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A Briefing on the Ekpyrotic/Cyclic Universe

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Abstract

This is a short overview of the ekpyrotic/cyclic model of the universe, an alternative to the standard big bang inflationary paradigm.

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... However, it is important to ask to what extent the predictions of inflation are unique, and whether there are additional frameworks (observationally distinguishable from inflation) which can also solve the standard problems. In the past, this has led to various proposed alternatives to inflationary cosmology, for example, pre-big bang cosmology [24,25], string gas cosmology [26][27][28][29][30][31] and the ekpyrotic scenario [2,[32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51]. ...
... Here the on the correlation functions indicate that we have removed the momentumconserving delta functions from both sides. 44 43 An important subtlety, which we completely glossed over is that it is important that π → constant as k → 0 in order to identify the free field power spectrum with that of the interacting theory and to translate back and forth between different pictures at late times [144]. Strictly speaking, we should define a new field π ≡ tπ, which does go to a constant at late times, and repeat the analysis, but this does not change anything. ...
Article
In this dissertation, we introduce and investigate a general framework to describe the dynamics of the early universe. This mechanism is based on spontaneously broken conformal symmetry; we find that spectator fields in the theory can acquire a scale invariant spectrum of perturbations under generic conditions. Before introducing the conformal mechanism, we first consider the landscape of cosmologies involving a single scalar field which can address the canonical early universe puzzles. We find that, generically, single field non-inflationary solutions become strongly-coupled. We are therefore led to consider theories with multiple fields. We introduce the conformal mechanism via specific examples before constructing the most general effective theory for the conformal mechanism by utilizing the coset construction familiar from particle physics to construct the lagrangian for the Goldstone field of the broken conformal symmetry. This theory may be observationally distinguished from inflation by considering the non-linearly realized conformal symmetries. We systematically derive the Ward identities associated to the non-linearly realized symmetries, which relate (N+1)-point correlation functions with a soft external Goldstone to N-point functions, and discuss observational implications, which cannot be mimicked by inflation. Finally, we consider violating the null energy condition (NEC) within the general framework considered. We show that the DBI conformal galileons, derived from the world-volume theory of a 3-brane moving in an Anti-de Sitter bulk, admit a background which violates the NEC. Unlike other known examples of NEC violation, such as ghost condensation and conformal galileons, this theory also admits a stable, Poincaré-invariant vacuum. However, perturbations around deformations of this solution propagate superluminally.
... Finally, we consider the possibility that the universe smoothly bounces from a contracting to an expanding phase. Such a possibility has been considered previously in the literature (for a review, see Ref. [9]), but in previous treatments the bounce was not under theoretical control. We construct a completely smooth bouncing solution with no instability. ...
... In this controllable setup it is easy to follow the evolution of cosmological perturbations from the contracting to the expanding phase. This is crucial for the experimental viability of ekpyrotic/cyclic scenarios [9]. Since our bounce satisfies the general hypotheses of Ref. [22], it unfortunately leads to a non scale-invariant prediction for density perturbations. ...
Article
We present a consistent effective theory that violates the null energy condition (NEC) without developing any instabilities or other pathological features. The model is the ghost condensate with the global shift symmetry softly broken by a potential. We show that this system can drive a cosmological expansion with dH/dt > 0. Demanding the absence of instabilities in this model requires dH/dt <~ H^2. We then construct a general low-energy effective theory that describes scalar fluctuations about an arbitrary FRW background, and argue that the qualitative features found in our model are very general for stable systems that violate the NEC. Violating the NEC allows dramatically non-standard cosmological histories. To illustrate this, we construct an explicit model in which the expansion of our universe originates from an asymptotically flat state in the past, smoothing out the big-bang singularity within control of a low-energy effective theory. This gives an interesting alternative to standard inflation for solving the horizon problem. We also construct models in which the present acceleration has w < -1; a periodic ever-expanding universe and a model with a smooth ``bounce'' connecting a contracting and expanding phase. Comment: 27 pages, 3 figures; v2: comments added about blue spectrum of GWs from inflation and the relation between NEC and modification of gravity, JHEP published version
... where S 0 corresponds to x 0 = a,ψ is of the form Without loss of generality we shall assume c 0 = 1. Then N is ARW with respect to the future, if the metric is close to the Robertson-Walker metric (0. 5) ds 2 = e 2f {−dx 0 2 +σ ij (x)dx i dx j } near the singularity τ = b. By close we mean that the derivatives of arbitrary order with respect to space and time of the conformal metric e −2fg αβ in (0.1) should converge to the corresponding derivatives of the conformal limit metric in (0.5) when x 0 tends to b. ...
