Physics 129 LECTURE 9 February 4, 2014

Transcription

1 Physics 129 LECTURE 9 February 4, 2014 Introduction to Cosmology! The Expanding Universe! The Friedmann Equation! The Age of the Universe! Cosmological Parameters! History of Cosmic Expansion! The Benchmark Model! The Backward Lightcone! Cosmic Particle and Event Horizons! Distances in the Expanding Universe! Ned Wright s Cosmology Calculator The in-class open-book Midterm Exam will be Thursday February 13.

2 Reserve Books for Physics 129

3 ts will lead to tight constraints favouring = 0 ama75dark, energy equation-of-state parameter,k w = nd o).39, a32we note that we choose to use the R)+BOSS data combination in the likelihood Sect. 6. This choice includes the two most accueasurements and, since the e ective +2.8redshifts of es are widely separated, it should be 2.0a very good on to neglect correlations between the surveys ± ± The Expanding Universe < ± Edwin Hubble ± discovered the expansion of the universe by discovering a linear relation 5.34 between the expansion velocity v of a galaxy and its distance D: ±± bble constant v = H0D mpersult from the fits of the base CDM model to Planck ndard where constant of proportionality H0, called the Hubble constant or Hubble parameter, has a is the low value ofthe the Hubble which is ±constant, ology, ± 8.0 rained by CMB data alone in this model. From the l den- the value +highl analysis we find (according to Perkins) 0028, ± value for H, from the Planck satellite data plus much other astronomical ± Actually, the latest 0 nstant, 1.2) km s Mpc (68%; Planck+WP+highL).(51) direct data, is ± And the actual recession velocity of a galaxy is the sum of its expansion alue67.77 of H has been found in other CMB experisnotably from 3.30 ±analysis from the recent 3.25 WMAP-9 Fitting velocity and its peculiar velocity vp, generated mostly by local gravitational effects: precidm model, Hinshaw et al. (2012) ± find round ±ogical 2.2)53.0 km s 1 Mpc 1 (68%; WMAP-9), 49.7 ± 5.0 (52) v = H0D + vp ± Planck Collaboration: Cosmological parameters oured ith Eq. (51) to within 1. We emphasize here that tensor imates are highly model dependent. It is important +1.2 A galaxy s redshift is given by with 68% errors on m and Table 8. Approximate is no ± constraints H0 (in units of km s Mpc 1 ) from BAO, with!m and!b fixed B data, z =best-fit (λo λ e)/λe to the Planck+WP+highL values for the base CDM asses ± mical cosmology. where λ e and λo are the emitted and Planck uccess Planck errors are small and Planck s > Sample Measuring galaxy observed wavelengths. H0 m to the values for H0 and Ωm are rather in the redshifts is easy, but measuring their +3.2 different from some earlier ones 6dF...± cance, Edition) distancessdss is hard. Milton Humason and bration parameters, and addimodel. n the Explanatory Supplement others had <measured a number of galaxy WiggleZ... l rightsredshifts, reserved. BOSS but.hubble figured out... how to +2.8 ut permission from6df+sdss+boss+wigglez the publisher, 1 measure....using distances to galaxies 2.0 applicable copyright law. 6dF+SDSS(R)+BOSS SDSS(R) Cepheid variable stars.the He got relative 6dF+SDSS(R)+BOSS+WiggleZ... the.2 = the Planck spectra ± or distances more less right, although his e mean assuming therecalibrated model isas distance scale was later surements constrain parameters in the base CDM model, we ± variables were better understood. essescepheidin units of the disperform, ± 1.1 = (x x CDM )T C 2 BAO 1 BAO (x x CDM ), (50)

4 General Relativity CURVED SPACE TELLS MATTER HOW TO MOVE du µ ds + Γ µ αβ u α u β = 0 MATTER TELLS SPACE HOW TO CURVE xt xt Text Einstein Field Equations G µν R µν ½Rg µν = 8πGT µν Λg µν Here u α is the velocity 4-vector of a particle. The Riemann curvature tensor R λ µσν, Ricci curvature tensor R µν R λµσν g λσ, curvature scalar R R µν g µν, and affine connection Γ µ can αβ be calculated from the metric tensor g λσ. If the metric is just that of flat space, then Γ µ αβ = 0 and the first equation above just says that the particle is unaccelerated -- i.e., it satisfies the law of inertia (Newton s 1st law).

