DOCSLIB.ORG

Type Ia Supernova Observations Physics 121 Galaxy Rotation Curves Dark Matter & Dark Energy December 4, 2009 Scaling Factor A(T) Course Evaluations Fainter

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Type Ia Supernova Observations Physics 121 Galaxy Rotation Curves Dark Matter & Dark Energy December 4, 2009 Scaling Factor A(T) Course Evaluations Fainter Today Supernovae revisited Type Ia Supernova Observations Physics 121 Galaxy Rotation Curves Dark Matter & Dark Energy December 4, 2009 Scaling Factor a(t) Course Evaluations fainter brighter Distant supernovae are fainter than nearby, and their light curves (observed at Earth) evolve more slowly, due to time dilation and other relativistic effects. Supernova Results Supernova Results:The Context Hubble’s observations found that Galaxies at small The results of the supernova redshifts (Z~0.003) are streaming away from us, with velocities proportional to distances. The brightness measurements surprised fainter universe is expanding. everyone. They show that supernova in high redshift galaxies are fainter Observations of supernovae in more distant galaxies brighter than expected. Crudely speaking, this (Z~1.00) found that they are fainter than expected. The expansion of the universe is accelerating, that is, means that they are more distant than the speeds with which galaxies are moving away predicted by the Hubble law. from one another are increasing. This was a surprise, because gravitational forces between the galaxies should cause them to This implies that the expansion of the decelerate. We conclude that there must be some universe is accelerating. The fainter sort of anti-gravity force, which we call “dark expansion causes the supernova light energy” or “cosmological constant.” to be spread more thinly over space, brighter What is this dark energy? Let’s put that question so that less of it is captured by the aside for the moment and look at yet another kind of telescope. mysterious stuff. 1 Galaxy Rotation Curves Galaxy Rotation Curves, continued Measurement of Doppler shifts of stars in our measured Galaxy (and therefore their speeds) give a surprising result: speeds do not get smaller as distance from the center gets larger. Stars in the outer reaches of our Galaxy are expected moving faster around their orbits than expected. R v Something must be wrong with our model of the galaxy: the net force on a given star is larger than that expected from the gravitational attraction of other stars in the Galaxy. Conclusion: there must be additional matter in our galaxy, pulling on stars and causing them to Stars in spiral galaxies such as our own Milky Way In reality, the net force on a star results from follow nearly circular orbits around the centers of the gravitational interactions with all the other stars in the move more quickly in their orbits. We call this galaxies. Consider a star at distance R from the galaxy, so it is a complicated thing to calculate. r v stuff “dark matter:” center of the galaxy, moving at speed v. We expect v Nevertheless, because most of the stars are relatively to be large for small R and small for large R. close to the center, if motion of stars is due only to • “dark:” we can’t see it gravity from other stars, then we expect velocity, v, to This is easy to prove if the force on each star is • “matter:” it interacts via the gravitational be smaller for stars at larger distance, R. dominated by gravity from a single massive lump in force, just like normal matter. the center of the galaxy. Milky Way rotation curve: Sofue et al 1999, Astrophysical Journal 523: 136 Galaxy Rotation Curves, continued Measurements in other galaxies give similar results: velocities are more-or- less constant as distance from the center of the galaxy is increased. In a typical spiral galaxy, stars can be found as far as 10-15 kiloparsecs from the center. Beyond that, there are no stars, but there are clouds of gas which allow us to measure rotation curves to Left: Expected drop-off of speed with distance from center of galaxy larger distances. The rotation curves Right: Constant speed at all distances from center of galaxy remain flat to remarkably large distances (The animation might work better in a browser than in PowerPoint.) Animation from: http://find.uchicago.edu/~pryke/dasi/documents/200309_plancourse/mgp00034.html Rotation curves: Bosma 1978 PhD Thesis via Faber & Gallagher 1979 Annual Review of Astronomy and Astrophysics 2 What is Dark Matter? The story so far MACHOS: MAssive Compact Halo Objects? Too much gravity on small scales (<~0.1 Megaparsec) In and around galaxies, things are moving too fast. The attractive forces “Machos” is basically a fancy name for “dead stars, which you can’t see any more:” “Massive” because they are heavy, “compact” because dead stars are very dense, “Halo” because