Causal Structures in Cosmology

C. R. Acad. Sci. Paris, t. 0, Serie´ IV, p. 1–??, 2014 - PXHY????.TEX - Rubrique/Heading (Sous-rubrique/Sub-Heading) INFLATION: THEORETICAL AND OBSERVATIONAL STATUS Causal structures in inflation Structures causales de l’inflation George F.R. Ellis a, Jean-Philippe Uzanb,c a Mathematics and Applied Mathematics Department, University of Cape Town, Rondebosch, Cape Town 7701, South Africa, E-mail: [email protected] b Institut d’Atrophysique de Paris, UMR-7095 du CNRS, Universite´ Pierre et Marie Curie, 98 bis, boulevard Arago, 75014 Paris, France, c Sorbonne Universites,´ Institut Lagrange de Paris, 98 bis, boulevard Arago, 75014 Paris, France. E-mail: [email protected] (Received le blank, accepted on ) Abstract. This article reviews the properties and limitations associated with the existence of particle, visual, and event horizons in cosmology in general and in inflationary universes in particular, carefully distinguishing them from ‘Hubble horizons’. It explores to what extent one might be able to probe conditions beyond the visual horizon (which is close in size to the present Hubble radius), thereby showing that visual horizons place major limits on what are observationally testable aspects of a multiverse, if such exists. Indeed these limits largely prevent us from observationally proving a multiverse either does or does not exist. We arXiv:1612.01084v1 [gr-qc] 4 Dec 2016 emphasize that event horizons play no role at all in observational cosmology, even in the multiverse context, despite some claims to the contrary in the literature. Cet article definit puis passe en revue les proprietes et les limites associees a` l’existence d’horizons (des particules, des evenements et visuels) en cosmologie en general et pendant l’inflation en particulier, en insistant sur leurs differences avec les ‘horizons de Hubble’. Il discute la possibilite de tester les conditions physiques de l’univers au-dela` de notre horizon visuel (qui, en taille, est proche du rayon de Hubble) et dmontre que l’existence d’horizons visuels impose des limites strictes sur ce qui est potentiellement testable dans les scenarios´ de type multivers, si ils existent. Ces limites nous interdisent de prouver observationnelle- ment l’existence ou la non-existence de ces multivers. Il est aussi demontr´ e´ que les horizons Note presented by Jean-Philippe UZAN S1296-2147(01)0????-?/FLA c 2014 Academie´ des sciences. 1 G.F.R. Ellis and J.-P. Uzan INFLATION: THEORETICAL AND OBSERVATIONAL STATUS des evenements ne jouent aucun roleˆ en cosmologie observationnelle, mme dans le contexte des modeles` de multivers. c 2014 Academie´ des sciences Cosmology/ Inflation/Causal structures 2 INFLATION: THEORETICAL AND OBSERVATIONAL STATUS Causal structures in cosmology of a global direction of time. The expansion of the 1. Introduction universe, and associated physically meaningful horizons, occur relative to the future direction of time. The causal structure of spacetimes plays a ma- Secondly, in the cosmological case, we assume jor role in the understanding of the physics of black existence everywhere of a family of fundamental ob- holes and in cosmology. In particular these space- servers with 4-velocity u : uµu = 1, defin- time possess horizons. A horizon is a frontier that µ µ ing a preferred cosmological restframe at− each point bounds causality, or separates observable events from [7, 8]. This implies that the worldlines of fundamen- non-observable ones. In cosmology they limit the tal observers never intersect. observational possibilities, and they have to be dis- tinguished from the natural scales fixed by the cos- 2.1. Past Light Cone mic expansion rate. The way they do so differs in Given a spacetime with metric gµν , one can non-inflationary and inflationary cosmology. M define for any event p the past lightcone C−(p) as These hypersurfaces play different roles in their the set of events q such that there exists a future di- two main contexts, the physics of black holes and rected null geodesic joining q to p. It characterizes cosmology [1, 2, 3, 4, 5, 6]. We focus here on their the set of events that can be observed by an observer role in cosmology, but contrast this with the black at event p by electromagnetic radiation, irrespective hole case. There appears to be substantial confusion of its wavelength. Technically, for any event q on about this in some of the current literature on infla- µ C−(p), there exists a null geodesic x (λ) parame- tionary cosmology, where in particular event hori- terized by the affine parameter λ 0 (chosen neg- zons are claimed to play a significant physical role; ative so that increasing λ corresponds≤ to the future but this is not the case. direction of time), such that xµ(0) = p and there is The article is organized as follows: Section 2. de- µ a value