a ߙfb@sPddlZddlmZddlmZmZmZmZmZm Z m Z m Z m Z ddl mZddlZddlmZddlmZddlmZddlmZddlmZdd lmZdd lmZdd l m!Z!m"Z"m#Z#dd l$m%Z%m&Z&dd l'm(Z(m)Z)erddl*m+Z+ddl,m-Z-m.Z.nddZ+ddZ-ddZ.gdZ/ddgiZ0ej1ej2ej34dej2Z5ej24ej6Z7dde ej8j9j:Z;e dZej?dj9j:Z@ejA4ejBejCZDGdd d e!ZEGd!d"d"e"ZFGd#d$d$eEZGGd%d&d&eFeGZHGd'd(d(eEZIGd)d*d*eFeIZJGd+d,d,eEZKGd-d.d.eFeKZLGd/d0d0eEZMGd1d2d2eEZNdS)3N)abstractmethod) acoscosexpfloorinflogpisinsqrt)Number) HAS_SCIPY) lazyproperty)AstropyUserWarning)scalar_inv_efuncs)units) CosmologyFlatCosmologyMixin Parameter)_validate_non_negative_validate_with_unit)aszarrvectorize_redshift_method)quad) ellipkinchyp2f1cOs tddS)Nz!No module named 'scipy.integrate'ModuleNotFoundErrorargskwargsr"5lib/python3.9/site-packages/astropy/cosmology/flrw.pyrsrcOs tddSNzNo module named 'scipy.special'rrr"r"r#rsrcOs tddSr$rrr"r"r#r!sr) FLRW LambdaCDM FlatLambdaCDMwCDMFlatwCDMw0waCDM Flatw0waCDMwpwaCDMw0wzCDM FlatFLRWMixin*Zscipy?g#Ag@cseZdZdZeddddZedddZed d dZed d d ddZedddZ edde ddZ eddZ de jdde jdfdddfdd Ze jddZ eddZed d!Zed"d#Zed$d%Zed&d'Zed(d)Zed*d+Zed,d-Zed.d/Zed0d1Zed2d3Zd4d5Zd6d7Z d8d9Z!d:d;Z"dd?Z$d@dAZ%dBdCZ&dDdEZ'dFdGZ(dHdIZ)dJdKZ*dLdMZ+dNdOZ,dPdQZ-dRdSZ.dTdUZ/dVdWZ0dXdYZ1dZd[Z2d\d]Z3d^d_Z4e5d`daZ6dbdcZ7dddeZ8dfdgZ9e5dhdiZ:djdkZ;dldmZdtduZ?dvdwZ@dxdyZAdzd{ZBd|d}ZCd~dZDe5ddZEddZFddZGddZHddZIddZJddZKddZLZMS)r%a A class describing an isotropic and homogeneous (Friedmann-Lemaitre-Robertson-Walker) cosmology. This is an abstract base class -- you cannot instantiate examples of this class, but must work with one of its subclasses, such as :class:`~astropy.cosmology.LambdaCDM` or :class:`~astropy.cosmology.wCDM`. Parameters ---------- H0 : float or scalar quantity-like ['frequency'] Hubble constant at z = 0. If a float, must be in [km/sec/Mpc]. Om0 : float Omega matter: density of non-relativistic matter in units of the critical density at z=0. Note that this does not include massive neutrinos. Ode0 : float Omega dark energy: density of dark energy in units of the critical density at z=0. Tcmb0 : float or scalar quantity-like ['temperature'], optional Temperature of the CMB z=0. If a float, must be in [K]. Default: 0 [K]. Setting this to zero will turn off both photons and neutrinos (even massive ones). Neff : float, optional Effective number of Neutrino species. Default 3.04. m_nu : quantity-like ['energy', 'mass'] or array-like, optional Mass of each neutrino species in [eV] (mass-energy equivalency enabled). If this is a scalar Quantity, then all neutrino species are assumed to have that mass. Otherwise, the mass of each species. The actual number of neutrino species (and hence the number of elements of m_nu if it is not scalar) must be the floor of Neff. Typically this means you should provide three neutrino masses unless you are considering something like a sterile neutrino. Ob0 : float or None, optional Omega baryons: density of baryonic matter in units of the critical density at z=0. If this is set to None (the default), any computation that requires its value will raise an exception. name : str or None (optional, keyword-only) Name for this cosmological object. meta : mapping or None (optional, keyword-only) Metadata for the cosmology, e.g., a reference. Notes ----- Class instances are immutable -- you cannot change the parameters' values. That is, all of the above attributes (except meta) are read only. For details on how to create performant custom subclasses, see the documentation on :ref:`astropy-cosmology-fast-integrals`. z7Hubble constant as an `~astropy.units.Quantity` at z=0.z km/(s Mpc)Zscalar)docunit fvalidatez5Omega matter; matter density/critical density at z=0.z non-negativer4r6?Omega dark energy; dark energy density/critical density at z=0.floatz;Temperature of the CMB as `~astropy.units.Quantity` at z=0.ZKelvinz0.4g)r4r5fmtr6z%Number of effective neutrino species.zMass of neutrino species.eV)r4r5Z equivalenciesr:z>Omega baryon; baryonic matter density/critical density at z=0.)r4RQ@Nnamemetacs.tj|| d||_||_||_||_||_||_||_|durFdn |j |j |_ |j j d|_tj|j tj|_|j j t} tt| tj|_t| d} t| tjtjd|_t|j|_ d|_!|j dkr|j"j dkr|j|j |_#|j$}|j%rL|j dkr"|j |_&d|_'n(d|_!d|_&|j |_'|j t()|j |_*n|j +dkrdt,d|j -dkr|j |_&d|_'nhd|_!t.||j krd } t,| t.t(/|j dkd|_&|j |j&|_'t(/|j dkd} || |_*|j"j dkrtt0|j"j d |jj |_1d |j"|_2|j!r`|j*t3|j2}|j |_4|j45|_6|j1|7d|_8nd |j|j1|_8nd |_1d tj9|_2d |_8|j dks|j2j dkrd|_$nP|j!st(:|j&}n,|j&dkr|j*}nt(;t(:|j&|j*j }||j$j<>|_$d|j |j=|j1|j8|_>|j?|_@d|_AdS)Nr?gY@r1FrTz,Invalid (negative) neutrino mass encounteredz$Unexpected number of neutrino massesr3grv}+?C?r=r0r")Bsuper__init__H0Om0Ode0Tcmb0Neffm_nuOb0_Om0_Ob0_Odm0_H0value_hconstctouMpc_hubble_distanceH0units_to_invsQuantity sec_to_GyrGyr _hubble_timecritdens_constgcm_critical_density0r_NeffZ _nneutrinos _massivenu_Tcmb0 _neff_per_nuZ_m_nuZisscalar _nmasslessnuZ _nmassivenunponesZ_massivenu_massmin ValueErrormaxlenZnonzeroa_B_c2_Ogamma0_Tnu0kB_evK_nu_ytolist _nu_y_listnu_relative_density_Onu0Kzerosappendr5_Ode0_Ok0 inv_efunc_inv_efunc_scalar_inv_efunc_scalar_args)selfrFrGrHrIrJrKrLr@rAZH0_sZcd0valueZerrstrwZnu_ym __class__r"r#rEs          z FLRW.__init__cCs.|dur |St|||}||jkr*td|S)zCValidate baryon density to None or positive float > matter density.Nz=baryonic density can not be larger than total matter density.)rrGrj)r~ZparamrQr"r"r#rLs   zFLRW.Ob0cCs|jS)z?Omega dark matter; dark matter density/critical density at z=0.)rOr~r"r"r#Odm0sz FLRW.Odm0cCs|jS)zIOmega curvature; the effective curvature density/critical density at z=0.)rzrr"r"r#Ok0$szFLRW.Ok0cCs|jS)zKTemperature of the neutrino background as `~astropy.units.Quantity` at z=0.)rorr"r"r#Tnu0)sz FLRW.Tnu0cCs|jjdkrdS|jS)z?Does this cosmology have at least one massive neutrino species?rF)rorQrcrr"r"r#has_massive_nu.s zFLRW.has_massive_nucCs|jS)z:Dimensionless Hubble constant: h = H_0 / 100 [km/sec/Mpc].)rRrr"r"r#h5szFLRW.hcCs|jS)z)Hubble time as `~astropy.units.Quantity`.)r]rr"r"r# hubble_time:szFLRW.hubble_timecCs|jS)z-Hubble distance as `~astropy.units.Quantity`.)rXrr"r"r#hubble_distance?szFLRW.hubble_distancecCs|jS)z5Critical density as `~astropy.units.Quantity` at z=0.)rarr"r"r#critical_density0DszFLRW.critical_density0cCs|jS)ztfdd|D}td|Stjdt |dd}td|SdS)a{Evaluates the redshift dependence of the dark energy density. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- I : ndarray or float The scaling of the energy density of dark energy with redshift. Returns `float` if the input is scalar. Notes ----- The scaling factor, I, is defined by :math:`\rho(z) = \rho_0 I`, and is given by .. math:: I = \exp \left( 3 \int_{a}^1 \frac{ da^{\prime} }{ a^{\prime} } \left[ 1 + w\left( a^{\prime} \right) \right] \right) The actual integral used is rewritten from [1]_ to be in terms of z. It will generally helpful for subclasses to overload this method if the integral can be done analytically for the particular dark energy equation of state that they implement. References ---------- .. [1] Linder, E. (2003). Exploring the Expansion History of the Universe. Phys. Rev. Lett., 90, 091301. cs&g|]}tjdtd|dqS)rr)rrr).0redshiftrr"r# sz)FLRW.de_density_scale..r1rr0N) r isinstancer rgZgenericZarrayrrrr)r~rZivalr"rr#rs)zFLRW.de_density_scalecCsj|jr|jd||}n |j|j}t|d}t|d|||j||j|j | |S)a\Function used to calculate H(z), the Hubble parameter. