DOCSLIB.ORG

Solving the Flatness Problem with an Anisotropic Instanton in Hořava

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Solving the Flatness Problem with an Anisotropic Instanton in Hořava PHYSICAL REVIEW D 97, 043512 (2018) Solving the flatness problem with an anisotropic instanton in Horava-Lifshitzˇ gravity Sebastian F. Bramberger,1 Andrew Coates,2 João Magueijo,3 Shinji Mukohyama,4,5 Ryo Namba,6 and Yota Watanabe5,4 1Max Planck Institute for Gravitational Physics (Albert Einstein Institute), 14476 Potsdam-Golm, Germany 2School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom 3Theoretical Physics, Blackett Laboratory, Imperial College, London SW7 2BZ, United Kingdom 4Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, 606-8502 Kyoto, Japan 5Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan 6Department of Physics, McGill University, Montr´eal, Qu´ebec H3A 2T8, Canada (Received 5 October 2017; published 12 February 2018) In Horava-Lifshitzˇ gravity a scaling isotropic in space but anisotropic in spacetime, often called “anisotropic scaling,” with the dynamical critical exponent z ¼ 3, lies at the base of its renormalizability. This scaling also leads to a novel mechanism of generating scale-invariant cosmological perturbations, solving the horizon problem without inflation. In this paper we propose a possible solution to the flatness problem, in which we assume that the initial condition of the Universe is set by a small instanton respecting the same scaling. We argue that the mechanism may be more general than the concrete model presented here. We rely simply on the deformed dispersion relations of the theory, and on equipartition of the various forms of energy at the starting point. DOI: 10.1103/PhysRevD.97.043512 I. INTRODUCTION the cosmological constant problem and we simply assume that Λ has the observed value. The slowest decaying In general relativity a homogeneous and isotropic component on the right-hand side of the Friedmann Universe is described by the Friedmann equation equation is the spatial curvature term −3K/a2 and is the 3 source of the flatness problem in the standard cosmology. 3 2 8π ρ − K Λ ρ H ¼ G 2 þ ; ð1Þ Inflation, once it occurs, makes almost constant for an a extended period in the early Universe so that even the curvature term decays faster than ρ. The initial condition of where H is the Hubble expansion rate; G is Newton’s the standard cosmology is thus set at the end of inflation in constant; ρ is the energy density; K ¼ 0; 1; −1 is the such a way that the curvature term is sufficiently smaller curvature constant of a maximally symmetric 3-space; a than 8πGρ. Subsequently, the ratio of the curvature term to Λ is the scale factor; and is the cosmological constant. 8πGρ grows but the initial value of the ratio at the end of ρ The asymptotic value of at late times can be set to zero inflation is so small that the Universe reaches the current Λ ρ by redefinition of . In the standard cosmology, then epoch before the ratio becomes order unity. This is how 4 includes energy densities of radiation (∝ 1/a ) and pres- inflation solves the flatness problem. sureless matter (∝ 1/a3). The fact that all but Λ decay as the If a theory of quantum gravity predicts that the ratio Universe expands is the source of the cosmological con- ð3K/a2Þ/ð8πGρÞ were sufficiently small at the beginning of stant problem. The present paper does not intend to solve the Universe then this could be an alternative solution to the flatness problem. The purpose of the present paper is to propose such a solution based on the projectable version of Published by the American Physical Society under the terms of Horava-Lifshitzˇ (HL) gravity [1,2], which has recently been the Creative Commons Attribution 4.0 International license. proved to be renormalizable [3,4] and thus is a good Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, candidate for a quantum gravity theory. Since our proposal and DOI. Funded by SCOAP3. is solely based on a fundamental principle called “anisotropic 2470-0010=2018=97(4)=043512(10) 043512-1 Published by the American Physical Society SEBASTIAN F. BRAMBERGER et al. PHYS. REV. D 97, 043512 (2018) scaling,” which is respected by all versions of the HL theory, anisotropic scaling and the semiclassical evolution of it is expected that the same idea can be implemented in HL theory, the curvature is sufficiently suppressed to other versions of HL gravity. solve the flatness problem without the need for inflation. One of the fundamental principles of HL gravity is the The solution may be more general than the concrete model so-called anisotropic scaling, or Lifshitz scaling, presented here, as argued in Sec. IV, where we show that on dimensional grounds we can always predict the modifica- t → bzt; x⃗→ bx;⃗ ð2Þ tions to (1) from