DOCSLIB.ORG

Spatial Homogeneity and Redshift-Distance Laws (Galaxies/Cosmology/Hubble Law/Lundmark Law) J

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Spatial Homogeneity and Redshift-Distance Laws (Galaxies/Cosmology/Hubble Law/Lundmark Law) J Proc. Natl. Acad. Sci. USA Vol. 79, pp. 3913-3917, June 1982 Astronomy Spatial homogeneity and redshift-distance laws (galaxies/cosmology/Hubble law/Lundmark law) J. F. NICOLL* AND 1. E. SEGALt *Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742; and tDepartment of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Contributed by I. E. Segal, March 11, 1982 ABSTRACT Spatial homogeneity in the radial direction of the low redshift region and may be subject to various biases. low-redshift galaxies is subjected to Kafka-Schmidt V/Vm tests Here radial homogeneity is treated by the Kafka-Schmidt V/ using well-documented samples. Homogeneity is consistent with Vm test (see ref. 7), modified by the imposition oflimits on red- the assumption of the Lundmark (quadratic redshift-distance) shifts. The lower limit is required because of peculiar motions, law, but large deviations from homogeneity are implied by the and the upper limit because errors ofmeasurement place a red- assumption of the Hubble (linear redshift-distance) law. These shift limit on completeness (see ref. 2). The V/Vm test retains deviations are similar to what would be expected on the basis of its important property ofimmunity to the observational cutoffs the Lundmark law. Luminosity functions are obtained foreach law when such limits are imposed. In addition, it becomes capable by a nonparametric statistically optimal method that removes the ofbeing quite discriminatory between different redshift-distance observational cutoff bias in complete samples. Although the Hub- the without such limits, the V/Vm test ble law correlation ofabsolute magnitude with redshift is reduced laws, which is not case considerably by elimination of the bias, computer simulations then being substantially equivalent to the analysis of the N(m) show that its bias-free value is nevertheless at a statistically quite relationship. Such limits also tend to produce a sample that is significant level, indicating the self-inconsistency of the law. The subject to fewer and smaller errors. corresponding Lundmark law correlations are quite satisfactory The primary sample used here for statistical analysis is that statistically. The regression ofredshift on magnitude also involves of Visvanathan (2). This appears particularly appropriate be- radial spatial homogeneity and, according to R. Soneira, has slope cause of its homogeneity, completeness, whole sky character, determining the redshift-magnitude exponent independently of and relative freedom from corrections that may be model de- the luminosity function. We have, however, rigorously proved the pendent. It consists of -300 elliptical and similar galaxies; for material dependence of the regression on this function and here comparison, a sample ofde Vaucouleurs (3) that consists largely exemplifyour treatment by using the bias-free functions indicated, of spiral galaxies is also treated, including its angular diameter with results consistent with the foregoing argument. relations. As a consequence ofthe completeness in diameter of the Nilson sample (4) and the inclusion of these measurements For small redshifts z that are beyond the range of influence of in the work ofde Vaucouleurs et al. (5), from which the sample peculiar motions, the observed correlation ofapparent magni- (3) of de Vaucouleurs largely derives, it is to be expected that tude with redshift suggests that the redshift z varies approxi- this sample is likewise complete, down to a limiting diameter, mately as a power p ofthe "distance" r, p being a constant. The here taken conservatively large, at least in the northern galactic distance itselfis not observable and the empirically meaningful hemisphere. Subsamples defined by further limitations on mag- content of such a law derives from its combination with laws nitude, redshift, and galactic latitude are also considered. A relating other observable quantities such as luminosities and wide discrepancy from homogeneity, on the assumption of the angular sizes to distance. Elimination ofthe distance then leads Hubble law, is found in virtually all cases, in contrast to the to predicted relations involving only observed quantities. homogeneity implied by the Lundmark law, and has a character Such predictions are predicated on hypothesized uniformi- similar to that found earlier for a sample oflarger-redshift bright ties ofone type or another. For studies ofthe magnitude-redshift cluster galaxies studied by Peterson (6) in an analysis by Segal relation, uniformity in luminosity is involved; i.e., it is assumed (7). Moreover the departures from homogeneity are quite sim- that the absolute luminosities of galaxies in space have a fre- ilar to those predicted by the Lundmark law. quency distribution that is independent ofthe distance, for gal- Another observable relation, predicated in part on radial spa- axies in the class observed. (Observational