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by: Hailey Halvorson


Hailey Halvorson
GPA 3.8


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Class Notes
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This 57 page Class Notes was uploaded by Hailey Halvorson on Thursday October 22, 2015. The Class Notes belongs to PHYS 133 at University of California Santa Barbara taught by Staff in Fall. Since its upload, it has received 31 views. For similar materials see /class/227144/phys-133-university-of-california-santa-barbara in Physics 2 at University of California Santa Barbara.




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Date Created: 10/22/15
tiny fraction of a secend Lecture 6 January 29 2007 Previously The universe is a 4 dimensional manifold as in General Relativity The universe is homogeneous and isotropic This implies that space and time can be separated so that we can de ne a cosmic time t There are only three possible geometries for the universe Their metric is the RobertsonWalker metric In the RW nonstatic universe redshift is a measure of distance The dynamics of the Universe is described by Friedmann Equation Newtonian analog Outline Dynamics of the Universe Friedmann Equation Fluid Equation Acceleration Equation Equation of State Cosmological Constant Dynamics of the Universe Friedmann Equation Limitations of Newtonian analog Newtonian What s outside the sphere Inhomogeneous Isotropic Friedmann Equation 1922 is the correct form with energy instead of mass and curvature instead of internal energy Blackboard 1888 1925 Dynamics of the Universe Fluid equation We have 1 equation and two functions at and at What is the connection between the two Adiabatic expansion Black board Dynamics of the Universe Acceleration equation Another convenient form not independent is the equation of motion with the second derivative Black board 0 Examples of Equation of state blackboard Fate of the universe in curvature mass models AK open o e Universe at c m 7 The cosmological constant Soon after the completion of general relativity 1916 people used it to describe the universe However with only matter there was no way to obtain a static solution which at that time was the prejudice Einstein added the cosmological constant to his equations to nd a static solution Blackboard The cosmological constant Unfortunately the static solution is unstable And when Hubble announced his discovery of the expansion unnecessary So the cosmological constant remained on the outskirts of cosmology for a long time Now it s back Cosmological constant or dark energy Nowadays people prefer to talk in terms of dark energy instead of cosmological constant The classic cosmological constant is a modi cation of Einstein s equation Black board Dark energy is interpreted as something with negative pressure lling space Is it some sort of vacuum energy We really don t know Dynamics of the Universe Summary Friedmann Equation describes the evolution of at depending on content and geometry of the universe Fluid equations describe the evolution of the content of the universe with at The cosmological constant or dark energy is an extra term in F riedmann Equation and can induce acceleration Currently supported by observations tiny fraction of a secend Lecture 9 February 2 2007 Previously I Curvature only at linear in time tOHO1 Horizon in nite Flat With w at scales as t to the power of 233W tOHO231W Horizon can be nite Previously II Matter only at scales as t to the power of 23 tOHO23 Horizon nite Radiation only at scales as t to the power of 12 tOHO12 Horizon nite Previously III A only at exponential int In nite Age In nite horizon Outline Modeling the Universe Let s get real Generalized F riedrnann Equation Multiple components Matter Curvature Matter A Multiple component Universes Introduction The universe is more complicated than this We know for sure that there are at least two components They are What happens when we have multiple components Generalized Friedmann Equation Ham H3 Elma 1 yaw3 95 173 m Ham H5 E L 3 391 j Email 3393 mu 1 2quot 11 ml 39I39 gnu 39I39 n RA 1 Blackboard Matter curvature New phenomena appear For example the big crunch Black board Matter A umHzmnm Mn xuma Illlllulllmlkl Mmqum th m Alquot39rl 391ll lllk Mnmu Jnalillwuxtuw CHILL constant can induce a r am acceleration The big chill Close to our model of the universe Black board Matter Curvature A New phenomena For example the Big Bounce Black board Big bounce Matter Curvature A New phenomena For example the Big Bounce Black board Big bounce Multiple component Universes Summary Generalized Friedmann Equation Four free parameters They are Matter Curvature Big Crunch Matter A Big Chill simplest model close to concordance cosmology tiny fraction of a secend Lecture 11 February 7 2007 Previously Matter Curvature A Big Bounce or various scenarios Radiation Matter Good for the early universe The current best guess Concordance cosmology or benchmark model Another interesting paper wwwarXivorgabsgrqcO612146 Outline Measuring cosmological parameters Intro A brief history and notation Proper distance Luminosity distance A good reference for some of today s stuff XXXlanl govabs astroph 99051 16 Kinematics and Mass da 1 Pa 2 3 IN 1tuEftH t tDH g 12m t l g Ot 1 111 10 10 Em qu E a 111le 1 I 10529u 111393m 1 10 QM EQWU 91 The kinematics of the universe measure its content In the good old days with no A HO and q0 were enough Seriously how do we measure the kinematics of the Universe What s the relation between redshift and proper distance locally Hubble s law Blackboard This allows us to measure one parameter maybe But proper distance is hard to measure Proper distance and cosmological parameters Blackboard More convenient distances Luminosity distance We can measure