the modified dispersion relations of the theory. Together with equipartition of energy at the initial where t is the time coordinate, x⃗are the spatial coordinates point, evolution in this regime enforces the necessary and z is a number called the “dynamical critical exponent.” suppression of the curvature. In Appendixes A and B we In 3 þ 1 dimensions the anisotropic scaling with z ¼ 3 in discuss the generation of scale-invariant perturbations and the ultraviolet (UV) regime is the essential reason for evolution after the instanton based on the concrete setup of renormalizability. The anisotropic scaling with z ¼ 3 also Sec. III. Appendix C then discusses further generalization leads to a novel mechanism of generating scale-invariant of the already general scenarios of Sec. IV. cosmological perturbations, solving the horizon problem without inflation [5]. II. PROJECTABLE HL GRAVITY In the context of quantum cosmology, the initial conditions The basic variables of the projectable version of HL of the Universe are typically set by quantum tunneling gravity are described by an instanton, i.e. a classical solution to some Euclidean equations of motion with suitable boundary lapse∶ NðtÞ; shift∶ Niðt; x⃗Þ; 3d metric∶ g ðt; x⃗Þ: conditions. In relativistic theories, where z ¼ 1, quantum ij tunneling is thought to be dominated by an Oð4Þ symmetric ð4Þ instanton, implying that T ¼ L, where T and L are the Euclidean time and length scales, respectively. After analytic The theory respects the so-called “foliation preserving continuation to the real time evolution, this causes the diffeomorphisms,” flatness problem unless inflation follows. Setting z ¼ 3, however, the story is completely different. t → t0ðtÞ; x⃗→ x⃗0ðt; x⃗Þ: ð5Þ An instanton should lead to T ∝ L3 and thus Adopting the notation of [6], the action of the gravity sector T ≃ M2L3; ð3Þ is then given by Z 2 ffiffiffi where T and L are again the Euclidean time and length MPl p 3 ij 2 I ¼ Ndt gd x⃗ðK K − λK − 2Λ þ R þ L 1Þ; scales, respectively, and M is the scale above which the g 2 ij z> 3 anisotropic scaling (2) with z ¼ becomes important. If ð6Þ the theory is UV complete then the scaling (3) is expected to apply to any kind of instanton deep in the UV regime, where i.e. for L ≪ 1/M. If the size of the instanton L is indeed much smaller than 1/M then this implies that T ≪ L and 2 MPl i jk i j k i L 1 c1D R D R c2D RD R c3R R R thus the instanton has a highly anisotropic shape. We thus 2 z> ¼ð i jk þ i þ i j k “ ” call this kind of instanton an anisotropic instanton. If the j i 3 j i 2 c4RR R c5R c6R R c7R : 7 creation of the Universe is dominated by a small anisotropic þ i j þ Þþð i j þ Þ ð Þ instanton then in the real time Universe after analytic Here, Kij ¼ð∂tgij − DiNj − DjNiÞ/ð2NÞ is the extrinsic continuation, the spatial curvature length scale will be ij ik jl much greater than the cosmological time scale. In this way curvature of the constant t hypersurfaces, K ¼ g g Kkl, ij j ij j the anisotropic instanton may solve the flatness problem K ¼ g Kij, Ni ¼ gijN , g is the inverse of gij, Di and Ri are the covariant derivative and the Ricci tensor constructed without inflation. pffiffiffiffiffiffiffiffiffi i 1 π The rest of the present paper is organized as follows. from gij, R ¼ Ri is the Ricci scalar of gij, MPl ¼ / 8 G is In Sec. II we review projectable HL theory, obtaining the the Planck scale, and λ and cn (n ¼ 1; …; 7) are constants. equivalent of Friedmann’s equation (1) in this theory. New In HL gravity, as already stated in (4), a spacetime curvature-dependent terms are found, which will be essen- geometry is described by a family of spatial metrics tial for the solution to the flatness problem proposed here. parametrized by the time coordinate t, together with the In Sec. III we examine a quantum state inspired by the no- lapse function and the shift vector. The 3D space at each t boundary proposal: the idea that the Universe nucleated can have nontrivial topology and may consist of several from nothing, as represented by Euclidean evolution connected pieces, Σα (α ¼ 1; ÁÁÁ), each of which is dis- replacing the big bang singularity. We find that under connected from the others. In this situation, we have a 043512-2 SOLVING THE FLATNESS PROBLEM WITH AN … PHYS. REV. D 97, 043512 (2018) common lapse function and a set of shift vectors and a set 3λ − 1 ∂ α 3 α 2 2 tHα 3 2 3Kα 2Kα − Kα Λ þ Hα ¼ 6 þ 4 2 þ : of spatial metrics parametrized by not only (continuous) t 2 N aα aα aα but also (discrete) α,as ð14Þ i i ⃗ α ⃗ ⃗∈ Σ N ¼ Nαðt; xÞ;gij ¼ gijðt; xÞ; ðx αÞ: ð8Þ Integrating this equation once, we obtain 3 3λ − 1 α 3 3α 2 3 The equation of motion for NðtÞ then leads to a global ð Þ 2 Cα − 3Kα − 2Kα − Kα Λ Hα ¼ 3 6 4 2 þ ; ð15Þ Hamiltonian constraint of the form 2 aα aα aα aα X Z where Cα is an integration constant.