... N ∪N can be regarded as a cyclic universe with a contracting part N = {x 0 < 0} and an expanding partN = {x 0 > 0} which are joined at the singularity {x 0 = 0}, cf. [5,6] for similar ideas. ...
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We consider spacetimes $N$ satisfying some structural conditions, which are still fairly general, and prove convergence results for the leaves of an inverse mean curvature flow. Moreover, we define a new spacetime $\hat N$ by switching the light cone and using reflection to define a new time function, such that the two spacetimes $N$ and $\hat N$ can be pasted together to yield a smooth manifold having a metric singularity, which, when viewed from the region $N$ is a big crunch, and when viewed from $\hat N$ is a big bang. The inverse mean curvature flows in $N$ \resp $\hat N$ correspond to each other via reflection. Furthermore, the properly rescaled flow in $N$ has a natural smooth extension of class $C^3$ across the singularity into $\hat N$. With respect to this natural, globally defined diffeomorphism we speak of a transition from big crunch to big bang.
... Theories advocating preferred frames and ether-like models have been lately proposed in mathematically rigorous contexts: diverse Einstein-ethers 7 , MOND 8 , quintessence 9,10 , scalar-tensor theory 11,12 , dark fluid 13 , Chameleon scalar field 14,15 , TeVeS 16 , and the general notion of condensed vacuum states 17 in quantum mechanical (QM) theories of condensed-matter, specifically in super fluid Helium III 18 . With the advent of stringy universe-on-a-membrane models 19,20,21 , space being a medium has entered the mainstream: The membrane is the new medium, woven from strings. Naïve substantivalism cannot be the final word; the nature of space-time is relational on principle 22 , but we also cannot anymore ignore that GR is likely emergent. ...
... One could think of the SV being nurtured by gravitationally repulsive dark energy: especially phantom energy 36 that would eventually pull apart any bound system in a "big rip" 37,38 , would be "feeding" the reproduction. Considering universe-on-a-membrane models 19,20,21 , one could imagine that the SV are due to the strings that the membrane is made from. The slowness of inflation is the fact that string excitations could travel the string length P l ≅ about 10 8 times during every generation, i.e. while reproducing just one more unit of about the string's size. ...
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Theories with ingredients like the Higgs mechanism, gravitons, and inflaton fields rejuvenate the idea that relativistic kinematics is dynamically emergent. Eternal infiation treats the Hubble constant H as depending on location. Microscopic dynamics implies that H is over much smaller lengths than pocket universes to be understood as a local space reproduction rate. We illustrate this via discussing that even exponential inflation in TeV-gravity is slow on the relevant time scale. In our on small scales inhomogeneous cosmos, a reproduction rate H depends on position. We therefore discuss Einstein-Strauss vacuoles and a Lindquist-Wheeler like lattice to connect the local rate properly with the scaling of an expanding cosmos. Consistency allows H to locally depend on Weyl curvature similar to vacuum polarization. We derive a proportionality constant known from Kepler's third law and discuss the implications for the finiteness of the cosmological constant.
... It is worth emphasizing, however, that at least the ekpyrotic universe still suffers from (a mild version of) the big bang singularity. Khoury (2004) concedes that a formal proof that the transition through the big bang can successfully be completed is still missing. The regularity of the dynamical evolution of the universe also at the big bang constitutes an assumption of the scenario, rather than a formally derived result. ...
... Cf.Gasperini and Veneziano (2003) for a detailed review of the pre-big bang model.3 The original article on the ekpyrotic model isKhoury et al. (2001), a more recent review can be found inKhoury (2004). ...