5 General Relativity and Cosmology CURVED SPACE TELLS MATTER HOW TO MOVE du µ ds + Γ µ αβ u α u β = 0 xt MATTER TELLS SPACE HOW TO CURVE Text Einstein Field Equations G µν R µν ½Rg µν = 8πGT µν Λg µν Einstein s Cosmological Principle: on large scales, space is uniform and isotropic. COBE-Copernicus Theorem: If all observers observe a nearly-isotropic Cosmic Background Radiation (CBR), then the universe is locally nearly homogeneous and isotropic i.e., is approximately described by the Friedmann-Robertson-Walker metric ds 2 = dt 2 a 2 (t) [dr 2 (1 kr 2 ) 1 + r 2 dω 2 ] with curvature constant k = 1, 0, or +1. Substituting this metric into the Einstein equations above, we get the Friedmann equations. Here r is the comoving coordinate, and the expansion factor a(t) = 1/(1+z), where z is the redshift. At the present epoch t = t0, a0 = a(t0) = 1 and z(t0) = 0. The distance D(t) = a(t) r. [Perkins R(t) = a(t).]

6 Friedmann-Robertson-Walker Metric (homogeneous, isotropic universe) Friedmann equation at t0, with a(t0)=1 (note that p0 = 0) deceleration parameter age of the universe

7 at t0, with a(t0)=1 (note that p0 = 0) deceleration parameter age of the universe = 13.97h70 1 Gyr p = wρ, k = 0 ρ a 3(1+w)

8 Cosmological Parameters (observations and simulations) (1-ΩΛ) Why a cosmological constant corresponds to negative pressure: When gas pushes the piston out it does work pdv and the internal energy of the gas is reduced. But when the vacuum expands, the energy increases by ρv dv. Hence p = ρv, so w = 1.

