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LetSubtracting the measured distance modulus from the measured apparent magnitude givesFor the Hubble flow, defineThe Hubble law, with the Hubble constant parametrized by $\theta =5{\log }_{10}h$, gives $r_i\sim N(M_0-\theta ,H_i)$. After integrating over each intrinsic absolute magnitude and each unobserved true distance modulus, independence therefore gives the likelihood function

In the homogeneous background, $\mathbf{v}=0$ and the physical velocity is the Hubble flowThe comoving coordinate $\mathbf{x}$ is held fixed by the time derivative in the fluid equation, soMoreover, spatial homogeneity gives $\nabla \bar{P}=0$. The Euler equation therefore reduces toTaking the divergence in three spatial dimensions givesConsistency with the Poisson equation,then requiresThis is the pressureless, zero-cosmological-constant Raychaudhuri equation, obtained by the Newtonian fluid derivation of the Raychaudhuri equation.

The elementary Newtonian derivation of the Friedmann equation follows a test particle on the boundary of a finite expanding sphere. It therefore introduces a distinguished centre and can misleadingly suggest that all matter expands away from one origin.

The fluid derivation is local: the Euler equation and Poisson equation must be compatible in every comoving neighbourhood. It consequently describes a genuinely homogeneous infinite universe, in which every comoving observer sees the same Hubble flow, without selecting a physical centre. This is the improvement captured by the Newtonian fluid derivation of the Raychaudhuri equation.

Write proper position as $\mathbf{r}=a(t)\mathbf{x}$ and proper velocity asSubstitute this decomposition into the proper-coordinate Euler equations for an inviscid fluid, use ${\nabla }_r=a^{-1}{\nabla }_x$, and subtract the homogeneous-background acceleration. With $\rho =\bar{\rho }(1+\delta )$, one obtains the comoving peculiar-velocity equationThe peculiar gravitational potential $\Phi$ is the total Newtonian potential with the potential of the exactly homogeneous expanding background subtracted. Its gradient therefore generates only accelerations relative to the Hubble flow.

For a pressureless fluid, linearization discards the quadratic advection term, leavingIn the early Einstein-de Sitter universe, the growing mode has $\Phi (\mathbf{x},t)=[D(a)/a]{\Phi }_i(\mathbf{x})$. Integration gives

The linear growth factor obeysSince $\bar{\rho }={\bar{\rho }}_i/a^3$ after choosing $a_i=1$, this is equivalent toIt follows that $a\mathbf{v}=-a^2\dot{D}\nabla {\Phi }_i/(4\pi G{\bar{\rho }}_i)$. Since $\dot{\mathbf{x}}=\mathbf{v}/a$, one final integration gives the Zeldovich approximationwhere an additive initial displacement has been absorbed into ${\mathbf{x}}_i$.

An Einstein-de Sitter universe has $a\propto t^{2/3}$, $H=2/(3t)$, and homogeneous density $\rho =1/(6\pi G{t}^2)$. HenceBefore the perturbation appreciably changes the motion, every shell follows the Hubble flow,

By Newton's shell theorem, only matter currently inside a shell contributes to its radial acceleration. Labelling matter by initial radius givesBefore shell crossing, the enclosed mass of a given shell is $M_i$. Its initial specific energy isAt the turnaround radius, $E_i=-G{M}_i/r_{\ast }$, soThe radial Kepler solution from the Big Bang to apocenter givesAfter turnaround a shell falls inward. Shell crossing makes its enclosed mass time dependent; shells then oscillate through the center, form caustics near successive apocenters, and phase mix into a halo.

For self-similar secondary infall, writeSubstitution in the shell equation givesUsing $G{t}_{\ast }^2/r_{\ast }^3={\pi }^2/(8M_i)$,

Now imposeThe case $ϵ=0$ is a scale-independent fractional overdensity, for which all shells turn around together in the ideal model. The case $ϵ=1$ has constant excess mass $\delta M_i=M_0$, corresponding to a central point-mass seed outside its core.

Since $r_{\ast }/r_i=(M_i/M_0{)}^{ϵ}$, while $r_i^3=(9/2)G{t}_i^2{M}_i$, the turnaround formulas giveThe shell turning at time $t$ therefore hasFor a shell with $\tau =t/t_{\ast }(M_i)$,Consequently its autonomous similarity equation isThe exponent $2/(3ϵ)$ follows directly from the preceding mass ratio; the printed $2ϵ/3$ in the final displayed equation is inconsistent with the supplied initial condition and preceding requested result. The corrected nonlinear equation is integrated numerically after turnaround.

The peculiar gravitational potential is the Newtonian potential after subtracting the potential of the homogeneous expanding background. In comoving coordinates it obeys a Poisson equation sourced by the density contrast, so its gradient accelerates matter relative to the Hubble flow.

For a fluid in an expanding universe, the peculiar-velocity equation is the Euler equations for an inviscid fluid after decomposing physical velocity into Hubble flow and peculiar velocity. It contains the Hubble-drag term $H\mathbf{v}$ and force $-\nabla \Phi /a$ from the peculiar gravitational potential.