Cosmology Lecture 23

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Quantum Field Quickie

To understand inflation, we need Quantum Field Theory. Fields are generalizations of classical particles. Here are some properties of classical particles and their generalizations as fields:

${\displaystyle {\begin{matrix}Classical\ Particles:&&Fields:&\\Particle:&x(t)&Field(scalar):&\phi (x^{\mu }),\overbrace {\mu =0,1,2,3} ^{1\ time \atop 3\ spatial}\\Action:&S=\int {dt\,L}&&S=\int {d^{4}x{\mathfrak {L}}\overbrace {\sqrt {-g}} ^{determ. \atop of\ g^{\mu \nu }}}\\Lagrangian:&L={\frac {1}{2}}m{\dot {x}}^{2}-V(x)&Lagrangian&{\mathfrak {L}}={\frac {1}{2}}\partial _{\mu }\phi \partial ^{\mu }\phi -V(\phi )\\&&Density:&{\mathfrak {L}}={\frac {1}{2}}{\dot {\phi }}^{2}-{\frac {1}{2}}(\nabla \phi )^{2}-V(\phi )\\Energy\ (Hamiltonian):&E=p\cdot {\dot {x}}-L={\frac {1}{2}}m{\dot {x}}^{2}+V(x)&&T^{\mu \nu }=\partial ^{\mu }\phi \partial ^{\nu }\phi -g^{\mu \nu }{\mathfrak {L}}\\Euler-Lagrange\ Eq.&{\frac {d}{dt}}{\partial L \over \partial {\dot {x}}}={\partial L \over \partial x}&&\partial _{\mu }\left[{\partial {\mathfrak {L}} \over \partial (\partial _{\mu }\phi )}\right]={\partial {\mathfrak {L}} \over \partial \phi }\\of\ Motion\ (\partial S=0):&m{\ddot {x}}=-{dV \over dx}&&\partial _{\mu }\partial ^{\mu }\phi =-{dV \over d\phi }\\&&&{\ddot {\phi }}-\nabla ^{2}\phi \underbrace {3{{\dot {a}} \over a}{\dot {\phi }}} _{{due\ to \atop expansion} \atop {\sqrt {-g}}=a^{3}}=-{dV \over d\phi }\\\end{matrix}}\,\!}$

where ${\displaystyle \partial _{\mu }\equiv {\partial \over \partial x^{\mu }}}$, repeated indices are summed over, and ${\displaystyle g^{\mu \nu }}$ is the metric tensor defined by:

${\displaystyle ds^{2}=g^{\mu \nu }dx_{\mu }dx_{\nu }\,\!}$

Inflation

Quantum Field Theory is relevant to inflation because the scalar field ${\displaystyle \phi }$ contributes energy to the universe. Including the energy density associated with this field in the Friedmann Equation:

${\displaystyle H^{2}={8\pi \over 3}G(\rho _{m}+\rho _{r}+\rho _{\phi }+\dots )\,\!}$

We can express the energy density for the scalar field as the (0,0) component of the energy momentum tensor:

{\displaystyle {\begin{aligned}T^{\mu \nu }&=\partial ^{\mu }\phi \partial ^{\nu }\phi -g^{\mu \nu }{\mathfrak {L}}\\T^{00}&\equiv \rho _{\phi }={\dot {\phi }}^{2}-{\mathfrak {L}}\\&={\frac {1}{2}}\,\phi ^{2}+{\frac {1}{2}}(\nabla \phi )^{2}+V(\phi )\\\end{aligned}}\,\!}

where we used ${\displaystyle g^{00}=1}$. The pressure associated with ${\displaystyle \phi }$ is given by:

${\displaystyle P_{\phi }=T^{ii}={1 \over 3}(\nabla \phi )^{2}+{\mathfrak {L}}={\frac {1}{2}}{\dot {\phi }}^{2}-{1 \over 6}(\nabla \phi )^{2}-V(\phi )\,\!}$

where in this case the ${\displaystyle i}$’s are not summed over. Note the negative sign on ${\displaystyle V(\phi )}$. For a slowly varying, spatially homogeneous ${\displaystyle \phi }$:

${\displaystyle {\begin{matrix}{\dot {\phi }}^{2}&\ll V(\phi )\\(\nabla \phi )^{2}&\ll V(\phi )\\\end{matrix}}\Rightarrow {\begin{matrix}\rho _{\phi }&\approx V(\phi )\\P_{\phi }&\approx -V(\phi )\\\end{matrix}}\,\!}$

Then using that ${\displaystyle P_{\phi }=-\rho _{\phi }\equiv w\rho _{\phi }}$, we find that:

${\displaystyle {w=-1}\,\!}$

So we have a very viable candidate for dark energy ${\displaystyle \rho _{\Lambda }}$. If ${\displaystyle \rho _{\phi }\gg \rho _{m},\rho _{r}}$, then we have exponential expansion. Next time we’ll investigate ${\displaystyle V(\phi )}$, which will aid us in understanding why this exponential expansion should dominate for the very beginning of the universe.