9 D [µk 2 ] Planck Collaboration: P. A. R. Ade 90, N. Aghanim 63, C. Armitage-Caplan 96, M. Arnaud 77, M. Ashdown 74,6, F. Atrio-Barandela 19, J. Aumont 63, C. Baccigalupi 89, The A. J. Banday main 99,10 Planck, R. B. Barreiro 70, J. G. Bartlett 1,72, E. Battaner 102, K. BenabedPlanck 64,98, A. Benoît errors 61, A. Benoit-Lévy are 26,64,98, J.-P. Bernard 10, M. Bersanelli 37,53, P. Bielewicz 99,10,89, J. Bobin 77, J. J. Bock 72,11, A. Bonaldi 73, J. R. Bond 9, J. Borrill 14,93, F. R. Bouchet 64,98, M. Bridges anomaly 74,6,67, M. Bucher 1, is C. the Burigana 52,35, R. C. Butler 52, E. Calabrese 96, B. Cappellini small 53, J.-F. Cardoso and Planck s 78,1,64, A. Catalano 79,76, A. Challinor 67,74,12 low, A. Chamballu amplitudes 77,16,63, R.-R. Chary 60, X. Chen 60, L.-Y Chiang 66, H. C. Chiang 29,7, P. R. Christensen 85,40, S. Church 95, values for D. L. Clements 59, S. Colombi 64,98, L. P. L. Colombo 25,72, F. Couchot 75, A. Coulais 76, B. P. Crill 72,86, A. CurtoH0 and 6,70, F. Cuttaia Ωm 52, L. Danese 89, R. D. Davies 73, R. J. at Davis l 73, P. de Bernardis 36, A. de Rosa 52, G. de Zotti 49,89, J. Delabrouille 1, J.-M. are Delouis rather 64,98, F.-X. different Désert 56, C. Dickinson 73, J. M. Diego 70, K. Dolag 101,82, H. Dole 63,62, S. Donzelli 53, O. Doré 72,11, M. Douspis 63, J. Dunkley 96, X. Dupac 43, G. Efstathiou 67, F. Elsner 64,98, T. A. Enßlin 82, H. K. Eriksen 68, F. Finelli 52,54, O. Forni 99,10, M. Frailis 51, A. A. Fraisse 29, E. Franceschi 52, T. from C. Gaier WMAP s 72, S. Galeotta 51, S. Galli 64, K. Ganga 1, M. Giard Planck 99,10, G. Giardino Collaboration: 44, Y. Giraud-Héraud Cosmological 1, E. Gjerløw 68, J. González-Nuevo parameters 70,89, K. M. Górski 72,104, S. Gratton 74,67, A. Gregorio 38,51, A. Gruppuso 52, J. E. Gudmundsson 29, J. Haissinski 75, J. Hamann 97, F. K. Hansen 68, D. Hanson 83,72,9, D. Harrison 67,74, S. Henrot-Versillé 75, C. Hernández-Monteagudo 13,82, D. Herranz 70, S. R. Hildebrandt 11, E. Hivon 64,98, M. Hobson 6, W. A. Holmes 72, A. Hornstrup 17, Z. Hou 31, W. Hovest 20 82, K. M. Hu enberger , T. R. Ja e 40 99,10, A. H. Ja e 50 59, J. Jewell 72, W. C. Jones 29, M. Juvela 28, E. Keihänen 28, R. Keskitalo 23,14, T. S. Kisner Maximum 81, R. Kneissl multipole 42,8, J. Knoche moment, 82, L. Knox 31, M. Kunz 18,63,3, H. Kurki-Suonio 28,47, G. Lagache 63, A. Lähteenmäki 2,47, J.-M. Lamarre 76, A. Lasenby 6,74, M. Lattanzi 35, R. J. Laureijs 44, max C. R. Lawrence 72, S. Leach 89, J. P. Leahy 73, R. Leonardi 43, J. León-Tavares Fig. 16. Comparison of H 0 measurements, with estimates of Planck+WP 45,2, J. Lesgourgues 97,88, A. Lewis Planck+WP+highL 27, M. Liguori 34, P. B. Lilje 68, M. Linden-Vørnle Planck+lensing+WP+highL 17, M. López-Caniego 70, P. M. Lubin 32, ±1 errors, from Planck+WP+highL+BAO a number of techniques (see text for details). J. F. Macías-Pérez 79, B. Ma ei 73, D. Maino 37,53, N. Mandolesi 52,5,35, M. Maris 51, D. J. Marshall 77, P. G. Martin 9, E. Martínez-González These are compared 70, with the spatially-flat CDM model constraints from 36, Planck and WMAP-9. S. Masi 36, S. Matarrese 34, F. Matthai 82, P. Mazzotta 39, P. R. Meinhold 32, A. Melchiorri 36,55, J.-B. Melin 16, L. Mendes 43, E. Menegoni A. Mennella 37,53, M. Migliaccio 67,74, M. Millea 31, S. Mitra 58,72, M.-A. Miville-Deschênes 63,9, A. Moneti 64, L. Montier 99,10, G. Morgante 52, The results of this section show that BAO measurements are D. Mortlock 59, A. Moss 91, D. Munshi 90, P. Naselsky 85,40, F. Nati 36, P. Natoli 35,4,52, C. B. Netterfield 21, H. U. Nørgaard-Nielsen 17, F. Noviello an extremely valuable 73, complementary data set to Planck. The D. Novikov 59, I. Novikov 85, I. J. O Dwyer 72, S. Osborne 95, C. A. Oxborrow 17, F. Paci 89, L. Pagano 36,55, F. Pajot 63, D. Paoletti 52,54 measurements, B. Partridge are 46, basically geometrical and free from complex F. Pasian 51, G. Patanchon systematic e ects that plague many other types of astrophysical first-year 1, D. Pearson papers (Hinshaw 72, T. J. Pearson et al. 11,60, H. V. Peiris 2003; Spergel 26, O. Perdereau et al. 2003) and 75, L. Perotto 79, F. Perrotta 89, V. Pettorino 18, F. Piacentini 36, M. Piat measurements. The results are consistent from survey to survey acted 1, E. Pierpaoli as motivation 25, D. Pietrobon to fit an 72, S. Plaszczynski inflation model 75, P. Platania with a step-like 71, E. Pointecouteau feature (Peiris et al. 2003). Similar investigations have been carried 99,10, G. Polenta 4,50, N. Ponthieu 63,56, and are of comparable precision to Planck. In addition, BAO measurements can be used to break parameter degeneracies that M. Reinecke 82, M. Remazeilles limit analyses based purely on CMB data. For example, from out by a number 63,1, C. Renault of authors, 79, S. Ricciardi (see e.g., 52, T. Riller Mortonson 82, I. Ristorcelli et al. 2009, 99,10, G. Rocha and 72,11, C. Rosset 1, G. Roudier 1,76,72, the excellent agreement with the base CDM model evident in references therein). At these low multipoles, the Planck spectrum is in excellent agreement with the WMAP nine-year spec- Fig. 15, we can infer that the combination of Planck and BAO M. D. Sei ert departure 72,11, E. P. S. Shellard 12, L. D. Spencer 90, J.