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A Big Rip is a finite-future-time cosmological singularity at which the scale factor, Hubble parameter, and phantom energy density diverge.

For a null vector, the perfect fluid in general relativity tensor givesbecause $g_{\alpha \beta }{V}^{\alpha }{V}^{\beta }=0$. For phantom energy, $\rho +P=(1+w)\rho <0$. A nonzero $u_{\alpha }{V}^{\alpha }$ therefore makes the contraction negative, so phantom energy violates the null energy condition.

For example, choose $w=-5/3$. A schematic solution can join the pressureless matter law $a\propto t^{2/3}$ to the phantom-dominated law $a\propto (t^{\ast }-t{)}^{-1}$. It begins at the Big Bang, bends down during matter domination, changes to accelerated expansion when phantom energy takes over, and has a vertical asymptote at the finite Big Rip time $t^{\ast }$.

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.

Phantom energy with $w=-4/3$ has ${\rho }_P\propto a$, so flatness givesFor $b^2=1+{\beta }^{2}a$, integration yieldsSolving gives the displayed scale factorFor $1-\Omega \ll 1$ and times near $t_0$, this approaches de Sitter growth $e^{H_0\sqrt{\Omega }(t-t_0)}$. The denominator vanishes atEven a tiny phantom fraction therefore ends expansion in finite time; its divergent density ultimately disrupts loosely bound systems.