An introduction to models of open universes originating from bubbles,
including a summary of recent theoretical results for the power spectrum. To
appear in the proceedings of the XXXIth Moriond meeting, "Microwave Background
Anisotropies."Comment: LaTeX file, uses epsf.tex, 3 figures, 8 pages, minor wording changes
  to clarify discussion of coordinate system

Cohn, J. D.

English

arXiv

Cohn, J.D.

eScholarship - University of California

Lawrence Berkeley National LaboratoryRecent WorkTitleOpen Universes from Bubbles: An Introduction and UpdatePermalinkhttps://escholarship.org/uc/item/8vg857ckAuthorCohn, J.D.Publication Date1996-09-10eScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaLBNL-38960 UC-414 ERNEST ORLANDO LAWRENCE BERKELEY NATIONAL LABORATORY Open Universes from Bubbles: An Introduction and Update J.D. Cohn Physics Division September 1996 Presented at the XXX!th Moriond Meeting, Microwave Background Anisotropies, Les Arc, France, Maroh:·1~23.; .. 1996, afi(fio~h~~pub~shed"iii. the Proceedings ::c 1'1'1 (") "TT ..... 0!'1'1 , 0 ::c 0 (I) 1'1'1 s:: Ill z ..... (") Ill Z!Tl t+O (I) t+ (") 0 "0 < I tD z I (") I 0 w "0 CD '< ~ 0\ ..... lSI DISCLAIMER This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor the Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or the Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof or the Regents of the University of California. LBNL-38960 UC-414 Open Universes from Bubbles: An Introduction and Update J.D. Cohn Department of Physics University of California, Berkeley and Theoretical Physics Group Physics Division Ernest Orlando Lawrence Berkeley National Laboratory University of California Berkeley, California 94 720 September 1996 I This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics, of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098 and by the National Science Foundation under Grant PHY-90-21139. 0 Recycled Paper September 10, 1996 LBNL-38960 UCB-PTH-96/24 astro-ph/9606052 Open universes from bubbles: an introduction and update J.D. Cohn1 Department of Physics, University of California, and Theoretical Physics Group, Lawrence Berkeley National Laboratory University of California Berkeley, CA 94720 Abstract This is an introduction to models of open universes originating from bubbles, including a summary of recent theoretical results for the power spectrum. To appear in the Proceedings of the XXXfh Moriond meeting, "Microwave Background Anisotropies." PACS numbers 98.80Bp, 98.80Cq A flat (f2tot = 1) universe has long been considered a generic prediction of inflationary [1] models. It has recently been demonstrated that inflation can also produce viable open (n < 1) universes [2]. As the data on n has not yet converged, further investigation of these models is worthwhile. In the following, why and how bubbles give open universes is reviewed, the main features of the new models are outlined and current results and questions are summarized. Due to space limitations, this is necessarily just an overview, and readers are referred to the cited papers and their references for more in depth discussion and comprehensive referencing. The basic reason bubbles give an open universe can be illustrated with empty Minkowski space, which does not have a unique coordinate system leading to a metric of the form (1) 1 jdc@asterix.lbl.gov 1 In empty space (p = 0), with zero cosmological constant (A = 0), the curvature k obeys (2) Consider the two coordinate systems for Minkowski space shown in figure one. On the left, lines of constant time are shown for a coordinate system with metric (x2 = Xt +X~+ X~) (3) so one can read off that a(t) = 1. As a result, for this coordinate system, ( a(t))2 = o = -k a(t) ' (4) the universe has k = 0 and is therefore spatially flat. A second coordinate system (T, x) is shown on the right. Here, lines of constant timeT are shown. In terms of (x, t), we have T = )t2 - x2 , tanh x = xjt. The metric becomes ds2 = dT2 - T 2dx 2 . (5) So in this case, we identify a(T) = T. Then, (6) a spatially open universe. In empty space, there is no reason to prefer either coordinate system. How-ever, if there is matter present, one or the otp.er might be preferred. For example, if the background matter distribution is homogeneous in space only in a partic-ular coordinate system, the metric has the form eqn.