-L. Starck 77, V. Stolyarov 6,74,94, R. Stompor 1, R. Sudiwala 90, R. Sunyaevmeasurements 82,92, F. Sureau will 77, lead to tight constraints favouring K = 0 (Sect. 6.2) and a dark energy equation-of-state parameter, w = M. Tucci 18,75, J. Tuovinen trum 84 (Planck, M. Türler Collaboration 57, G. Umana 48 XV, L. Valenziano 2013), so 52 it, J. isvaliviita unlikely 47,28,68 that, B. any Van Tent 80, P. Vielva 70, F. Villa 52, N. 1 (Sect. Vittorio 6.5). 39, L. A. Wade 72, B. D. of Wandelt the features 64,98,33, I. such K. Wehus as the 72, M. lowwhite quadrupole 30, S. D. M. or White dip 82 in, A. the Wilkinson multipole range are caused by instrumental e ects or Galactic 73, D. Yvon 16, A. Zacchei 51, and A. Finally, Zonca 32 we note that we choose to use the 6dF+SDSS(R)+BOSS data combination in the likelihood analysis of Sect. 6. This choice includes the two most accurate BAO measurements and, since the e ective redshifts of foregrounds. These are (A real liations features can of be the found CMB after anisotropies. the references) these samples are widely separated, it should be a very good ns Best-fit amplitude, A g. 39. Left: Planck TT spectrum at low multipoles with 68% ranges on the posteriors. The rainbow band show the best fits to entire Planck+WP likelihood for the base CDM cosmology, colour-coded according to the value of the scalar spectral index. Right: Limits (68% and 95%) on the relative amplitude of the base CDM fits to the Planck+WP likelihood fitted only to the anck TT likelihood over the multipole range 2 apple ` apple `max. Parameter Best fit 68% limits Best fit 68% limits Best fit 68% limits Best fit 68% limits b h ± ± ± ± We find the following notable results using CMB data alone: c h L. Popa , T. Poutanen ± ,28,2, G. W. Pratt , G. Prézeau ,72, S. Prunet ± ,98, J.-L. Puget 63, J. P. Rachen 22,82, W. T. Reach ± , R. Rebolo 69,15, , ± The deviation of the scalar spectral index from unity is robust to the addition of tensor modes and to changes in the matter content of the Universe. For example, adding a tensor component we find n s = ± , a 5.5 from n s = 1. A 95% upper limit on the tensor-to-scalar ratio of r < The combined contraints on n s and r are on the borderline of compatibility with single-field inflation with a quadratic potential (Fig. 23). A 95% upper limit on the summed neutrino mass of P m < 0.66 ev. A determination of the e ective number of neutrino-like relativistic degrees of freedom of N e = 3.36±0.34, compatible with the standard value of The results from Planck are consistent with the results of standard big bang nucleosynthesis. In fact, combining the CMB data with the most recent results on the deuterium abundance, leads to the constraint N e = 3.02 ± 0.27, again compatible with the standard value of New limits on a possible variation of the fine-structure constant, dark matter annihilation and primordial magnetic fields. Planck 2013 results. XVI. Cosmological parameters 100 MC M Rowan-Robinson ± , J. A. Rubiño-Martín ,41, B Rusholme 60 ±, M Sandri 52, D. Santos , M. Savelainen ,47, G. ± Savini , D. Scott 24, ± D. Sutton 67, , A.-S Suur-Uski 28,47, J.-F. Sygnet , J. A. Tauber , D. Tavagnacco ,38, L. Terenzi , L. To olatti ,70, M Tomasi 53, M. Tristram , ± n s ± ± ± ± ln(10 10 A s ) ± ± ± March 2013 The Planck data, however, constrain the parameters of the base CDM model to such high precision ABSTRACT that there is little re- PS ± ± maining flexibility to fit the low-multipole ± The Hubble constant ± part of the spectrum ± % and 95% limits on the relative amplitude ± of the base CDM ± ± z re ± ± ± ± 1.1 H H 0 = 67.3 ± km s± Mpc 1, and a high value of the matter density 67.3 parameter, ± 1.2 m = ± These values 67.9 are in± tension 1.0 with recent direct ± ± ± ± ± r drag ± ± ± ± 0.45 CDM model. Beam and calibration parameters, a in our Universe. Interpretation of large-scale anomalies (includ Table data5. to set Best-fit limits on a possible values variationand of the fine-structure 68% confidence constant, dark matter annihilation limits and for primordial the magnetic base fields. ΛCDM Despite themodel. success approximation to neglect correlations between the surveys. Abstract: This paper presents To illustrate the first this cosmological point, the results right-hand based on panel Planck of measurements Fig. 39 shows of the cosmic microwave background (CMB) temperature and lensing-potential