Load more

Recommended publications
  • Lecture 8: the Big Bang and Early Universe
    Astr 323: Extragalactic Astronomy and Cosmology Spring Quarter 2014, University of Washington, Zeljkoˇ Ivezi´c Lecture 8: The Big Bang and Early Universe 1 Observational Cosmology Key observations that support the Big Bang Theory • Expansion: the Hubble law • Cosmic Microwave Background • The light element abundance • Recent advances: baryon oscillations, integrated Sachs-Wolfe effect, etc. 2 Expansion of the Universe • Discovered as a linear law (v = HD) by Hubble in 1929. • With distant SNe, today we can measure the deviations from linearity in the Hubble law due to cosmological effects • The curves in the top panel show a closed Universe (Ω = 2) in red, the crit- ical density Universe (Ω = 1) in black, the empty Universe (Ω = 0) in green, the steady state model in blue, and the WMAP based concordance model with Ωm = 0:27 and ΩΛ = 0:73 in purple. • The data imply an accelerating Universe at low to moderate redshifts but a de- celerating Universe at higher redshifts, consistent with a model having both a cosmological constant and a significant amount of dark matter. 3 Cosmic Microwave Background (CMB) • The CMB was discovered by Penzias & Wil- son in 1965 (although there was an older mea- surement of the \sky" temperature by McKel- lar using interstellar molecules in 1940, whose significance was not recognized) • This is the best black-body spectrum ever mea- sured, with T = 2:73 K. It is also remark- ably uniform accross the sky (to one part in ∼ 10−5), after dipole induced by the solar mo- tion is corrected for. • The existance of CMB was predicted by Gamow in 1946.
    [Show full text]
  • Arxiv:Hep-Ph/9912247V1 6 Dec 1999 MGNTV COSMOLOGY IMAGINATIVE Abstract
    IMAGINATIVE COSMOLOGY ROBERT H. BRANDENBERGER Physics Department, Brown University Providence, RI, 02912, USA AND JOAO˜ MAGUEIJO Theoretical Physics, The Blackett Laboratory, Imperial College Prince Consort Road, London SW7 2BZ, UK Abstract. We review1 a few off-the-beaten-track ideas in cosmology. They solve a variety of fundamental problems; also they are fun. We start with a description of non-singular dilaton cosmology. In these scenarios gravity is modified so that the Universe does not have a singular birth. We then present a variety of ideas mixing string theory and cosmology. These solve the cosmological problems usually solved by inflation, and furthermore shed light upon the issue of the number of dimensions of our Universe. We finally review several aspects of the varying speed of light theory. We show how the horizon, flatness, and cosmological constant problems may be solved in this scenario. We finally present a possible experimental test for a realization of this theory: a test in which the Supernovae results are to be combined with recent evidence for redshift dependence in the fine structure constant. arXiv:hep-ph/9912247v1 6 Dec 1999 1. Introduction In spite of their unprecedented success at providing causal theories for the origin of structure, our current models of the very early Universe, in partic- ular models of inflation and cosmic defect theories, leave several important issues unresolved and face crucial problems (see [1] for a more detailed dis- cussion). The purpose of this chapter is to present some imaginative and 1Brown preprint BROWN-HET-1198, invited lectures at the International School on Cosmology, Kish Island, Iran, Jan.
