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 [itex]10^25[/itex]) 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 [itex]10^25[/itex]) 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 [itex]\sim g\bar{\psi}\phi \psi[/itex], where [itex]\psi[/itex] is the fermion and [itex]\phi[/itex] the scalar field, and [itex]g[/itex] is the coupling strength. Like all fields, the fermions contribute to the stress-energy tensor, [itex]T_{\mu \nu}[/itex] as follows:
[tex]T_{\mu \nu} = \frac{2}{\sqrt{-g(x)}}\frac{\delta S}{\delta g^{\mu \nu}(x)}[/tex]
where [itex]g_{\mu \nu}(x)[/itex] is the metric (and [itex]g(x)[/itex] its determinant) and the action [itex]S=\int d^4x \mathcal{L}[/itex], where the Lagrangian density for the fermion field is
[tex]\mathcal{L}=\frac{1}{2}i[\bar{\psi}\gamma^\alpha\psi_{,\alpha} - \bar{\psi}_{,\alpha}\gamma^\alpha \psi]-m\bar{\psi}\psi[/tex]
Putting all that together gives
[tex]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][/tex]

An excellent cosmology reference is Kolb and Turner.

 

[sardar]

for the question. As a scientist, it is important to consider all aspects and components of the early universe in our calculations and theories. Fermions, as one of the fundamental particles in the Standard Model, do indeed have a significant impact on the dynamics of the early universe. However, there are several reasons why they are often ignored in our current understanding of the early universe.

Firstly, fermions are typically much less abundant in the early universe compared to bosons, such as the inflaton field. This is due to their fermionic nature, which makes them subject to Pauli exclusion principle and thus limits their numbers. Therefore, their overall contribution to the energy density of the universe is relatively small and can be neglected in many cases.

Secondly, fermions are also subject to different interactions compared to bosons, such as the strong and weak nuclear forces. These interactions can significantly affect their behavior and dynamics in the early universe, making their study more complex and challenging.

However, there are scenarios where fermions in the early universe can play a crucial role. For example, in models that consider fermionic dark matter, the abundance and dynamics of fermions in the early universe can have a significant impact on the evolution of the universe. Moreover, in certain inflationary models, fermionic fields can also contribute to the dynamics of inflation and affect the production of primordial gravitational waves.

In conclusion, while fermions are often ignored in our calculations of the early universe, they should not be completely disregarded. Further research and studies on their behavior and interactions in the early universe can provide valuable insights and enhance our understanding of the early universe. As scientists, it is important to continue exploring and considering all components of the universe in our theories and models.