Populating the Landscape: a Top-Down Approach
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PHYSICAL REVIEW D 73, 123527 (2006) Populating the landscape: A top-down approach S. W. Hawking1 and Thomas Hertog2 1DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK 2Physics Department, Theory Division, CERN, CH-1211 Geneva 23, Switzerland (Received 20 February 2006; published 23 June 2006) We put forward a framework for cosmology that combines the string landscape with no boundary initial conditions. In this framework, amplitudes for alternative histories for the universe are calculated with final boundary conditions only. This leads to a top-down approach to cosmology, in which the histories of the universe depend on the precise question asked. We study the observational consequences of no boundary initial conditions on the landscape, and outline a scheme to test the theory. This is illustrated in a simple model landscape that admits several alternative inflationary histories for the universe. Only a few of the possible vacua in the landscape will be populated. We also discuss in what respect the top-down approach differs from other approaches to cosmology in the string landscape, like eternal inflation. DOI: 10.1103/PhysRevD.73.123527 PACS numbers: 98.80.Qc, 11.25.ÿw, 98.80.Cq I. INTRODUCTION the universe, and there is certainly no opportunity for observing multiple copies of the universe. It seems likely that string theory contains a vast en- In fact if one does adopt a bottom-up approach to semble of stable and metastable vacua, including some cosmology, one is immediately led to an essentially clas- with a small positive effective cosmological constant [1] sical framework, in which one loses all ability to explain and the low energy effective field theory of the standard cosmology’s central question—why our universe is the model. Recent progress on the construction of metastable way it is. In particular a bottom-up approach to cosmology de Sitter vacua [2] lends further support to the notion of a either requires one to postulate an initial state of the uni- string landscape [3], and a statistical analysis gives an idea verse that is carefully fine-tuned [10]—as if prescribed by of the distribution of some properties among the vacua [4]. an outside agency—or it requires one to invoke the notion But it has remained unclear what is the correct framework of eternal inflation [11], which prevents one from predict- for cosmology in the string landscape. There are good ing what a typical observer would see. reasons to believe, however, that a proper understanding Here we put forward a different approach to cosmology of the cosmological dynamics will be essential for the in the string landscape, based not on the classical idea of a landscape to be predictive [5]. single history for the universe but on the quantum sum over In particle physics, one usually computes S-matrix ele- histories [12]. We argue that the quantum origin of the ments. This is useful to predict the outcome of laboratory universe naturally leads to a framework for cosmology experiments, where one prepares the initial state and mea- where amplitudes for alternative histories of the universe sures the final state. It could be viewed as a bottom-up are computed with boundary conditions at late times only. approach to physics, in which one evolves forward in time We thus envision a set of alternative universes in the land- a particular initial state of the system. The predictivity of scape, with amplitudes given by the no boundary path this approach arises from and relies upon the fact that one integral [13]. has control over the initial state, and that experiments can The measure on the landscape provided by no boundary be repeated many times to gain statistically significant initial conditions allows one to derive predictions for ob- results. servations. This is done by evaluating probabilities for But cosmology poses questions of a very different char- alternative histories that obey a set of constraints at late acter. In our past there is an epoch of the early universe times. The constraints provide information that is supple- when quantum gravity was important. The remnants of this mentary to the fundamental laws and act as a selection early phase are all around us. The central problem in principle. In particular, they select the subclass of histories cosmology is to understand why these remnants are what that contribute to the amplitude of interest. One then they are, and how the distinctive features of our universe emerged from the big bang. Clearly it is not an S-matrix identifies alternatives within this subclass that have prob- that is the relevant observable1 for these predictions, since abilities near one. These include, in particular, predictions we live in the middle of this particular experiment. of future observations. The framework we propose is thus Furthermore, we have no control over the initial state of more like a top-down approach to cosmology, where the histories of the universe depend on the precise question asked. We illustrate our framework in a model landscape that 1See [6–9] for recent work on the existence and the construc- admits several distinct classes of inflationary histories for tion of observables in cosmological spacetimes. the universe. In this model, we predict several properties of 1550-7998=2006=73(12)=123527(9) 123527-1 © 2006 The American Physical Society S. W. HAWKING AND THOMAS HERTOG PHYSICAL REVIEW D 73, 123527 (2006) the subclass of histories that are three-dimensional, ex- have been found in various minisuperspace approximations panding and approximately flat at late times. We also [14]. It has been shown, however, using several different discuss in general terms the predictions of top-down cos- techniques, that solutions of this kind are unstable [15,16]. mology in more complicated models like the string In particular, one finds that generic small perturbations at landscape. early times (or merely taking in account the remaining Finally we discuss in what respect the top-down ap- degrees of freedom) dramatically change the evolution proach differs from other (bottom-up) approaches to cos- near the transition. Rather than evolving towards an ex- mology in the string landscape, such as eternal inflation or panding semiclassical universe at late times, one generi- pre-big bang cosmology. cally produces a strong curvature singularity. Hence the evolution of pre-big bang cosmologies always includes a II. QUANTUM STATE genuinely quantum gravitational phase, unless the initial state is extremely fine-tuned. It is therefore more appro- In cosmology one is generally not concerned with ob- priate to describe these cosmologies by a path integral in servables at infinity or with properties of the entire four- quantum cosmology, and not in terms of a single semiclas- geometry, but with alternatives in some finite region in the sical trajectory. The universe will not have a single history interior of the spacetime. The amplitudes for these more but every possible history, each with its own probability. restricted sets of observables are obtained from the ampli- In fact, the quantum state of the universe at late times is tudes of four dimensional metric and matter field configu- likely to be independent of the state on the initial surface. rations, by integrating over the unobserved quantities.2 A This is because there are geometries in which the initial particularly important case is the amplitude of finding a surface is in one universe and the final surface in a separate 3 compact spacelike surface S with induced three-metric gij disconnected universe. Such metrics exist in the Euclidean and matter field configuration , regime, and correspond to the quantum annihilation of one Z universe and the quantum creation of another. Moreover, 3 iSg; g ; DgDe : (1) because there are so many different possible universes, C these geometries dominate the path integral. Therefore Here the path integral is taken over the class C of space- even if the path integral had an initial boundary in the 3 times which agree with gij and on a compact boundary infinite past, the state on a surface S at late times would be S. The quantum state of the universe is determined by the independent of the state on the initial surface. It would be remaining specification of the class C. given by a path integral over all metric and matter field Usually one sums over histories that have an initial and a configurations whose only boundary is the final surface S. final boundary. This is useful for the computation of S- But this is precisely the no boundary quantum state [13] matrix elements to predict the outcome of laboratory ex- Z periments, where one prepares the initial state and mea- 3 ÿSEg; g ; DgDe ; (2) sures the final state. It is far from clear, however, that this is C the appropriate setup for cosmology, where one has no control over the initial state, and no opportunity for ob- where the integral is taken over all regular geometries serving multiple copies of the universe. In fact, if one does bounded only by the compact three-geometry S with in- 3 apply this approach to cosmology one is naturally led to an duced metric gij and matter field configuration . The essentially classical picture, in which one simply assumes 3 Euclidean action SE is given by the universe began and evolved in a way that is well- defined and unique. Z Z q 1 4 p 3 3 S ÿ d x g R L g; ÿ d x g K; (3) Pre-big bang cosmologies [10] are examples of models E 2 that are based on a bottom-up approach. In these models S one specifies an initial state on a surface in the infinite past where L g; is the matter Lagrangian.