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My dissertation studies the foundations of loop quantum gravity (LQG), a candidate for a quantum theory of gravity based on classical general relativity. At the outset, I discuss two---and I claim separate---questions: first, do we need a quantum theory of gravity at all; and second, if we do, does it follow that gravity should or even must be quantized? My evaluation of different arguments either way suggests that while no argument can be considered conclusive, there are strong indications that gravity should be quantized. LQG attempts a canonical quantization of general relativity and thereby provokes a foundational interest as it must take a stance on many technical issues tightly linked to the interpretation of general relativity. Most importantly, it codifies general relativity's main innovation, the so-called background independence, in a formalism suitable for quantization. This codification pulls asunder what has been joined together in general relativity: space and time. It is thus a central issue whether or not general relativity's four-dimensional structure can be retrieved in the alternative formalism and how it fares through the quantization process. I argue that the rightful four-dimensional spacetime structure can only be partially retrieved at the classical level. What happens at the quantum level is an entirely open issue. Known examples of classically singular behaviour which gets regularized by quantization evoke an admittedly pious hope that the singularities which notoriously plague the classical theory may be washed away by quantization. This work scrutinizes pronouncements claiming that the initial singularity of classical cosmological models vanishes in quantum cosmology based on LQG and concludes that these claims must be severely qualified. In particular, I explicate why casting the quantum cosmological models in terms of a deterministic temporal evolution fails to capture the concepts at work adequately. Finally, a scheme is developed of how the re-emergence of the smooth spacetime from the underlying discrete quantum structure could be understood.
... Plot of the function F (1, x 2 , x 3 ) x 2 2 x 2 3 for non-Gaussianities generated by higher derivative interactions(12) and in the DBI model of inflation[20,21]. The figure is normalized to have value 1 for equilateral configurations x 2 = x 3 = 1 and set to zero outside the region 1 − x 2 ≤ x 3 ≤ x 2 . ...
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... The problem of finding a smooth transition from a spacetime N with a big crunch to a spacetimeN with a big bang singularity has been the focus of some recent works in general relativity, see e.g., [6,8] and the references therein. For abstract spacetimes, i.e., for spacetimes that are not embedded in a bulk space, it is even a non-trivial question how to define a smooth transition. ...
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We consider branes $N=I\times\so$, where $\so$ is an $n$\ndash dimensional space form, not necessarily compact, in a Schwarzschild-AdS_{(n+2)} bulk $\mc N$. The branes have a big crunch singularity. If a brane is an ARW space, then, under certain conditions, there exists a smooth natural transition flow through the singularity to a reflected brane $\hat N$, which has a big bang singularity and which can be viewed as a brane in a reflected Schwarzschild-AdS_{(n+2)} bulk $\hat{\mc N}$. The joint branes $N\uu \hat N$ can thus be naturally embedded in $R^2\times \so$, hence there exists a second possibility of defining a smooth transition from big crunch to big bang by requiring that $N\uu\hat N$ forms a $C^\infty$-hypersurface in $R^2\times \so$. This last notion of a smooth transition also applies to branes that are not ARW spaces, allowing a wide range of possible equations of state.
... Galileon models thus generically predict a late-time contracting phase. This could naturally match onto an ekpyrotic [30, 31, 32, 33, 34], New Ekpyrotic [35, 36, 37, 38, 39] or cyclic [40, 41] contracting phase, for instance. See [42] for a review of these models.Figure 5 shows the effective BD parameter ω eff BD , introduced in (4.15), for the whole evolution, with r c = 10 Gpc (dotted curve), 15 Gpc (dash-dotted) and 20 Gpc (dashed). ...
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... Alternatively, flatness and homogeneity can be achieved during an ekpyrotic phase [2, 3, 4, 5, 6, 7], a period of ultra-slow contraction before the big bang, driven by a stiff fluid with w > 1; the ekpyrotic phase also suppresses chaotic mixmaster behavior [8, 9, 10, 11]. See [12, 13] for reviews. In both cases, phases with nearly constant w can be achieved with a single canonical scalar field with appropriately chosen potential V (φ). ...
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... The scaling solution will be accelerating when P > 1 (or c 2 < 1) and U is stabilized at a positive value [8,9]. However, the scaling solution will be ekpyrotic when P < 1 /3 (or c 2 > 3) and U is stabilized at a negative value [10]. ...
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... Searching for ekpyrotic solutions is mathematically very similar to the search for accelerating scaling solutions. The same exponential potential (2) could be used for ekpyrosis if it is very steep (P < 1 /3) and U is stabilised at a negative value [35] in order to avoid growing anisotropies at the singularity [36,37]. ...
