6.4 Newtonian limit

The Newtonian limit is defined by spatial scales much smaller than the horizon radius, peculiar velocities small compared to the speed of light, and a gravitational potential that is both much smaller than unity (in geometric units) and slowly varying in time. A comprehensive review of the theory of cosmological perturbations can be found in [124].

6.4.1 Dark and baryonic matter equations

The appropriate perturbation equations in this limit are easily derived for a background FLRW expanding model, assuming a metric of the form

ds2 = − (1 + 2Φ) dt2 + a(t)2(1 − 2Φ)γij dxi dxj, (66 )
where
( 2) −2 γ = δi 1 + kr-- , (67 ) ij j 4
and k = − 1, 0, + 1 for open, flat and closed Universes. Also, a ≡ 1∕(1 + z) is the cosmological scale factor, z is the redshift, and Φ is the comoving inhomogeneous gravitational potential.

The governing equations in the Newtonian limit are the hydrodynamic conservation equations,

∂&tidle;ρb ∂ i ˙a ----+ ---i(&tidle;ρbvb) + 3-ρ&tidle;b = 0, (68 ) j ∂t ∂x a ∂(&tidle;ρbvb)- -∂-- i j a˙ j -1-∂p&tidle;- &tidle;ρb-∂Φ&tidle;- ∂t + ∂xi (&tidle;ρbvbvb) + 5a &tidle;ρbvb + a2 ∂xj + a2 ∂xj = 0, (69 ) ∂&tidle;e ∂ ∂vi ˙a ---+ --i(&tidle;evib) + &tidle;p---bi + 3-(&tidle;e + &tidle;p) = S&tidle;cool, (70 ) ∂t ∂x ∂x a
the geodesic equations for collisionless dust or dark matter (in comoving coordinates),
dxi ---d= vid, (71 ) dt dvid ˙a-i -1-∂Φ&tidle;- dt = − 2avd − a2 ∂xi, (72 )
Poisson’s equation for the gravitational potential,
2 2 ∇ &tidle;Φ = 4πGa (ρ&tidle;b + ρ&tidle;d − &tidle;ρ0), (73 )
and the Friedman equation for the cosmological scale factor,
[ ]1∕2 da-= H0 Ωm (1-− 1) + Ω Λ(a2 − 1) + 1 . (74 ) dt a
Here &tidle;ρd, &tidle;ρb, &tidle;p and &tidle;e are the dark matter density, baryonic density, pressure and internal energy density in the proper reference frame, xi and vi b are the baryonic comoving coordinates and peculiar velocities, i xd and i vd are the dark matter comoving coordinates and peculiar velocities, 2 3 &tidle;ρ0 = 3H 0Ω0 ∕(8πGa ) is the proper background density of the Universe, Ω0 is the total density parameter, Ωm = Ωb + Ωd is the density parameter including both baryonic and dark matter contributions, Ω Λ = Λ ∕(3H2 ) 0 is the density parameter attributed to the cosmological constant Λ, H = 100 h km s− 1 Mpc −1 0 is the present Hubble constant with 0.5 < h < 1, and &tidle;S cool represents microphysical radiative cooling and heating rates which can include Compton cooling (or heating) due to interactions of free electrons with the CMBR, bremsstrahlung, and atomic and molecular line cooling. Notice that ‘tilded’ (‘non-tilded’) variables refer to proper (comoving) reference frame attributes.

