# I Different forms of energy density in inflation

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1. Jan 13, 2017

### shinobi20

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. $V = \frac{1}{2}m^2\phi^2$)?

2. Jan 16, 2017

### Mordred

I'm not quite sure what Your trying to do but if I'm not mistaken
$$V_(\phi)$$ is your potential

$$\rho=\frac{1}{2}\dot{\phi}^2+V(\phi)$$
$$p=\frac{1}{2}\dot{\phi}^2-V(\phi)$$

Looks like you dropped the pressure term of $$M_p$$

$$H^2=\frac{8\pi G_n}{3}\rho\rightarrow \frac{\dot{a}}{a}=\sqrt{\frac{8\pi G_nV(\phi)}{3}}$$

Last edited: Jan 16, 2017
3. Jan 21, 2017

### shinobi20

How could $ρ$ suddenly become $V(φ)$ in the last equation? Also, there is no pressure term in the second friedmann equation.

4. Jan 21, 2017

### Mordred

see equation 1.2 and 1.3 in the article above

Start with the equation of state $$w=p/\rho$$
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

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

Last edited: Jan 22, 2017
5. Jan 22, 2017

### Mordred

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. Jan 22, 2017

### shinobi20

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. Jan 22, 2017

### Mordred

Would it help to know the definition of pressure is force per unit volume?

So $$\frac {8\pi G_n}{3}$$ 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 $$\dot {R}$$ in order to derive H^2 or alternately the two time derivitaves of $$\frac {\dot {a}}{a}$$ 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
$$R =\frac {8\pi G_n}{3}\rho$$
$$\dot {R }=\dot {\frac{{8\pi G_n}}{3}}\dot {\rho}$$ where $$\frac {8\pi G_n}{3}$$ is in essence the force per unit volume of time derivitave R

$$H^2=\frac {\dot {R}}{R}$$ 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

$$\dot {\phi}$$ 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

Last edited: Jan 22, 2017
8. Jan 22, 2017

### bapowell

There is no pressure term in this equation.

9. Jan 22, 2017

### Staff: Mentor

Pressure is force per unit area, not volume.

10. Jan 22, 2017

### Mordred

Doh right lol my bad working too late in am

11. Jan 22, 2017

### shinobi20

That is what I'm wondering about, as Mordred posted.

12. Jan 23, 2017

### Mordred

I should have said derive not give. Did you look at equations 1.1 to 1.3 of the first link yet?
1.2
$$H^2=(\frac{\dot{a}}{a})^2=\frac{8\pi G}{3}\rho-\frac{k}{a^2}$$
1.3
$$\frac{\ddot{a}}{a}=-\frac{-4\pi G}{3}(\rho+3p)$$

13. Jan 23, 2017

### shinobi20

Yes, I'm using (1.2) and there is no pressure term there.

14. Jan 23, 2017

### Mordred

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. Jan 23, 2017

### shinobi20

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. Jan 23, 2017

### Mordred

what does the subscript p stand for on M?

17. Jan 23, 2017

### shinobi20

$M_p$ stands for the reduced Planck mass.

18. Jan 23, 2017

### Mordred

right and your reduced planck mass formula is?
$$m_p=\sqrt{\frac{\hbar c}{8\pi G}}$$

19. Jan 23, 2017

### shinobi20

Yes, and $\hbar = c = 1$.

20. Jan 23, 2017

### Mordred

and in your OP you want your temperature right ? ie$$\rho=T+V$$ right?

How do you solve the temperature without knowing the pressure?

Particularly since your starting with the scalar field equations