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For constant $P=w\rho$ and velocity potential $v_i=i{k}_i\theta$, stress-energy conservation gives the displayed density equation in synchronous gauge in cosmology. Here $h$ is the trace of the spatial metric perturbation, and primes denote conformal time. In particular cold baryons obey ${\delta }_b^{\prime }=k^2{\theta }_b-h^{\prime }/2$. A positive sign in front of $k^2{\theta }_b$ on the left contradicts this potential convention.

In a spatially flat FLRW metric, the supremum of the comoving radial distance from which a signal emitted at the initial time can eventually reach a fixed comoving observer is the total conformal time interval. At an intermediate time $t$, this equals its comoving particle horizon plus the comoving radius of its cosmological event horizon, when both integrals converge. This is a visibility bound for emission arbitrarily close to the initial boundary; it does not assert that a transparent universe or a source existed there.

The unrestricted spatial coordinate unwraps the spatial circle of two-dimensional de Sitter spacetime. The conformal time $\eta =\arcsin (\tanh t)$ lies in $(-\pi /2,\pi /2)$, and null geodesics have slopes $d\xi /d\eta =\pm 1$. Constant-$\eta$ lines are Cauchy hypersurfaces: along any causal curve $|d\xi /d\eta |\le 1$, so a curve cannot escape to spatial infinity in a bounded conformal-time interval. The covering spacetime is therefore globally hyperbolic, despite its observer-dependent cosmological horizons. Identifying $\xi$ modulo $2\pi$ gives the de Sitter cylinder, also globally hyperbolic.

In linear cosmological perturbation theory, an unsourced vector cosmological perturbation of a perfect fluid in synchronous gauge obeys stress-energy conservation in the form ${\mathbf{q}}^{\prime }+4\mathcal{H}\mathbf{q}=0$, where $\mathbf{q}=(\bar{\rho }+\bar{P}){\mathbf{v}}_{\perp }$ is the momentum density, $\mathcal{H}$ the conformal Hubble parameter, and the prime a conformal time derivative. Its conserved quantity is $a^4\mathbf{q}$. For constant equation-of-state parameter $w\ne -1$, the background cosmological continuity equation gives ${\mathbf{v}}_{\perp }\propto a^{3w-1}$. Thus a pressureless matter vector velocity falls as $a^{-1}$, while a radiation vector velocity is constant. This statement assumes vanishing vector anisotropic stress and no momentum exchange with another component.

For a minimally coupled massless scalar field in a flat radiation-dominated universe, conformal time gives $a=\alpha \tau$ and $(a\varphi {)}^{\prime \prime }-{\nabla }^2(a\varphi )=0$. Every nonzero Fourier mode is therefore a free oscillator divided by $a$. The zero-wavenumber equation instead gives $a{\varphi }_0=A+B\tau$, hence the displayed constant-plus-decaying solution. Substituting $k=0$ into a fixed-amplitude oscillatory formula retains only the $1/a$ term and loses the constant field solution. A truly general solution must include this zero mode or explicitly exclude a homogeneous background.