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We consider a cosmological scenario in which a scale-invariant spectrum of curvature perturbations is generated by a rapidly-evolving equation of state on a slowly expanding background. This scenario generalizes the "adiabatic ekpyrotic" mechanism proposed recently in arXiv:0910.2230. Whereas the original proposal assumed a slowly contracting background, the present work shows that the mechanism works equally well on an expanding background. This greatly expands the realm of broader cosmological scenarios in which this mechanism can be embedded. We present a phase space analysis and show that both the expanding and contracting versions of the scenario are dynamical attractors, with the expanding branch having a broader basin of attraction. In both cases, a finite range of scale invariant modes can be generated within the regime of validity of perturbation theory.
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For a single scalar field with unit sound speed minimally coupled to Einstein gravity, there are exactly three distinct cosmological solutions which produce a scale invariant spectrum of curvature perturbations in a dynamical attractor background, assuming vacuum initial conditions: slow-roll inflation; a slowly contracting adiabatic ekpyrotic phase, described by a rapidly-varying equation of state; and an adiabatic ekpyrotic phase on a slowly expanding background. Of these three, only inflation remains weakly coupled over a wide range of modes, the other scenarios can produce at most 12 e-folds of scale invariant and gaussian modes. In this paper, we investigate how allowing the speed of sound of fluctuations to evolve in time affects this classification. While in the presence of a variable sound speed there are many more scenarios which are scale invariant at the level of the two-point function, they generically suffer from strong coupling problems similar to those in the canonical case. There is, however, an exceptional case with superluminal sound speed, which suppresses non-gaussianities and somewhat alleviates strong coupling issues. We focus on a particular realization of this limit and show these scenarios are constrained and only able to produce at most 28 e-folds of scale invariant and gaussian perturbations. A similar bound should hold more generally --- the condition results from the combined requirements of matching the observed amplitude of curvature perturbations, demanding that the Hubble parameter remain sub-Planckian and keeping non-gaussianities under control. We therefore conclude that inflation remains the unique scenario, assuming a single degree of freedom on an attractor background, capable of producing arbitrarily many scale invariant modes while remaining weakly coupled. Alternative mechanisms must inevitably be unstable or rely on multiple degrees of freedom.
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We study the transition from an Emergent Galileon condensate phase of the early universe to a later expanding radiation phase. This "defrosting" or "preheating" transition is a consequence of the excitation of matter fluctuations by the coherent Galileon condensate, in analogy to how preheating in inflationary cosmology occurs via the excitation of matter fluctuations through coupling of matter with the coherent inflaton condensate. We show that the "minimal" coupling of matter (modeled as a massless scalar field) to the Galileon field introduced by Creminelli, Nicolis and Trincherini in order to generate a scale-invariant spectrum of matter fluctuations is sufficient to lead to efficient defrosting, provided that the effects of the non-vanishing expansion rate of the universe are taken into account. If we neglect the effects of expansion, an additional coupling of matter to the Galileon condensate is required. We study the efficiency of the defrosting mechanism in both cases.
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The pseudo-conformal scenario is an alternative to inflation in which the early universe is described by an approximate conformal field theory on flat, Minkowski space. Some fields acquire a time-dependent expectation value, which breaks the flat space so(4,2) conformal algebra to its so(4,1) de Sitter subalgebra. As a result, weight-0 fields acquire a scale invariant spectrum of perturbations. The scenario is very general, and its essential features are determined by the symmetry breaking pattern, irrespective of the details of the underlying microphysics. In this paper, we apply the well-known coset technique to derive the most general effective lagrangian describing the Goldstone field and matter fields, consistent with the assumed symmetries. The resulting action captures the low energy dynamics of any pseudo-conformal realization, including the U(1)-invariant quartic model and the Galilean Genesis scenario. We also derive this lagrangian using an alternative method of curvature invariants, consisting of writing down geometric scalars in terms of the conformal mode. Using this general effective action, we compute the two-point function for the Goldstone and a fiducial weight-0 field, as well as some sample three-point functions involving these fields.
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In cyclic theory of universe the scenario is just inverse of the theory of inflation. The cyclic theory model almost solves the problem of homogeneity, isotropy and flatness. It also has relevant cause and effect scenarios of different phenomenon like reheating, contraction of branes and it also shows the thermodynamic relevance. The basic approach of the paper is to satisfy the thermodynamics laws and to formulate the situation of ultimate fate of universe. Each and every phenomena that has been described in cyclic theory of universe has been taken into account while drawing the hypothetical graphs of thermodynamics. While treating thermodynamics with cosmology and astrophysics, many factors have been neglected like shape of universe, the cosmological constant, invariant scalar factor and factors related to the depth of cosmology. The paper also shows the behaviour of pressure, volume and temperature by using arbitrary values. The time and space has been neglected in graphical representation.