imitations may have tial homogeneity and in part on luminosity uniformity, is the the consequence that this physical uniformity may not be di- redshift-magnitude regression. Claims ofSoneira (8) to the con- rectly observed as such, of course, appropriate procedures trary notwithstanding, it has been proved by Nicoll and Segal being required to elicit it.) (9) that this regression depends on the luminosity function and A logically quite distinct type ofuniformity is that ofthe spa- not merely, apart from zero point, on the redshift-magnitude tial distribution ofgalaxies. This spatial uniformity may be fur- exponent p. The latter work is here exemplified by a quanti- ther subdivided into angular and radial homogeneity. The sub- tative determination of the Hubble and Lundmark law predic- ject is classical (1) and, from early on, only radial homogeneity tions ofthis regression, for the Visvanathan sample in the same has appeared tenable on the scales here involved, and only this conservative magnitude and redshift ranges as in the V/Vm type is considered here. This is, briefly, the assumption that tests. the number of galaxies up to distance r varies as r3 in the low- The ROBUSTdeterminations ofluminosity functions treated redshift region, say up to redshifts of 0.01. Classical study has by-Nicoll et at (10) required to this end show independently of is nondiscrim- any assumption of radial spatial homogeneity the self-inconsis- treated the N(m) relationship, which, however, tency of the Hubble law by virtue of its improbably large cor- inatory between different redshift-distance exponents p in relation ofabsolute magnitude with redshift after removal ofthe observational cutoff bias by the ROBUST analysis. This sup- The publication costs ofthis article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertise- ment" in accordance with 18 U. S. C. §1734 solely to indicate this fact. Abbreviations: LF, luminosity function; OCB, observational cutoffbias. IQI .R Downloaded by guest on September 29, 2021 3914 Astronomy: Nicoll and Segal Proc. Natl. Acad. Sci. USA 79 (1982) plements and confirms the interpretation ofthe nonparametric C. 8>0 B<0 cross-testing of the Hubble and Lundmark laws reported ear- 0 lier. The Lundmark law is itself entirely self-consistent in the cr 1.0 indicated and other respects. 0 lot a0 00 Because of its dependence on two distinct uniformity hy- O. 0.8 _ at potheses, the redshift-magnitude regression is more difficult 0 a6 o D 0.6 0 6 o to analyze probabilistically than either the magnitude-redshift 0 a0 a o0 .0 0 8 u or V/Vm relations. Despite significant departures from both of 0 0g '- OA a 0 these uniformities, due to partial cancellation of their effects, 1[ p zo // the redshift-magnitude regression predicted by the Hubble law happens not to appear very strongly deviant from the empirical r 0.2 10 relationship, but systematic departures from it are suggested 0 0 E 0, , 02 0 8 1. and not found in the case of the closer-fitting prediction of the U 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 1.0 Lundmark law. V/Vm The V/Vm tests FIG. 2. V/VYi relations for the same sample as in Fig. 1 in the northern (Left) and southern(Right) galactic hemispheres (118 and 65 The sample of Visvanathan (2) is now considered up to its re- galaxies, respectively). Symbols are as in Fig. 1. ported limiting apparent magnitude of 12.4 and in the conser- vative redshift range 500 < cz < 2,250. The results of V/Vm natural question from a systematic methodological standpoint tests on this sample are shown in Fig. 1. There appear to be ofthe extent to which the Lundmark law explains the deviations systematic deviations of the Hubble law values from expecta- from the Hubble law is addressed by comparison of the V/Vi tion, assuming as throughout unless otherwise stated that the distribution actually observed using the Hubble law analysis galaxy distribution is homogeneous in the radial direction, but with the Lundmark law prediction of the results of the V/Vm the Lundmark law values appear to fit reasonably well. Statis- test when predicated on the Hubble law. As shown in Fig. 1, tical analysis confirms this suggestion from the visual appear- such predictions are in excellent agreement with the results ance of the curve. The Kolmogorov-Smirnov statistic D, de- actually observed. In view of these features, an initial pre- fined as the maximum deviation between the empirical cumu- sumption ofreal physical inhomogeneity, assuming the Hubble lative distribution function of the V/Vm value and that for a law, cannot be sustained consistently with general scientific uniform distribution on the unit interval [0,1], has the values principles. D, = 0.2475 and D2 = 0.0925, where the subscripts 1 and 2 From the standpoint ofthe Hubble law, a number of obser- refer, respectively, to the Hubble and Lundmark laws. The vational or statistical features present themselves as possible corresponding probabilities P of deviations as large as these, explications of this unexpected result; the major ones will be assuming the correctness ofthe respective laws, are < 10' and taken up seriatim.