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- E : ndarray or float The redshift scaling of the Hubble constant. Returns `float` if the input is scalar. Defined such that :math:`H(z) = H_0 E(z)`. Notes ----- It is not necessary to override this method, but if de_density_scale takes a particularly simple form, it may be advantageous to. rr0rB) rcrnrtrurrgr rMrzryrr~rZOrzp1r"r"r#efuncs  "z FLRW.efunccCsh|jr|jd||}n |j|j}t|d}|d|||j||j|j||dS)a]Inverse of ``efunc``. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- E : ndarray or float The redshift scaling of the inverse Hubble constant. Returns `float` if the input is scalar. rr0rB) rcrnrtrurrMrzryrrr"r"r#r{s  zFLRW.inv_efunccCs|j|g|jR|dS)aIntegrand of the lookback time (equation 30 of [1]_). Parameters ---------- z : float Input redshift. Returns ------- I : float The integrand for the lookback time. References ---------- .. [1] Hogg, D. (1999). Distance measures in cosmology, section 11. arXiv e-prints, astro-ph/9905116. r0)r|r}rr"r"r#_lookback_time_integrand_scalarsz$FLRW._lookback_time_integrand_scalarcCst|}|||dS)aIntegrand of the lookback time (equation 30 of [1]_). Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- I : float or array The integrand for the lookback time. References ---------- .. [1] Hogg, D. (1999). Distance measures in cosmology, section 11. arXiv e-prints, astro-ph/9905116. r0rr{rr"r"r#lookback_time_integrandszFLRW.lookback_time_integrandcCs$|j}|dd|j|g|RS)aIntegrand of the absorption distance [1]_. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- X : float The integrand for the absorption distance. References ---------- .. [1] Hogg, D. (1999). Distance measures in cosmology, section 11. arXiv e-prints, astro-ph/9905116. r0rB)r}r|)r~rr r"r"r#_abs_distance_integrand_scalarsz#FLRW._abs_distance_integrand_scalarcCst|}|dd||S)aIntegrand of the absorption distance [1]_. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- X : float or array The integrand for the absorption distance. References ---------- .. [1] Hogg, D. (1999). Distance measures in cosmology, section 11. arXiv e-prints, astro-ph/9905116. r0rBrrr"r"r#abs_distance_integrand&szFLRW.abs_distance_integrandcCs|j||S)aMHubble parameter (km/s/Mpc) at redshift ``z``. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- H : `~astropy.units.Quantity` ['frequency'] Hubble parameter at each input redshift. )rPrrr"r"r#H;s zFLRW.HcCsdt|dS)aScale factor at redshift ``z``. The scale factor is defined as :math:`a = 1 / (1 + z)`. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- a : ndarray or float Scale factor at each input redshift. Returns `float` if the input is scalar. r0)rrr"r"r# scale_factorJszFLRW.scale_factorcCs ||S)a)Lookback time in Gyr to redshift ``z``. The lookback time is the difference between the age of the Universe now and the age at redshift ``z``. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- t : `~astropy.units.Quantity` ['time'] Lookback time in Gyr to each input redshift. See Also -------- z_at_value : Find the redshift corresponding to a lookback time. )_lookback_timerr"r"r# lookback_time\szFLRW.lookback_timecCs|j||S)aLookback time in Gyr to redshift ``z``. The lookback time is the difference between the age of the Universe now and the age at redshift ``z``. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- t : `~astropy.units.Quantity` ['time'] Lookback time in Gyr to each input redshift. )r]_integral_lookback_timerr"r"r#rrszFLRW._lookback_timecCst|jd|dS)aLookback time to redshift ``z``. Value in units of Hubble time. The lookback time is the difference between the age of the Universe now and the age at redshift ``z``. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- t : float or ndarray Lookback time to each input redshift in Hubble time units. Returns `float` if input scalar, `~numpy.ndarray` otherwise. r)rrrr"r"r#rszFLRW._integral_lookback_timecCs||tjtjS)a The lookback distance is the light travel time distance to a given redshift. It is simply c * lookback_time. It may be used to calculate the proper distance between two redshifts, e.g. for the mean free path to ionizing radiation. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- d : `~astropy.units.Quantity` ['length'] Lookback distance in Mpc )rrSrTrUrVrWrr"r"r#lookback_distanceszFLRW.lookback_distancecCs ||S)aAge of the universe in Gyr at redshift ``z``. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- t : `~astropy.units.Quantity` ['time'] The age of the universe in Gyr at each input redshift. See Also -------- z_at_value : Find the redshift corresponding to an age. )_agerr"r"r#ageszFLRW.agecCs|j||S)aAge of the universe in Gyr at redshift ``z``. This internal function exists to be re-defined for optimizations. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- t : `~astropy.units.Quantity` ['time'] The age of the universe in Gyr at each input redshift. )r] _integral_agerr"r"r#rsz FLRW._agecCst|j|tjdS)aEAge of the universe at redshift ``z``. Value in units of Hubble time. Calculated using explicit integration. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- t : float or ndarray The age of the universe at each input redshift in Hubble time units. Returns `float` if input scalar, `~numpy.ndarray` otherwise. See Also -------- z_at_value : Find the redshift corresponding to an age. r)rrrgrrr"r"r#rszFLRW._integral_agecCs|j||dS)aVCritical density in grams per cubic cm at redshift ``z``. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- rho : `~astropy.units.Quantity` Critical density in g/cm^3 at each input redshift. rB)rarrr"r"r#critical_densityszFLRW.critical_densitycCs |d|S)aComoving line-of-sight distance in Mpc at a given redshift. The comoving distance along the line-of-sight between two objects remains constant with time for objects in the Hubble flow. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- d : `~astropy.units.Quantity` ['length'] Comoving distance in Mpc to each input redshift. r)_comoving_distance_z1z2rr"r"r#comoving_distanceszFLRW.comoving_distancecCs |||S)a$ Comoving line-of-sight distance in Mpc between objects at redshifts ``z1`` and ``z2``. The comoving distance along the line-of-sight between two objects remains constant with time for objects in the Hubble flow. Parameters ---------- z1, z2 : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshifts. Returns ------- d : `~astropy.units.Quantity` ['length'] Comoving distance in Mpc between each input redshift. ) _integral_comoving_distance_z1z2r~z1z2r"r"r#rszFLRW._comoving_distance_z1z2rB)ZnincCst|j|||jddS)a` Comoving line-of-sight distance between objects at redshifts ``z1`` and ``z2``. Value in Mpc. The comoving distance along the line-of-sight between two objects remains constant with time for objects in the Hubble flow. Parameters ---------- z1, z2 : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshifts. Returns ------- d : float or ndarray Comoving distance in Mpc between each input redshift. Returns `float` if input scalar, `~numpy.ndarray` otherwise. )r r)rr|r}rr"r"r#'_integral_comoving_distance_z1z2_scalarsz,FLRW._integral_comoving_distance_z1z2_scalarcCs|j|||S)a Comoving line-of-sight distance in Mpc between objects at redshifts ``z1`` and ``z2``. The comoving distance along the line-of-sight between two objects remains constant with time for objects in the Hubble flow. Parameters ---------- z1, z2 : Quantity-like ['redshift'] or array-like Input redshifts. Returns ------- d : `~astropy.units.Quantity` ['length'] Comoving distance in Mpc between each input redshift. )rXrrr"r"r#r2sz%FLRW._integral_comoving_distance_z1z2cCs |d|S)aComoving transverse distance in Mpc at a given redshift. This value is the transverse comoving distance at redshift ``z`` corresponding to an angular separation of 1 radian. This is the same as the comoving distance if :math:`\Omega_k` is zero (as in the current concordance Lambda-CDM model). Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- d : `~astropy.units.Quantity` ['length'] Comoving transverse distance in Mpc at each input redshift. Notes ----- This quantity is also called the 'proper motion distance' in some texts. r)"_comoving_transverse_distance_z1z2rr"r"r#comoving_transverse_distanceEsz!FLRW.comoving_transverse_distancecCsx|j}|||}|dkr|Stt|}|j}|dkrV||t||j|jS||t||j|jSdS)a Comoving transverse distance in Mpc between two redshifts. This value is the transverse comoving distance at redshift ``z2`` as seen from redshift ``z1`` corresponding to an angular separation of 1 radian. This is the same as the comoving distance if :math:`\Omega_k` is zero (as in the current concordance Lambda-CDM model). Parameters ---------- z1, z2 : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshifts. Returns ------- d : `~astropy.units.Quantity` ['length'] Comoving transverse distance in Mpc between input redshift. Notes ----- This quantity is also called the 'proper motion distance' in some texts. rN) rzrr absrXrgZsinhrQr )r~rrrZdcZsqrtOk0dhr"r"r#r]s  z'FLRW._comoving_transverse_distance_z1z2cCst|}|||dS)aAngular diameter distance in Mpc at a given redshift. This gives the proper (sometimes called 'physical') transverse distance corresponding to an angle of 1 radian for an object at redshift ``z`` ([1]_, [2]_, [3]_). Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- d : `~astropy.units.Quantity` ['length'] Angular diameter distance in Mpc at each input redshift. References ---------- .. [1] Weinberg, 1972, pp 420-424; Weedman, 1986, pp 421-424. .. [2] Weedman, D. (1986). Quasar astronomy, pp 65-67. .. [3] Peebles, P. (1993). Principles of Physical Cosmology, pp 325-327. r0rrrr"r"r#angular_diameter_distance~szFLRW.angular_diameter_distancecCst|}|d||S)aLuminosity distance in Mpc at redshift ``z``. This is the distance to use when converting between the bolometric flux from an object at redshift ``z`` and its bolometric luminosity [1]_. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- d : `~astropy.units.Quantity` ['length'] Luminosity distance in Mpc at each input redshift. See Also -------- z_at_value : Find the redshift corresponding to a luminosity distance. References ---------- .. [1] Weinberg, 1972, pp 420-424; Weedman, 1986, pp 60-62. r0rrr"r"r#luminosity_distanceszFLRW.luminosity_distancecCsNt|t|}}t||kr:td|d|dt||||dS)aAngular diameter distance between objects at 2 redshifts. Useful for gravitational lensing, for example computing the angular diameter distance between a lensed galaxy and the foreground lens. Parameters ---------- z1, z2 : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshifts. For most practical applications such as gravitational lensing, ``z2`` should be larger than ``z1``. The method will work for ``z2 < z1``; however, this will return negative distances. Returns ------- d : `~astropy.units.Quantity` The angular diameter distance between each input redshift pair. Returns scalar if input is scalar, array else-wise. zSecond redshift(s) z2 (z%) is less than first redshift(s) z1 (z).r0)rrganywarningswarnrrrr"r"r#angular_diameter_distance_z1z2s z#FLRW.angular_diameter_distance_z1z2cCst|jd|dS)a6Absorption distance at redshift ``z``. This is used to calculate the number of objects with some cross section of absorption and number density intersecting a sightline per unit redshift path ([1]_, [2]_). Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- d : float or ndarray Absorption distance (dimensionless) at each input redshift. Returns `float` if input scalar, `~numpy.ndarray` otherwise. References ---------- .. [1] Hogg, D. (1999). Distance measures in cosmology, section 11. arXiv e-prints, astro-ph/9905116. .. [2] Bahcall, John N. and Peebles, P.J.E. 1969, ApJ, 156L, 7B r)rrrr"r"r#absorption_distanceszFLRW.absorption_distancecCs,dtt||jd}t|tjS)aFDistance modulus at redshift ``z``. The distance modulus is defined as the (apparent magnitude - absolute magnitude) for an object at redshift ``z``. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- distmod : `~astropy.units.Quantity` ['length'] Distance modulus at each input redshift, in magnitudes. See Also -------- z_at_value : Find the redshift corresponding to a distance modulus. g@g9@)rgZlog10rrrQrVrZZmag)r~rvalr"r"r#distmodsz FLRW.distmodcCs|j}|dkr$dt||dS|jj}||j}dt|dd|tjd}||t d|||d}t t |||}|dkr||dt t |t |S||dt t |t |Sd S) a=Comoving volume in cubic Mpc at redshift ``z``. This is the volume of the universe encompassed by redshifts less than ``z``. For the case of :math:`\Omega_k = 0` it is a sphere of radius `comoving_distance` but it is less intuitive if :math:`\Omega_k` is not. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- V : `~astropy.units.Quantity` Comoving volume in :math:`Mpc^3` at each input redshift. rgUUUUUU?r1g@@rrBr0N) rzr rrXrQrrVrWrgr rarcsinharcsin)r~rrrdmZterm1Zterm2Zterm3r"r"r#comoving_volumes """zFLRW.comoving_volumecCs(||}|j|d||tj>S)aDifferential comoving volume at redshift z. Useful for calculating the effective comoving volume. For example, allows for integration over a comoving volume that has a sensitivity function that changes with redshift. The total comoving volume is given by integrating ``differential_comoving_volume`` to redshift ``z`` and multiplying by a solid angle. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- dV : `~astropy.units.Quantity` Differential comoving volume per redshift per steradian at each input redshift. r)rrXrrVZ steradian)r~rrr"r"r#differential_comoving_volume#s z!FLRW.differential_comoving_volumecCs||tjttjS)a Separation in transverse comoving kpc corresponding to an arcminute at redshift ``z``. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- d : `~astropy.units.Quantity` ['length'] The distance in comoving kpc corresponding to an arcmin at each input redshift. )rrUrVkpcarcmin_in_radiansarcminrr"r"r#kpc_comoving_per_arcmin:s zFLRW.kpc_comoving_per_arcmincCs||tjttjS)a Separation in transverse proper kpc corresponding to an arcminute at redshift ``z``. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- d : `~astropy.units.Quantity` ['length'] The distance in proper kpc corresponding to an arcmin at each input redshift. )rrUrVrrrrr"r"r#kpc_proper_per_arcminMs zFLRW.kpc_proper_per_arcmincCstj||tjtS)a Angular separation in arcsec corresponding to a comoving kpc at redshift ``z``. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- theta : `~astropy.units.Quantity` ['angle'] The angular separation in arcsec corresponding to a comoving kpc at each input redshift. )rVarcsecrrUrarcsec_in_radiansrr"r"r#arcsec_per_kpc_comoving`szFLRW.arcsec_per_kpc_comovingcCstj||tjtS)a Angular separation in arcsec corresponding to a proper kpc at redshift ``z``. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- theta : `~astropy.units.Quantity` ['angle'] The angular separation in arcsec corresponding to a proper kpc at each input redshift. )rVrrrUrrrr"r"r#arcsec_per_kpc_propersszFLRW.arcsec_per_kpc_proper)N__name__ __module__ __qualname____doc__rrFrGrHrIrJrVZ mass_energyrKrLrvr;rEZ validatorpropertyrrrrrrrrrrrrrrrrrrrrrrtrrrr{rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr __classcell__r"r"rr#r%9s;               B2   !  