power spectra. We find that the Planck spectra at high multipoles (` > 40) are extremely well described A striking result from the fits of the base CDM model to Planck power by the spectra standard is the low value of the Hubble constant, which is spatially-flat six-parameter CDM cosmology with a power-law spectrum of adiabatic scalar perturbations. Within the context of tightly this cosmology, constrained by CMB data alone in this model. From the the Planck data determine model the cosmological (sampling parameters the chainstoconstrained high precision: bythethe angular full likelihood) size of the sound horizon at recombination, Planck+WP+highL the physical densities of baryons and cold fitted darkonly matter, toand the the Planck scalartt spectral likelihood index areover estimated the multipole to be = ( ± ) 10 range 2, b h 2 = ± , analysis we find H 0 = (67.3±1.2) km s 1 Mpc 1 (68%; Planck+WP+highL).(51) c h 2 = ± , and n s = ± , respectively (68% errors). For this cosmology, we find a low value of the Hubble constant, 2 apple ` apple `max. From multipoles `max 25 to multipoles `max 35, we see more than a 2 departure from values of unity. (The maximum deviation from unity is 2.7 at ` = 30.) It is di cult to know what to make of this result, and we present it here as a A low value of H 0 has been found in other CMB experi- constraints most notably from from the recent WMAP-9 analysis. measurements of H 0 and the magnitude-redshift relation for Type Ia supernovae, but are in excellent agreement with geometricalments, Fitting Age/Gyr baryon acoustic oscillation ± (BAO) surveys Including curvature, we find ± that the Universe is consistent with spatial flatness ± to percent the base level CDM preci model, Hinshaw et al. (2012) ± find0.037 sion using Planck CMB data alone. We use high-resolution CMB data together with Planck to provide greater control on extragalactic foreground H 0 = (70.0 ± 2.2) km s 1 Mpc 1 (68%; WMAP-9), (52) components in an investigation curiosity of extensions that needs to the further six-parameter investigation. CDM The model. Planck We present temperature additional data astrophysical are remarkably data sets consistent in addition with to Planck the predictions and high-resolution of the CMB data. None of these models consistent are favoured with Eq. (51) to within 1. We emphasize here that selected results from a large grid of cosmological models, using a range of over the standard six-parameter base CDM CDMmodel cosmology. at highthe multipoles, deviation ofbut theitscalar is also spectral conceivable index from unity is insensitive to the addition the CMBof estimates tensor are highly model dependent. It is important modes and ) results, to changes in that the the matter CDM contentcosmology of the Universe. failswe atfind lowa 95% multipoles. upper limit There of r are< 0.11 on the tensor-to-scalar ratio. There is no evidence for additionalother neutrino-like indications, relativistic fromparticles both WMAP beyond and the three Planck 30 families dataof for neutrinos anomalies forat thelow e ective multipoles number (Planck of relativistic Collaboration degrees of freedom, XXIII and 2013), an upper that limit of 0.23 ev for the sum of neutrino masses. in the standard model. Using BAO and CMB data, we find N e = 3.30 ± 0.27 Our results are in excellent may agreement be indicative with big of new bang physics nucleosynthesis operating and on the the standard largest value scales of N e = We find no evidence for dynamical dark energy; using BAO and CMB data, the dark energy equation of state parameter is constrained to be w = We also use the Planck We also find a number of marginal (around 2 rhaps indicative of internal tension within the Planck data. amples include the preference of the (phenomenological) sing Best-fit parameter for values greater and than unity 68% (A L = confidence 1.23±0.11; limits for the base. 44) and for negative running (dn s /d ln k = 0.015±0.09; Eq. b). In Planck Collaboration XXII (2013), the Planck of the six-parameter data indite a preference for anti-correlated isocurvature temperature modes and power forspectrum theoretical at low multipoles. framework. The The unusual problem shapehere of the isspectrum assessing the multipole role of range 20 < ` < 40 was seen previously in the CDM ing the model results in describing shown in thefig. Planck 39) data is at dihigh cult multipoles, in the absence we note that of athis cosmology does not provide a good fit to the dels Tuesday, with afebruary truncated4, power 14 spectrum on large WMAP scales. data and None is a real a feature posteriori of thechoices, primordial i.e., CMB that anisotropies. inconsistencies The poorattract fit to the our spectrum atten-at low multipoles is not of decisive significance,