(l) only for that choice of coordinates. A bubble is said to create an open universe because inside it the (T, x) coordinate system is preferred. To get a bubble, start with a system (here described by a field ¢) stuck in a false vacuum. A bubble forms when a region of space, the bubble interior, FLAT cootrdinates OPEN coordinates ____________ _, ______________ X Figure 1: Lines of constant time in two coordinate systems for Minkowski space 2 v (phi) t phi_F phi_T Figure 2: On the left, a potential with a false and true vacuum. On the right, lines of constant <P after the bubble nucleates. tunnels to the true vacuum. An example of a potential where this could happen is shown on the left of figure two. After tunneling, there is a true vacuum interior, where ¢ = <PT, a bubble (domain) wall, and a false vacuum exterior where </> = <J>p. The lowest energy solution for tunneling in Minkowski space [3] has spherical 0(4) symmetry,¢ = ¢(x2 - t 2). As the matter (the field ¢) is constant on surfaces of constant t 2 - x 2 = T 2 , it is natural to use the open coordinates [4], where the matter is homogeneous in space. On the right side of figure two is a spacetime diagram showing lines of constant ¢. The dashed line is the bubble wall, and the horizontal axis is the nucleation time. As a function of the flat coordinate time t, the bubble wall moves out. Classically the energy difference between the false and true vacua (the latent heat) goes into accelerating the wall outwards. Inside the future light cone of the center of the bubble is a patch with open coordinates. As the light cone is T = 0 in this coordinate system, and the bubble wall in this example is exterior to the light cone (approaching it at large times), the bubble wall is "before" T = 0. This picture can be used to generate a candidate for the early universe by making some modifications. Gravity must be included, the field ¢ is taken to be part of the inflationary potential, and the bubble nucleates in a vacuum, the de Sitter vacuum, ¢ ...., const. Nucleating a bubble in de Sitter space also means that the global structure of spacetime is more complicated-because the spacetime is expanding in the false vacuum, the future light cone of the bubble does not eventually cover the whole future of the space (see, e.g. [5]). Many of the calculations require normalizing on a Cauchy surface (a spacelike hypersur-face which every non-spacelike curve intersects exactly once. [6]) and thus use information exterior to the open universe. The interior of the vacuum bubble has energy density depending on the true vacuum potential, V( <I>T ). If V( ¢T) ~ 0, then after</> has tunneled, the universe is effectively empty. All the energy density is in the bubble wall and not in the resulting open universe, which has n "' 0 and stays that way. Under our 3 Figure 3: A potential with the features that give a viable open universe infla-tionary model. assumptions, to get a universe with something in it means that V( <Pr) =P 0. If V( <Pr) =P 0, then the resulting vacuum is also an expanding de Sitter space, that is, it is still inflating. So one needs a bubble plus inflation before and after nucleation. If one has inflation but no bubble, the amount of inflation providing sufficient homogeneity to agree with observation drives n -- 1 (for a nice summary in the context of bubbles, see [7]). Bucher, Goldhaber and Turok [2] combined bubble formation with inflation to give an open universe. They use four stages, corresponding to potential features sketched in figure 3: • First, on the far left, the system is trapped in a potential well (at <P = <P F), a false vacuum. Inflation occurs, driving the system to the Bunch-Davies vacuum (the natural vacuum here, respecting the symmetries and going over at short distances to the Minkowski space vacuum) and wiping out inhomogeneities. • Then <P tunnels to <P "' ¢n, resetting Q "' 0; • Next, on the approximately linear part of the potential, from <P "' <Pn to <P .....- <Pr' slow roll inflation drives n up to a specific value less than or equal to one. • Finally, on the far right, at <P "' ¢r, ¢ decays, leading to the usual reheating after inflation. The endpoint of the tunneling <P "' ¢n will be called the true vacuum (since <P .....- const.), even though it is also the starting point of the slow roll inflationary stage. 