Flux If we know the luminosity we have a distance FL47t d12 But what distance is this Blackboard Standard candle Summary Measuring kinematics of the universe determines cosmological parameters Proper distance depends on redshift Via the Hubble constant to rst order Higher order terms of the kinematics are needed to obtain other cosmological parameters Proper distance is not appropriate We need stuff we can measure Luminosity distance Proper distance Hz for a at universe 7 tiny fraction of a secend Lecture 16 February 26 2007 Previously Energy is preserved during the expansion locally There is dark matter Galaxies rotation curves Clusters Virial theorem and hydrostatic equilibrium We do not know What it is It cannot be hidden baryons It could be new exotic particles Outline Gravitational lensing intro Gravitational lensing theory Strong Weak Micro Ryden 84 Detecting dark matter Gravitational lensing Mass concentrations perturb spacetirne altering the propagation of light true position of Star A apparent position at Star A true and apparent position of Star E Copyright XII Adamo quotquotquotquot 6y Under special circumstances the distortion is so strong that creates two images of a background object This is called strong lensing Image 1 aha Distant Object Dbsenrer V rd quot39 De e ur quot39 Image 2 JO73728453216185 J095629775100066 J120540434910293 J125028250523490 J140228216321335 J162746440053575 J163028154520362 J23212093 0939102 Einstein Ring Gravitational Lenses Hubble Space Telescope Advanced Camera for Surveys NASA ESA A Bolton Harvafd Smllhsonian CIA and the SLACS Team STSCIPRCOS 32 Detecting dark matter Why is it called lensing The physics is very Source Quadruple similar to that of common optical lenses In fact many of the features of gravitational lensing can be reproduced by common optical dCViCCS Einstein Ring Double Figure courtesy of Phil Marshall Clusters are also strong lenses The blue objects here are distorted images of the same object Gravitational Lens Galaxy Cluster 00241654 Hubble Space Telescope WFPC2 The image separation gives us a direct measurement of the mass enclosed by the images It is arguably the most precise measurement of mass that we can make And there are other applications too we ll see i 1 3th Gravitational Lens Galaxy Cluster 00241654 Hubble Spa Telescope WFPC2 Detecting dark matter Weak lensing Even when the gravitational eld is not strong enough to produce multiple images the large scale structure perturbs space time PATH Of UGH39I39 AROUND DARK MATIER This alters the shape of observed galaxies in the sky shearing and magnifying them in a a i 7 5quot measurable way Detecting dark matter Weak lensing mass maps Optical image Dark matter mass Lensing a little math Lensing Basics I Thin Screen Approximation Surface mass density 35 95 2 dz De ection angle 39 Rx 24G E 015 6 2 Wf M 40623 5 we 2w 0 mews Dds 7 l d I d l 17 at D Reduced de ection angfes I I 398 399 aim Lens equatlon 1 02 DH 2 D E E DdDds Critical density Lensing Basics 11 useful general relations 2D potential gt d5 Dd zdz d s Vg Ddngb g d quot511 dz 52 u 2D Poisson Equa uon 2 deds 2 DdDds 2093 2 2 Vggb 62 DE V ltIgtdz 62 DE 4 er 2 Ear 2me 3 9 9 9r2 AE 5 6a 6 M l 39339 6611693 525 W i ma J acobian matrix Mcaustic s 7 Lensing Basics III Caustics and Critical Lines QLiti9a1lines r r y r r r V r r 391 quot39r quotvquot Saha amp Williams 2003 0047 281 a g Koopmans amp Treu 2003 The time delay surface and Fermat s principle The time delay surface and Fermat s principle lT V910 0 Lens equation 6 10 Q2 lb Extrema 0f the 9 5 time delay surface Time delay surface 0 12 1Dgs 16 tgoom tgrav adimensional 6209 ae aej oc 61 my A Multiple images form at the extrema 0f the time delay surface Gravitational lensing Meaning of the time delay surface Gravitational elds not only bend light but the also slow down time For this reason as light travels close to a lens it takes longer than it should in normal geometry As we will see we can use this effect to measure the Hubble constant Summary Lensing basics Mass concentrations distort the images of objects in on the sky in a way similar to that of optical images Strong and weak lensing provide very accurate mass maps of objects in projection Gravitational elds also slow down light Strong lensing can be formulated in terms of Fermat s principle tiny fraction 01 a second Lecture 8 April 15 2009 Previously Friedmann Equation Fluid Equation and an Equation of State describes the evolution of at depending on content and geometry of the universe The cosmological constant a avor of dark energy is an extra term and can induce acceleration Currently supported by observations Outline What kind of Universe do we live in ewe or ewe Modeling the Universe Single component 7 Curvature only 7 Flat universe Matter only Radiation only 7 Lambda only Cosmological Models Friedmann Eqn Fluid Eqn and EOS What is the connection between at and at for any constant w Given appropiiate boundary conditions we can solve these equations for at at and Pt for all times Fluid equation tells us how at evolves with the expansion described by at 12 8WG H02 2 a 302 R602 3 lt P0 l P we What kind of Universe 8 may have many components We have a framework but the solutions may not be analytic when all components are included Let s investigate models made from single components and tted to the observed expansion rate H0 Comparison with observations will tell us the answer 74 Dark Energy 1 YA What kind of Universe Relativistic e from CMB and Stars CMB photons dominate over other photons e g starlight QCMB505 X 10395 relic om era when universe was hot and a 116 EEK 393 dense enough to be opaque to m Photons g if w w v 1 ww quot fem5 8 11 L t0 0007 eVcm3 K on QCMByo 003 M


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