λ1 such that x (λ1) = q. Its tangent vector fines the different notions of horizons, and Section 3. kµ dxµ/dλ satisfies the null geodesic equation focuses on the use of conformal diagrams. Section 4. ≡ µ µ ν discusses the case of standard cosmology, while Sec- k kµ = 0; k µk = 0: (2) r tions 5. and 6. consider inflationary cosmology and The past lightcone is a 3-dimensional null surface alternative models, including small universes and so- that can be parameterized by 2 angles (θ; φ) repre- me multiverse proposals. senting the direction of observation in the sky and a 2. Different notions of horizons redshift z that characterizes the distance “down” the lightcone, and is defined as Very different concepts of horizons have to be con- sidered. In particular, one needs to distinguish be- ( kµu ) µ source (3) 1 + z − µ tween local and non-local (or global) notions of hori- ≡ ( k uµ)obs zons, respectively defined in 2.2. and 2.3.. − µ In order to introduce all thesex notions,x we firstly where u is the tangent vector to the observer and assume that the spacetimes under consideration are source worldlines. These quantities are defined with globally hyperbolic so that they can be foliated by respect to the fundamental observers. By construc- a continuous family of spacelike three-dimensional tion, C−(p) depends on the event p so that two different observers have different lightcones, and any hypersurfaces, Σt [3]. This means that there exists a smooth function tôn whose gradient never van- specific observer’s lightcone changes over time. In ishes and is timelike soM that each hypersurface is a cosmology, this latter effect is at the origin of the surface of constant t^, time drift of observed redshift [9]. Let us recall an important property. Consider two Σt = p ; t^(p) = t ; (1) spacetimes whose metrics gµν and g~µν are confor- f 2 M g 2 mal, i.e. gµν = Ω g~µν . Any null geodesic of gµν µν t I R and g t^µt^ν < 0 where I is a max- 8 2 ⊂ with affine parameter λ is a null geodesic of g~µν imal subset of R so that Σt covers all . Such ~ 2 ~ M with affine parameter λ where dλ = Ω dλ, so that spacetimes represent most spacetimes of astrophysi- k~µ = Ω2kµ; see e.g. Ref. [10]. cal and cosmological interest. They imply existence 3 G.F.R. Ellis and J.-P. Uzan INFLATION: THEORETICAL AND OBSERVATIONAL STATUS 2.2. Non-local Horizons [B] Visual horizons (VH). If the past lightcone intersects a spatial section We can define several types of non-local horizons, Σtd before which the spacetime is opaque to elec- namely particle horizons [2], visual horizons [11, tromagnetic radiation, this leads to the existence of 12] and event horizons [2]. They are non-local in the a visual horizon for electromagnetic radiation, de- sense that they depend on the large-scale geodesic fined as the set of fundamental worldlines passing structure of the spacetime. They are defined relative through Σp := C−(p) Σtd . We cannot see fur- to the future direction of time. ther out matter by any form\ of electromagnetic radiation, and the observable universe for p is the space- [A] Particle Horizons (PH). time region delimited by the past light cone C−(p) Particle horizons are defined as the worldlines of up to Σp (see Fig. 1). A typical example of such a the limiting fundamental particles that can affect an visual horizon is the last scattering surface in cos- observer O at a spacetime position p [2], provided mology which characterizes how far we can see by such a limit exists. For an observer O at time t0, the electromagnetic radiation [11, 12] so that it is a key particle horizon is the timelike hypersurface that at concept in the discussion of our observational limits time t = t0 divides all particles in the Universe into (see below). The 2-sphere Σp delineates the furthest two non-empty families: the ones that can have al- observable matter in the universe, for p; it is what ready been observed or been in causal contact with we observe by means of cosmic background radi- O at the time t0, and the ones that cannot have been ation observations, such as by COBE, Planck, and observed or in causal contact then. For each time Bicep2. At later times p0, the corresponding horizon t0, the particle horizon is determined as the inter- will move out and Σp0 will lie outside Σp. Visual section between the limiting geodesics of the most horizons can also appear in black hole physics, when distant comoving particles that can be causally inter- it is surrounded by an accretion disk. acted with (they lie on the past light cone of C−(p) It follows from the definition that the visual hori- of p), with the past hypersurface Σt1 as t1 is taken zon contains the worldlines of all matter that we can to the limits of the boundary of the spacetime (t1 observe by photons of any wavelength.

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