Load more

Recommended publications
  • THE HOMOGENEITY SCALE of the UNIVERSE 1 Introduction On
    THE HOMOGENEITY SCALE OF THE UNIVERSE P. NTELIS Astroparticle and Cosmology Group, University Paris-Diderot 7 , 10 rue A. Domon & Duquet, Paris , France In this study, we probe the cosmic homogeneity with the BOSS CMASS galaxy sample in the redshift region of 0.43 <z<0.7. We use the normalised counts-in-spheres estimator N (<r) and the fractal correlation dimension D2(r) to assess the homogeneity scale of the universe. We verify that the universe becomes homogenous on scales greater than RH 64.3±1.6 h−1Mpc, consolidating the Cosmological Principle with a consistency test of ΛCDM model at the percentage level. Finally, we explore the evolution of the homogeneity scale in redshift. 1 Introduction on Cosmic Homogeneity The standard model of Cosmology, known as the ΛCDM model is based on solutions of the equations of General Relativity for isotropic and homogeneous universes where the matter is mainly composed of Cold Dark Matter (CDM) and the Λ corresponds to a cosmological con- stant. This model shows excellent agreement with current data, be it from Type Ia supernovae, temperature and polarisation anisotropies in the Cosmic Microwave Background or Large Scale Structure. The main assumption of these models is the Cosmological Principle which states that the universe is homogeneous and isotropic on large scales [1]. Isotropy is well tested through various probes in different redshifts such as the Cosmic Microwave Background temperature anisotropies at z ≈ 1100, corresponding to density fluctuations of the order 10−5 [2]. In later cosmic epochs, the distribution of source in surveys the hypothesis of isotropy is strongly supported by in X- ray [3] and radio [4], while the large spectroscopic galaxy surveys such show no evidence for anisotropies in volumes of a few Gpc3 [5].As a result, it is strongly motivated to probe the homogeneity property of our universe.
    [Show full text]
  • Shallow Water Waves on the Rotating Sphere
    PHYSICAL RBVIEW E VOLUME SI, NUMBER S MAYl99S Shallowwater waves on the rotating sphere D. MUller andJ. J. O'Brien Centerfor Ocean-AtmosphericPrediction Studies,Florida State Univenity, TallahDSSee,Florida 32306 (Received8 September1994) Analytical solutions of the linear wave equation of shallow waters on the rotating, spherical surface have been found for standing as weD as for inertial waves. The spheroidal wave operator is shown to playa central role in this wave equation. Prolate spheroidal angular functions capture the latitude- longitude anisotropy, the east-westasymmetry, and the Yoshida inhomogeneity of wave propagation on the rotating spherical surface. The identification of the fundamental role of the spheroidal wave opera- tor permits an overview over the system's wave number space. Three regimes can be distinguished. High-frequency gravity waves do not experience the Yoshida waveguide and exhibit Cariolis-induced east-westasymmetry. A second, low-frequency regime is exclusively populated by Rossby waves. The familiar .8-plane regime provides the appropriate approximation for equatorial, baroclinic, low- frequency waves. Gravity waves on the .8-plane exhibit an amplified Cariolis-induced east-westasym- metry. Validity and limitations of approximate dispersion relations are directly tested against numeri- caDy calculated solutions of the full eigenvalueproblem. PACSnumber(s): 47.32.-y. 47.3S.+i,92.~.Dj. 92.10.Hm L INTRODUcnON [4] in the framework of the .B-pJaneapproximation. In this Cartesian approximation of the spherical geometry The first theoretical investigation of the large-scale near the equator Matsuno identified Rossby, mixed responseof the ocean-atmospheresystem to external tidal Rossby-gravity, as well as gravity waves, including the forces was conducted by Laplace [I] in terms of the KelVin wave.
