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For a constant equation-of-state parameter $w=P/\rho$, the cosmological perfect-fluid continuity equation integrates to $\rho \propto a^{-3(1+w)}$. Comparing the given powers yieldsThis constant-equation-of-state density scaling describes a cosmic string network and a cosmological domain wall network. An intuitive check is that the energy in a fixed comoving cell grows in proportion to the total string length, $a$, or total wall area, $a^2$, while the physical cell volume grows as $a^3$. Thus the densities scale as $a^{-2}$ and $a^{-1}$ respectively. These effective equations of state use the noninteracting, statistically isotropic network assumptions of the problem.

Separate stress-energy conservation and the constant equation-of-state parameter $w$ give the constant-equation-of-state density scalingBecause $w<-1$, the phantom energy density grows with the scale factor, whereas the pressureless matter density decays as $a^{-3}$. Phantom energy therefore dominates at late times. With $\Lambda =0$, the Friedmann equation becomesIntegrating and absorbing the integration constant into the finite time $t^{\ast }$ givesThe exponent is negative, so $a(t)\to \infty$ as $t\to t^{\ast -}$. This is a Big Rip.

The cosmological perfect-fluid continuity equation isFor constant equation-of-state parameter $w=p/\rho$, integration giveswhere $a_0=1$. Define the present cosmological density parameterThe Friedmann equation then givesSpatial curvature may be included as an effective $w=-1/3$ component. Once the parameters are specified, the scale factor follows from the first-order equation $\dot{a}=aH(a)$, or equivalently

Reheating transfers the inflaton energy into other particles after cosmic inflation, eventually establishing a hot thermal bath. It requires interactions in addition to a background scalar potential. The duration and averaged equation-of-state parameter of reheating affect the number of e-folds between a chosen scale's horizon exit and the end of inflation.

A flat universe dominated by a canonical homogeneous scalar field with constant equation-of-state parameter $w>-1$ has $\rho ={\rho }_0(a/a_0{)}^{-3(1+w)}$ and ${\dot{\varphi }}^2=(1+w)\rho$. The positive rolling branch gives the displayed relation and $V(\varphi )=(1-w){\rho }_0\exp [-\sqrt{24\pi G(1+w)}(\varphi -{\varphi }_0)]/2$. The opposite rolling branch reverses the exponential slope. For $w=-1$ the field is constant; $w<-1$ cannot be realized by a real scalar with this positive kinetic term. An exponential slope $\lambda \sqrt{8\pi G}$ yields $w={\lambda }^2/3-1$, with accelerating nonphantom expansion when $0<{\lambda }^2<2$.

A homogeneous canonical scalar field with positive scalar potential has energy density ${\rho }_{\varphi }={\dot{\varphi }}^2/2+V$ and pressure $P_{\varphi }={\dot{\varphi }}^2/2-V$. When its potential energy dominates its kinetic energy, the equation-of-state parameter approaches $-1$. If other components and spatial curvature are negligible, the Friedmann acceleration equation gives accelerating expansion. Small initial kinetic energy is not enough to ensure long cosmic inflation; the field must also remain on a sufficiently flat part of the scalar potential, as measured by the potential slow-roll parameters.

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.