=\sqrt{\frac{\hbar c}{8\pi G}}[/tex]
Yes, and ##\hbar = c = 1##.
 
  • #20
shinobi20 said:
Yes, and ##\hbar = c = 1##.
and in your OP you want your temperature right ? ie[tex]\rho=T+V[/tex] right?

How do you solve the temperature without knowing the pressure?

Particularly since your starting with the scalar field equations
 
  • #21
Mordred said:
and in your OP you want your temperature right ? ie[tex]\rho=T+V[/tex] right?

How do you solve the temperature without knowing the pressure?
The kinetic ##T## and potential energy ##V## of the radiation density; but aside from that I have one more question, the equation of motion of inflation is
$$\frac{d^2\phi}{dt^2} + 3H\frac{d\phi}{dt} + V,_\phi = 0$$

where ##ϕ## is the inflaton, ##H## is the Hubble parameter, and ##V,_ϕ## is the derivative of the potential with respect to ##ϕ##. To solve this differential equations, we need two initial value conditions, ##\phi(0)\,## and ##\dot \phi(0)\,##. I'm using Mathematica to solve for ##\phi##, and in order to solve the above DE we need to eliminate the unknown function ##H## and from the Friedmann equations, $$H^2 = \frac{1}{3M_p^2}\Big(\frac{1}{2}\dot \phi^2 + V\Big)$$

For a potential given by ##V = \frac{1}{2}m^2\phi^2##, we have
$$\ddot \phi^2 + 3\sqrt{\frac{1}{6M_p^2}\Big(\dot \phi^2 + \phi^2\Big)} \dot \phi + m^2\phi = 0$$

I can set ##\phi(0) \approx M_p## and ##\dot \phi(0) \approx 0.1## (any small value just to get started).
But in order to get an exact numerical value for ##\phi(t)## after solving this DE numerically, I need to set some ##t##. But I'm not sure about what ##t##'s to set during the inflationary stage, or in other words, I want to know the proper ##t## to use during the evolution of ##\phi##?
 
  • #22
Mordred said:
I'm not quite sure what Your trying to do

lol glad I stated previously the above in your OP. I misread the T for temp not kinetic energy.

I'm not sure what the appropriate values for t on your time derivatives are most appropriate in this case. Bapowell will probably best answer that
 
  • #23
When you solve a DE with ##t## as the independent variable, you need to state at what ##t## you want to solve for the dependent variable right? It's just the same case with what I'm doing, here the dependent variable is ##\phi##.
 
  • #24
I'm still looking for your t question. There is a section that applies in the first link I supplied see chapter 2.with regards to the rates involved in inflation in terms from the potentials. I can't quite remember this so have to dig deeper than the article I just mentioned.
 
  • #25
Mordred said:
I'm still looking for your t question. There is a section that applies in the first link I supplied see chapter 2.with regards to the rates involved in inflation in terms from the potentials. I can't quite remember this so have to dig deeper than the article I just mentioned.
I'm not sure if ##t## should be the Hubble time which during inflation, the Hubble parameter is fairly constant or ##t## should be the time period for inflation.
 
  • #26
Ok found what I was looking for. In essence acceleration occurs when the potential energy of the field dominates over the its kinetic term. Inflation stops when the kinetic term has grown to become comparable to the potential energy. You have those equations already. I found you a good lecture from arxiv to cover all the details you need.

http://www.google.ca/url?sa=t&sourc...g98i0LHSrs3IMtjnQ&sig2=CPyOjqYaqy26D-fdmDKBrQ
 
  • #27
Mordred said:
Ok found what I was looking for. In essence acceleration occurs when the potential energy of the field dominates over the its kinetic term. Inflation stops when the kinetic term has grown to become comparable to the potential energy. You have those equations already. I found you a good lecture from arxiv to cover all the details you need.
Yes, that is the standard scenario for inflation. I'm thinking about the ##t## I need fro the DE.
 
  • #28
shinobi20 said:
Yes, that is the standard scenario for inflation. I'm thinking about the ##t## I need fro the DE.
Usually the overdot denotes t_now not sure if this is the case in inflation actually I'm fairly sure it is t_now for the overdot as its observer based. ie your the observer. Hopefully Bapowell will clarify though I can't see it not being t_now.

Though I could always reread his dissertation lol
 
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  • #29
Mordred, there is no pressure term in the first FE. I think you're really confusing the op.
 
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  • #30
I already stated I was the one confused.
 
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FAQ: Different forms of energy density in inflation

1. What is energy density in the context of inflation?

Energy density refers to the amount of energy per unit volume in a given system. In the context of inflation, it refers to the energy density of the inflaton field, which is responsible for driving the rapid expansion of the universe during the inflationary period.

2. How does the energy density of the inflaton field change during inflation?

The energy density of the inflaton field decreases as the universe expands during inflation. This is because the energy of the inflaton field is converted into the energy of the expanding universe.

3. What are the different forms of energy density in inflation?

There are two main forms of energy density in inflation: the potential energy of the inflaton field and the kinetic energy of the inflaton field. The potential energy is responsible for driving the expansion of the universe, while the kinetic energy is associated with the motion of the inflaton field.

4. How does the energy density affect the rate of inflation?

The energy density of the inflaton field is directly related to the rate of inflation. A higher energy density leads to a faster expansion of the universe, while a lower energy density results in a slower expansion. This is because the energy density determines the strength of the gravitational force that drives inflation.

5. Can the energy density in inflation be measured?

While it is not possible to directly measure the energy density of the inflaton field, it can be indirectly inferred from observations of the cosmic microwave background radiation and the large-scale structure of the universe. These observations provide valuable insights into the energy density and other properties of inflation.

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