Astronomy, Astrophysics, and Cosmology Luis A. Anchordoqui Department of Physics and Astronomy Lehman College, City University of New York Lesson VI March 15, 2016 arXiv:0706.1988 L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 3-15-2016 1 / 26 L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 3-15-2016 2 / 26 Table of Contents 1 Expansion of the Universe Friedmann-Robertson-Walker cosmologies L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 3-15-2016 3 / 26 Expansion of the Universe Friedmann-Robertson-Walker cosmologies In 1917 Einstein presented model of universe which describing geometrically symmetric (spherical) space with finite volume but no boundary In accordance with Cosmological Principle model is homogeneous and isotropic It is also static + volume of space does not change To obtain static model Einstein introduced new repulsive force in his equations Size of this cosmological term is given by L Einstein presented model before redshifts of galaxies were known taking universe to be static was then reasonable When expansion of universe was discovered argument in favor of cosmological constant vanished Most recent observations seem to indicate that non-zero L has to be present L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 3-15-2016 4 / 26 22 CHAPTER 16 COSMOLOGY EXAMPLE 16.2 Critical Density of the Universe We can estimate the critical mass density of the Universe, Using H ϭ 23 ϫ 10Ϫ3 m/(s · lightyear), where 1 light- 15 Ϫ11 2 2 ␳c, using classical energy considerations. The result turns year ϭ 9.46 ϫ 10 m and G ϭ 6.67 ϫ 10 N · m /kg , out to be in agreement with the rigorous predictions of yields a present value of the critical density ␳c ϭ 1.1 ϫ general relativity because of the simplifying assumption 10Ϫ26 kg/m3. As the mass of a hydrogen atom is 1.67 ϫ Ϫ27 that the mass of the Universe is uniformly distributed. 10 kg, ␳c corresponds to about 7 hydrogen atoms per cubic meter, an incredibly low density. Solution Figure 16.16 shows a large section of the Uni- verse with radius R with the critical density, containing a total mass M, where M consists of the total mass of matter Expansion of the Universe Friedmann-Robertson-Walker cosmologies 2 plus the effective mass of radiation with energy E, E/c . A ~v galaxy of mass m and speed v at R will just escape to infin- v ity with zero speed if the sum of its kinetic energy and m gravitational potential energy is zero. Thus, 1 2 GmM Etotal ϭ 0 ϭ K ϩ U ϭ 2 mv Ϫ R ⇢m,M 4 3 1 Gm 3␲R ␳c mv2 ϭ R 2 R 8␲G v2 ϭ R2␳ 3 c Because the galaxy of mass m obeys the Hubble law, v ϭ HR, the preceding equation becomes Figure 16.16 (Example 16.2) A galaxy escaping from a 8␲G 3H 2 Consider sphericallarge cluster regioncontained of within galaxies radius R. Only the mass H 2 ϭ ␳ or ␳ ϭ 3 c c 8␲G with largerwithin radiusR slows than the distancemass m. between clusters of galaxies but smaller radius than any distance characterizing the universe Assume + L = 0 16.6 FREIDMANNMass of this MODELS sphere AND THE AGE OF THE UNIVERSE 4 p R3 M = r (1) Freidmann’s work established the foundation for describing3 them time evolu- tion of the Universe based on general relativity. General relativity must be used in cosmologicalConsider calculations the motion because of ait galaxycorrectly atdescribes edge ofgravity, spherical the region most important force determining the Universe’s structure, over immense cos- mologicalL. distances. A. Anchordoqui Newtonian (CUNY) theoryAstronomy, can lead Astrophysics, to errors andwhen Cosmology applied to the 3-15-2016 5 / 26 Universe as a whole because it assumes that the force of gravity is always attrac- tive and is instantaneously transmitted. Although Freidmann did consider models both with and without Einstein’s repulsive form of gravity (cosmologi- cal constant), it is easiest to see the general form of Big Bang behavior without introducing repulsive gravitational forces at this point. Freidmann found three types of time-dependent universes, which may be described in terms of the universal expansion scaling factor a(t). Figure 16.17 shows a(t )(the separation between galaxies) as a function of time for the three cases labeled open universe, flat universe, and closed universe. Note that a(t) alone has a value of zero at the lower-left corner of the graph, not t, and that the three curves start at different times in the past in order to give the same scaling factor at the present time, denoted t 0. Open universes have less mass and energy than that needed to halt the