r%csDeZdZdZedddZfddZfddZfd d ZZ S) r.a Mixin class for flat FLRW cosmologies. Do NOT instantiate directly. Must precede the base class in the multiple-inheritance so that this mixin's ``__init__`` proceeds the base class'. Note that all instances of ``FlatFLRWMixin`` are flat, but not all flat cosmologies are instances of ``FlatFLRWMixin``. As example, ``LambdaCDM`` **may** be flat (for the a specific set of parameter values), but ``FlatLambdaCDM`` **will** be flat. r8T)r4Zderivedcs"td|jjvrtddS)NrHz>subclasses of `FlatFLRWMixin` cannot have `Ode0` in `__init__`)rD__init_subclass__Z_init_signature parameters TypeError)clsrr"r#rs  zFlatFLRWMixin.__init_subclass__cs4tj|i|d|j|j|j|_d|_dS)Nr0r=)rDrErMrnruryrz)r~r kwrr"r#rEszFlatFLRWMixin.__init__cs|ttrtSddjddD}j|vr@tStjtjkovt fddjDovj dk}|S)a>flat-FLRW equivalence. Use ``.is_equivalent()`` for actual check! Parameters ---------- other : `~astropy.cosmology.FLRW` subclass instance The object in which to compare. Returns ------- bool or `NotImplemented` `True` if 'other' is of the same class / non-flat class (e.g. ``FlatLambdaCDM`` and ``LambdaCDM``) has matching parameters and parameter values. `False` if 'other' is of the same class but has different parameters. `NotImplemented` otherwise. cSs"g|]}t|tr|tur|qSr") issubclassr%)rrTr"r"r#rsz+FlatFLRWMixin.__equiv__..rNc3s(|] }tt|t|kVqdS)N)rgallgetattr)rrotherr~r"r# sz*FlatFLRWMixin.__equiv__..r=) rr.rD __equiv__rmroNotImplementedsetZ__all_parameters__rr)r~rZcomparable_classesZ params_eqrrr#rs   zFlatFLRWMixin.__equiv__) rrrrrrHrrErrr"r"rr#r.s   r.cseZdZdZdejddejdfdddfdd Zdd Zd d Z d d Z ddZ ddZ ddZ ddZddZddZddZddZddZd d!Zd"d#Zd$d%Zd&d'ZZS)(r&aKFLRW cosmology with a cosmological constant and curvature. This has no additional attributes beyond those of FLRW. Parameters ---------- H0 : float or scalar quantity-like ['frequency'] Hubble constant at z = 0. If a float, must be in [km/sec/Mpc]. Om0 : float Omega matter: density of non-relativistic matter in units of the critical density at z=0. Ode0 : float Omega dark energy: density of the cosmological constant in units of the critical density at z=0. Tcmb0 : float or scalar quantity-like ['temperature'], optional Temperature of the CMB z=0. If a float, must be in [K]. Default: 0 [K]. Setting this to zero will turn off both photons and neutrinos (even massive ones). Neff : float, optional Effective number of Neutrino species. Default 3.04. m_nu : quantity-like ['energy', 'mass'] or array-like, optional Mass of each neutrino species in [eV] (mass-energy equivalency enabled). If this is a scalar Quantity, then all neutrino species are assumed to have that mass. Otherwise, the mass of each species. The actual number of neutrino species (and hence the number of elements of m_nu if it is not scalar) must be the floor of Neff. Typically this means you should provide three neutrino masses unless you are considering something like a sterile neutrino. Ob0 : float or None, optional Omega baryons: density of baryonic matter in units of the critical density at z=0. If this is set to None (the default), any computation that requires its value will raise an exception. name : str or None (optional, keyword-only) Name for this cosmological object. meta : mapping or None (optional, keyword-only) Metadata for the cosmology, e.g., a reference. Examples -------- >>> from astropy.cosmology import LambdaCDM >>> cosmo = LambdaCDM(H0=70, Om0=0.3, Ode0=0.7) The comoving distance in Mpc at redshift z: >>> z = 0.5 >>> dc = cosmo.comoving_distance(z) r=r>Nr?c stj||||||||| d |jjdkrbtj|_|j|j|j f|_ |j dkrX| q|j |_ nV|jstj|_|j|j|j |j|jf|_ n*tj|_|j|j|j |j|j|j|jf|_ dSN rFrGrHrIrJrKrLr@rAr)rDrErdrQrZlcdm_inv_efunc_norelr|rMryrzr}_optimize_flat_norad _elliptic_comoving_distance_z1z2rrcZlcdm_inv_efunc_nomnurnruZlcdm_inv_efuncrerfrs) r~rFrGrHrIrJrKrLr@rArr"r#rEs(       zLambdaCDM.__init__cCsd|jdkr$|j|_|j|_|j|_n<|jdkrH|j|_|j|_|j |_n|j |_|j |_|j |_dS)z>Set optimizations for flat LCDM cosmologies with no radiation.rrN) rM_dS_comoving_distance_z1z2r_dS_ager_dS_lookback_timer_EdS_comoving_distance_z1z2_EdS_age_EdS_lookback_time&_hypergeometric_comoving_distance_z1z2 _flat_age_flat_lookback_timerr"r"r#rs    zLambdaCDM._optimize_flat_noradcCs&t|}dt|dr t|jndS)aReturns dark energy equation of state at redshift ``z``. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- w : ndarray or float The dark energy equation of state. Returns `float` if the input is scalar. Notes ----- The dark energy equation of state is defined as :math:`w(z) = P(z)/\rho(z)`, where :math:`P(z)` is the pressure at redshift z and :math:`\rho(z)` is the density at redshift z, both in units where c=1. Here this is :math:`w(z) = -1`. rr0rrrgrhrrr"r"r#r/sz LambdaCDM.wcCs"t|}t|drt|jSdS)a*Evaluates the redshift dependence of the dark energy density. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- I : ndarray or float The scaling of the energy density of dark energy with redshift. Returns `float` if the input is scalar. Notes ----- The scaling factor, I, is defined by :math:`\rho(z) = \rho_0 I`, and in this case is given by :math:`I = 1`. rr0rrr"r"r#rGszLambdaCDM.de_density_scalec Csjzt||\}}Wn.tyB}ztd|WYd}~n d}~00|jdksb|jdksb|jdkrn|||Sd|jd|j|jd}|t|}|dksd|krZdd}t||d t ||dd }d ||d |d}t |d|d} d t | } d| |d d|d | } ||j|j||| |} ||j|j||| |} nd|kr.|dkr.|j|jkr.d d}t t d |d}t dt t d |d}d d ||}d d d|}d d ||}dt ||} ||||} ||j|j|||} ||j|j|||} n |||S|j t t|j}|| t| | t| | S)aComoving transverse distance in Mpc between two redshifts. This value is the transverse comoving distance at redshift ``z`` corresponding to an angular separation of 1 radian. This is the same as the comoving distance if :math:`\Omega_k` is zero. For :math:`\Omega_{rad} = 0` the comoving distance can be directly calculated as an elliptic integral [1]_. Not valid or appropriate for flat cosmologies (Ok0=0). Parameters ---------- z1, z2 : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshifts. Returns ------- d : `~astropy.units.Quantity` ['length'] Comoving distance in Mpc between each input redshift. References ---------- .. [1] Kantowski, R., Kao, J., & Thomas, R. (2000). Distance-Redshift in Inhomogeneous FLRW. arXiv e-prints, astro-ph/0002334. z1 and z2 have different shapesNrg+rBr1cSsFt|d|t|||||d|t||||SNr0)rgZarccosr)rGrkappay1Arr"r"r#phi_zs"z9LambdaCDM._elliptic_comoving_distance_z1z2..phi_zrUUUUUU?rr3cSs,tt|||d|t||Sr)rgrr r)rGrry2rr"r"r#rs)rgbroadcast_arraysrjrMryrzrrpowr rrr rXr)r~rrebrrZv_krrr_Zk2Zphi_z1Zphi_z2ZybZycrZy3 prefactorr"r"r#r]s>   "  " z*LambdaCDM._elliptic_comoving_distance_z1z2c CsRzt||\}}Wn.tyB}ztd|WYd}~n d}~00|j||S)a Comoving line-of-sight distance in Mpc between objects at redshifts ``z1`` and ``z2`` in a flat, :math:`\Omega_{\Lambda}=1` cosmology (de Sitter). The comoving distance along the line-of-sight between two objects remains constant with time for objects in the Hubble flow. The de Sitter case has an analytic solution. Parameters ---------- z1, z2 : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshifts. Must be 1D or scalar. Returns ------- d : `~astropy.units.Quantity` ['length'] Comoving distance in Mpc between each input redshift. rNrgr rjrX)r~rrr r"r"r#rs  z$LambdaCDM._dS_comoving_distance_z1z2c Csjzt||\}}Wn.tyB}ztd|WYd}~n d}~00d|j}||dd|ddS)a Comoving line-of-sight distance in Mpc between objects at redshifts ``z1`` and ``z2`` in a flat, :math:`\Omega_M=1` cosmology (Einstein - de Sitter). The comoving distance along the line-of-sight between two objects remains constant with time for objects in the Hubble flow. For :math:`\Omega_M=1`, :math:`\Omega_{rad}=0` the comoving distance has an analytic solution. Parameters ---------- z1, z2 : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshifts. Must be 1D or scalar. Returns ------- d : `~astropy.units.Quantity` ['length'] Comoving distance in Mpc between each input redshift. rNrBr0rr)r~rrr r r"r"r#rs   z%LambdaCDM._EdS_comoving_distance_z1z2c Cszt||\}}Wn.tyB}ztd|WYd}~n d}~00d|j|jd}|jt||j}||||d|||dS)a Comoving line-of-sight distance in Mpc between objects at redshifts ``z1`` and ``z2``. The comoving distance along the line-of-sight between two objects remains constant with time for objects in the Hubble flow. For :math:`\Omega_{rad} = 0` the comoving distance can be directly calculated as a hypergeometric function [1]_. Parameters ---------- z1, z2 : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshifts. Returns ------- d : `~astropy.units.Quantity` ['length'] Comoving distance in Mpc between each input redshift. References ---------- .. [1] Baes, M., Camps, P., & Van De Putte, D. (2017). Analytical expressions and numerical evaluation of the luminosity distance in a flat cosmology. MNRAS, 468(1), 927-930. rNrrr0)rgr rjrMrXr _T_hypergeometric)r~rrr sr r"r"r#rs z0LambdaCDM._hypergeometric_comoving_distance_z1z2cCs"dt|tddd|d S)aCompute value using Gauss Hypergeometric function 2F1. .. math:: T(x) = 2 \sqrt(x) _{2}F_{1}\left(\frac{1}{6}, \frac{1}{2}; \frac{7}{6}; -x^3 \right) Notes ----- The :func:`scipy.special.hyp2f1` code already implements the hypergeometric transformation suggested by Baes et al. [1]_ for use in actual numerical evaulations. References ---------- .. [1] Baes, M., Camps, P., & Van De Putte, D. (2017). Analytical expressions and numerical evaluation of the luminosity distance in a flat cosmology. MNRAS, 468(1), 927-930. rBgUUUUUU?g?g?r1)rgr r)r~xr"r"r#rszLambdaCDM._T_hypergeometriccCs(t|trtntj|ttd}|j|S)aAge of the universe in Gyr at redshift ``z``. The age of a de Sitter Universe is infinite. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- t : `~astropy.units.Quantity` ['time'] The age of the universe in Gyr at each input redshift. )Zdtype)rr rrgZ full_liker9r])r~rtr"r"r#rszLambdaCDM._dS_agecCsd|jt|ddS)Age of the universe in Gyr at redshift ``z``. For :math:`\Omega_{rad} = 0` (:math:`T_{CMB} = 0`; massless neutrinos) the age can be directly calculated as an elliptic integral [1]_. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- t : `~astropy.units.Quantity` ['time'] The age of the universe in Gyr at each input redshift. References ---------- .. [1] Thomas, R., & Kantowski, R. (2000). Age-redshift relation for standard cosmology. PRD, 62(10), 103507. UUUUUU?r0g)r]rrr"r"r#r/szLambdaCDM._EdS_agecCsVd|jtjd|j}ttjd|jddt|dd}||jS)rrryr0r1)r]rgZemathr rMrrreal)r~rr argr"r"r#rFs0zLambdaCDM._flat_agecCs|d||S)aLookback time in Gyr to redshift ``z``. The lookback time is the difference between the age of the Universe now and the age at redshift ``z``. For :math:`\Omega_{rad} = 0` (:math:`T_{CMB} = 0`; massless neutrinos) the age can be directly calculated as an elliptic integral. The lookback time is here calculated based on the ``age(0) - age(z)``. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- t : `~astropy.units.Quantity` ['time'] Lookback time in Gyr to each input redshift. r)rrr"r"r#raszLambdaCDM._EdS_lookback_timecCs|jtt|dS)aLookback time in Gyr to redshift ``z``. The lookback time is the difference between the age of the Universe now and the age at redshift ``z``. For :math:`\Omega_{rad} = 0` (:math:`T_{CMB} = 0`; massless neutrinos) the age can be directly calculated. .. math:: a = exp(H * t) \ \text{where t=0 at z=0} t = (1/H) (ln 1 - ln a) = (1/H) (0 - ln (1/(1+z))) = (1/H) ln(1+z) Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- t : `~astropy.units.Quantity` ['time'] Lookback time in Gyr to each input redshift. r0)r]rgrrrr"r"r#rwszLambdaCDM._dS_lookback_timecCs|d||S)aLookback time in Gyr to redshift ``z``. The lookback time is the difference between the age of the Universe now and the age at redshift ``z``. For :math:`\Omega_{rad} = 0` (:math:`T_{CMB} = 0`; massless neutrinos) the age can be directly calculated. The lookback time is here calculated based on the ``age(0) - age(z)``. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- t : `~astropy.units.Quantity` ['time'] Lookback time in Gyr to each input redshift. r)rrr"r"r#rszLambdaCDM._flat_lookback_timecCs`|jr|jd||}n |j|j}t|d}t|d|||j||j|j S)Function used to calculate H(z), the Hubble parameter. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- E : ndarray or float The redshift scaling of the Hubble constant. Returns `float` if the input is scalar. Defined such that :math:`H(z) = H_0 E(z)`. r0rB) rcrnrtrurrgr rMrzryrr"r"r#rs   zLambdaCDM.efunccCs^|jr|jd||}n |j|j}t|d}|d|||j||j|jdS)Function used to calculate :math:`\frac{1}{H_z}`. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- E : ndarray or float The inverse redshift scaling of the Hubble constant. Returns `float` if the input is scalar. Defined such that :math:`H_z = H_0 / E`. rr0rBr)rcrnrtrurrMrzryrr"r"r#r{s   zLambdaCDM.inv_efunc)rrrrrVrvr;rErrrrrrrrrrrrrrrr{rr"r"rr#r&s,8J&r&csNeZdZdZdejddejdfdddfdd Zdd Zd d Z Z S) r'aFLRW cosmology with a cosmological constant and no curvature. This has no additional attributes beyond those of FLRW. Parameters ---------- H0 : float or scalar quantity-like ['frequency'] Hubble constant at z = 0. If a float, must be in [km/sec/Mpc]. Om0 : float Omega matter: density of non-relativistic matter in units of the critical density at z=0. Tcmb0 : float or scalar quantity-like ['temperature'], optional Temperature of the CMB z=0. If a float, must be in [K]. Default: 0 [K]. Setting this to zero will turn off both photons and neutrinos (even massive ones). Neff : float, optional Effective number of Neutrino species. Default 3.04. m_nu : quantity-like ['energy', 'mass'] or array-like, optional Mass of each neutrino species in [eV] (mass-energy equivalency enabled). If this is a scalar Quantity, then all neutrino species are assumed to have that mass. Otherwise, the mass of each species. The actual number of neutrino species (and hence the number of elements of m_nu if it is not scalar) must be the floor of Neff. Typically this means you should provide three neutrino masses unless you are considering something like a sterile neutrino. Ob0 : float or None, optional Omega baryons: density of baryonic matter in units of the critical density at z=0. If this is set to None (the default), any computation that requires its value will raise an exception. name : str or None (optional, keyword-only) Name for this cosmological object. meta : mapping or None (optional, keyword-only) Metadata for the cosmology, e.g., a reference. Examples -------- >>> from astropy.cosmology import FlatLambdaCDM >>> cosmo = FlatLambdaCDM(H0=70, Om0=0.3) The comoving distance in Mpc at redshift z: >>> z = 0.5 >>> dc = cosmo.comoving_distance(z) r=r>Nr?c stj||d||||||d |jjdkrJtj|_|j|jf|_ | nN|j srtj |_|j|j|j |jf|_ n&tj|_|j|j|j |j|j|jf|_ dS)Nr=rr)rDrErdrQrZflcdm_inv_efunc_norelr|rMryr}rrcZflcdm_inv_efunc_nomnurnruZflcdm_inv_efuncrerfrs) r~rFrGrIrJrKrLr@rArr"r#rEs$   zFlatLambdaCDM.