    [Show full text]
  • Dark Energy and the Homogeneous Universe 暗能量及宇宙整體之膨脹
    Dark Energy and the Homogeneous Universe 暗能量及宇宙整體之膨脹 Lam Hui 許林 Columbia University This is a series of 3 talks. Today: Dark energy and the homogeneous universe July 11: Dark matter and the large scale structure of the universe July 18: Inflation and the early universe Outline Basic concepts: a homogeneous, isotropic and expanding universe. Measurements: how do we measure the universe’s expansion history? Dynamics: what determines the expansion rate? Dark energy: a surprising recent finding. There is by now much evidence for homogeneity and isotropy. There is also firm evidence for an expanding universe. scale ∼ 1023 cm Blandton & Hogg scale ∼ 1024 cm Hubble deep field scale ∼ 1025 cm CfA survey Sloan Digital Sky Survey scale ∼ 1026 cm Ned Wright There is by now much evidence for homogeneity and isotropy. There is also firm evidence for an expanding universe. There is by now much evidence for homogeneity and isotropy. There is also firm evidence for an expanding universe. But how should we think about it? Frequently Asked Questions: 1. What is the universe expanding into? 2. Where is the center of expansion? 3. Is our galaxy itself expanding? Frequently Asked Questions: 1. What is the universe expanding into? Nothing. 2. Where is the center of expansion? Everywhere and nowhere. 3. Is our galaxy itself expanding? No. C C A B A B dAC = 2 dAB ∆d ∆d AC = 2 AB Therefore: ∆t ∆t Hubble law: Velocity ∝ Distance Supernova 1998ba Supernova Cosmology Project (Perlmutter, et al., 1998) (as seen from Hubble Space Telescope) 3 Weeks Supernova Before Discovery (as seen from telescopes on Earth) Difference Supernova 超新星as standard candle FAINTER (Farther) (Further back in time) Perlmutter, et al.
    [Show full text]
  • On Some Consequences of the Canonical Transformation in the Hamiltonian Theory of Water Waves
    579 On some consequences of the canonical transformation in the Hamiltonian theory of water waves Peter A.E.M. Janssen Research Department November 2008 Series: ECMWF Technical Memoranda A full list of ECMWF Publications can be found on our web site under: http://www.ecmwf.int/publications/ Contact: [email protected] c Copyright 2008 European Centre for Medium-Range Weather Forecasts Shinfield Park, Reading, RG2 9AX, England Literary and scientific copyrights belong to ECMWF and are reserved in all countries. This publication is not to be reprinted or translated in whole or in part without the written permission of the Director. Appropriate non-commercial use will normally be granted under the condition that reference is made to ECMWF. The information within this publication is given in good faith and considered to be true, but ECMWF accepts no liability for error, omission and for loss or damage arising from its use. On some consequences of the canonical transformation in the Hamiltonian theory of water waves Abstract We discuss some consequences of the canonical transformation in the Hamiltonian theory of water waves (Zakharov, 1968). Using Krasitskii’s canonical transformation we derive general expressions for the second order wavenumber and frequency spectrum, and the skewness and the kurtosis of the sea surface. For deep-water waves, the second-order wavenumber spectrum and the skewness play an important role in understanding the so-called sea state bias as seen by a Radar Altimeter. According to the present approach, but in contrast with results obtained by Barrick and Weber (1977), in deep-water second-order effects on the wavenumber spectrum are relatively small.