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A prescription is developed for matching general relativistic perturbations across singularities of the type encountered in the ekpyrotic and cyclic scenarios, i.e., a collision between orbifold planes. We show that there exists a gauge in which the evolution of perturbations is locally identical to that in a model space-time (compactified Milne mod Z2) where the matching of modes across the singularity can be treated using a prescription previously introduced by two of us. Using this approach, we show that long wavelength, scale-invariant, growing-mode perturbations in the incoming state pass through the collision and become scale-invariant growing-mode perturbations in the expanding hot big-bang phase.
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We consider cosmological models with a scalar field with equation of state $w\ge 1$ that contract towards a big crunch singularity, as in recent cyclic and ekpyrotic scenarios. We show that chaotic mixmaster oscillations due to anisotropy and curvature are suppressed, and the contraction is described by a homogeneous and isotropic Friedmann equation if $w>1$. We generalize the results to theories where the scalar field couples to p-forms and show that there exists a finite value of $w$, depending on the p-forms, such that chaotic oscillations are suppressed. We show that $Z_2$ orbifold compactification also contributes to suppressing chaotic behavior. In particular, chaos is avoided in contracting heterotic M-theory models if $w>1$ at the crunch.
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We construct a simple nonsingular cosmological model in which the currently observed expansion phase was preceded by a contraction. This is achieved, in the framework of pure general relativity, by means of a radiation fluid and a free scalar field having negative energy. We calculate the power spectrum of the scalar perturbations that are produced in such a bouncing model under the assumption of initial vacuum state for the quantum field associated with the hydrodynamical perturbation. The matching conditions applying to this bouncing model are derived and shown to be different from those in the case of a sharp transition. We show that if our bounce transition is smoothly connected to a slowly contracting phase, this provides a new way to generate a scale invariant power spectrum.
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In the ekpyrotic scenario the Universe is initially collapsing, the energy density coming from a scalar field with a negative exponential potential. On the basis of a calculation ignoring the gravitational back-reaction the authors of the scenario claim that during collapse the vacuum fluctuation creates a perturbation in the comoving curvature, which has a flat spectrum in accordance with observation. In this Letter the back-reaction is included, and it is found that the spectrum during collapse is strongly scale-dependent with negligible magnitude.
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We present a nearly model-independent estimate that yields the predictions of a class of simple inflationary and ekpyrotic or cyclic models for the spectral tilt of the primordial density inhomogeneities that enables us to compare the two scenarios. Remarkably, we find that the two produce an identical result, n(s) approximately 0.95. For inflation, the same estimate predicts a ratio of tensor to scalar contributions to the low l multipoles of the microwave background anisotropy of T/S approximately 20%; the tensor contribution is negligible for ekpyrotic or cyclic models, as shown in earlier papers.
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For a 4-dimensional spatially-flat Friedmann-Robertson-Walker universe with a scalar field $\phi(x)$, potential $V(\phi)$ and constant equation of state $w=p/\rho$, we show that an expanding solution characterized by $\epsilon=3(1+w)/2$ produces the same scalar perturbations as a contracting solution with $\hat{\epsilon}=1/\epsilon$. The same symmetry applies to both the dominant and subdominant scalar perturbation modes. This result admits a simple physical interpretation and generalizes to $d$ spacetime dimensions if we define $\epsilon \equiv [(2d-5)+(d-1)w]/(d-2)$. Comment: 9 pages, 2 figures, 1 table
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We investigate both analytically and numerically the evolution of scalar perturbations generated in models which exhibit a smooth transition from a contracting to an expanding Friedmann universe. We find that the resulting spectral index in the late radiation dominated universe depends on which of the $\Psi$ or \$zeta$ variables passes regularly through the transition. The results can be parameterized through the exponent $q$ defining the rate of contraction of the universe. For $q \geq -1/2$ we find that there are no stable cases where both variables are regular during the transition. In particular, for $0<q\ll 1$, we find that the resulting spectral index is close to scale invariant if $\Psi$ is regular, whereas it has a steep blue behavior if $\zeta$ is regular. We also show that as long as $q\leqslant 1$, perturbations arising from the Bardeen potential remain small during contraction in the sense that there exists a gauge in which all the metric and matter perturbation variables are small. Comment: 30 pages, 16 figures. Version to appear in Phys. Rev. D. Slight modifications, but no change in the conclusion
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We study string theory in supersymmetric time-dependent backgrounds. In the framework of general relativity, supersymmetry for spacetimes without flux implies the existence of a covariantly constant null vector, and a relatively simple form of the metric. As a result, the local nature of any such spacetime can be easily understood. We show that we can view any such geometry as a sequence of solutions to lower-dimensional Euclidean gravity. If we choose the lower-dimensional solutions to degenerate at some light-cone time, we obtain null singularities, which may be thought of as generalizations of the parabolic orbifold singularity. We find that in string theory, many such null singularities get repaired by $\alpha'$-corrections - in particular, by worldsheet instantons. As a consequence, the resulting string theory solutions do not suffer from any instability. Even though the CFT description of these solutions is not always valid, they can still be well understood after taking the effects of light D-branes into account; the breakdown of the worldsheet conformal field theory is purely gauge-theoretic, not involving strong gravitational effects.
