ber ▁ro pe ▁model " ▁one ▁imag ines ▁an ▁ant ▁( ide al ized ▁as ▁point like ) ▁craw ling ▁at ▁a ▁constant ▁speed ▁on ▁a ▁perfectly ▁el astic ▁ro pe ▁which ▁is ▁constantly ▁stretch ing . ▁If ▁we ▁stretch ▁the ▁ro pe ▁in ▁accord ance ▁with ▁the ▁ Λ CD M ▁scale ▁factor ▁and ▁think ▁of ▁the ▁ant ' s ▁speed ▁as ▁the ▁speed ▁of ▁light , ▁then ▁this ▁anal ogy ▁is ▁numer ically ▁accurate ▁– ▁the ▁ant ' s ▁position ▁over ▁time ▁will ▁match ▁the ▁path ▁of ▁the ▁red ▁line ▁on ▁the ▁embedding ▁diagram ▁above . ▁ ▁In ▁the ▁" r ub ber ▁sheet ▁model " ▁one ▁rep laces ▁the ▁ro pe ▁with ▁a ▁flat ▁two - dimensional ▁rub ber ▁sheet ▁which ▁exp ands ▁uniformly ▁in ▁all ▁directions . ▁The ▁addition ▁of ▁a ▁second ▁spatial ▁dimension ▁ra ises ▁the ▁possibility ▁of ▁showing ▁local ▁perturb ations ▁of ▁the ▁spatial ▁geometry ▁by ▁local ▁curv ature ▁in ▁the ▁sheet . ▁ ▁In ▁the ▁" bal lo on ▁model " ▁the ▁flat ▁sheet ▁is ▁replaced ▁by ▁a ▁spher ical ▁bal lo on ▁which ▁is ▁infl ated ▁from ▁an ▁initial ▁size ▁of ▁zero ▁( re present ing ▁the ▁big ▁b ang ). ▁A ▁bal lo on ▁has ▁positive ▁Gaussian ▁curv ature ▁while ▁observations ▁suggest ▁that ▁the ▁real ▁universe ▁is ▁spatial ly ▁flat , ▁but ▁this ▁incons ist ency ▁can ▁be ▁elimin ated ▁by ▁making ▁the ▁bal lo on ▁very ▁large ▁so ▁that ▁it ▁is ▁locally ▁flat ▁to ▁within ▁the ▁limits ▁of ▁observation . ▁This ▁anal ogy