they between masses are larger than we expect. We infer that there must be dark would form a halo around the galaxy. matter: massive, gravitationally interacting stuff which we cannot otherwise see. There are, in fact, dead stars in the galaxy, but are there enough to make up the dark matter? The answer turns out to be no. Dead stars will occasionally come close to the line-of-sight between us and ordinary stars. Their gravity of these dead stars will distort the space through Anti-gravity on large scales (~1000 Megaparsec) which light travels from the ordinary star to us. This is called a “microlensing event.” Such Galaxies are rushing away from each other, and this process is accelerating. events have been seen, but not enough to account for the dark matter. Our usual explanation is that there must be a force or phenomenon (such as Einstein’s cosmological constant) which is not part of our standard physical WIMPS: Weakly Interacting Massive Particles? laws. This is sometimes called dark energy. We can think of dark energy as a These would be tiny particles (tiny in the sense that protons and electrons are tiny). They could peculiar sort of stuff which has mass (so it does gravitationally interact) but not interact through the electromagnetic force; otherwise we would see them. Hence they must which also interacts through a new, not-yet-understood, repulsive (anti-gravity) interact through the weak force (and gravity). Physicists are actively searching for WIMPs, but force. none have been found. Something else entirely? Perhaps the most exciting possibility is that neither of these is right, and that there are as-yet-discovered physical processes, different from Newton/Einstein gravity, contributing to motions around galaxies. So What’s the Universe Made of Today? So What’s the Universe Made of Today? Only 5% of the content of the universe Dark Energy 72% (measured by energy content) is “normal matter,” stuff we can see. The other 95% is dark energy and dark matter. Radiation (Light) <0.01% In this jar, 95% of the jellybeans are black. Imagine how different this would look if we Normal Matter 4.5% could only see the colored jellybeans. Dark Matter 23% Image Credit: Fermilab. To see this jar constructed, search on “jelly bean universe” on YouTube. 3 So What’s the Universe Made of Today? The Metric for an Evolving Universe The table shows the energy content of the universe, tabulated using all available evidence. Refence: M. Fukugita and P. J. E. Peebles, 2004, “The Cosmic Energy Inventory”, Astrophysical Journal, 616: 643. For a copy of the full paper, do a web search on “astro-ph/0406095”. We can describe the expansion of the universe by defining a “scale function”, a(t). By definition, a=1 right now. At earlier times t, when the universe was smaller, a(t) was smaller. As the universe expands, a(t) is getting larger. We can take the flat-space metric, Δs2 = Δt2 – (Δx2 + Δy2 + Δz2) And modify it to allow for an expanding (or contracting) universe: Δs2 = Δt2 – a2(t)(Δx2 + Δy2 + Δz2) We model the universe as homogenous, so a(t) is the same everywhere. The universe is not in steady state, so a(t) changes with time. [Note: in an earlier handout, I called the scale factor R(t). Starting today, I am changing notation and calling it a(t) instead.] normal matter or dark matter w = 0 The scale factor a(t) changes over time. a˙˙ 4#G 2 = " $(1+ 3w) light w = 1/3 da d a a 2 Write its first and time derivatives as: a˙ = a˙˙ = 3c cosmological constant (dark energy) w = –1 dt dt2 General relativity gives a formula for the acceleration of the scale factor. It depends on ! two characteristics of the stuff which! causes! the acceleration, which we label ε and w: a˙˙ 4#G = " $(1+ 3w) a 3c2 Here G is Newton’s gravitation constant, G=6.67×10-11 m3/kg s2. Typically in physics we write! “ρ” (greek letter rho) to denote density, that is, mass divided by volume. However, when we talk about stuff that doesn’t have mass in the usual sense (dark energy) we need to do something different. So we use “ε” to represent energy density. Thus ε is the amount of energy in a given volume of space. For normal matter at rest, E=mc2, so ε = energy/volume=mc2/V. The variable “w” is the “equation of state” parameter. It is related to how the pressure of something changes with density. Here are the rules: normal matter or dark matter w = 0 light (photons) w = 1/3 dark energy w = –1 4 a˙˙ 4#G normal matter or dark matter w = 0 = " $(1+ 3w) light w = 1/3 a 3c2 cosmological constant (dark energy) w = –1 How does this fit our picture of the evolution of the Universe? ! In the early Universe (but after inflation...more about that later), the Universe was dense, and was dominated by normal matter. Plugging w=0 into the above equation, we see that ä is negative, so the scale factor, a(t), decelerates over time.