4 There are constraints on the potential so that this picture is consistent. The potential V ( 4>) has to be such that the tunneling rate is rare, so that tunneling from 4> ...., 4> F only occurs after a long period of inflation has erased initial conditions. The false vacuum well cannot be too flat or 4> will stochastically go over the barrier rather than tunneling through it (the Hawking-Moss instanton) [8], leading to density perturbations which are too large [9]. The potential V( 4>) must be tuned to within a few percent so that at the end of the slow roll stage 0.1 ~ n ~ o.9. A simple polynomial potential (up to order ¢ 4 ) does not satisfy these con-ditions [10], although in principle an effective potential could arise in a super-gravity theory with the right features. One can also use more fields, driving inflation by one field while the other one tunnels. Some of the two-field variants of the model lose the ability to predict a definite value of n [11], and lead to more questions which have been studied extensively but are outside the scope of this summary. The density perturbations in these models are a modification of those in Q = 1 inflationary models. For Q = 1 inflationary models, inflation drives the universe to the vacuum, and the quantum fluctuations of the inflaton in this vacuum ·become seeds for structure formation once they re-enter the horizon. In these open universe bubble models, there are vacuum fluctuations (from the false vacuum before tunneling) outside the bubble wall. These evolve through the bubble wall, a time dependent background, to reach the open universe. There is a continuum of these modes and sometimes (depending on properties of the wall and the mass in the false vacuum) a discrete mode. The bubble wall itself also has fluctuations. The calculations incorporate methods for bubbles in Minkowski space [12, 13], and techniques for including gravity as in [14]; see these papers and references therein for background. Various choices for the true and false vacuum masses and the wall profile have been considered. Calculations so far have been for vacuum fluctuations due to one field, in two field models the second field has been frozen out. Thus one solves ( 62 V(¢) ) 0 - 6¢2 l<t>=<l>bkgd 6¢ = 0 (7) The gauge invariant gravitational potential produced by the continuous modes has been found assuming a constant value of H in the resulting de Sitter space. The power spectrum has been calculated for a thin wall with arbitrary false vac-uum mass M 2 ;::: 2H2 and zero radius [15] and nonzero radius [16, 17]. It has. also been found for a large false vacuum mass and varying wall profile (with the restriction that the wall is completely exterior to the light cone of the bubble cen-ter, the open universe) [16]. In all of these cases, the contribution to the fluctua-tions is within an envelope between coth 7rpj2 and tanh 1rpj2 times a scale invari-ant spectrum. (The correspondence with wavelength is p = k/(HoJ1- Oo).) Thus, except at very large scales, where cosmic variance interferes with their measurement, these spectra all coincide. In open de Sitter spacetime, when there is no bubble present, the Bunch Davies vacuum fluctuations of a field of mass M 2 < 2H2 has a continous spec-trum of fluctuations plus a discrete mode [18]. This discrete mode was shown 5 to be part of the complete basis of states [18] and is normalizable on a Cauchy surface, but not inside the open universe (which does not contain a Cauchy surface). In the presence of a bubble, this discrete 'supercurvature' mode with -1 :::; p2 :::; 0 may remain, depending upon the false vacuum mass and the wall profile. If the field providing density perturbations does not change its mass across the wall (as can happen in two field models), and has M 2 < 2H2 , this mode appears [19, 20]. Its contribution to density fluctuations has been found and used to constrain models when combined with the CMB for various cases [19, 20, 21, 22]. It does not seem to give a very strong constraint. However, it has been argued that this state's contribution can be enhanced significantly when the ratio of false to true vacuum energy is large, ruling out some models [22]. There is not yet complete agreement on this. If there is a mass change across the wall [16], an analogue of this state may persist. When matched across the bubble wall, the original vacuum discrete mode does not automatically remain normalizable, but a mode with some other value of -1 :::; p2 < 0 may become normalizable instead. The effects of this new mode have been found in a two field model for a thin wall [16]. This mode disappears if the false vacuum mass is large enough (as argued in [16], the false vacuum mass is necessarily large for most one field models considered so far, so that it does not appear there). There are also fluctuations of the wall itself, contributing mostly at large scales, and considered in [23, 10, 13, 24, 21, 16]. Requiring these fluctuations to be small