An alternative total energy conservative form of the hydrodynamics equations that allows high resolution Godunov-type shock capturing techniques is

∂ρb 1 ∂ ----+ ----i(ρb&tidle;vib) = 0, (75 ) j ∂t a∂x ∂(ρb&tidle;vb)- 1--∂-- i j ˙a- j ρb-∂-&tidle;Φ- ∂t + a ∂xi(ρb&tidle;vb &tidle;vb + p δij) + aρb&tidle;vb + a ∂xj = 0, (76 ) ∂E 1 ∂ 2a˙ ρ &tidle;vi∂ &tidle;Φ ----+ -----(E&tidle;vib + p&tidle;vib) + ---E + -b-b----= Scool, (77 ) ∂t a∂xi a a ∂xi
with the corresponding particle and gravity equations
i i dxd-= &tidle;vd, (78 ) dt a d&tidle;vid ˙a 1 ∂Φ&tidle; ----= − -&tidle;vid − ----i, (79 ) dt a a ∂x ∇2 &tidle;Φ = 4πG-(ρb + ρd − ρ0), (80 ) a
where ρb is the comoving density, ρ0 = a3&tidle;ρ0, &tidle;vi b is the proper frame peculiar velocity, p is the comoving pressure, E = ρ &tidle;v2∕2 + p∕ (γ − 1 ) b b is the total peculiar energy per comoving volume, and &tidle;Φ is the gravitational potential.

6.4.2 Primordial chemistry

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The baryonic equations from the previous section are easily extended to include reactive chemistry of both atomic and molecular species (e.g., H, H+, He, He+, He++, H−, H+2, H2, and e−) by assuming a common flow field, supplementing the total mass conservation equation (68View Equation) with

∂ &tidle;ρj ∂ i a˙ ∑ ∑ ∑ ----+ --i(&tidle;ρjvb) + 3--&tidle;ρj = kil(T)&tidle;ρi&tidle;ρl + Ii&tidle;ρi (81 ) ∂t ∂x a i l i
for each of the species, and including the effects of non-equilibrium radiative cooling and consistent coupling to the hydrodynamics equations. The k (T) il are rate coefficients for the two body reactions and are generally incorporated in numerical codes as tabulated functions of the gas temperature T. The Ii are integrals evaluating the photoionization and photodissociation of the different species.

A fairly complete chemical network system useful for primordial gas phase compositions, including hydrogen molecules, consists of the following collisional, photoionization, and photodissociation chains

∙ Collisional reactions (primordial chain):
       
(1) H + e  →  H+ + 2e
       
(2) H + e  →  H+ + 2e
       
(3) H + e  →  H+ + 2e
       
(4) H + e  →  H+ + 2e
       
(5) He+ + e  →  He++ + 2e
       
(6) He++ + e  →  He+ + γ
       
∙ Collisional reactions (H 2 molecular chain):
       
(7) H + e  →  H− + γ
       
(8) H + e  →  H− + γ
       
(9) H + e  →  H− + γ
       
(10) H + e  →  H− + γ
       
(11) H + e  →  H− + γ
       
(12) H + e  →  H− + γ
       
(13) H + e  →  − H + γ
       
(14) H + e  →  − H + γ
       
(15) H + e  →  − H + γ
       
(16) H + e  →  − H + γ
       
(17) H + e  →  − H + γ
       
(18) + − H 2 + H  →  H2 + H
       
(19) + − H 2 + H  →  H2 + H
       
∙ Photoionization reactions (primordial chain):
       
(20) H + γ  →  + H + e
       
(21) He + γ  →  + He + e
       
(22) + He + γ  →  ++ He + e
       
∙ Photodissociation/ionization reactions (molecular chain):
       
(23) H− + γ  →  H + e
       
(24) H− + γ  →  H + e
       
(25) H− + γ  →  H + e
       
(26) H− + γ  →  H + e
       
(27) H2 + γ  →  2H
       

For a comprehensive description of the chemistry and explicit formulas modeling the kinetic and cooling rates relevant for cosmological calculations, the reader is referred to [92144541Jump To The Next Citation Point21Jump To The Next Citation Point]. This reactive network is by no means complete, and in fact, ignores important coolants and contaminants (e.g., HD, LiH, and their intermediary products [1517848]) expected to form through non-equilibrium reactions at low temperatures and high densities. Although it is certainly possible to include even in three dimensional simulations, the inclusion of more complex reactants demands either more computational resources (to resolve both the chemistry and cooling scales) or an increasing reliance on equilibrium assumptions which can be very inaccurate.