    [Show full text]
  • 2.Homogeneity & Isotropy Fundamental
    11/2/03 Chris Pearson : Fundamental Cosmology 2: Homogeneity and Isotropy ISAS -2003 HOMOGENEITY & ISOTROPY FuFundndaammententaall CCoosmsmoolologygy:: 22..HomHomogeneityogeneity && IsotropyIsotropy “When I trace at my pleasure the windings to and fro of the heavenly bodies, I no longer touch the earth with my feet: I stand in the presence of Zeus himself and take my fill of ambrosia, food of the gods.”!! Ptolemy (90-168(90-168 BC) 1 11/2/03 Chris Pearson : Fundamental Cosmology 2: Homogeneity and Isotropy ISAS -2003 HOMOGENEITY & ISOTROPY 2.2.1:1: AncientAncient CosCosmologymology • Egyptian Cosmology (3000B.C.) Nun Atum Ra 結婚 Shu (地球) Tefnut (座空天国) 双子 Geb Nut 2 11/2/03 Chris Pearson : Fundamental Cosmology 2: Homogeneity and Isotropy ISAS -2003 HOMOGENEITY & ISOTROPY 2.2.1:1: AncientAncient CosCosmologymology • Ancient Greek Cosmology (700B.C.) {from Hesoid’s Theogony} Chaos Gaea (地球) 結婚 Gaea (地球) Uranus (天国) 結婚 Rhea Cronus (天国) The Titans Zeus The Olympians 3 11/2/03 Chris Pearson : Fundamental Cosmology 2: Homogeneity and Isotropy ISAS -2003 HOMOGENEITY & ISOTROPY 2.2.1:1: AncientAncient CosCosmologymology • Aristotle and the Ptolemaic Geocentric Universe • Ancient Greeks : Aristotle (384-322B.C.) - first Cosmological model. • Stars fixed on a celestial sphere which rotated about the spherical Earth every 24 hours • Planets, the Sun and the Moon, moved in the ether between the Earth and stars. • Heavens composed of 55 concentric, crystalline spheres to which the celestial objects were attached and rotated at different velocities (w = constant for a given sphere) • Ptolemy (90-168A.D.) (using work of Hipparchus 2 A.D.) • Ptolemy's great system Almagest.
    [Show full text]
  • 3 Classical Symmetries and Conservation Laws
    3 Classical Symmetries and Conservation Laws We have used the existence of symmetries in a physical system as a guiding principle for the construction of their Lagrangians and energy functionals. We will show now that these symmetries imply the existence of conservation laws. There are different types of symmetries which, roughly, can be classi- fied into two classes: (a) spacetime symmetries and (b) internal symmetries. Some symmetries involve discrete operations, hence called discrete symme- tries, while others are continuous symmetries. Furthermore, in some theories these are global symmetries,whileinotherstheyarelocal symmetries.The latter class of symmetries go under the name of gauge symmetries.Wewill see that, in the fully quantized theory, global and local symmetries play different roles. Spacetime symmetries are the most common examples of symmetries that are encountered in Physics. They include translation invariance and rotation invariance. If the system is isolated, then time-translation is also a symme- try. A non-relativistic system is in general invariant under Galilean trans- formations, while relativistic systems, are instead Lorentz invariant. Other spacetime symmetries include time-reversal (T ), parity (P )andcharge con- jugation (C). These symmetries are discrete. In classical mechanics, the existence of symmetries has important conse- quences. Thus, translation invariance,whichisaconsequenceofuniformity of space, implies the conservation of the total momentum P of the system. Similarly, isotropy implies the conservation of the total angular momentum L and time translation invariance implies the conservation of the total energy E. All of these concepts have analogs in field theory. However, infieldtheory new symmetries will also appear which do not have an analog in the classi- 52 Classical Symmetries and Conservation Laws cal mechanics of particles.