__init__cCsV|jr|jd||}n |j|j}t|d}t|d|||j|jS)rrr0r1) rcrnrtrurrgr rMryrr"r"r#r's   zFlatLambdaCDM.efunccCsT|jr|jd||}n |j|j}t|d}|d|||j|jdS)rr0r1r)rcrnrtrurrMryrr"r"r#r{@s   zFlatLambdaCDM.inv_efunc rrrrrVrvr;rErr{rr"r"rr#r's4r'csleZdZdZedddZddejddejdfddd fd d Z d d Z ddZ ddZ ddZ ZS)r(a FLRW cosmology with a constant dark energy equation of state and curvature. This has one additional attribute beyond those of FLRW. Parameters ---------- H0 : float or scalar quantity-like ['frequency'] Hubble constant at z = 0. If a float, must be in [km/sec/Mpc]. Om0 : float Omega matter: density of non-relativistic matter in units of the critical density at z=0. Ode0 : float Omega dark energy: density of dark energy in units of the critical density at z=0. w0 : float, optional Dark energy equation of state at all redshifts. This is pressure/density for dark energy in units where c=1. A cosmological constant has w0=-1.0. Tcmb0 : float or scalar quantity-like ['temperature'], optional Temperature of the CMB z=0. If a float, must be in [K]. Default: 0 [K]. Setting this to zero will turn off both photons and neutrinos (even massive ones). Neff : float, optional Effective number of Neutrino species. Default 3.04. m_nu : quantity-like ['energy', 'mass'] or array-like, optional Mass of each neutrino species in [eV] (mass-energy equivalency enabled). If this is a scalar Quantity, then all neutrino species are assumed to have that mass. Otherwise, the mass of each species. The actual number of neutrino species (and hence the number of elements of m_nu if it is not scalar) must be the floor of Neff. Typically this means you should provide three neutrino masses unless you are considering something like a sterile neutrino. Ob0 : float or None, optional Omega baryons: density of baryonic matter in units of the critical density at z=0. If this is set to None (the default), any computation that requires its value will raise an exception. name : str or None (optional, keyword-only) Name for this cosmological object. meta : mapping or None (optional, keyword-only) Metadata for the cosmology, e.g., a reference. Examples -------- >>> from astropy.cosmology import wCDM >>> cosmo = wCDM(H0=70, Om0=0.3, Ode0=0.7, w0=-0.9) The comoving distance in Mpc at redshift z: >>> z = 0.5 >>> dc = cosmo.comoving_distance(z) zDark energy equation of state.r9r7rr=r>Nr?c  stj|||||||| | d ||_|jjdkrPtj|_|j|j |j |j f|_ n^|j stj|_|j|j |j |j|j|j f|_ n.tj|_|j|j |j |j|j|j|j|j f|_ dSr)rDrEw0rdrQrZwcdm_inv_efunc_norelr|rMryrz_w0r}rcZwcdm_inv_efunc_nomnurnruZwcdm_inv_efuncrerfrs) r~rFrGrHrrIrJrKrLr@rArr"r#rEs*     z wCDM.__init__cCs(t|}|jt|dr"t|jndS)aReturns dark energy equation of state at redshift ``z``. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- w : ndarray or float The dark energy equation of state Returns `float` if the input is scalar. Notes ----- The dark energy equation of state is defined as :math:`w(z) = P(z)/\rho(z)`, where :math:`P(z)` is the pressure at redshift z and :math:`\rho(z)` is the density at redshift z, both in units where c=1. Here this is :math:`w(z) = w_0`. rr0)rrrrgrhrrr"r"r#rszwCDM.wcCst|ddd|jS)aZEvaluates the redshift dependence of the dark energy density. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- I : ndarray or float The scaling of the energy density of dark energy with redshift. Returns `float` if the input is scalar. Notes ----- The scaling factor, I, is defined by :math:`\rho(z) = \rho_0 I`, and in this case is given by :math:`I = \left(1 + z\right)^{3\left(1 + w_0\right)}` r0@)rrrr"r"r#rszwCDM.de_density_scalecCsr|jr|jd||}n |j|j}t|d}t|d|||j||j|j |dd|j S)rr0rBr) rcrnrtrurrgr rMrzryrrr"r"r#rs  "z wCDM.efunccCsp|jr|jd||}n |j|j}t|d}|d|||j||j|j|dd|jdS)rr0rBrr) rcrnrtrurrMrzryrrr"r"r#r{s  zwCDM.inv_efunc)rrrrrrrVrvr;rErrrr{rr"r"rr#r(Ws>   r(csPeZdZdZddejddejdfdddfdd Zd d Zd d Z Z S) r)a FLRW cosmology with a constant dark energy equation of state and no spatial curvature. This has one additional attribute beyond those of FLRW. Parameters ---------- H0 : float or scalar quantity-like ['frequency'] Hubble constant at z = 0. If a float, must be in [km/sec/Mpc]. Om0 : float Omega matter: density of non-relativistic matter in units of the critical density at z=0. w0 : float, optional Dark energy equation of state at all redshifts. This is pressure/density for dark energy in units where c=1. A cosmological constant has w0=-1.0. Tcmb0 : float or scalar quantity-like ['temperature'], optional Temperature of the CMB z=0. If a float, must be in [K]. Default: 0 [K]. Setting this to zero will turn off both photons and neutrinos (even massive ones). Neff : float, optional Effective number of Neutrino species. Default 3.04. m_nu : quantity-like ['energy', 'mass'] or array-like, optional Mass of each neutrino species in [eV] (mass-energy equivalency enabled). If this is a scalar Quantity, then all neutrino species are assumed to have that mass. Otherwise, the mass of each species. The actual number of neutrino species (and hence the number of elements of m_nu if it is not scalar) must be the floor of Neff. Typically this means you should provide three neutrino masses unless you are considering something like a sterile neutrino. Ob0 : float or None, optional Omega baryons: density of baryonic matter in units of the critical density at z=0. If this is set to None (the default), any computation that requires its value will raise an exception. name : str or None (optional, keyword-only) Name for this cosmological object. meta : mapping or None (optional, keyword-only) Metadata for the cosmology, e.g., a reference. Examples -------- >>> from astropy.cosmology import FlatwCDM >>> cosmo = FlatwCDM(H0=70, Om0=0.3, w0=-0.9) The comoving distance in Mpc at redshift z: >>> z = 0.5 >>> dc = cosmo.comoving_distance(z) rr=r>Nr?c stj||d||||||| d |jjdkrHtj|_|j|j|j f|_ nV|j sttj |_|j|j|j |j|j f|_ n*tj|_|j|j|j |j|j|j|j f|_ dS)Nr=) rFrGrHrrIrJrKrLr@rAr)rDrErdrQrZfwcdm_inv_efunc_norelr|rMryrr}rcZfwcdm_inv_efunc_nomnurnruZfwcdm_inv_efuncrerfrs) r~rFrGrrIrJrKrLr@rArr"r#rEK s(   zFlatwCDM.__init__cCsh|jr|jd||}n |j|j}t|d}t|d|||j|j|dd|j S)rr0r1rr) rcrnrtrurrgr rMryrrr"r"r#rb s  zFlatwCDM.efunccCsf|jr|jd||}n |j|j}t|d}|d|||j|j|dd|jdS)aFunction used to calculate :math:`\frac{1}{H_z}`. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- E : ndarray or float The inverse redshift scaling of the Hubble constant. Returns `float` if the input is scalar. Defined such that :math:`H(z) = H_0 E(z)`. r0r1rr)rcrnrtrurrMryrrr"r"r#r{z s  zFlatwCDM.inv_efuncrr"r"rr#r) s;r)csjeZdZdZedddZedddZdddejddej d fd d d fd d Z d dZ ddZ Z S)r*ao FLRW cosmology with a CPL dark energy equation of state and curvature. The equation for the dark energy equation of state uses the CPL form as described in Chevallier & Polarski [1]_ and Linder [2]_: :math:`w(z) = w_0 + w_a (1-a) = w_0 + w_a z / (1+z)`. Parameters ---------- H0 : float or scalar quantity-like ['frequency'] Hubble constant at z = 0. If a float, must be in [km/sec/Mpc]. Om0 : float Omega matter: density of non-relativistic matter in units of the critical density at z=0. Ode0 : float Omega dark energy: density of dark energy in units of the critical density at z=0. w0 : float, optional Dark energy equation of state at z=0 (a=1). This is pressure/density for dark energy in units where c=1. wa : float, optional Negative derivative of the dark energy equation of state with respect to the scale factor. A cosmological constant has w0=-1.0 and wa=0.0. Tcmb0 : float or scalar quantity-like ['temperature'], optional