The Zero Point Field In Friedmann Cosmology

[Dubbelosix]

The normal procedure to talk about the fluctuations in spacetime is by introducing the Einstein relationship of spacetime curvature to the gravitational metric:

 

[math]S(0) = - \frac{1}{16 \pi G} \int\ dx\ \sqrt{-g}R[/math]

 

Sakharov considers the hypothesis that identifies the action with the change in the field of the quantum fluctuations of space in the presence of curvature: Though a hypothesis, there has been a lot of work on the area of research since he wrote his paper and has garnered interest in its implications over the field of physics: One such implication was the non-conservation of particles in curved spacetimes. 

 

The field theory explains such fluctuations as part of the stress energy tensor, [math]T^{i}_{k}(0)[/math] - it had been thought though that the forth power over the momenta of the particles would effectively be zero

 

[math]\hbar c \int k\ dk^3 = 0[/math]

 

But it was soon realized by some bright minds that the gravitational interaction interaction could lead to small perturbations and would result in a finite cosmological constant, something we believe it to be today, very small, positive and finite. Sakharov was interested in the dependence of those fluctuations in space when curvature was present, so he expanded the Langrange function in a series of powers of the curvature, and he got

 

[math]\mathcal{L} = \mathcal{L}(0) + A \int k\ dk \cdot R + B \int \frac{dk}{k} R^2 + ...[/math]

 

Or we can rewrite this in the following way, more suitable to the physical dynamics we wish to be talking about:

 

[math]\mathcal{L} = C \times \hbar c R \int k\ dk + \higher\ \terms[/math]

 

Where [math]C[/math] denotes some constant and [math]k[/math] is the wave number. If we consider the self energy of a fluctuation to be [math]\hbar c k[/math] then the density of a confined region of [math]k^3[/math] is

 

[math]\rho = \hbar c \int k\ dk^3[/math]

 

So how do we even begin to describe the fluctuations of spacetime in the context of the universe itself? I show it is possible, but to do it, we need to modify the Friedmann equation. The curvature term in a Friedmann equation looks like

 

[math]\frac{kc^2}{a^2}[/math]

 

That curvature expression can be modified to reveal a new term with other dynamics, 

 

[math](\frac{\dot{R}}{R})^2 = \frac{8 \pi G}{3}\rho + \frac{2E}{mR^2}[/math]

 

This will become part of our tool when we recover a form of the Friedmann equation which takes into respect the modes of fluctuations. Since the energy of the zero point modes is

 

[math]E = \hbar c R \int\ k\ dk[/math] 

 

Plugging in

 

[math](\frac{\dot{R}}{R})^2 = \frac{8 \pi G}{3}\rho + \frac{2\hbar c}{mR^2} R \int\ k\ dk[/math]

 

[math](\frac{\dot{R}}{R})^2 = \frac{8 \pi G}{3}\rho + \frac{2\hbar c}{mR} \int\ k\ dk[/math]

 

and again, rearranging we find

 

[math]m\dot{R}^2 = \frac{8 \pi GmR^2}{3}\rho + 2\hbar c R \int\ k\ dk[/math]

 

Which recovers the Sakharov zero point field term. So it seems, that after some trivial manipulations, it wasn't too difficult to get what looks like, an identical Sakharov term. Retrieving  the more recognizable form of the Friedmann equation:

 

[math](\frac{\dot{R}}{R})^2 = \frac{8 \pi G}{3}\rho + 2\frac{\hbar c}{mR} \int\ k\ dk[/math]

 

Of course, the story doesn't end there in the physics-rich Friedmann cosmology that divulges in gravitational-fluctuation non-equilibrium in the presence of spacetime curvature. For instance, for adiabatic (non-conserved systems) you have to use a modified first thermodynamic law!

 

The modified law for irreversible particle production we have

 

[math]dE = dQ - PdV + (\frac{\rho + P}{n}) dN[/math]

 

and

 

[math]\dot{E} = \dot{Q} - P\dot{V} + (\frac{\rho + P}{n}) \dot{N}[/math]

 

Here, [math]n[/math] is the number density and [math]N[/math] particle number. This equation works for [math]\rho[/math] being an energy density. 

 

 

It is surprisingly a little bit tricky to implement this law into the structure of our modified Friedmann equation for Sakharov fluctuations: It shouldn't be done haphazardly. 

 

If we define the fluid expansion as

 

 

[math]\Theta^{\mu;}_{\mu'} = \frac{\dot{n}}{n}[/math]

 

this acts on the entire equation like so:

 

[math](R \ddot{R} = \frac{ 8 \pi G R^2}{3} \rho) \Theta^{\mu}_{\mu}[/math]
 

 

 

[math]\rightarrow \dot{R}\ddot{R} =  \frac{8 \pi G R^2}{3}(\frac{\rho}{N})\dot{N}[/math] 

 

 

So it is possible to describe this gravitational theory of fluctuations in an expanding space, but to do so fully, we need to understand how the sakharov term in the modified Friedmann equation interacts over large distances and how significant those effects really are. It's been shown how the Sakharov term may enter the Friedmann equation as a significant parameter perhaps during the early conditions of the universe when curvature will have been more significant. We have also shown how a modified first law can lead us to an adiabatic modified Friedmann equation that will satisfy the non-zero particle production in the presence of curved space. 

 

 

 

 

 

 

 

 

 

 

ref.

 

Sakharov's paper.

 

https://www.scribd.com/document/320426041/Sakharov-Vacuum-Quantum-Fluctuations-in-Curved-Space-pdf

Edited May 26, 2017 by Dubbelosix