    [Show full text]
  • Cosmic Homogeneity: a Spectroscopic and Model-Independent Measurement
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Portsmouth University Research Portal (Pure) MNRAS 475, L20–L24 (2018) doi:10.1093/mnrasl/slx202 Advance Access publication 2017 December 20 Cosmic homogeneity: a spectroscopic and model-independent measurement R. S. Gonc¸alves,1‹ G. C. Carvalho,1,2 C. A. P. Bengaly Jr,1,3 J. C. Carvalho,1 A. Bernui,1 J. S. Alcaniz1,4 and R. Maartens3,5 1Observatorio´ Nacional, 20921-400, Rio de Janeiro - RJ, Brasil 2Departamento de Astronomia, Universidade de Sao˜ Paulo, 05508-090 Sao˜ Paulo - SP, Brasil 3Department of Physics and Astronomy, University of the Western Cape, Cape Town 7535, South Africa 4Physics Department, McGill University, Montreal QC, H3A 2T8, Canada 5Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth PO1 3FX, UK Accepted 2017 December 11. Received 2017 November 14; in original form 2017 October 20 ABSTRACT Cosmology relies on the Cosmological Principle, i.e. the hypothesis that the Universe is homogeneous and isotropic on large scales. This implies in particular that the counts of galaxies should approach a homogeneous scaling with volume at sufficiently large scales. Testing homogeneity is crucial to obtain a correct interpretation of the physical assumptions underlying the current cosmic acceleration and structure formation of the Universe. In this letter,weuse the Baryon Oscillation Spectroscopic Survey to make the first spectroscopic and model- independent measurements of the angular homogeneity scale θ h. Applying four statistical estimators, we show that the angular distribution of galaxies in the range 0.46 < z < 0.62 is consistent with homogeneity at large scales, and that θ h varies with redshift, indicating a smoother Universe in the past.
    [Show full text]
  • Lecture 12  Lagrangian of System of Particles (, , ) →
    PH101 Lecture 12 ¬È¿Ä¹¿Æ» ż ¨»·ÉÊ ¹Ê¿ÅÄ Lagrangian of system of particles ͆(ͥ%, ͥʖ%, ͨ) → Ê̋ Action integral ǹ ¨ ÇÀ,Çʖ À,Ê ºÊ Ê̊ A mechanical system will evolve in time in such that action integral is stationary Hamilton’s Principle of Least Action → Ê̋ Ê̋ Stationary ǹ ¨ ÇÀ,Çʖ À,Ê ºÊ î ǹ ¨ ÇÀ,Çʖ À, Ê ºÊ = ̉ Ê̊ Ê̊ Stationary condition of Action integral ͘ "͆ "͆ Lagrange’s equation from Variational − = 0 principle ͨ͘ "ͥʖ% "ͥ% symmetries and conservation laws Cyclic coordinates, symmetries and conservation laws Cyclic coordinate ( Corresponding generalized (canonical) ͥ%) momentum ( i conserved ͤ% = ) i,ʖĝ is independent of the particular ͆ cyclic coordinate ( . System is symmetric under translation ͥ%) has no effect on in generalized coordinate ͥ% → ͥ% + ͥ% ͆. ÇÀ Symmetry in generalized coordinate gives rise to a conserved canonical momentum. Example : For planetary motion 1 ͇́͡ ͆ = ͡ ͦʖ ͦ + ͦͦʖ ͦ + is independent of rotation angle (cyclic2 coordinate), the systemͦ has rotational symmetry ͆ [symmetric under translation in , the system remains the same after change in . → + ) As is cyclic, corresponding generalized (canonical) momentum ͦ =Constant; ͤW = ͦ͡ ʖ which is nothing but angular momentum. Conclusion : Conservation of angular momentum is related to rotational symmetry of the system. Translational symmetry and homogeneity of space Homogeneity of space : Space is such that in a particular direction, all the points are equivalent. Z Homogeneous in X-direction Potential at → every point along the line ͆ ͬ, ͭ, ͮ = ͆(ͬɑ, ͭ, ͮ) Î̊ Î̎ Y Homogeneous along X-direction is invariant under → ͆ translation along X-direction conserved X → ͤ3 Homogeneous along Y-direction is invariant under → ͆ translation along Y-direction conserved.