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We consider the set of controlled time-dependent backgrounds of general relativity and string theory describing ``bubbles of nothing'', obtained via double analytic continuation of black hole solutions. We analyze their quantum stability, uncover some novel features of their dynamics, identify their causal structure and observables, and compute their particle production spectrum. We present a general relation between squeezed states, such as those arising in cosmological particle creation, and nonlocal theories on the string worldsheet. The bubble backgrounds have various aspects in common with de Sitter space, Rindler space, and moving mirror systems, but constitute controlled solutions of general relativity and string theory with no external forces. They provide a useful theoretical laboratory for studying issues of observables in systems with cosmological horizons, particle creation, and time-dependent string perturbation theory. Comment: 38 pages, harvmac big, 6 figures
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We consider the four-dimensional effective field theory which has been used in previous studies of perturbations in the Ekpyrotic Universe, and discuss the spectrum of cosmological fluctuations induced on large scales by quantum fluctuations of the bulk brane. By matching cosmological fluctuations on a constant energy density hypersurface we show that the growing mode during the very slow collapsing pre-impact phase couples only to the decaying mode in the expanding post-impact phase, and that hence no scale-invariant spectrum of adiabatic fluctuations is generated. Note that our conclusions may not apply to improved toy models for the Ekpyrotic scenario. Comment: 8 pages, few sentences added. Conclusions unchanged. Added references. Missing name added to Ref. 50
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We respond to the criticisms by Kallosh, Kofman and Linde concerning our proposal of the ekpyrotic universe scenario. We point out a number of errors in their considerations and argue that, at this stage, the ekpyrotic model is a possible alternative to inflationary cosmology as a description of the very early universe.
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At low energy, the four-dimensional effective action of the ekpyrotic model of the universe is equivalent to a slightly modified version of the pre big bang model. We discuss cosmological perturbations in these models. In particular we address the issue of matching the perturbations from a collapsing to an expanding phase in full generality. We show that, generically, one obtains $n=0$ for the spectrum of scalar perturbations in the original pre big model (with vanishing potential). When an exponential potential for the dilaton is included, a scale invariant spectrum ($n=1$) of adiabatic scalar perturbations is produced under very generic matching conditions, both in a modified pre big bang and ekpyrotic scenario. We also derive general results valid for power law scale factors matched to a radiation dominated era. Comment: 11 pages, 1 figure, revised version with small corrections to match version in print. Results and conclusions unchanged
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We study big-crunch big-bang cosmologies that correspond to exact world-sheet superconformal field theories of type II strings. The string theory spacetime contains a big crunch and a big bang cosmology, as well as additional “whisker” asymptotic and intermediate regions. Within the context of free string theory, we compute, unambiguously, the scalar fluctuation spectrum in all regions of spacetime. Generically, the big crunch fluctuation spectrum is altered while passing through the bounce singularity. The change in the spectrum is characterized by a function Δ, which is momentum and time dependent. We compute Δ explicitly and demonstrate that it arises from the whisker regions. The whiskers are also shown to lead to “entanglement” entropy in the big bang region. Finally, in the Milne orbifold limit of our superconformal vacua, we show that Δ→1 and, hence, the fluctuation spectrum is unaltered by the big-crunch big-bang singularity. We comment on, but do not attempt to resolve, subtleties related to gravitational back reaction and light winding modes when interactions are taken into account.