    [Show full text]
  • Homework 2: Classical Cosmology
    Homework 2: Classical Cosmology Due Mon Jan 21 2013 You may find Hogg astro-ph/9905116 a useful reference for what follows. Ignore radiation energy density in all problems. Problem 1. Distances. a) Compute and plot for at least three sets of cosmological parameters of your choice the fol- lowing quantities as a function of redshift (up to z=10): age of the universe in Gyrs; angular size distance in Gpc; luminosity distance in Gpc; angular size in arcseconds of a galaxy of 5kpc in intrin- sic size. Choose one of them to be the so-called concordance cosmology (Ωm, ΩΛ, h) = (0.3, 0.7, 0.7), one of them to have non-zero curvature and one of them such that the angular diameter distance becomes negative. What does it mean to have negative angular diameter distance? [10 pts] b) Consider a set of flat cosmologies and find the redshift at which the apparent size of an object of given intrinsic size is minimum as a function of ΩΛ. [10 pts] 2. The horizon and flatness problems 1) Compute the age of the universe tCMB at the time of the last scattering surface of the cosmic microwave background (approximately z = 1000), in concordance cosmology. Approximate the horizon size as ctCMB and get an estimate of the angular size of the horizon on the sky. Patches of the CMB larger than this angular scale should not have been in causal contact, but nonetheless the CMB is observed to be smooth across the entire sky. This is the famous ”horizon problem”.
    [Show full text]
  • Or Causality Problem) the Flatness Problem (Or Fine Tuning Problem
    4/28/19 The Horizon Problem (or causality problem) Antipodal points in the CMB are separated by ~ 1.96 rhorizon. Why then is the temperature of the CMB constant to ~ 10-5 K? The Flatness Problem (or fine tuning problem) W0 ~ 1 today coupled with expansion of the Universe implies that |1 – W| < 10-14 when BB nucleosynthesis occurred. Our existence depends on a very close match to the critical density in the early Universe 1 4/28/19 Theory of Cosmic Inflation Universe undergoes brief period of exponential expansion How Inflation Solves the Flatness Problem 2 4/28/19 Cosmic Inflation Summary Standard Big Bang theory has problems with tuning and causality Inflation (exponential expansion) solves these problems: - Causality solved by observable Universe having grown rapidly from a small region that was in causal contact before inflation - Fine tuning problems solved by the diluting effect of inflation Inflation naturally explains origin of large scale structure: - Early Universe has quantum fluctuations both in space-time itself and in the density of fields in space. Inflation expands these fluctuation in size, moving them out of causal contact with each other. Thus, large scale anisotropies are “frozen in” from which structure can form. Some kind of inflation appears to be required but the exact inflationary model not decided yet… 3 4/28/19 BAO: Baryonic Acoustic Oscillations Predict an overdensity in baryons (traced by galaxies) ~ 150 Mpc at the scale set by the distance that the baryon-photon acoustic wave could have traveled before CMB recombination 4 4/28/19 Curves are different models of Wm A measure of clustering of SDSS Galaxies of clustering A measure Eisenstein et al.
    [Show full text]
  • Anisotropy of the Cosmic Background Radiation Implies the Violation Of
    YITP-97-3, gr-qc/9707043 Anisotropy of the Cosmic Background Radiation implies the Violation of the Strong Energy Condition in Bianchi type I Universe Takeshi Chiba, Shinji Mukohyama, and Takashi Nakamura Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-01, Japan (January 1, 2018) Abstract We consider the horizon problem in a homogeneous but anisotropic universe (Bianchi type I). We show that the problem cannot be solved if (1) the matter obeys the strong energy condition with the positive energy density and (2) the Einstein equations hold. The strong energy condition is violated during cosmological inflation. PACS numbers: 98.80.Hw arXiv:gr-qc/9707043v1 18 Jul 1997 Typeset using REVTEX 1 I. INTRODUCTION The discovery of the cosmic microwave background (CMB) [1] verified the hot big bang cosmology. The high degree of its isotropy [2], however, gave rise to the horizon problem: Why could causally disconnected regions be isotropized? The inflationary universe scenario [3] may solve the problem because inflation made it possible for the universe to expand enormously up to the presently observable scale in a very short time. However inflation is the sufficient condition even if the cosmic no hair conjecture [4] is proved. Here, a problem again arises: Is inflation the unique solution to the horizon problem? What is the general requirement for the solution of the horizon problem? Recently, Liddle showed that in FRW universe the horizon problem cannot be solved without violating the strong energy condition if gravity can be treated classically [5]. Actu- ally the strong energy condition is violated during inflation.
