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For a spatially homogeneous minimally coupled scalar field with zero scalar potential, the stress-energy tensor has $\rho =P={\dot{\varphi }}^2/2$. Its field equation is $\ddot{\varphi }+3H\dot{\varphi }=0$, so $a^3\dot{\varphi }$ is constant and its energy density decreases as $a^{-6}$. Thus its homogeneous stress is precisely that of a stiff fluid. Spatial gradients would introduce directional stresses, so this equivalence requires homogeneity.

For a homogeneous canonical massless scalar field, the flat Friedmann equation and field equation in conformal time arePut $C=\sqrt{3/2}$. On the negative-time branch, $\mathcal{H}=1/(2\tau )$, ${\varphi }_0^{\prime }=C/\tau$ and ${\varphi }_0^{\prime \prime }=-C/{\tau }^2$. ThereforeBoth equations are satisfied. Since $a^{\prime }<0$, this is cosmological contraction with a free homogeneous scalar. Its energy density and pressure are equal, so it is a stiff fluid, and the scale factor tends to zero as $\tau \uparrow 0$.

With zero scalar potential and a homogeneous scalar field, the preceding perfect fluid identification givesThis equation of state is that of a stiff fluid. The field equation integrates to $a^3\dot{\varphi }=K$, since ${\partial }_t(a^3\dot{\varphi })=0$. Hence $\rho =K^2/(2a^6)$, agreeing with the cosmological continuity equation for $w=1$. If this nonzero component dominates a flat universe, the Friedmann equation giveson the expanding branch. Integrating once more, $a^3=3\sqrt{4\pi G/3}|K|(t-t_B)$. Choosing the Big Bang at $t_B=0$ yieldsThe homogeneous free scalar as a stiff fluid construction requires $K\ne 0$; a constant zero-potential field has zero energy density and cannot dominate this expansion.