    [Show full text]
  • Interaction of Linear Modulated Waves and Unsteady Dispersive Hydrodynamic States with Application to Shallow Water Waves
    J. Fluid Mech. (2019), vol. 875, pp. 1145–1174. c The Author(s) 2019 1145 This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited. doi:10.1017/jfm.2019.534 . Interaction of linear modulated waves and unsteady dispersive hydrodynamic states with application to shallow water waves T. Congy1, †, G. A. El1 and M. A. Hoefer2 1Department of Mathematics, Physics and Electrical Engineering, Northumbria University, https://www.cambridge.org/core/terms Newcastle upon Tyne NE1 8ST, UK 2Department of Applied Mathematics, University of Colorado, Boulder, CO 80309-0526, USA (Received 14 December 2018; revised 27 June 2019; accepted 27 June 2019) A new type of wave–mean flow interaction is identified and studied in which a small-amplitude, linear, dispersive modulated wave propagates through an evolving, nonlinear, large-scale fluid state such as an expansion (rarefaction) wave or a dispersive shock wave (undular bore). The Korteweg–de Vries (KdV) equation is considered as a prototypical example of dynamic wavepacket–mean flow interaction. Modulation equations are derived for the coupling between linear wave modulations and a nonlinear mean flow. These equations admit a particular class of solutions that describe the transmission or trapping of a linear wavepacket by an unsteady , subject to the Cambridge Core terms of use, available at hydrodynamic state. Two adiabatic invariants of motion are identified that determine the transmission, trapping conditions and show that wavepackets incident upon smooth expansion waves or compressive, rapidly oscillating dispersive shock waves exhibit so-called hydrodynamic reciprocity recently described in Maiden et al.
    [Show full text]
  • Cosmic Microwave Background (CMB) Cosmic Microwavepenzias and Wilson Background (1965)
    an orientation and an invitation we are here photo Courtesy: A Frolov, Aspen, CO 2017 Image Credit: Hubble Ultra-Deep Field, NASA/ESA we are here Length Scales ? few kpc ⇠ few light hours ⇠ few m ⇠ 9 few thousand km Gpc = 10 pc Mpc = 106 pc kpc = 103 pc pc 3.26 lyrs ⇡ few Mpc ⇠ Image Credit: Hubble Ultra-Deep Field, NASA/ESA large scale distribution of galaxies few Gpc ⇠ Image Credit: Hubble Ultra-Deep Field, NASA/ESA large scale isotropy and homogeneity (+ inhomogeneities) few Gpc ⇠ cosmological principle: no special location or directionImage Credit: Hubble Ultra-Deep Field, NASA/ESA dynamics: what are distant galaxies doing ? Image Credit: Hubble Ultra-Deep Field, NASA/ESA 1930’s interpreted as the universe is expanding separation a(t2)∆x separation a(t1)∆x time a(t)=scalefactor our cosmic story 1 1 a˙(today) = H 68 km s− Mpc− 0 ⇡ g expandin 1 10 age of universe H− 10 yrs ⇠ 0 ⇠ credit: adapted from NASA early universe was hotter expanding (filled with radiationhot + matter) cold credit: adapted from NASA too hot for atoms to exist - - + - + + + - + - - - + - + - + - - + + + - + - Cosmic Microwave Background (CMB) cosmic microwavePenzias and Wilson background (1965) T0 =2.73K Credit: Roger Ressmeyer cosmic microwave background T =2.726 0.001 K ± COBE/FIRAS Mather et. al 1994 Fixen et. al 1996 Cosmic Microwave Background temperature anisotropies 5 δT/T 10− ⇠ Planck 2015 today the deviations from homogeneity are larger! Image Credit: Hubble Ultra-Deep Field, NASA/ESA our cosmic story lumpier homogeneous g expandin hot (filled with radiation
    [Show full text]
  • DIMENSIONAL ANALYSIS Ain A