6.4.3 Numerical methods

Update

Many numerical techniques have been developed to solve the hydrodynamic and collisionless particle equations. For the hydrodynamic equations, the methods range from Lagrangian SPH algorithms with artificial viscosity [7288], to high resolution shock capturing Eulerian techniques on single static meshes [142134], nested grids [19], moving meshes [82], and adaptive mesh refinement [51]. For the dark matter equations, the canonical choices are treecodes [159] or PM and P3M methods [9068Jump To The Next Citation Point], although many variants have been developed to optimize computational performance and accuracy, including adaptive mesh, particle, and smoothing kernel refinement methods [4577Jump To The Next Citation Point130]. An efficient method for solving non-equilibrium, multi-species chemical reactive flows together with the hydrodynamic equations in a background FLRW model is described in [121].

It is beyond the scope of this review to discuss algorithmic details of the different methods and their strengths and weaknesses. Instead, the reader is referred to [103Jump To The Next Citation Point77Jump To The Next Citation Point] for thorough comparisons of various numerical methods applied to problems of structure formation. Kang et al. [103] compare (by binning data at different resolutions) the statistical performance of five codes (three Eulerian and two SPH) on the problem of an evolving CDM Universe on large scales using the same initial data. The results indicate that global averages of physical attributes converge in rebinned data, but that some uncertainties remain at small levels. Frenk et al. [77] compare twelve Lagrangian and Eulerian hydrodynamics codes to resolve the formation of a single X-ray cluster in a CDM Universe. The study finds generally good agreement for both dynamical and thermodynamical quantities, but also shows significant differences in the X-ray luminosity, a quantity that is especially sensitive to resolution [17].

6.4.4 Linear initial data

The standard Zel’dovich solution [16568] is commonly used to generate initial conditions satisfying observed or theoretical power spectra of matter density fluctuations. Comoving physical displacements and velocities of the collisionless dark matter particles are set according to the power spectrum realization

||δρ ||2 ||--(k )|| ∝ knT 2(k), (82 ) | ρ |
where the complex phases are chosen from a gaussian random field, T (k) is a transfer function [27Jump To The Next Citation Point] appropriate to a particular structure formation scenario (e.g., CDM), and n = 1 corresponds to the Harrison–Zel’dovich power spectrum. The fluctuations are normalized with top hat smoothing using
∫ σ2 = 1- = ∞4πk2P (k )W 2(k)dk, (83 ) 8 b2 0
where b is the bias factor chosen to match present observations of rms density fluctuations in a spherical window of radius −1 Rh = 8 h Mpc. Also, P(k) is the Fourier transform of the square of the density fluctuations in Equation (82View Equation), and
W (k ) = --3----(sin (kRh ) − (kRh )cos(kRh )) (84 ) (kRh )3
is the Fourier transform of a spherical window of radius R h.

Overdensity peaks can be filtered on specified spatial or mass scales by Gaussian smoothing the random density field [27]

( ) 1 ∫ δρ ′ |ro − r′|2 3 ′ σ(ro) = -----2-3∕2- ---(r)exp − -----2--- d r (85 ) (2πR f) ρ 2R f
on a comoving scale Rf centered at r = ro (for example, Rf = 5h− 1 Mpc with a filtered mass of M ∼ 1015M f ⊙ over cluster scales). N σ peaks are generated by sampling different random field realizations to satisfy the condition ν = σ(ro)∕σ(Rf) = N, where σ (Rf ) is the rms of Gaussian filtered density fluctuations over a spherical volume of radius Rf.

Bertschinger [44] has provided a useful and publicly available package of programs called COSMICS for computing transfer functions, CMB anisotropies, and gaussian random initial conditions for numerical structure formation calculations. The package solves the coupled linearized Einstein, Boltzman, and fluid equations for scalar metric perturbations, photons, neutrinos, baryons, and collisionless dark matter in a background isotropic Universe. It also generates constrained or unconstrained matter distributions over arbitrarily specifiable spatial or mass scales.


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