constrains models. Schematically, the allowed models have a high barrier between the false and true vacuum, producing a large surface tension and making it energetically unfavorable for the wall to fluctuate. In summary, the continuum modes reproduce a scale invariant spectrum starting at very large scales. The vacuum supercurvature mode or its analogue (when present) and the wall fluctuation (also a supercurvature mode) have been used so far to constrain possible potentials by requiring that their contribution be small. Many fits to the data [25] have been made for open universes assuming a scale invariant [26] spectrum, and, except at large scales, the models considered so far also have this spectrum, and so carry over. Many things remain to be done, including adding gravity waves and tilt, and motivating models from particle theory. Acknowledgement: I thank F. Bouchet for the chance to speak and for or-ganizing the conference, and the participants (D. Bond, P. Ferreira, A. Liddle, N. Turok and M. White in particular) for conversations. I also thank K. Ben-son, M. Bucher, and V. Periwal for conversations and D. Scott and M. White for comments on the draft. This work was supported. in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY-90-21139. 6 References [1] A. Guth, Phys. Rev. D23(1981)347; A. Guth and E. Weinberg, Nucl. Phys. B212(1983) 321; Peebles, P.J .E., Ap.J. 284(1984) 439; Turner, M.S., Steigman, G., Krauss, L.M. Phys. Rev. Lett. 52(1984) 2090 [2] M. Bucher, A. Goldhaber, N.Turok, Phys. Rev. D52(1995), 3314 [3] S. Coleman, Phys. Rev. D15(1977)2929; 16(1977)1248(E) [4] J. R. Gott III, Nature 295 (1982) 304; J.R.Gott and T. Statler, Phys. Lett. 136B(1984)157; S. Coleman and F. DeLuccia, Phys. Rev. D21(1980) 3305 [5] B. Allen, Phys. Rev. D51 (1985) 3136 [6] See, for example, S. Hawking, G.F.R. Ellis, The Large Scale Structure of Spacetime (Cambridge: Cambridge University Press,1973) [7] J. Garcia-Bellido, Contribution to TAUP'95 Conference proceedings, astro-ph/9511078 [8] S.W.Hawking and I.G.Moss, Phys. Lett. 110B(1982)35 [9] For a discussion and references see A. Linde, Particle Physics and Infla-tionary Cosmology, Harwood Academic Publishers (1993:USA) [10] A. Linde, A. Mezhlumian, Phys. Rev. D52 (1995) 6789 [11] A. Linde, Phys. Lett. B351(1995) 99 [12] T. Tanaka, M.Sasaki, K. Yamamoto, Phys. Rev.D49(1994) 1039; M. Sasaki, T. Tanaka, K. Yamamoto, J. Yokoyama, Prog. Theor. Phys. 90(1993) 1019; Phys. Lett. B317(1993) 510; K. Yamamoto, T. Tanaka, M. Sasaki, Phys. Rev. D 51(1995) 2968 [13] T. Hamazaki, M. Sasaki, T. Tanaka, and K. Yamamoto, Phys. Rev. D53(1996) 2045 [14] T. Tanaka and M. Sasaki, Phys. Rev. D50(1994) 6444; Prog. Theor. Phys. 88(1992), 503 [15] M. Bucher, N. Turok, Phys. Rev. D52(1995),5538 [16] K. Yamamoto, M. Sasaki, T. Tanaka, "Quantum fluctuations and CMB anisotropies in one-bubble open inflation models", astro-ph/9605104 [17] J.D. Cohn, "Open universes from finite radius bubbles", astro-ph/9605132 [18] M. Sasaki, T. Tanaka, K. Yamamoto, Phys. Rev. D51(1995)2979 [19] K. Yamamoto, M. Sasaki, T. Tanaka, ApJ. 455 (1995) 412 7 [20] E.F. Bunn, K. Yamamoto, "Observational Tests of One-Bubble Open In-flationary Cosmological Models", KUNS-1357, CfPA-95-TH-16, ApJ. 464 in press [21] J. Garcia-Bellido, 'Density Perturbations from Quantum TUnneling in Open Inflation', SUSSEX-AST 95/10-1, astro-ph/9510029 [22] Sasaki and Tanaka, "Can the Simplest Two-Field Model of Open Inflation Survive?", OU-TAP-35, astro-ph/9605104 . [23] J. Garriga and A. Vilenkin, Phys. Rev. D44 (1991), 1007; Phys. Rev. D45(1992), 3469 [24] J. Garriga, "Bubble fluctuations in n < 1 inflation", gr-qc/9602025 [25] See R. Stompor, these proceedings, and for one recent summary, A. Lid-dle, D. Lyth, D. Roberts, P. Viana, "Open cold dark matter models", SUSSEX-AST 95/6-2,astro-ph/9506091 and references therein. For tilted open models the COBE four year normalization is in M. White and D. Scott, "The impact of the cosmic microwave background on large scale structure", astro-ph/9601170, to appear in Comm. on Astrophysics, v .8, No.5 [26] D. Lyth, E. Stewart, Phys. Lett. B252 (1990) 336; B. Ratra, P.J .E. Peebles, Phys. Rev. D52 (1995) 1837; Astrophys. J.Lett. 423 (1994) 15 (1995) astro-ph/9508090 8 ~fl~l¥4:-f\J' ~ l!a\tA•Jj~lili @lillfli:!!ii!!L@'I ~ ~ .l§li93 ~ rr ®l!lfi3#11Liiii!\?o ~ ~ 

Open Universes from Bubbles: An Introduction and Update

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Open universes from bubbles: an introduction and update

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