Load more

Recommended publications
  • AST 541 Lecture Notes: Classical Cosmology Sep, 2018
    ClassicalCosmology | 1 AST 541 Lecture Notes: Classical Cosmology Sep, 2018 In the next two weeks, we will cover the basic classic cosmology. The material is covered in Longair Chap 5 - 8. We will start with discussions on the first basic assumptions of cosmology, that our universe obeys cosmological principles and is expanding. We will introduce the R-W metric, which describes how to measure distance in cosmology, and from there discuss the meaning of measurements in cosmology, such as redshift, size, distance, look-back time, etc. Then we will introduce the second basic assumption in cosmology, that the gravity in the universe is described by Einstein's GR. We will not discuss GR in any detail in this class. Instead, we will use a simple analogy to introduce the basic dynamical equation in cosmology, the Friedmann equations. We will look at the solutions of Friedmann equations, which will lead us to the definition of our basic cosmological parameters, the density parameters, Hubble constant, etc., and how are they related to each other. Finally, we will spend sometime discussing the measurements of these cosmological param- eters, how to arrive at our current so-called concordance cosmology, which is described as being geometrically flat, with low matter density and a dominant cosmological parameter term that gives the universe an acceleration in its expansion. These next few lectures are the foundations of our class. You might have learned some of it in your undergraduate astronomy class; now we are dealing with it in more detail and more rigorously. 1 Cosmological Principles The crucial principles guiding cosmology theory are homogeneity and expansion.
    [Show full text]
  • Large Scale Structure Signals of Neutrino Decay
    Large Scale Structure Signals of Neutrino Decay Peizhi Du University of Maryland Phenomenology 2019 University of Pittsburgh based on the work with Zackaria Chacko, Abhish Dev, Vivian Poulin, Yuhsin Tsai (to appear) Motivation SM neutrinos: We have detected neutrinos 50 years ago Least known particles in the SM 1 Motivation SM neutrinos: We have detected neutrinos 50 years ago Least known particles in the SM What we don’t know: • Origin of mass • Majorana or Dirac • Mass ordering • Total mass 1 Motivation SM neutrinos: We have detected neutrinos 50 years ago Least known particles in the SM What we don’t know: • Origin of mass • Majorana or Dirac • Mass ordering D1 • Total mass ⌫ • Lifetime …… D2 1 Motivation SM neutrinos: We have detected neutrinos 50 years ago Least known particles in the SM What we don’t know: • Origin of mass • Majorana or Dirac probe them in cosmology • Mass ordering D1 • Total mass ⌫ • Lifetime …… D2 1 Why cosmology? Best constraints on (m⌫ , ⌧⌫ ) CMB LSS Planck SDSS Huge number of neutrinos Neutrinos are non-relativistic Cosmological time/length 2 Why cosmology? Best constraints on (m⌫ , ⌧⌫ ) Near future is exciting! CMB LSS CMB LSS Planck SDSS CMB-S4 Euclid Huge number of neutrinos Passing the threshold: Neutrinos are non-relativistic σ( m⌫ ) . 0.02 eV < 0.06 eV “Guaranteed”X evidence for ( m , ⌧ ) Cosmological time/length ⌫ ⌫ or new physics 2 Massive neutrinos in structure formation δ ⇢ /⇢ cdm ⌘ cdm cdm 1 ∆t H− ⇠ Dark matter/baryon 3 Massive neutrinos in structure formation δ ⇢ /⇢ cdm ⌘ cdm cdm 1 ∆t H− ⇠ Dark matter/baryon
    [Show full text]
  • Measuring the Velocity Field from Type Ia Supernovae in an LSST-Like Sky
    Prepared for submission to JCAP Measuring the velocity field from type Ia supernovae in an LSST-like sky survey Io Odderskov,a Steen Hannestada aDepartment of Physics and Astronomy University of Aarhus, Ny Munkegade, Aarhus C, Denmark E-mail: [email protected], [email protected] Abstract. In a few years, the Large Synoptic Survey Telescope will vastly increase the number of type Ia supernovae observed in the local universe. This will allow for a precise mapping of the velocity field and, since the source of peculiar velocities is variations in the density field, cosmological parameters related to the matter distribution can subsequently be extracted from the velocity power spectrum. One way to quantify this is through the angular power spectrum of radial peculiar velocities on spheres at different redshifts. We investigate how well this observable can be measured, despite the problems caused by areas with no information. To obtain a realistic distribution of supernovae, we create mock supernova catalogs by using a semi-analytical code for galaxy formation on the merger trees extracted from N-body simulations. We measure the cosmic variance in the velocity power spectrum by repeating the procedure many times for differently located observers, and vary several aspects of the analysis, such as the observer environment, to see how this affects the measurements. Our results confirm the findings from earlier studies regarding the precision with which the angular velocity power spectrum can be determined in the near future. This level of precision has been found to imply, that the angular velocity power spectrum from type Ia supernovae is competitive in its potential to measure parameters such as σ8.