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The question of how perturbations evolve through a bounce in the Cyclic and Ekpyrotic models of the Universe remains a topical one. Issues concerning singularities at the background level and at the level of perturbation theory continue to be a matter of debate. In this report we hope to demonstrate a nonsingular collision between the boundary branes at the background level, and circumstances under which all perturbation variables remain bounded through the collision. As expected, we find most collisions to be singular even in the full 5D formalism, where first order perturbation theory breaks down for at least one perturbation variable. Only in the case that the boundary branes approach each other with constant velocity shortly before the bounce, can a consistent, nonsingular solution be found. It is then possible to follow the perturbations explicitly until the actual collision. In this case, we find that if a scale-invariant spectrum developed on the hidden brane, it will get transferred to the visible brane during the bounce.
Article
By considering a simplified but exact model for realizing the ekpyrotic scenario, we clarify various assumptions that have been used in the literature. In particular, we discuss the new ekpyrotic prescription for passing the perturbations through the singularity which we show to provide a spectrum depending on a nonphysical normalization function. We also show that this prescription does not reproduce the exact result for a sharp transition. Then, more generally, we demonstrate that, in the only case where a bounce can be obtained in Einstein general relativity without facing singularities and/or violation of the standard energy conditions, the bounce cannot be made arbitrarily short. This contrasts with the standard (inflationary) situation where the transition between two eras with different values of the equation of state can be considered as instantaneous. We then argue that the usually conserved quantities are not constant on a typical bounce time scale. Finally, we also examine the case of a test scalar field (or gravitational waves) where similar results are obtained. We conclude that the full dynamical equations of the underlying theory should be solved in a nonsingular case before any conclusion can be drawn.
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We present a novel scenario where a scalar field acquires a mass which depends on the local matter density: the field is massive on Earth, where the density is high, but is essentially free in the solar system, where the density is low. All existing tests of gravity are satisfied. We predict that near-future satellite experiments could measure an effective Newton's constant in space different from that on Earth, as well as violations of the equivalence principle stronger than currently allowed by laboratory experiments.
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We address the perturbation power spectrum generated in the recently proposed ekpyrotic scenario by Khoury et al. The issue has been raised recently by Lyth who used the conventional method based on a conserved variable in the large-scale limit, and derived different results from Khoury et al. The calculation is straightforward in the uniform-curvature gauge where the generated blue spectrum with suppressed amplitude survives as the final spectrum. Whereas, although the metric fluctuations become unimportant and a scale-invariant spectrum is generated in the zero-shear gauge the mode does not survive the bounce, thus with the same final result. Therefore, an exponential potential leads to a power-law expansion/contraction $a \propto |t|^p$, and the power $p$ dictates the final power spectra of both the scalar and tensor structures. If $p \ll 1$ as one realization of the ekpyrotic scenario suggests, the results are $n_S -1 \simeq 2 \simeq n_T$ and the amplitude of the scalar perturbation is suppressed relative to the one of the gravitational wave by a factor $\sqrt{p}/2$. Both results confirm Lyth's. An observation is made on the constraint on the dynamics of the seed generating stage from the requirement of scale-invariant spectrum.
Article
Time dependent orbifolds with spacelike or null singularities have recently been studied as simple models of cosmological singularities. We show that their apparent simplicity is an illusion: the introduction of a single particle causes the spacetime to collapse to a strong curvature singularity (a Big Crunch), even in regions arbitrarily far from the particle. Comment: 16 pages. References and comments added. Discussion of Milne with shift corrected
Article
We analyze the general conditions on the equation of state $w$ required for quantum fluctuations of a scalar field to produce a scale-invariant spectrum of density perturbations, including models which (in the four dimensional effective description) bounce from a contracting to an expanding phase. We show that there are only two robust cases: $w\approx -1$ (inflation) and $w \gg 1$ (the ekpyrotic/cyclic scenario). All other cases, including the $w \approx 0$ case considered by some authors, require extreme fine-tuning of initial conditions and/or the effective potential. For the ekpyrotic/cyclic ($w \gg 1$) case, we also analyze the small deviations from scale invariance. Comment: 6 pages, no figures
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