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For nearby redshifts in a homogeneous expanding universe, the comoving radial distance increment is $c\delta z/H(z)$. Multiplying by the scale factor at the absorber's epoch gives the displayed proper distance in cosmology. The conversion neglects peculiar velocities; observed redshift differences can also contain such velocities and errors in the quasar emission redshift.

For pressureless homologous spherical collapse, let $\varphi$ be its cycloid parameter. The density contrast equation becomes ${\delta }_{\varphi \varphi }-\tan (\varphi /2){\delta }_{\varphi }-3\delta /(1+\cos \varphi )=0$. One solution is ${\delta }_g=\sin \varphi /(1+\cos \varphi {)}^2$. Reduction of order gives an independent solution that decays at collapse,For $ϵ=\pi -\varphi$, the integral is ${ϵ}^5/40+O({ϵ}^7)$, while ${\delta }_g\sim 4{ϵ}^{-3}$. Therefore ${\delta }_d\sim {ϵ}^2/10$. Since $R\sim R_0{ϵ}^2/4$, the displayed powers follow. The growing branch is unique up to scale and addition of the decaying branch; many mixed solutions grow, so it is not literally the only function with growth. Finite perturbations eventually leave the linear regime.

For a flat single-fluid background with constant barotropic equation of state, $w\ne -1/3$, adiabatic perturbations and zero anisotropic stress, $a\propto {\eta }^{2/(1+3w)}$ and $a^2\bar{\rho }\propto {\eta }^{-2}$. The scalar potential obeys ${\varphi }^{\prime \prime }+[6(1+w)/(1+3w)]{\varphi }^{\prime }/\eta -w{\nabla }^2\varphi =0$. Substituting the Poisson constraint $\Delta \propto {\eta }^2{\nabla }^2\varphi$ gives the displayed equation. The homogeneous mode and the degenerate vacuum fluid $w=-1$ require separate interpretation.

During matter domination in a flat universe, the growing comoving-gauge density contrast is proportional to $a\propto {\eta }^2$. Its Newtonian potential is constant by the Poisson constraint. The momentum constraint gives $v=-2\varphi /(3\mathcal{H})=-\eta \varphi /3$. Thus the peculiar velocity potential grows as $a^{1/2}$ in the convention where the peculiar velocity is $\nabla v$.

For adiabatic scalar perturbations without anisotropic stress, define $A=4\pi G{a}^2(\bar{\rho }+\bar{P})={\mathcal{H}}^2-{\mathcal{H}}^{\prime }$ and $c_a^2={\bar{P}}^{\prime }/{\bar{\rho }}^{\prime }$. In conformal Newtonian gauge, $\mathcal{R}=-\varphi -\mathcal{H}({\varphi }^{\prime }+\mathcal{H}\varphi )/A$. Background conservation gives $A^{\prime }/A=-\mathcal{H}(1+3c_a^2)$. Differentiating the expression for $\mathcal{R}$ and using the scalar-potential equation gives the displayed identity. For regular adiabatic modes with nonzero background enthalpy, evolution is suppressed on scales $k\ll \mathcal{H}$.

Two elements commute when their two products in opposite orders agree. A particular pair of matrices may commute even when matrix multiplication is not a commutative operation on the whole set of matrices.

Two elements $a,b$ of a commutative ring are associates if $a=ub$ for some unit $u$. This is an equivalence relation: reflexivity uses $1$, symmetry uses $u^{-1}$, and transitivity uses a product of units.

In $\mathbb{Z}$, the units are $1,-1$, so associates differ only by sign. In the polynomial ring $k[X]$ over a field, two nonzero polynomials are associates exactly when one is a nonzero scalar multiple of the other. Unique factorization identifies factors up to this unavoidable ambiguity. In an integral domain, nonzero elements generate the same principal ideal exactly when they are associates.

Functions on a finite group form a Hopf algebra under pointwise multiplication, with $(\Delta f)(g,h)=f(gh)$, $\epsilon (f)=f(1)$ and $(Sf)(g)=f(g^{-1})$. The coproduct is cocommutative exactly when the group is abelian, since delta functions distinguish distinct products $gh,hg$.

For prime $p$, a graph with $m>(p-1)n$ edges and maximum degree at most $2p-1$ contains a nonempty $p$-regular subgraph. Over ${\mathbb{F}}_p$, apply the Combinatorial Nullstellensatz on the Boolean edge-variable cube toIts full squarefree coefficient is nonzero because the first product has degree below $m$. A nonzero value cannot occur at the zero vector; at any other Boolean vector the second product vanishes, and the first forces all selected degrees to be multiples of $p$. The maximum-degree condition makes every positive selected degree exactly $p$.

Suppose $|A|>|B|>0$ in ${\mathbb{F}}_p$. Put $a=|A|$ and $b^{\prime }=\min (|B|,p+2-a)$, and choose $B^{\prime }\subseteq B$ of size $b^{\prime }$. Let $r=a+b^{\prime }-2$. If the restricted sumset fits in a set $D$ of size $r-1$, then $(x-y){\prod }_{d\in D}(x+y-d)$ vanishes on $A\times B^{\prime }$. Its coefficient of $x^{a-1}{y}^{b^{\prime }-1}$ isAll factorial arguments are below $p$ and $0<a-b^{\prime }<p$. The Combinatorial Nullstellensatz gives a contradiction. Since $r=\min (p,|A|+|B|-2)$, this includes the saturated case as well as the small-set case.