10 History of Cosmic Expansion for General Ω M & Ω Λ

11 History of Cosmic Expansion for General Ω M & Ω Λ

12 History of Cosmic Expansion for Ω Λ = 1- Ω M With Ω Λ = 0 the age of the decelerating universe would be only 9 Gyr, but Ω Λ = 0.7, Ω m = 0.3 gives an age of 14 Gyr, consistent with stellar and radioactive decay ages now past future Saul Perlmutter, Physics Today, Apr 2003

13 LCDM Benchmark Cosmological Model: Ingredients & Epochs Barbara Ryden, Introduction to Cosmology (Addison-Wesley, 2003)

14 Benchmark Model: Scale Factor vs. Time Barbara Ryden, Introduction to Cosmology (Addison-Wesley, 2003)

15 Age of the Universe t 0 in FRW Cosmologies = a(t) Benchmark Model Benchmark Model Ω m

16 Age t 0 of the Double Dark Universe Ω Λ,0 Age in Gyr Ω m,0 Calculated for k=0 and h=0.7. For any other value of the Hubble parameter h, multiply the age by (h/0.7).

17 Age of the Universe and Lookback Time Gyr Redshift z These are for the Benchmark Model Ω m,0 =0.3, Ω Λ,0 =0.7, h=0.7.

18 Distances in the Expanding Universe: Ned Wright s Javascript Calculator H 0 D L (z=0.83) =17.123/13.97 =1.23 Web app iphone app