Musings of a mad scientist (inflation as cosmological constant)

[Mordred]

 My interest in Cosmology started back in the late 70's. However in the early 80's when Alllen Guth's first published his false vacuum inflation model accelerated that interest even further. That interest has been a primary focus of my studies ever since. Back then inflation often involved quantum tunnelling from a false vacuum state to a true vacuum state. These models typically had the energy density graph transitions as in a similar fashion to 

https://www.wolframalpha.com/input?i2d=true&i=plot+Power[x%2C2]

thoough they would often include a higher potential of the same curve to represent the stable region of the higher potential false vacuum state. Those graphs then were employed to define quantum tunneling from the two potential VeV's through the separation potential barrier between their corresponding stable regions.

 Modern models however use the Mexican hat potential as per the inflaton and the Higgs field. Their effective equations of state are close matches. Today there is strong supportive evidence that Higgs inflation is highly viable. However the inflaton is also equally viable. The latter includes chaotic eternal inflation leading to pocket multiverses defined by separate expansion regions with the causal connection of the same inflation mechanism that  rapidly expanded our universe in early times.

 I had also had to ask myself the question.

 Is there a connection between inflation and the cosmological constant ? "

 given the behavior of inflation I found that this is highly likely. I will support this with the relevant equations however it will take time to place them into the thread. ( I will be saving often during edits)So starting from our hot dense state at 10^{-43} seconds. (prior to this leads to infinite blueshift as well as other infinity problems. ( The Big Bang mathematical singularity conditions)

so first we need our metric LCDM

FLRW Metric equations

\[d{s^2}=-{c^2}d{t^2}+a({t^2})[d{r^2}+{S,k}{(r)^2}d\Omega^2]\]

\[S\kappa(r)= \begin{cases} R sin(r/R &(k=+1)\\ r &(k=0)\\ R sin(r/R) &(k=-1) \end {cases}\]

\[\rho_{crit} = \frac{3c^2H^2}{8\pi G}\]

\[H^2=(\frac{\dot{a}}{a})^2=\frac{8 \pi G}{3}\rho+\frac{\Lambda}{3}-\frac{k}{a^2}\]

setting \[T^{\mu\nu}_\nu=0\] gives the energy stress mometum tensor as 

\[T^{\mu\nu}=pg^{\mu\nu}+(p=\rho)U^\mu U^\nu)\]

\[T^{\mu\nu}_\nu\sim\frac{d}{dt}(\rho a^3)+p(\frac{d}{dt}(a^3)=0\]

which describes the conservation of energy of a perfect fluid in commoving coordinates describes by the scale factor a with curvature term K=0.

the related GR solution the the above will be the Newton approximation.

\[G_{\mu\nu}=\eta_{\mu\nu}+H_{\mu\nu}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}\]

Thermodynamics

Tds=DU+pDV Adiabatic and isentropic fluid (closed system)

equation of state

\[w=\frac{\rho}{p}\sim p=\omega\rho\]

\[\frac{d}{d}(\rho a^3)=-p\frac{d}{dt}(a^3)=-3H\omega(\rho a^3)\]

as radiation equation of state is

\[p_R=\rho_R/3\equiv \omega=1/3 \]

radiation density in thermal equilibrium is therefore

\[\rho_R=\frac{\pi^2}{30}{g_{*S}=\sum_{i=bosons}gi(\frac{T_i}{T})^3+\frac{7}{8}\sum_{i=fermions}gi(\frac{T_i}{T})}^3 \]

\[S=\frac{2\pi^2}{45}g_{*s}(at)^3=constant\]

temperature scales inversely to the scale factor giving

\[T=T_O(1+z)\]

with the density evolution of radiation, matter and Lambda given as a function of z

\[H_z=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{rad}(1+z)^4+\Omega_{\Lambda}}\]

the above typically has the equation of states cosmology given by 

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

where the scalar field is the given equation of state for the cosmological constant.

\[w=\frac{\frac{1}{2}V\dot{\varphi^2}-V(\varphi)}{\frac{1}{2}V\dot{\varphi^2}+V(\varphi)}\]

 

 

Higgs Inflation Single scalar field Modelling.

\[S=\int d^4x\sqrt{-g}\mathcal{L}(\Phi^i\nabla_\mu \Phi^i)\]

g is determinant

Einstein Hilbert action in the absence of matter.

