Scalar field phase dynamics in preheating

We study the model of a massive inflaton field $\phi$ coupled to another scalar filed $\chi$ with interaction term $g^2\phi^2\chi^2$ for the first stage of preheating. We obtain the the behavior of the phase in terms of the iteration of a simple family of circle maps. When expansion of the universe is taken into account the qualitative behavior of the phase and growth number evolution is reminiscent of the behavior found in the case without expansion.Comment: 4 pages, 4 figures, LaTeX; submitted to the Proceedings of Eleventh Marcel Grossmann Meetin

9/11: Seven Years into History

The after-effects of the 9/11 attacks continue to reverberate around the world. It is much too soon to draw any definitive conclusions about its long-term impact, let alone its place in history. But five years is long enough to ask some preliminary questions and suggest some tentative answers. This article addresses only two of the many possible questions. First, did 9/11 represent a "Revolution in Terrorism Affairs"? That is, did it amount to such a profound break with the past practice of terrorism that the world now confronts an unprecedented threat? Second, did it "alter the course of history"? Did it initiate any significant events or have consequences which otherwise would not have occurred? The development of terrorism over the last three decades calls into question the notion that al-Qaeda and 9/11 marked a ‘revolutionary’ change in the nature of terrorism. In many respects, they seem to constitute a ‘paradigm shift’ rather than a ‘quantum leap’ – more evolutionary than revolutionary. There are plenty of factors that suggest 9/11 was a ‘world-changing event’: the launching of a "Global War on Terrorism"; the consequent invasions and occupations of Iraq and Afghanistan; the rise of major insurgencies in both countries; radicalization of expatriate Muslim communities; the fracturing of the trans-Atlantic alliance; the adoption of harsh anti-terrorism measures by liberal democratic states; and finally, the American adoption of the strategy of ‘preventive war.’ While it is tempting to suggest that none of this would have happened without 9/11, the truth of the matter is less clear-cut than one might think. Granted that five years is still much too close to place an event in its proper historical context, the evidence thus far suggests first, that 9/11 may have been less earth-shaking in its strategic consequences than first imagined, and second, that reactions to it contain some ‘genetic markers’ of longer-term trends that pre-date that event. Its strategic significance will probably be determined by three factors: the outcome of the war in Iraq; the extent to which ‘pre-emptive war’ becomes an accepted model of international crisis management; and the ability of democracies to balance security and civil liberties in the face of a prolonged war. At this juncture, perhaps all that we can say with any certainty is that history is a continuum and that 9/11 represents neither a beginning nor an end