Temperature of the CMB z=0. If a float, must be in [K]. Default: 0 [K]. Setting this to zero will turn off both photons and neutrinos (even massive ones). Neff : float, optional Effective number of Neutrino species. Default 3.04. m_nu : quantity-like ['energy', 'mass'] or array-like, optional Mass of each neutrino species in [eV] (mass-energy equivalency enabled). If this is a scalar Quantity, then all neutrino species are assumed to have that mass. Otherwise, the mass of each species. The actual number of neutrino species (and hence the number of elements of m_nu if it is not scalar) must be the floor of Neff. Typically this means you should provide three neutrino masses unless you are considering something like a sterile neutrino. Ob0 : float or None, optional Omega baryons: density of baryonic matter in units of the critical density at z=0. If this is set to None (the default), any computation that requires its value will raise an exception. name : str or None (optional, keyword-only) Name for this cosmological object. meta : mapping or None (optional, keyword-only) Metadata for the cosmology, e.g., a reference. Examples -------- >>> from astropy.cosmology import w0waCDM >>> cosmo = w0waCDM(H0=70, Om0=0.3, Ode0=0.7, w0=-0.9, wa=0.2) The comoving distance in Mpc at redshift z: >>> z = 0.5 >>> dc = cosmo.comoving_distance(z) References ---------- .. [1] Chevallier, M., & Polarski, D. (2001). Accelerating Universes with Scaling Dark Matter. International Journal of Modern Physics D, 10(2), 213-223. .. [2] Linder, E. (2003). Exploring the Expansion History of the Universe. Phys. Rev. Lett., 90, 091301. %Dark energy equation of state at z=0.r9r7>Negative derivative of dark energy equation of state w.r.t. a.rr=r>Nr?c  stj||||||| | | d ||_||_|jjdkrZtj|_|j |j |j |j |j f|_nf|jstj|_|j |j |j |j|j|j |j f|_n2tj|_|j |j |j |j|j|j|j|j |j f |_dSr)rDrErwardrQrZw0wacdm_inv_efunc_norelr|rMryrzr_war}rcZw0wacdm_inv_efunc_nomnurnruZw0wacdm_inv_efuncrerfrs) r~rFrGrHrrrIrJrKrLr@rArr"r#rE s.     zw0waCDM.__init__cCs t|}|j|j||dS)aReturns dark energy equation of state at redshift ``z``. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- w : ndarray or float The dark energy equation of state Returns `float` if the input is scalar. Notes ----- The dark energy equation of state is defined as :math:`w(z) = P(z)/\rho(z)`, where :math:`P(z)` is the pressure at redshift z and :math:`\rho(z)` is the density at redshift z, both in units where c=1. Here this is :math:`w(z) = w_0 + w_a (1 - a) = w_0 + w_a \frac{z}{1+z}`. r0)rrr rr"r"r#r sz w0waCDM.wcCs@t|}|d}|dd|j|jtd|j||S)aEvaluates the redshift dependence of the dark energy density. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- I : ndarray or float The scaling of the energy density of dark energy with redshift. Returns `float` if the input is scalar. Notes ----- The scaling factor, I, is defined by :math:`\rho(z) = \rho_0 I`, and in this case is given by .. math:: I = \left(1 + z\right)^{3 \left(1 + w_0 + w_a\right)} \exp \left(-3 w_a \frac{z}{1+z}\right) r0r1r)rrr rgrr~rrr"r"r#r szw0waCDM.de_density_scale)rrrrrrrrVrvr;rErrrr"r"rr#r* sJ  r*csBeZdZdZdddejddejdfdddfdd ZZS) r+a FLRW cosmology with a CPL dark energy equation of state and no curvature. The equation for the dark energy equation of state uses the CPL form as described in Chevallier & Polarski [1]_ and Linder [2]_: :math:`w(z) = w_0 + w_a (1-a) = w_0 + w_a z / (1+z)`. Parameters ---------- H0 : float or scalar quantity-like ['frequency'] Hubble constant at z = 0. If a float, must be in [km/sec/Mpc]. Om0 : float Omega matter: density of non-relativistic matter in units of the critical density at z=0. w0 : float, optional Dark energy equation of state at z=0 (a=1). This is pressure/density for dark energy in units where c=1. wa : float, optional Negative derivative of the dark energy equation of state with respect to the scale factor. A cosmological constant has w0=-1.0 and wa=0.0. Tcmb0 : float or scalar quantity-like ['temperature'], optional Temperature of the CMB z=0. If a float, must be in [K]. Default: 0 [K]. Setting this to zero will turn off both photons and neutrinos (even massive ones). Neff : float, optional Effective number of Neutrino species. Default 3.04. m_nu : quantity-like ['energy', 'mass'] or array-like, optional Mass of each neutrino species in [eV] (mass-energy equivalency enabled). If this is a scalar Quantity, then all neutrino species are assumed to have that mass. Otherwise, the mass of each species. The actual number of neutrino species (and hence the number of elements of m_nu if it is not scalar) must be the floor of Neff. Typically this means you should provide three neutrino masses unless you are considering something like a sterile neutrino. Ob0 : float or None, optional Omega baryons: density of baryonic matter in units of the critical density at z=0. If this is set to None (the default), any computation that requires its value will raise an exception. name : str or None (optional, keyword-only) Name for this cosmological object. meta : mapping or None (optional, keyword-only) Metadata for the cosmology, e.g., a reference. Examples -------- >>> from astropy.cosmology import Flatw0waCDM >>> cosmo = Flatw0waCDM(H0=70, Om0=0.3, w0=-0.9, wa=0.2) The comoving distance in Mpc at redshift z: >>> z = 0.5 >>> dc = cosmo.comoving_distance(z) References ---------- .. [1] Chevallier, M., & Polarski, D. (2001). Accelerating Universes with Scaling Dark Matter. International Journal of Modern Physics D, 10(2), 213-223. .. [2] Linder, E. (2003). Exploring the Expansion History of the Universe. Phys. Rev. Lett., 90, 091301. rr=r>Nr?c  stj||d||||||| | d |jjdkrNtj|_|j|j|j |j f|_ n^|j s~tj |_|j|j|j|j|j |j f|_ n.tj|_|j|j|j|j|j|j|j |j f|_ dS)Nr=) rFrGrHrrrIrJrKrLr@rAr)rDrErdrQrZfw0wacdm_inv_efunc_norelr|rMryrr r}rcZfw0wacdm_inv_efunc_nomnurnruZfw0wacdm_inv_efuncrerfrs) r~rFrGrrrIrJrKrLr@rArr"r#rEz s*   zFlatw0waCDM.__init__) rrrrrVrvr;rErr"r"rr#r+2 s G r+cseZdZdZedddZedddZedejdZ dd d ejd e j d d e j d fd d d fd d Z ddZddZZS)r,a FLRW cosmology with a CPL dark energy equation of state, a pivot redshift, and curvature. The equation for the dark energy equation of state uses the CPL form as described in Chevallier & Polarski [1]_ and Linder [2]_, but modified to have a pivot redshift as in the findings of the Dark Energy Task Force [3]_: :math:`w(a) = w_p + w_a (a_p - a) = w_p + w_a( 1/(1+zp) - 1/(1+z) )`. Parameters ---------- H0 : float or scalar quantity-like ['frequency'] Hubble constant at z = 0. If a float, must be in [km/sec/Mpc]. Om0 : float Omega matter: density of non-relativistic matter in units of the critical density at z=0. Ode0 : float Omega dark energy: density of dark energy in units of the critical density at z=0. wp : float, optional Dark energy equation of state at the pivot redshift zp. This is pressure/density for dark energy in units where c=1. wa : float, optional Negative derivative of the dark energy equation of state with respect to the scale factor. A cosmological constant has wp=-1.0 and wa=0.0. zp : float or quantity-like ['redshift'], optional Pivot redshift -- the redshift where w(z) = wp Tcmb0 : float or scalar quantity-like ['temperature'], optional Temperature of the CMB z=0. If a float, must be in [K]. Default: 0 [K]. Setting this to zero will turn off both photons and neutrinos (even massive ones). Neff : float, optional Effective number of Neutrino species. Default 