    1 The Physical Basis of DIMENSIONAL ANALYSIS Ain A. Sonin Second Edition 2 Copyright © 2001 by Ain A. Sonin Department of Mechanical Engineering MIT Cambridge, MA 02139 First Edition published 1997. Versions of this material have been distributed in 2.25 Advanced Fluid Mechanics and other courses at MIT since 1992. Cover picture by Pat Keck (Untitled, 1992) 3 Contents 1. Introduction 1 2. Physical Quantities and Equations 4 2.1 Physical properties 4 2.2 Physical quantities and base quantities 5 2.3 Unit and numerical value 10 2.4 Derived quantities, dimension, and dimensionless quantities 12 2.5 Physical equations, dimensional homogeneity, and physical constants 15 2.6 Derived quantities of the second kind 19 2.7 Systems of units 22 2.8 Recapitulation 27 3. Dimensional Analysis 29 3.1 The steps of dimensional analysis and Buckingham’s Pi-Theorem 29 Step 1: The independent variables 29 Step 2: Dimensional considerations 30 Step 3: Dimensional variables 32 Step 4: The end game and Buckingham’s P -theorem 32 3.2 Example: Deformation of an elastic sphere striking a wall 33 Step 1: The independent variables 33 Step 2: Dimensional considerations 35 Step 3: Dimensionless similarity parameters 36 Step 4: The end game 37 3.2 On the utility of dimensional analysis and some difficulties and questions that arise in its application 37 Similarity 37 Out-of-scale modeling 38 Dimensional analysis reduces the number of variables and minimizes work. 38 4 An incomplete set of independent quantities may destroy the analysis 40 Superfluous independent quantities complicate the result unnecessarily 40 On the importance of simplifying assumptions 41 On choosing a complete set of independent variables 42 The result is independent of how one chooses a dimensionally independent subset 43 The result is independent of the type of system of units 43 4.
    [Show full text]
  • Cosmology and Time
    Cosmology and Time Claus Beisbart Seminar Philosophy of Space and Time (Prof. J. Earman, Prof. J. D. Norton) Pitt, 18-March-2009 Overview Physics: 1. General Relativistic Cosmology 2. Inflation Philosophy: 1. The notion of Cosmic Time 2. Does Time have a beginning? 3. The Cosmic Arrow of Time 4. Cosmology and the Past Hypothesis General Theory of Relativity Potential models of the Theory: <M, g, T> M space-time manifold mn 2 m n g/g metric field tensor: ds = gmn d x d x T/Tmn momentum energy tensor Actual models fulfill Einstein’s field equations: G + g = 8 T Cosmological Constant G Einstein curvature tensor, derived from g Cosmological Models History: Einstein (1917), Sitzungsberichte: Kosmologische Betrachtungen zur Allgemeinen Relativitätstheorie -> Einstein model/cosmos De Sitter (1917), KNAW Proceedings 19 II: de Sitter model Cosmological Models? Terminology: Matter/energy: “Cosmic fluid” Constraints on cosmological models: 1. Realistic matter model for cosmic fluid: constrains/fixes T. Simplest model: Dust model: Every fluid element characterized in terms of density r and four-velocity um pressure-less fluid: Tmn = r um un Idealization! 2. <M,g> faithfully captures the geometrical structure of the Universe at large scales. Cosmological Standard Models Obey the Cosmological Principle. First shot: (CP) The Universe is spatially homogeneous and spatially isotropic around every observer comoving with the cosmic fluid. Covariance?! Define spatial homogeneity and spatial isotropy in a covariant way. Homogeneity and isotropy < M, g > is spatially homogeneous, iff there is a foliation of space-time into space-like hypersurfaces Ht with the following property: For each t and each pair of events p, q Ht there is an isometry (metric-preserving map) that maps p into q.
    [Show full text]