    [Show full text]
  • New Varying Speed of Light Theories
    New varying speed of light theories Jo˜ao Magueijo The Blackett Laboratory,Imperial College of Science, Technology and Medicine South Kensington, London SW7 2BZ, UK ABSTRACT We review recent work on the possibility of a varying speed of light (VSL). We start by discussing the physical meaning of a varying c, dispelling the myth that the constancy of c is a matter of logical consistency. We then summarize the main VSL mechanisms proposed so far: hard breaking of Lorentz invariance; bimetric theories (where the speeds of gravity and light are not the same); locally Lorentz invariant VSL theories; theories exhibiting a color dependent speed of light; varying c induced by extra dimensions (e.g. in the brane-world scenario); and field theories where VSL results from vacuum polarization or CPT violation. We show how VSL scenarios may solve the cosmological problems usually tackled by inflation, and also how they may produce a scale-invariant spectrum of Gaussian fluctuations, capable of explaining the WMAP data. We then review the connection between VSL and theories of quantum gravity, showing how “doubly special” relativity has emerged as a VSL effective model of quantum space-time, with observational implications for ultra high energy cosmic rays and gamma ray bursts. Some recent work on the physics of “black” holes and other compact objects in VSL theories is also described, highlighting phenomena associated with spatial (as opposed to temporal) variations in c. Finally we describe the observational status of the theory. The evidence is slim – redshift dependence in alpha, ultra high energy cosmic rays, and (to a much lesser extent) the acceleration of the universe and the WMAP data.
    [Show full text]
  • The Rh = Ct Universe Without Inflation
    A&A 553, A76 (2013) Astronomy DOI: 10.1051/0004-6361/201220447 & c ESO 2013 Astrophysics The Rh = ct universe without inflation F. Melia Department of Physics, The Applied Math Program, and Department of Astronomy, The University of Arizona, Tucson, AZ 85721, USA e-mail: [email protected] Received 26 September 2012 / Accepted 3 April 2013 ABSTRACT Context. The horizon problem in the standard model of cosmology (ΛDCM) arises from the observed uniformity of the cosmic microwave background radiation, which has the same temperature everywhere (except for tiny, stochastic fluctuations), even in regions on opposite sides of the sky, which appear to lie outside of each other’s causal horizon. Since no physical process propagating at or below lightspeed could have brought them into thermal equilibrium, it appears that the universe in its infancy required highly improbable initial conditions. Aims. In this paper, we demonstrate that the horizon problem only emerges for a subset of Friedmann-Robertson-Walker (FRW) cosmologies, such as ΛCDM, that include an early phase of rapid deceleration. Methods. The origin of the problem is examined by considering photon propagation through a FRW spacetime at a more fundamental level than has been attempted before. Results. We show that the horizon problem is nonexistent for the recently introduced Rh = ct universe, obviating the principal motivation for the inclusion of inflation. We demonstrate through direct calculation that, in this cosmology, even opposite sides of the cosmos have remained causally connected to us – and to each other – from the very first moments in the universe’s expansion. −35 −32 Therefore, within the context of the Rh = ct universe, the hypothesized inflationary epoch from t = 10 sto10 s was not needed to fix this particular “problem”, though it may still provide benefits to cosmology for other reasons.