    [Show full text]
  • AST4220: Cosmology I
    AST4220: Cosmology I Øystein Elgarøy 2 Contents 1 Cosmological models 1 1.1 Special relativity: space and time as a unity . 1 1.2 Curvedspacetime......................... 3 1.3 Curved spaces: the surface of a sphere . 4 1.4 The Robertson-Walker line element . 6 1.5 Redshifts and cosmological distances . 9 1.5.1 Thecosmicredshift . 9 1.5.2 Properdistance. 11 1.5.3 The luminosity distance . 13 1.5.4 The angular diameter distance . 14 1.5.5 The comoving coordinate r ............... 15 1.6 TheFriedmannequations . 15 1.6.1 Timetomemorize! . 20 1.7 Equationsofstate ........................ 21 1.7.1 Dust: non-relativistic matter . 21 1.7.2 Radiation: relativistic matter . 22 1.8 The evolution of the energy density . 22 1.9 The cosmological constant . 24 1.10 Some classic cosmological models . 26 1.10.1 Spatially flat, dust- or radiation-only models . 27 1.10.2 Spatially flat, empty universe with a cosmological con- stant............................ 29 1.10.3 Open and closed dust models with no cosmological constant.......................... 31 1.10.4 Models with more than one component . 34 1.10.5 Models with matter and radiation . 35 1.10.6 TheflatΛCDMmodel. 37 1.10.7 Models with matter, curvature and a cosmological con- stant............................ 40 1.11Horizons.............................. 42 1.11.1 Theeventhorizon . 44 1.11.2 Theparticlehorizon . 45 1.11.3 Examples ......................... 46 I II CONTENTS 1.12 The Steady State model . 48 1.13 Some observable quantities and how to calculate them . 50 1.14 Closingcomments . 52 1.15Exercises ............................. 53 2 The early, hot universe 61 2.1 Radiation temperature in the early universe .
    [Show full text]
  • Arxiv:0910.5224V1 [Astro-Ph.CO] 27 Oct 2009
    Baryon Acoustic Oscillations Bruce A. Bassett 1,2,a & Ren´ee Hlozek1,2,3,b 1 South African Astronomical Observatory, Observatory, Cape Town, South Africa 7700 2 Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, Cape Town, South Africa 7700 3 Department of Astrophysics, University of Oxford Keble Road, Oxford, OX1 3RH, UK a [email protected] b [email protected] Abstract Baryon Acoustic Oscillations (BAO) are frozen relics left over from the pre-decoupling universe. They are the standard rulers of choice for 21st century cosmology, provid- ing distance estimates that are, for the first time, firmly rooted in well-understood, linear physics. This review synthesises current understanding regarding all aspects of BAO cosmology, from the theoretical and statistical to the observational, and includes a map of the future landscape of BAO surveys, both spectroscopic and photometric . † 1.1 Introduction Whilst often phrased in terms of the quest to uncover the nature of dark energy, a more general rubric for cosmology in the early part of the 21st century might also be the “the distance revolution”. With new knowledge of the extra-galactic distance arXiv:0910.5224v1 [astro-ph.CO] 27 Oct 2009 ladder we are, for the first time, beginning to accurately probe the cosmic expansion history beyond the local universe. While standard candles – most notably Type Ia supernovae (SNIa) – kicked off the revolution, it is clear that Statistical Standard Rulers, and the Baryon Acoustic Oscillations (BAO) in particular, will play an increasingly important role. In this review we cover the theoretical, observational and statistical aspects of the BAO as standard rulers and examine the impact BAO will have on our understand- † This review is an extended version of a chapter in the book Dark Energy Ed.
