Cosmology

Cosmology: the expansion of a homogeneous universe.

ScaleFactorElement(label, visual=None, *, a=1.0, a_d1=0.0, H0=1.0, omega_matter=0.3, omega_lambda=0.7)

Bases: Element

Homogeneous universe whose size is the scale factor a, a state of order 2 with the expansion rate a_d1.

Parameters:

Name	Type	Description	Default
label	str	The label of the universe.	required
visual	VisualSpec | None	How the universe is drawn, or None for none.	None
a	Annotated[float, StateSpec(domain=open(0, oo), order=2)]	The initial scale factor.	1.0
a_d1	float	The initial rate of change of the scale factor.	0.0
H0	Annotated[float, ParameterSpec(domain=open(0, oo))]	The Hubble constant.	1.0
omega_matter	Annotated[float, ParameterSpec(domain=closed_open(0, oo))]	The density parameter of matter.	0.3
omega_lambda	Annotated[float, ParameterSpec()]	The density parameter of dark energy.	0.7

Examples:

>>> from skcomplex.models.physics.cosmology import ScaleFactorElement
>>> ScaleFactorElement('universe', H0=2.0).H0
2.0

Source code in src/skcomplex/models/physics/cosmology/_expansion.py

def __init__(
    self,
    label: str,
    visual: VisualSpec | None = None,
    *,
    a: Annotated[float, StateSpec(domain=RealSet.open(0, oo), order=2)] = 1.0,
    a_d1: float = 0.0,
    H0: Annotated[float, ParameterSpec(domain=RealSet.open(0, oo))] = 1.0,
    omega_matter: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.3,
    omega_lambda: Annotated[float, ParameterSpec()] = 0.7,
) -> None:
    super().__init__(label, visual=visual)
    self.a = a
    self.a_d1 = a_d1
    self.H0 = H0
    self.omega_matter = omega_matter
    self.omega_lambda = omega_lambda

friedmann_expansion(a=1.0, *, H0=1.0, omega_matter=0.3, omega_lambda=0.7)

Return the scale factor of a homogeneous universe of matter and dark energy, after Friedmann and Lemaître.

The acceleration equation a² a'' = -4 pi G M / 3 + Lambda a³ / 3 is written with the density parameters of matter Ω_m = 8 pi G M / (3 H0²) and of the cosmological constant Ω_Λ = Lambda / (3 H0²) in units of the Hubble constant H0 at a = 1, so a² a'' = -Ω_m H0² / 2 + Ω_Λ H0² a³. The initial expansion rate follows from the Friedmann equation a'² = H0² (Ω_m / a + Ω_Λ a² + 1 - Ω_m - Ω_Λ), which stays a first integral of the motion. Matter decelerates the expansion and dark energy accelerates it once a exceeds (Ω_m / (2 Ω_Λ))^(1/3).

Parameters:

Name	Type	Description	Default
a	float	The initial scale factor.	1.0
H0	float	The Hubble constant, the expansion rate at a = 1.	1.0
omega_matter	float	The density parameter of matter.	0.3
omega_lambda	float	The density parameter of the cosmological constant.	0.7

Returns:

Name	Type	Description
system	System	The model, with the scale factor universe.a of order 2 and its rate universe.a_d1.

Raises:

Type	Description
ValidationError	The universe does not expand at the initial scale factor.

Examples:

>>> from skcomplex.models.physics.cosmology import friedmann_expansion
>>> friedmann_expansion(omega_matter=1.0, omega_lambda=0.0).elements[0].a_d1
1.0

Source code in src/skcomplex/models/physics/cosmology/_expansion.py

@register_model(
    field='physics/cosmology',
    tags=('cosmology', 'ode', 'second order', 'gravity', 'dark energy'),
    references=('Friedmann 1922', 'Lemaître 1927', 'Satz 2022'),
    defaults={'until': 2.0, 'dt': 0.02},
)
def friedmann_expansion(
    a: float = 1.0,
    *,
    H0: float = 1.0,
    omega_matter: float = 0.3,
    omega_lambda: float = 0.7,
) -> System:
    """Return the scale factor of a homogeneous universe of matter and dark energy, after Friedmann and Lemaître.

    The acceleration equation `a² a'' = -4 pi G M / 3 + Lambda a³ / 3` is written with the density parameters of
    matter `Ω_m = 8 pi G M / (3 H0²)` and of the cosmological constant `Ω_Λ = Lambda / (3 H0²)` in units of the Hubble
    constant `H0` at `a = 1`, so `a² a'' = -Ω_m H0² / 2 + Ω_Λ H0² a³`. The initial expansion rate follows from the
    Friedmann equation `a'² = H0² (Ω_m / a + Ω_Λ a² + 1 - Ω_m - Ω_Λ)`, which stays a first integral of the motion.
    Matter decelerates the expansion and dark energy accelerates it once `a` exceeds `(Ω_m / (2 Ω_Λ))^(1/3)`.

    Args:
        a:
            The initial scale factor.

        H0:
            The Hubble constant, the expansion rate at `a = 1`.

        omega_matter:
            The density parameter of matter.

        omega_lambda:
            The density parameter of the cosmological constant.

    Returns:
        system:
            The model, with the scale factor `universe.a` of order 2 and its rate `universe.a_d1`.

    Raises:
        ValidationError:
            The universe does not expand at the initial scale factor.

    Examples:
        >>> from skcomplex.models.physics.cosmology import friedmann_expansion
        >>> friedmann_expansion(omega_matter=1.0, omega_lambda=0.0).elements[0].a_d1
        1.0
    """
    square = H0**2 * (omega_matter / a + omega_lambda * a**2 + 1 - omega_matter - omega_lambda)
    if square < 0:
        error_msg = (
            f'The Friedmann equation gives a negative squared expansion rate {square} at a = {a}. Choose a, '
            'omega_matter, and omega_lambda for which it is positive.'
        )
        raise ValidationError(error_msg)
    universe = ScaleFactorElement(
        'universe',
        a=a,
        a_d1=math.sqrt(square),
        H0=H0,
        omega_matter=omega_matter,
        omega_lambda=omega_lambda,
    )
    acceleration = GradientInteraction(
        'acceleration',
        target='universe',
        states='a',
        potential=_get_expansion_potential,
    )
    return System(elements=[universe], interactions=[acceleration])