Multiple Inflation, Cosmic String Networks and the String Landscape
  Motivated by the string landscape we examine scenarios for which inflation is a two-step process, with a comparatively short inflationary epoch near the string scale and a longer period at a much lower energy (like the TeV scale). We quantify the number of $e$-foldings of inflation which are required to yield successful inflation within this picture. The constraints are very sensitive to the equation of state during the epoch between the two inflationary periods, as the extra-horizon modes can come back inside the horizon and become reprocessed. We find that the number of $e$-foldings during the first inflationary epoch can be as small as 12, but only if the inter-inflationary period is dominated by a network of cosmic strings (such as might be produced if the initial inflationary period is due to the brane-antibrane mechanism). In this case a further 20 $e$-foldings of inflation would be required at lower energies to solve the late universe's flatness and horizon problems. 
C.P. Burgess, Richard Easther, Anupam Mazumdar, David F. Mota and   Tuomas Multamaki