\[S_H=\frac{M_{pl}^2}{2}\int d^4 x\sqrt{-g\mathbb{R}}\]

set spin zero inflaton as

\[\varphi\]

minimally coupled Langrangian as per General Covariance in canonical form. (kinetic term)

\[\mathcal{L_\varphi}=-\frac{1}{2}g^{\mu\nu}\nabla_\mu \varphi \nabla_\nu \varphi-V(\varphi)\]

where \[V(\varphi)\] is the potential term

integrate the two actions of the previous two equations for minimal scalar field gravitational couplings

\[S=\int d^4 x\sqrt{-g}[\frac{M_{pl}^2}{2}\mathbb{R}-\frac{1}{2}g^{\mu\nu}\nabla_\mu\varphi \nabla_\nu \varphi-V(\varphi)]\]

variations yield the Euler_Langrene

\[\frac{\partial \mathcal{L}}{\partial \Phi^i}-\nabla_\mu(\frac{\partial \mathcal{L}}{\partial[\nabla_\mu \Phi^i]})=0\]

using Euclidean commoving metric

\[ds^2-dt^2+a^2(t)(dx^2+dy^2=dz^2)\]

this becomes

\[\ddot{\varphi}+3\dot{\varphi}+V_\varphi=0\]

\[S=\int d^4 x\sqrt{-g}[\frac{M_{pl}^2}{2}\mathbb{R}-\frac{1}{2}g^{\mu\nu}\nabla_\mu\varphi \nabla_\nu \varphi-V(\varphi)]\]

and 

\[G_{\mu\nu}-\frac{1}{M_{pl}}T_{\mu\nu}\]

with flat commoving geometry of a perfect fluid gives the energy momentum for inflation as 

\[T^\mu_\nu=g^{\mu\lambda}\varphi_\lambda \varphi_\nu -\delta^\mu_\nu \frac{1}{2}g^{\rho \sigma} \varphi_\rho \varphi_\sigma V(\varphi)]\]

 

\[\rho=T^0_0=\frac{1}{2}\dot{\varphi}^2+V\]

\[p=T^i_i (diag)=\frac{1}{2}\dot{\varphi}^2-V\]

\[w=\frac{p}{\rho}\]

\[w=\frac{1-2 V/\dot{\varphi^2}}{1+2V/\dot{\varphi^2}}\]

***method by Fernando A. Bracho Blok Thesis paper.***

https://helda.helsinki.fi/bitstream/handle/10138/322422/Brachoblok_fernando_thesis_2020.pdf?sequence=2&isAllowed=y

Now any scalar field state dominated by the potential energy density will have negative pressure. a negative pressure to energy density ratio of w=-1 will describe a state that does not vary over time. However this also involves the critical density.

\[\rho_{crit} = \frac{3c^2H^2}{8\pi G}\]

 

the initial conditions will have a different critical density value at the false vacuum state prior to inflation. A critically dense universe is a k=0 flat universe. However it derivatives employ a pressure less w=0 equation of state to derive the critical density value. This corresponds to the equation of state for matter.

 in essence the universe is thermodynamically evolving from a hot dense state of a negative vacuum with opposing force relations given by replacement 

\[\rho_{(V\varphi)}\]

to 

\[V(\phi)

\[DU=\rho_{V(\phi)}DV\]

work is defined as

\[dW=\rho_{V(\phi)}dV\]

\[p_{V(\phi)}=-\rho_{V(\phi)}\]

the slower the roll from false vacuum potential to VeV today allows for greater number of e-folds. The greater the e-fold ratio the higher the number of pocket universes that can result in locally anisotropic regions during  inflation. Examinations under the eternal chaotic inflationary theory can lead up to an infinite amount of bubble or pocket universes. This arises with the result of a locally different rate in inflation to the global rate. 

  Once inflation slow rolls the VeV reduces on a gradual but not necessarily smooth rate, smaller metastable states arise due to thermal dropout of particle species as well as base elements such as hydrogen and lithium. The expansion and Cosmological redshift following the thermodynamic laws also contribute to the evolution to the scale factor.

I will add further detail later on (long work day)

 

 

 

Edited December 24, 2022 by Mordred