    [Show full text]
  • A Model-Independent Approach to Reconstruct the Expansion History of the Universe with Type Ia Supernovae
    A Model-Independent Approach to Reconstruct the Expansion History of the Universe with Type Ia Supernovae Dissertation submitted to the Technical University of Munich by Sandra Ben´ıtezHerrera Max Planck Institute for Astrophysics October 2013 TECHNISHCE UNIVERSITAT¨ MUNCHEN¨ Max-Planck Institute f¨urAstrophysik A Model-Independent Approach to Reconstruct the Expansion History of the Universe with Type Ia Supernovae Sandra Ben´ıtezHerrera Vollst¨andigerAbdruck der von der Fakult¨atf¨urPhysik der Technischen Uni- versitt M¨unchen zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigten Dissertation. Vorsitzende(r): Univ.-Prof. Dr. Lothar Oberauer Pr¨uferder Dissertation: 1. Hon.-Prof. Dr. Wolfgang Hillebrandt 2. Univ.-Prof. Dr. Shawn Bishop Die Dissertation wurde am 23.10.2013 bei der Technischen Universit¨atM¨unchen eingereicht und durch die Fakult¨atf¨urPhysik am 21.11.2013 angenommen. \Our imagination is stretched to the utmost, not, as in fiction, to imagine things which are not really there, but just to comprehend those things which are there." Richard Feynman \It is impossible to be a mathematician without being a poet in soul." Sofia Kovalevskaya 6 Contents 1 Introduction 11 2 From General Relativity to Cosmology 15 2.1 The Robertson-Walker Metric . 15 2.2 Einstein Gravitational Equations . 17 2.3 The Cosmological Constant . 19 2.4 The Expanding Universe . 21 2.5 The Scale Factor and the Redshift . 22 2.6 Distances in the Universe . 23 2.6.1 Standard Candles and Standard Rulers . 26 2.7 Cosmological Probes for Acceleration . 27 2.7.1 Type Ia Supernovae as Cosmological Test .
    [Show full text]
  • Arxiv:2002.11464V1 [Physics.Gen-Ph] 13 Feb 2020 a Function of the Angles Θ and Φ in the Sky Is Shown in Fig
    Flat Space, Dark Energy, and the Cosmic Microwave Background Kevin Cahill Department of Physics and Astronomy University of New Mexico Albuquerque, New Mexico 87106 (Dated: February 27, 2020) This paper reviews some of the results of the Planck collaboration and shows how to compute the distance from the surface of last scattering, the distance from the farthest object that will ever be observed, and the maximum radius of a density fluctuation in the plasma of the CMB. It then explains how these distances together with well-known astronomical facts imply that space is flat or nearly flat and that dark energy is 69% of the energy of the universe. I. COSMIC MICROWAVE BACKGROUND RADIATION The cosmic microwave background (CMB) was predicted by Gamow in 1948 [1], estimated to be at a temperature of 5 K by Alpher and Herman in 1950 [2], and discovered by Penzias and Wilson in 1965 [3]. It has been observed in increasing detail by Roll and Wilkinson in 1966 [4], by the Cosmic Background Explorer (COBE) collaboration in 1989{1993 [5, 6], by the Wilkinson Microwave Anisotropy Probe (WMAP) collaboration in 2001{2013 [7, 8], and by the Planck collaboration in 2009{2019 [9{12]. The Planck collaboration measured CMB radiation at nine frequencies from 30 to 857 6 GHz by using a satellite at the Lagrange point L2 in the Earth's shadow some 1.5 10 km × farther from the Sun [11, 12]. Their plot of the temperature T (θ; φ) of the CMB radiation as arXiv:2002.11464v1 [physics.gen-ph] 13 Feb 2020 a function of the angles θ and φ in the sky is shown in Fig.