3.04. m_nu : quantity-like ['energy', 'mass'] or array-like, optional Mass of each neutrino species in [eV] (mass-energy equivalency enabled). If this is a scalar Quantity, then all neutrino species are assumed to have that mass. Otherwise, the mass of each species. The actual number of neutrino species (and hence the number of elements of m_nu if it is not scalar) must be the floor of Neff. Typically this means you should provide three neutrino masses unless you are considering something like a sterile neutrino. Ob0 : float or None, optional Omega baryons: density of baryonic matter in units of the critical density at z=0. If this is set to None (the default), any computation that requires its value will raise an exception. name : str or None (optional, keyword-only) Name for this cosmological object. meta : mapping or None (optional, keyword-only) Metadata for the cosmology, e.g., a reference. Examples -------- >>> from astropy.cosmology import wpwaCDM >>> cosmo = wpwaCDM(H0=70, Om0=0.3, Ode0=0.7, wp=-0.9, wa=0.2, zp=0.4) The comoving distance in Mpc at redshift z: >>> z = 0.5 >>> dc = cosmo.comoving_distance(z) References ---------- .. [1] Chevallier, M., & Polarski, D. (2001). Accelerating Universes with Scaling Dark Matter. International Journal of Modern Physics D, 10(2), 213-223. .. [2] Linder, E. (2003). Exploring the Expansion History of the Universe. Phys. Rev. Lett., 90, 091301. .. [3] Albrecht, A., Amendola, L., Bernstein, G., Clowe, D., Eisenstein, D., Guzzo, L., Hirata, C., Huterer, D., Kirshner, R., Kolb, E., & Nichol, R. (2009). Findings of the Joint Dark Energy Mission Figure of Merit Science Working Group. arXiv e-prints, arXiv:0901.0721. z7Dark energy equation of state at the pivot redshift zp.r9r7rz$The pivot redshift, where w(z) = wp.)r4r5rr=r>Nr?c  stj|||||| | | | d ||_||_||_dd|jj} |jjdkrrtj |_ |j |j |j |j| |jf|_nj|jstj|_ |j |j |j |j|j|j| |jf|_n4tj|_ |j |j |j |j|j|j|j|j| |jf |_dS)Nrr0r)rDrEwprzp_zprQrdrZwpwacdm_inv_efunc_norelr|rMryrz_wpr r}rcZwpwacdm_inv_efunc_nomnurnruZwpwacdm_inv_efuncrerfrs)r~rFrGrHr#rr$rIrJrKrLr@rAapivrr"r#rE s2       zwpwaCDM.__init__cCs0dd|jj}|j|j|dt|dS)aReturns dark energy equation of state at redshift ``z``. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- w : ndarray or float The dark energy equation of state Returns `float` if the input is scalar. Notes ----- The dark energy equation of state is defined as :math:`w(z) = P(z)/\rho(z)`, where :math:`P(z)` is the pressure at redshift z and :math:`\rho(z)` is the density at redshift z, both in units where c=1. Here this is :math:`w(z) = w_p + w_a (a_p - a)` where :math:`a = 1/1+z` and :math:`a_p = 1 / 1 + z_p`. r0)r%rQr&r r)r~rr'r"r"r#r sz wpwaCDM.wcCsTt|}|d}dd|jj}|dd|j||jtd|j||S)aEvaluates the redshift dependence of the dark energy density. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- I : ndarray or float The scaling of the energy density of dark energy with redshift. Returns `float` if the input is scalar. Notes ----- The scaling factor, I, is defined by :math:`\rho(z) = \rho_0 I`, and in this case is given by .. math:: a_p = \frac{1}{1 + z_p} I = \left(1 + z\right)^{3 \left(1 + w_p + a_p w_a\right)} \exp \left(-3 w_a \frac{z}{1+z}\right) r0r)rr%rQr&r rgr)r~rrr'r"r"r#r# s zwpwaCDM.de_density_scale)rrrrrr#rcurr$rVrvr;rErrrr"r"rr#r, sT  r,csjeZdZdZedddZedddZdddejddej d fd d d fd d Z d dZ ddZ Z S)r-a FLRW cosmology with a variable dark energy equation of state and curvature. The equation for the dark energy equation of state uses the simple form: :math:`w(z) = w_0 + w_z z`. This form is not recommended for z > 1. Parameters ---------- H0 : float or scalar quantity-like ['frequency'] Hubble constant at z = 0. If a float, must be in [km/sec/Mpc]. Om0 : float Omega matter: density of non-relativistic matter in units of the critical density at z=0. Ode0 : float Omega dark energy: density of dark energy in units of the critical density at z=0. w0 : float, optional Dark energy equation of state at z=0. This is pressure/density for dark energy in units where c=1. wz : float, optional Derivative of the dark energy equation of state with respect to z. A cosmological constant has w0=-1.0 and wz=0.0. Tcmb0 : float or scalar quantity-like ['temperature'], optional Temperature of the CMB z=0. If a float, must be in [K]. Default: 0 [K]. Setting this to zero will turn off both photons and neutrinos (even massive ones). Neff : float, optional Effective number of Neutrino species. Default 3.04. m_nu : quantity-like ['energy', 'mass'] or array-like, optional Mass of each neutrino species in [eV] (mass-energy equivalency enabled). If this is a scalar Quantity, then all neutrino species are assumed to have that mass. Otherwise, the mass of each species. The actual number of neutrino species (and hence the number of elements of m_nu if it is not scalar) must be the floor of Neff. Typically this means you should provide three neutrino masses unless you are considering something like a sterile neutrino. Ob0 : float or None, optional Omega baryons: density of baryonic matter in units of the critical density at z=0. If this is set to None (the default), any computation that requires its value will raise an exception. name : str or None (optional, keyword-only) Name for this cosmological object. meta : mapping or None (optional, keyword-only) Metadata for the cosmology, e.g., a reference. Examples -------- >>> from astropy.cosmology import w0wzCDM >>> cosmo = w0wzCDM(H0=70, Om0=0.3, Ode0=0.7, w0=-0.9, wz=0.2) The comoving distance in Mpc at redshift z: >>> z = 0.5 >>> dc = cosmo.comoving_distance(z) rr9r7z9Derivative of the dark energy equation of state w.r.t. z.rr=r>Nr?c  stj||||||| | | d ||_||_|jjdkrZtj|_|j |j |j |j |j f|_nf|jstj|_|j |j |j |j|j|j |j f|_n2tj|_|j |j |j |j|j|j|j|j |j f |_dSr)rDrErwzrdrQrZw0wzcdm_inv_efunc_norelr|rMryrzr_wzr}rcZw0wzcdm_inv_efunc_nomnurnruZw0wzcdm_inv_efuncrerfrs) r~rFrGrHrr*rIrJrKrLr@rArr"r#rE s.     zw0wzCDM.__init__cCs|j|jt|S)aReturns dark energy equation of state at redshift ``z``. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- w : ndarray or float The dark energy equation of state. Returns `float` if the input is scalar. Notes ----- The dark energy equation of state is defined as :math:`w(z) = P(z)/\rho(z)`, where :math:`P(z)` is the pressure at redshift z and :math:`\rho(z)` is the density at redshift z, both in units where c=1. Here this is given by :math:`w(z) = w_0 + w_z z`. )rr+rrr"r"r#r sz w0wzCDM.wcCs<t|}|d}|dd|j|jtd|j|S)aEvaluates the redshift dependence of the dark energy density. Parameters ---------- z : Quantity-like ['redshift'], array-like, or `~numbers.Number` Input redshift. Returns ------- I : ndarray or float The scaling of the energy density of dark energy with redshift. Returns `float` if the input is scalar. Notes ----- The scaling factor, I, is defined by :math:`\rho(z) = \rho_0 I`, and in this case is given by .. math:: I = \left(1 + z\right)^{3 \left(1 + w_0 - w_z\right)} \exp \left(-3 w_z z\right) r0rr()rrr+rgrr"r"r"r#r szw0wzCDM.de_density_scale)rrrrrrr*rVrvr;rErrrr"r"rr#r-D sD   r-)OrabcrZmathrrrrrrr r r Znumbersr ZnumpyrgZastropy.constantsZ constantsrSZ astropy.unitsrrVZ"astropy.utils.compat.optional_depsr Zastropy.utils.decoratorsrZastropy.utils.exceptionsrr<rr)corerrrZ parameterrrZutilsrrZscipy.integraterZ scipy.specialrr__all__Z__doctest_requires__ZkmrrWrUrYr\r[GZcgsrQr^rrZsigma_sbrTrmZk_Br;rvrpr%r.r&r'r(r)r*r+r,r-r"r"r"r#sr ,          XC~9 a2