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Differentiate the Friedmann equation and use cosmological energy conservation:For an expanding universe $H>0$, division by $2H$ and substitution of the Friedmann equation give the Friedmann acceleration equationThe strong energy condition makes $\ddot{a}\le 0$. Since $aH=\dot{a}$ and $\dot{a}>0$,Thus the quantity is nonincreasing, with strict decrease when the energy condition is strict. Dividing the Friedmann equation by $H^2$ yieldsFor fixed nonzero curvature, the magnitude of $\Omega -1$ grows as expansion decelerates. The observed closeness to one therefore requires much tighter early-time closeness to one in the standard decelerating model: this is the flatness problem. Accelerated expansion during inflation reverses this evolution and suppresses the curvature contribution.

Differentiation of the Hubble parameter gives $\dot{H}+H^2=\ddot{a}/a$. The strong energy condition and the acceleration equation therefore implyAssume the present universe is expanding, $H_0>0$, and normalize the scale factor by $a(t_0)=1$. Since $\ddot{a}\le 0$, the past expanding branch has $\dot{a}(t)\ge \dot{a}(t_0)>0$. On this branch,Integrating from a past time $t$ to $t_0$ gives $H(t{)}^{-1}\le H_0^{-1}+t-t_0$. Thus, where the denominator is positive,The printed inequality needs this past-time qualification: for future times with $H>0$, the inequality reverses. Independently, a concave function lies below its tangent, soPositivity of $a$ prevents the past branch extending before the zero of this tangent. If $t_B$ is the beginning of the expanding universe, its age satisfies $\boxed{t_0-t_B\le H_0^{-1}}$. Without the normalization $a(t_0)=1$, the tangent bound instead reads $a(t)\le a(t_0)[1+H_0(t-t_0)]$.

The Hubble parameter is the fractional expansion rateHere $\rho$ is the homogeneous energy density, $k\in \{-1,0,1\}$ records the sign of the spatial curvature of an FLRW universe, and $R$ is the fixed curvature length scale used with the dimensionless scale factor $a(t)$.

Energy-momentum conservation in a homogeneous expanding universe gives the continuity equationDifferentiate the Friedmann equationUsing $\dot{a}/a=H$ and the continuity equation givesEliminating the curvature term with the Friedmann equation yieldsSince $\dot{H}+H^2=\ddot{a}/a$, this is the Raychaudhuri equation

Under the strong energy condition $\rho +3P\ge 0$, one has $\ddot{a}\le 0$. Thus $a(t)$ is concave, and for $t<t_0$ it lies below its tangent at the present time:The right side vanishes at $t=t_0-H_0^{-1}$. A positive scale factor therefore cannot extend regularly to that time; it must reach the Big Bang singularity at some $t_{\mathrm{BB}}\ge t_0-H_0^{-1}$. Equivalently,

The strong energy condition and Einstein's equation imply $R_{ab}{t}^a{t}^b\ge 0$. Since the congruence is hypersurface orthogonal, its vorticity vanishes. The tensor in the hint is the trace-free spatial shear, soBecause ${\nabla }_a{t}^a=-c$, part b gives