    [Show full text]
  • FRW Cosmology
    Physics 236: Cosmology Review FRW Cosmology 1.1 Ignoring angular terms, write down the FRW metric. Hence express the comoving distance rM as an function of time and redshift. What is the relation between scale factor a(t) and redshift z? How is the Hubble parameter related to the scale factor a(t)? Suppose a certain radioactive decay emits a line at 1000 A˚, with a characteristic decay time of 5 days. If we see it at z = 3, at what wavelength do we observe it, and whats the decay timescale? The FRW metric is given as 2 2 2 2 2 2 2 ds = −c dt + a(t) [dr + Sκ(r) dΩ ]: (1) Where for a given radius of curvature, R, 8 R sin(r=R) κ = 1 <> Sκ(r) = r κ = 0 :>R sinh(r=R) κ = −1 Suppose we wanted to know the instantaneous distance to an object at a given time, such that dt = 0. Ignoring the angular dependence, i.e. a point object for which dΩ = 0, we can solve for rM as Z dp Z rM ds = a(t) dr; 0 0 dp(t) = a(t)rM (t): (2) Where dp(t) is the proper distance to an object, and rM is the coming distance. Note that at the present time the comoving distance and proper distance are equal. Consider the rate of change of the proper time a_ d_ =ar _ = d : (3) p a p Hubble is famously credited for determining what is now called Hubbles law, which states that vp(t0) = H0dp(t0): (4) Relaxing this equation to now be a function of time we will let the Hubble constant now be the Hubble parameter.
    [Show full text]
  • Distances in Cosmology
    Distances in Cosmology REVIEW OF “WORLD MODELS” Simplify notation by adopting c=1, so that E=m. Friedmann's equation is then: . Substitute H(t)= a/a and recall that we can formally associate an energy density with the cosmological constant, i.e The index i refers to the type of particle fluid under consideration, e.g. matter or radiation. If the Universe is flat (k=0): Let us define the fraction of the critical density contributed by each component of the Universe : So that we have Ωm , Ωr and ΩΛ for matter, radiation and dark energy. These quantities are time-dependent, the values today are denoted as Ωm,0 We can rewrite the Friedmann eqn: If we define Ωk = -k/(aH)2, we can write: Flat FRW Cosmologies In the last lecture we showed that the density of matter evolves as: Set a0=1. The Friedmann equation becomes: or In a flat Universe, ΩΛ,0 + Ωm,0=1. Case 1: Λ>0 Use the substitution: to obtain Take the positive root: This can be integrated by completing the square in the u-integral and with substitutions v = u + 1 and cosh w = v: to yield the solution: Case 2, Λ < 0: Introduce Solution: CaseCase 33,, Λ=0 : This is now the Einstein-deSitter case which we have already encountered in the last lecture. A flat, pressureless universe with a small, but non-zero, cosmo- logical constant initially evolves as if it were Einstein-deSitter. Fo r Λ > 0, the second term on the right-hand side of these equations dominates at large values of t and the universe grows exponentially: A wide variety of world models are conceivable, depending on the values of the parameters Ω.
    [Show full text]
  • The Big Bang in a Nutshell
    Spring term 2014: Dark Matter – lecture 1/9 Torsten Bringmann ([email protected]) reading: any standard cosmology textbook • [here, I (mostly) follow the conventions of Weinberg] K. A. Olive, G. Steigman, T. P. Walker, “Primordial nucleosynthesis: The- • ory and observations,” Phys. Rept. 333, 389-407 (2000). [astro-ph/9905320]. 1 Recap – the big bang in a nutshell The three main building blocks of modern cosmology are general relativity, the cosmological principle and a description of matter as a perfect fluid. According to the cosmological principle,theuniverseishomogeneousand isotropic on large scales, i.e. it looks basically the same at all places and in all directions. This means that spacetime can be described by the Friedmann- Robertson-Walker (FRW) metric: dr2 ds2 = g dxµdx⌫ = dt2 + a2(t) + r2d⌦2 , (1) − µ⌫ − 1 kr2 − where k =0, 1, 1 for a flat, positively and negatively curved universe, re- spectively. For− the approximately flat geometry that we observe, the scale factor a(t)thusrelatesphysicaldistancesλphys to coordinate (or comoving) distances r via λphys = ar. Note that a change of coordinates would re-scale a,soitsnormalizationdoesnotcorrespondtoaphysicalquantity;conven- tionally one often sets its value today to a0 a(t0)=1. A perfect fluid can be completely characterized⌘ by its rest-frame energy density ⇢ and and isotropic pressure p (i.e. shear stresses, viscosity and heat conduction are assumed to be absent). In the rest frame of the fluid, its energy-momentum tensor takes the form T µ⌫ =diag(⇢, p, p, p); in a frame that is moving with 4-velocity v,itthentakestheform T µ⌫ =(⇢ + p)vµv⌫ + pgµ⌫ .
