NeutrinoComponent#

class astropy.cosmology.traits.NeutrinoComponent[source]#

Bases: object

The cosmology has attributes and methods for the neutrino density.

This trait handles both massless neutrinos (relativistic, radiation-like) and massive neutrinos (with complex evolution).

This is an abstract trait. Subclasses must implement:

has_massive_nu (property): Whether there are massive neutrinos
Onu0 (property): Neutrino density parameter at z=0
nu_relative_density (method): Neutrino-to-photon density ratio at redshift z

The parent class must provide Tcmb0 (CMB temperature) and Ogamma (method to compute photon density at redshift z).

Notes

The density in neutrinos is given by:

\[\begin{split}\rho_{\nu} \left(a\right) = 0.2271 \, N_{eff} \, f\left(m_{\nu} a / T_{\nu 0} \right) \, \rho_{\\gamma} \left( a \right)\end{split}\]

where

\[\begin{split}f \left(y\right) = \frac{120}{7 \pi^4} \int_0^{\\infty} \, dx \frac{x^2 \\sqrt{x^2 + y^2}} {e^x + 1}\end{split}\]

assuming that all neutrino species have the same mass.

If they have different masses, a similar term is calculated for each one.

Note that f has the asymptotic behavior \(f(0) = 1\). This method returns \(0.2271 f\) using an analytical fitting formula (Komatsu et al., 2011), ApJS, 192, 18.

The neutrino density evolution depends on whether neutrinos are massive or massless:

Massless neutrinos: Behave like radiation with density scaling as (1+z)^4. The density is simply proportional to the photon density with a constant ratio determined by Neff and Fermi-Dirac statistics.
Massive neutrinos: Have complex evolution that transitions from relativistic (radiation-like) at early times to non-relativistic (matter-like) at late times. The implementation typically uses the Komatsu fitting formula (Komatsu et al., 2011) for computational efficiency.

References

Komatsu et al. (2011), “Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation”, ApJS, 192, 18.

Examples

>>> import numpy as np
>>> import astropy.units as u
>>> from astropy.cosmology.traits import NeutrinoComponent
>>> NEUTRINO_FERMI_DIRAC_CORRECTION = 0.22710731766  # 7/8 (4/11)^4/3
>>>
>>> class ExampleNeutrinoCosmology(NeutrinoComponent):
...     def __init__(self):
...         self.Tcmb0 = 2.7255 * u.K
...         self.Neff = 3.046
...         self.Ogamma0 = 5e-5
...     @property
...     def has_massive_nu(self):
...         return False
...     @property
...     def Onu0(self):
...         return NEUTRINO_FERMI_DIRAC_CORRECTION * self.Neff * self.Ogamma0
...     def nu_relative_density(self, z):
...         return NEUTRINO_FERMI_DIRAC_CORRECTION * self.Neff * np.ones_like(np.asarray(z))
...     def Ogamma(self, z):
...         return self.Ogamma0 * (np.asarray(z) + 1.0) ** 4

Attributes Summary

`Onu0`	Omega nu; the density/critical density of neutrinos at z=0.
`Tnu0`	Temperature of the neutrino background as \|Quantity\| at z=0.
`has_massive_nu`	Does this cosmology have at least one massive neutrino species?

Methods Summary

`Onu`(z)	Return the density parameter for neutrinos at redshift `z`.
`Tnu`(z)	Return the neutrino temperature at redshift `z`.
`nu_relative_density`(z)	Neutrino density function relative to the energy density in photons.

Attributes Documentation

Onu0#

Omega nu; the density/critical density of neutrinos at z=0.

Returns:

Onu0python:float: The density parameter for neutrinos at z=0.

Notes

Subclasses must implement this property.

Tnu0[source]#

Temperature of the neutrino background as |Quantity| at z=0.

Returns:

Tnu0Quantity [:ref: ‘temperature’]: The neutrino temperature at z=0 in Kelvin.

Notes

The neutrino temperature is related to the CMB temperature by:

\[T_{\nu 0} = \left(\frac{4}{11}\right)^{1/3} T_{CMB}\]

This comes from the decoupling of neutrinos before electron-positron annihilation. See Weinberg ‘Cosmology’ p 154 eq (3.1.21).

has_massive_nu#

Does this cosmology have at least one massive neutrino species?

Returns:

has_massive_nubool: True if at least one neutrino species has non-zero mass.

Notes

Subclasses must implement this property.

Methods Documentation

Onu(z)[source]#

Return the density parameter for neutrinos at redshift z.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like: Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

Onundarray: The energy density of neutrinos relative to the critical density at each redshift. Note that this includes their kinetic energy (if they have mass), so it is not equal to the commonly used \(\sum \frac{m_{\nu}}{94 eV}\), which does not include kinetic energy.

Notes

The neutrino density parameter evolves with redshift according to:

\[\begin{split}\\Omega_{\\nu}(z) = \\Omega_{\\gamma}(z) \\times f(z)\end{split}\]

where f(z) is the neutrino-to-photon density ratio computed by nu_relative_density(z). For massless neutrinos, f(z) is constant. For massive neutrinos, f(z) evolves as neutrinos transition from relativistic to non-relativistic.

Tnu(z)[source]#

Return the neutrino temperature at redshift z.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like: Input redshift.

Changed in version 7.0: Passing z as a keyword argument is deprecated.

Returns:

TnuQuantity [:ref: ‘temperature’]: The temperature of the cosmic neutrino background in K.

Notes

The neutrino temperature scales with redshift as:

\[T_{\nu}(z) = T_{\nu 0} (1 + z)\]

This simple scaling applies to both massless and massive neutrinos, as the temperature depends only on the expansion of the universe.

abstractmethod nu_relative_density(z)[source]#

Neutrino density function relative to the energy density in photons.

Parameters:

zQuantity-like [‘redshift’], numpy:array_like: Input redshift.

Returns:

farray: The neutrino density scaling factor relative to the density in photons at each redshift.

Notes

Subclasses must implement this method. For massless neutrinos, this should return a constant. For massive neutrinos, this should use an appropriate fitting formula (e.g., Komatsu et al. 2011).