Fermions in Early Universe: Why We Ignore Them?

[Accidently]

When we talk about the inflation and other cosmological topics, we calculate the scalar dynamics in the early universe. But how do fermions behave? In principle they should carry same amount of energy and they can effect the evolution of scalars via interactions. Why we just ignore them? Is there a scenario where fermions in the early universe are important?

thx

 

[bapowell]

When we talk about the inflation and other cosmological topics, we calculate the scalar dynamics in the early universe. But how do fermions behave? In principle they should carry same amount of energy and they can effect the evolution of scalars via interactions. Why we just ignore them? Is there a scenario where fermions in the early universe are important?

Good question Accidentally. You are correct that a complete picture of early universe cosmology should involve fermions. The reason that scalars are generally relevant is because the inflaton is a scalar field. Scalars are the only fields that can possesses nonzero vacuum energy without breaking Lorentz invariance, and so they occupy an important place in early universe cosmology as sources of stress-energy that lead to accelerated expansion. During the early evolution of the universe before inflation, fermions were indeed in existence and contributed to the stress-energy of the universe. However, once inflation gets started, all pre-existing matter and energy gets massively redshifted (by a factor of at least $10^25$) by the exponential expansion. The result is that the only relevant stress-energy component during inflation is the inflaton itself -- hence the singular emphasis on scalar field dynamics.

That said, the fermions and everything else aren't out of the story for good, since we know they had to make their return somehow after inflation ended. In the theory of reheating, the inflaton decays into all of the matter and energy comprising the observable universe. In order for the inflaton to decay into fermions, there must be some coupling between them. So, you are indeed correct that in simple reheating models, there must be a fermion-inflaton coupling. But, due to the overwhelming energy density of the inflaton relative to all other species during inflation, the coupling is not important during this epoch and does not affect the dynamics of the inflaton field or the universe. It is after inflation, when the inflaton decays, that these couplings become important.

 

[Accidently]

Good question Accidentally. You are correct that a complete picture of early universe cosmology should involve fermions. The reason that scalars are generally relevant is because the inflaton is a scalar field. Scalars are the only fields that can possesses nonzero vacuum energy without breaking Lorentz invariance, and so they occupy an important place in early universe cosmology as sources of stress-energy that lead to accelerated expansion. During the early evolution of the universe before inflation, fermions were indeed in existence and contributed to the stress-energy of the universe. However, once inflation gets started, all pre-existing matter and energy gets massively redshifted (by a factor of at least $10^25$) by the exponential expansion. The result is that the only relevant stress-energy component during inflation is the inflaton itself -- hence the singular emphasis on scalar field dynamics.

That said, the fermions and everything else aren't out of the story for good, since we know they had to make their return somehow after inflation ended. In the theory of reheating, the inflaton decays into all of the matter and energy comprising the observable universe. In order for the inflaton to decay into fermions, there must be some coupling between them. So, you are indeed correct that in simple reheating models, there must be a fermion-inflaton coupling. But, due to the overwhelming energy density of the inflaton relative to all other species during inflation, the coupling is not important during this epoch and does not affect the dynamics of the inflaton field or the universe. It is after inflation, when the inflaton decays, that these couplings become important.

Thanks for you reply. But how to describe the fermions-scalar interacting system mathematically. And how do fermions contribute to the stress-energy tensor of the universe? Can you refer me come articles?

 

[bapowell]

Thanks for you reply. But how to describe the fermions-scalar interacting system mathematically. And how do fermions contribute to the stress-energy tensor of the universe? Can you refer me come articles?

The fermion-scalar interaction is typically governed by a Yukawa-type coupling $\sim g\bar{\psi}\phi \psi$, where $\psi$ is the fermion and $\phi$ the scalar field, and $g$ is the coupling strength. Like all fields, the fermions contribute to the stress-energy tensor, $T_{\mu \nu}$ as follows:
$$T_{\mu \nu} = \frac{2}{\sqrt{-g(x)}}\frac{\delta S}{\delta g^{\mu \nu}(x)}$$
where $g_{\mu \nu}(x)$ is the metric (and $g(x)$ its determinant) and the action $S=\int d^4x \mathcal{L}$, where the Lagrangian density for the fermion field is
$$\mathcal{L}=\frac{1}{2}i[\bar{\psi}\gamma^\alpha\psi_{,\alpha} - \bar{\psi}_{,\alpha}\gamma^\alpha \psi]-m\bar{\psi}\psi$$
Putting all that together gives
$$T_{\mu \nu} = \frac{1}{4}i[\bar{\psi}(\gamma_\mu \nabla_\nu + \gamma_\nu \nabla_\mu)\psi - (\nabla_\mu \bar{\psi}\gamma_\nu - \nabla_\nu \bar{\psi}\gamma_\mu)\psi]$$

An excellent cosmology reference is Kolb and Turner.