    [Show full text]
  • AST1100 Lecture Notes
    AST1100 Lecture Notes 23-24: Cosmology: models of the universe 1 The FRW-metric Cosmology is the study of the universe as a whole. In the lectures on cos- mology we will look at current theories of how the universe looked in the beginning, how it looks today, how it evolved from the beginning until today and how it will continue in the future. We learned i the lectures on general relativity that how an object moves depends on the geometry of space and time. The question of how the universe evolves is mainly a question about what happens to its ’ingredients’ as time goes by. How do the objects in the universe move with time? To answer this question, we thus need to find the geometry of the universe as a whole. We know that in order to find the geometry of spacetime, we need to specify the content of spacetime in the stress-energy tensor Tµν on the right hand side in the Einstein equation Gµν =8πTµν. But how can we specify the mass and energy content at every point in space- time of the whole universe? As always, we need to make some approximations and simplifying assumptions. One main assumption is the cosmological prin- ciple. The cosmological principle states that the universe is homogeneous and isotropic. The universe is assumed to, on average, have a similar compo- sition and density in all positions and to look the same in all directions. The two conditions, homogeneity and isotropy may, at first look, seem identical. It is easy to convince yourself that they are not: Imagine a universe filled with trees which are all aligned with each other, i.e.
    [Show full text]
  • Topics in Cosmology: Island Universes, Cosmological Perturbations and Dark Energy
    TOPICS IN COSMOLOGY: ISLAND UNIVERSES, COSMOLOGICAL PERTURBATIONS AND DARK ENERGY by SOURISH DUTTA Submitted in partial fulfillment of the requirements for the degree Doctor of Philosophy Department of Physics CASE WESTERN RESERVE UNIVERSITY August 2007 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the dissertation of ______________________________________________________ candidate for the Ph.D. degree *. (signed)_______________________________________________ (chair of the committee) ________________________________________________ ________________________________________________ ________________________________________________ ________________________________________________ ________________________________________________ (date) _______________________ *We also certify that written approval has been obtained for any proprietary material contained therein. To the people who have believed in me. Contents Dedication iv List of Tables viii List of Figures ix Abstract xiv 1 The Standard Cosmology 1 1.1 Observational Motivations for the Hot Big Bang Model . 1 1.1.1 Homogeneity and Isotropy . 1 1.1.2 Cosmic Expansion . 2 1.1.3 Cosmic Microwave Background . 3 1.2 The Robertson-Walker Metric and Comoving Co-ordinates . 6 1.3 Distance Measures in an FRW Universe . 11 1.3.1 Proper Distance . 12 1.3.2 Luminosity Distance . 14 1.3.3 Angular Diameter Distance . 16 1.4 The Friedmann Equation . 18 1.5 Model Universes . 21 1.5.1 An Empty Universe . 22 1.5.2 Generalized Flat One-Component Models . 22 1.5.3 A Cosmological Constant Dominated Universe . 24 1.5.4 de Sitter space . 26 1.5.5 Flat Matter Dominated Universe . 27 1.5.6 Curved Matter Dominated Universe . 28 1.5.7 Flat Radiation Dominated Universe . 30 1.5.8 Matter Radiation Equality . 32 1.6 Gravitational Lensing . 34 1.7 The Composition of the Universe .
    [Show full text]