    [Show full text]
  • Big Bang Nucleosynthesis Finally, Relative Abundances Are Sensitive to Density of Normal (Baryonic Matter)
    Big Bang Nucleosynthesis Finally, relative abundances are sensitive to density of normal (baryonic matter) Thus Ωb,0 ~ 4%. So our universe Ωtotal ~1 with 70% in Dark Energy, 30% in matter but only 4% baryonic! Case for the Hot Big Bang • The Cosmic Microwave Background has an isotropic blackbody spectrum – it is extremely difficult to generate a blackbody background in other models • The observed abundances of the light isotopes are reasonably consistent with predictions – again, a hot initial state is the natural way to generate these • Many astrophysical populations (e.g. quasars) show strong evolution with redshift – this certainly argues against any Steady State models The Accelerating Universe Distant SNe appear too faint, must be further away than in a non-accelerating universe. Perlmutter et al. 2003 Riese 2000 Outstanding problems • Why is the CMB so isotropic? – horizon distance at last scattering << horizon distance now – why would causally disconnected regions have the same temperature to 1 part in 105? • Why is universe so flat? – if Ω is not 1, Ω evolves rapidly away from 1 in radiation or matter dominated universe – but CMB analysis shows Ω = 1 to high accuracy – so either Ω=1 (why?) or Ω is fine tuned to very nearly 1 • How do structures form? – if early universe is so very nearly uniform Astronomy 422 Lecture 22: Early Universe Key concepts: Problems with Hot Big Bang Inflation Announcements: April 26: Exam 3 April 28: Presentations begin Astro 422 Presentations: Thursday April 28: 9:30 – 9:50 _Isaiah Santistevan__________ 9:50 – 10:10 _Cameron Trapp____________ 10:10 – 10:30 _Jessica Lopez____________ Tuesday May 3: 9:30 – 9:50 __Chris Quintana____________ 9:50 – 10:10 __Austin Vaitkus___________ 10:10 – 10:30 __Kathryn Jackson__________ Thursday May 5: 9:30 – 9:50 _Montie Avery_______________ 9:50 – 10:10 _Andrea Tallbrother_________ 10:10 – 10:30 _Veronica Dike_____________ 10:30 – 10:50 _Kirtus Leyba________________________ Send me your preference.
    [Show full text]
  • “The Constancy, Or Otherwise, of the Speed of Light”
    “Theconstancy,orotherwise,ofthespeedoflight” DanielJ.Farrell&J.Dunning-Davies, DepartmentofPhysics, UniversityofHull, HullHU67RX, England. [email protected] Abstract. Newvaryingspeedoflight(VSL)theoriesasalternativestotheinflationary model of the universe are discussed and evidence for a varying speed of light reviewed.WorklinkedwithVSLbutprimarilyconcernedwithderivingPlanck’s black body energy distribution for a gas-like aether using Maxwell statistics is consideredalso.DoublySpecialRelativity,amodificationofspecialrelativityto accountforobserverdependentquantumeffectsatthePlanckscale,isintroduced and it is found that a varying speed of light above a threshold frequency is a necessityforthistheory. 1 1. Introduction. SincetheSpecialTheoryofRelativitywasexpoundedandaccepted,ithasseemedalmost tantamount to sacrilege to even suggest that the speed of light be anything other than a constant.ThisissomewhatsurprisingsinceevenEinsteinhimselfsuggestedinapaperof1911 [1] that the speed of light might vary with the gravitational potential. Interestingly, this suggestion that gravity might affect the motion of light also surfaced in Michell’s paper of 1784[2],wherehefirstderivedtheexpressionfortheratioofthemasstotheradiusofastar whose escape speed equalled the speed of light. However, in the face of sometimes fierce opposition, the suggestion has been made and, in recent years, appears to have become an accepted topic for discussion and research. Much of this stems from problems faced by the ‘standardbigbang’modelforthebeginningoftheuniverse.Problemswiththismodelhave
    [Show full text]
  • (Ns-Tp430m) by Tomislav Prokopec Part III: Cosmic Inflation
    1 Lecture notes on Cosmology (ns-tp430m) by Tomislav Prokopec Part III: Cosmic Inflation In this and in the subsequent chapters we use natural units, that is we set to unity the Planck constant, ~ = 1, the speed of light, c = 1, and the Boltzmann constant, kB = 1. We do however maintain a dimensionful gravitational constant, and define the reduced Planck mass and the Planck mass as, M = (8πG )−1=2 2:4 1018 GeV ; m = (G )−1=2 1:23 1019 GeV : (1) P N ' × P N ' × Cosmic inflation is a period of an accelerated expansion that is believed to have occured in an early epoch of the Universe, roughly speaking at an energy scale, E 1016 GeV, for which the Hubble I ∼ parameter, H E2=M 1013 GeV. I ∼ I P ∼ The dynamics of a homogeneous expanding Universe with the line element, ds2 = dt2 a2d~x 2 (2) − is governed by the FLRW equation, a¨ 4πG Λ = N (ρ + 3 ) + ; (3) a − 3 P 3 from which it follows that an accelerated expansion, a¨ > 0, is realised either when the active gravitational energy density is negative, ρ = ρ + 3 < 0 ; (4) active P or when there is a positive cosmological term, Λ > 0. While we have no idea how to realise Λ in labo- ratory, a negative gravitational mass could be created by a scalar matter with a nonvanishing potential energy. Indeed, from the energy density and pressure in a scalar field in an isotropic background, 1 1 ρ = '_ 2 + ( ')2 + V (') ' 2 2 r 1 1 = '_ 2 + ( ')2 V (') (5) P' 2 6 r − we see that ρ = 2'_ 2 + ( ')2 2V (') ; (6) active r − can be negative, provided the potential energy is greater than twice the kinetic plus gradient energy, V > '_ 2 + ( ')2, = @~=a.
