Cosmology Lecture 23

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:

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

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

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

Inflation

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

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

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

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

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

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

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

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

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

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

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