Attach relators for every identity word of length at most $2K+2$, where $K$ is an integer fellow-traveller constant. For any null loop, connect its consecutive combing paths at integer heights by rungs of length at most $K$. Each resulting quadrilateral has perimeter at most $2K+2$ and thus is a relator. The bottom row is collapsed at the identity and the top row is the original loop, so these finitely many relators fill every null word and present the group.

If graphs $G_i$ cover the edges of an $n$-vertex graph with independence number at most $\alpha$, their coloring entropy weights satisfy the displayed bound. A random vertex's color is recorded in every covering graph containing it; independent dummy colors with matching marginal laws fill the other positions. Conditional on all recorded colors, compatible vertices form an independent set, leaving conditional entropy at most ${\log }_2\alpha$. The mutual information carried by the colors is at most $n^{-1}{\sum }_{i}w(G_i)$, proving the bound.

A colour singlet transforms in the trivial representation of the SU(3) colour group. Global colour rotations leave it invariant, and its colour generators annihilate it. A three-quark colour singlet uses the completely antisymmetric colour tensor; a quark–antiquark singlet contracts their colour indices.

Substitute the instantaneous Sine-Gordon kink profile ${\theta }_K(\gamma (x-X))$ and prescribed velocity $-\gamma \dot{X}{\theta }_K^{\prime }$ with $\gamma =(1-{\dot{X}}^2{)}^{-1/2}$ into the field density. For $A_1=0$, $A_0=xf(t)$, the integrals $\int ({\theta }_K^{\prime }{)}^2=8$ and $\int {\theta }_K^{\prime }=2\pi$ give the displayed collective-coordinate effective Lagrangian. Its equation is $d(8\gamma \dot{X})/dt=-2\pi f$. This is relativistic particle motion with mass $8$ and topological coupling charge $2\pi$. When $\gamma$ varies, the prescribed velocity omits a width-change contribution; acceleration correction for an instantaneous boosted kink describes the exact chain-rule distinction.

A coproduct is a colimit of a discrete diagram: it has injections from each summand and every family of maps out of the summands extends uniquely across it.

The dual universal family is a coend $Q$ with maps $i_C:T(C,C)\to Q$ satisfying $i_{C}T(f,1_C)=i_{D}T(1_D,f)$ on $T(D,C)$. Every compatible family to another object factors uniquely through $Q$. For small $\mathcal{C}$ in a cocomplete category, it is the coequalizer ${\coprod }_{f:C\to D}T(D,C)\rightrightarrows {\coprod }_{C}T(C,C)\to Q$.

For the regular growing mode in a purely matter-dominated flat background, $h=-2{\delta }_c$ and ${\delta }_c=C{\tau }^2$. The synchronous Einstein constraint $\mathcal{H}{h}^{\prime }+k^2{h}^{-}/3=3{\mathcal{H}}^2{\delta }_c$ with $\mathcal{H}=2/\tau$ gives $h^{-}=60C/k^2$. Hence $\zeta =h^{-}/6+{\delta }_c/3=10C/k^2+C{\tau }^2/3$, conserved to leading order on superhorizon scales. Identifying its early constant value gives the displayed density transfer; it includes a $k^2$ factor that is factored out of the conventionally normalized matter transfer function.

In the idealization that subhorizon cold matter freezes during radiation domination, early-entering modes have an entry density of order the primordial curvature, retain it until equality, and then share the same matter-era growth factor. Their density-per-curvature transfer becomes approximately independent of $k$. Matching this plateau to the matter-era $k^2$ transfer gives the displayed sharp-transition model, with its high-$k$ coefficient dependent on the matching convention. A scale-invariant curvature spectrum then gives density power proportional to $k$ below equality and $k^{-3}$ above it. Actual cold matter grows logarithmically in the radiation era, adding the familiar high-$k$ logarithms of the Mészáros effect.

The matter equation in the barotropic matter-string model contains no wavenumber. Growing initial data therefore evolve as ${\delta }_c(k,\eta )=A_{k}D(\eta )$ with the same $D$ for every mode. For regular adiabatic modes initially outside the horizon during matter domination, $A_k$ is proportional to $k^2$ times the primordial curvature amplitude. The density-per-curvature transfer is consequently proportional to $k^{2}D(\eta )$ for modes entering after equality. Factoring out both $k^2$ and $D$ gives a constant shape transfer on those scales. Comparing with continued matter-era growth instead introduces the common suppression $D(\eta )/\eta$, not a new wavenumber dependence.

Assume all relevant modes begin outside the horizon at common conformal time ${\tau }_i$, grow as ${\tau }^2$ until radiation-era horizon entry, then freeze until matter-radiation equality, and again grow as ${\tau }^2$ in matter domination. With horizon entry ${\tau }_h=2\pi /k$, $k_{\mathrm{eq}}=2\pi /{\tau }_{\mathrm{eq}}$ and final time ${\tau }_0$, direct multiplication of the growth factors gives the displayed transfer with $G=({\tau }_0/{\tau }_i{)}^2$. A primordial density power spectrum $P_i=Ak$ becomes $P_0=A{G}^{2}k$ below equality wavenumber and $A{G}^2{k}_{\mathrm{eq}}^4{k}^{-3}$ above it. This approximation neglects the true logarithmic radiation-era matter growth, baryonic acoustic structure and late acceleration; a conventionally normalized transfer divides out $G$.