    [Show full text]
  • Cosmic Microwave Background Anisotropies and Theories Of
    Cosmic Microwave Background Anisotropies and Theories of the Early Universe A dissertation submitted in satisfaction of the final requirement for the degree of Doctor Philosophiae SISSA – International School for Advanced Studies Astrophysics Sector arXiv:astro-ph/9512161v1 26 Dec 1995 Candidate: Supervisor: Alejandro Gangui Dennis W. Sciama October 1995 TO DENISE Abstract In this thesis I present recent work aimed at showing how currently competing theories of the early universe leave their imprint on the temperature anisotropies of the cosmic microwave background (CMB) radiation. After some preliminaries, where we review the current status of the field, we con- sider the three–point correlation function of the temperature anisotropies, as well as the inherent theoretical uncertainties associated with it, for which we derive explicit analytic formulae. These tools are of general validity and we apply them in the study of possible non–Gaussian features that may arise on large angular scales in the framework of both inflationary and topological defects models. In the case where we consider possible deviations of the CMB from Gaussian statis- tics within inflation, we develop a perturbative analysis for the study of spatial corre- lations in the inflaton field in the context of the stochastic approach to inflation. We also include an analysis of a particular geometry of the CMB three–point func- tion (the so–called ‘collapsed’ three–point function) in the case of post–recombination integrated effects, which arises generically whenever the mildly non–linear growth of perturbations is taken into account. We also devote a part of the thesis to the study of recently proposed analytic models for topological defects, and implement them in the analysis of both the CMB excess kurtosis (in the case of cosmic strings) and the CMB collapsed three–point function and skewness (in the case of textures).
    [Show full text]
  • Building an Inflationary Model of the Universe
    Imperial College London Department of Theoretical Physics Building an Inflationary Model of the Universe Dominic Galliano September 2009 Supervised by Dr. Carlo Contaldi Submitted in part fulfilment of the requirements for the degree of Master of Science in Theoretical Physics of Imperial College London and the Diploma of Imperial College London Abstract The start of this dissertation reviews the Big Bang model and its associated problems. Inflation is then introduced as a model which contains solutions to these problems. It is developed as an additional aspect of the Big Bang model itself. The final section shows how one can link inflation to the Large Scale Structure in the universe, one of the most important pieces of evidence for inflation. i Dedicated to my Mum and Dad, who have always supported me in whatever I do, even quitting my job to do this MSc and what follows. Thanks to Dan, Ali and Dr Contaldi for useful discussions and helping me understand the basics. Thanks to Sinead, Ax, Benjo, Jerry, Mike, James, Paul, Valentina, Mike, Nick, Caroline and Matthias for not so useful discussions and endless card games. Thanks to Lee for keeping me sane. ii Contents 1 Introduction 1 1.1 The Big Bang Model . 1 1.2 Inflation . 3 2 Building a Model of the Universe 5 2.1 Starting Principles . 5 2.2 Geometry of the Universe . 6 2.3 Introducing matter into the universe . 11 2.3.1 The Newtonian Picture . 11 2.3.2 Introducing Relativity . 14 2.4 Horizons and Patches . 19 2.5 Example: The de Sitter Universe .
    [Show full text]