expansion. They start with a scale factor of zero and grow without limit, any given galaxy approaching a limiting Copyright 2005 Thomson Learning, Inc. All Rights Reserved. Expansion of the Universe Friedmann-Robertson-Walker cosmologies According to Hubble’s law + v = HR Galaxy kinetic energy 1 1 K = mv2 = mH2R2 (2) 2 2 Potential energy at edge of sphere GMm 4pmR2r G U = − = − m (3) R 3 Total energy 1 4p E = K + U = mH2R2 − Gm R2r (4) 2 3 m has to remain constant as universe expands Rewrite (4) as 2E 8p = H2 − Gr (5) mR2 3 m L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 3-15-2016 6 / 26 Expansion of the Universe Friedmann-Robertson-Walker cosmologies Since we assume that universe is homogeneous H and rm cannot be functions of R LHS of 2E 8p = H2 − Gr (6) mR2 3 m cannot depend on chosen distance R to coordinate center However + 2E/(mR2) is time-dependent because galaxy-Earth distance changes as universe expands Since mass of test galaxy is arbitrary choose j2E/(mc2)j = 1 holds at an arbitrary moment (with E 6= 0) For different times + LHS of (6) scales as R−2 a˙ 2 8p kc2 = Gr − (7) m 2 2 a 3 a R0 L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 3-15-2016 7 / 26 Expansion of the Universe Friedmann-Robertson-Walker cosmologies Friedmann equation Since E is constant + k is constant too Actually + k = 0, ±1 is known as curvature constant We account for equivalence of mass and energy by including not only mass but also energy density 2 r = rmc + ··· (8) Friedmann equation (without L) in Newtonian limit a˙ 2 8p kc2 = Gr − (9) m 2 2 a 3 a R0 becomes a˙ 2 8p r kc2 2 ≡ = − (10) H G 2 2 a 3 c a(t)R0 (10) agrees exactly with the one derived from general relativity L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 3-15-2016 8 / 26 Expansion of the Universe Friedmann-Robertson-Walker cosmologies H0 fixes present density r0 for k = 0 as 3H2c2 r (k = 0) ≡ r = 0 (11) 0 c 8pG Most cosmological quantities depend on actual value of H0 One “hides” this dependence by introducing h −1 −1 H0 = 100 h km s Mpc (12) Critical density can be written in terms of Hubble parameter 11 2 3 rc = 2.77 × 10 h M /Mpc = 1.88 × 10−29h2g/cm3 = 1.05 × 10−5h2 GeV/cm3 (13) h ≈ 0.7 + flat universe requires energy density ∼ 10 protons/m3 L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 3-15-2016 9 / 26 Expansion of the Universe Friedmann-Robertson-Walker cosmologies Expansion of universe can be compared to motion of mass launched vertically from surface of celestial body Form of orbit depends on initial energy To compute complete orbit mass of main body and initial velocity have to be known In cosmology corresponding parameters are: G mean density G Hubble constant If density exceeds critial density expansion of any spherical region will turn to a contraction and it will collapse to a point This corresponds to closed Friedmann model If rm < rc + ever-expanding hyperbolic model is obtained These 3 models of universe are called standard models They are simplest relativistic cosmological models for L = 0 Models with L 6= 0 are mathematically more complicated but show same behaviour L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 3-15-2016 10 / 26 Expansion of the Universe Friedmann-Robertson-Walker cosmologies Define abundance Wi of different players in cosmology as their energy density relative to rc Examples r 8pG L c2 = m = = (14) Wm 2 rm and WL 2 rc 3H0 3H0 L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 3-15-2016 11 / 26 Expansion of the Universe Friedmann-Robertson-Walker cosmologies By solving Einstein equations Robertson and Walker showed that k-hypersurfaces + (hyper-sphere, hyper-plane, hyper-pseudosphere) are possible geometries for homegeneous and isotropic expanding universe FRW line element is most generally written in the form d$2 ds2 = c2dt2 − a2(t) + $2(dq2 + sin2 qdf2) (15) 1 − k$2/R2 Spatial component of the FRW metric + consists of uniformly curved space of radius R scaled by square of scale factor a(t) % ✓ R ✓ φ L. A. Anchordoqui (CUNY) Astronomy, Astrophysics, and Cosmology 3-15-2016 12 / 26 Expansion of the Universe Friedmann-Robertson-Walker cosmologies Equation of S2 of radius R 2 2 2 2 x1 + x2 + x3 = R (16) Line element in 3-dimensional Euclidean space 2 2 2 2 ds = dx1 + dx2 + dx3 (17) If x3 is taken as fictitious third spatial coordinate it can be eliminated from ds2 by use of (16) (x dx + x dx )2 ds2 = dx2 + dx2 + 1 1 2 2 . (18) 1 2 2 2 2 R − x1 − x2 Introduce coordinates $ and q defined in terms of x1 and x2 by x1 = $ cos q and x2 = $ sin q .