The Hubble Constant after GW170817 Jonathan Gair, Albert Einstein Institute Potsdam From Einstein and Eddington to LIGO: 100 years of gravitational light deflection, Principe, May 28th 2019 Talk outline ❖ Eddington and Cosmology ❖ Eddington and Gravitational Waves ❖ GW170817 ❖ Gravitational wave sources as cosmological probes ❖ GW170817: first gravitational wave constraint on H0; ❖ statistical H0 measurements with ground-based detectors; ❖ prospects for improved cosmological measurements using future observations; ❖ sources of systematics in GW constraints on cosmology. Cosmological models ❖ Standard cosmological model starts with homogeneous and isotropic line element 2 2 2 2 2 2 2 2 dr 2 2 2 2 ds = c d⌧ =dt a (t)d⌃ ,d⌃ = 2 + r d✓ +sin ✓dφ <latexit sha1_base64="i9PyHZkExmT8HqDxYCOmkNfxTJo=">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</latexit> − 1 kr − ❖ and stress-energy tensor of perfect fluid Tµ⌫ =(⇢ + p)uµu⌫ + pgµ⌫ ❖ Einstein’s equations then yield the (Friedmann) equations a˙ 2 k ⇤ 8⇡ + = ⇢ a a2 − 3 3 ✓ ◆ a¨ a˙ 2 k 8⇡ 2 + + ⇤ = p a a a2 − − 3 ✓ ◆ ❖ H =˙a/a The expansion rate <latexit sha1_base64="vpi2K3l7Fm7dzDJ6bMmMpIyyKV0=">AAAB8nicdVDLSgMxFM3UV62vqks3wSK4qjMitoJCwU2XFewDpkPJpJk2NJMMyR2hDP0MNy4UcevXuPNvTB+CzwOBwzn3kHtPmAhuwHXfndzS8srqWn69sLG5tb1T3N1rGZVqyppUCaU7ITFMcMmawEGwTqIZiUPB2uHoeuq375g2XMlbGCcsiMlA8ohTAlby61fdvoKMTE5Ir1hyy97FuVup4t/EK7szlNACjV7xzWZpGjMJVBBjfM9NIMiIBk4FmxS6qWEJoSMyYL6lksTMBNls5Qk+skofR0rbJwHP1K+JjMTGjOPQTsYEhuanNxX/8vwUomqQcZmkwCSdfxSlAoPC0/txn2tGQYwtIVRzuyumQ6IJBdtSwZbweSn+n7ROy55t5uasVLtc1JFHB+gQHSMPVVAN1VEDNRFFCt2jR/TkgPPgPDsv89Gcs8jso29wXj8AZCaRUA==</latexit> is called the Hubble parameter and its value today, the Hubble constant, is denoted H0. Einstein Static Universe ❖ Setting p = 0 (dust), k = 1 (closed Universe) and the conditions 1 ⇤ ⇤ = 2 ⇢ = <latexit sha1_base64="ywGQvIiYRfS8ZkMK1eFj4NXbk1M=">AAAB/3icdVDJSgNBFHwTtxi3UcGLl8YgeAozQUwEhYAXDx4imAUyY+jp9CRNeha6e4QwzsFf8eJBEa/+hjf/xs4iuBY0FFX1eK/LizmTyrLejdzc/MLiUn65sLK6tr5hbm41ZZQIQhsk4pFoe1hSzkLaUExx2o4FxYHHacsbno391g0VkkXhlRrF1A1wP2Q+I1hpqWvuOBc63MOnyPEFJqmdpfi6nHXNolWyj4+sShX9JnbJmqAIM9S75pvTi0gS0FARjqXs2Fas3BQLxQinWcFJJI0xGeI+7Wga4oBKN53cn6F9rfSQHwn9QoUm6teJFAdSjgJPJwOsBvKnNxb/8jqJ8qtuysI4UTQk00V+wpGK0LgM1GOCEsVHmmAimL4VkQHWPShdWUGX8PlT9D9plku2bubysFg7mdWRh13YgwOwoQI1OIc6NIDALdzDIzwZd8aD8Wy8TKM5YzazDd9gvH4Arz+V4g==</latexit> a <latexit sha1_base64="wpO2eIQCWoIjuYp0Ngfyfr2opLQ=">AAACBHicdVDLSgMxFM3UV62vUZfdBIvgqsyI2AoKBTcuXFSwD+gMJZPJtKGZZEgyQhlm4cZfceNCEbd+hDv/xvQh+DwQOJxzDzf3BAmjSjvOu1VYWFxaXimultbWNza37O2dthKpxKSFBROyGyBFGOWkpalmpJtIguKAkU4wOp/4nRsiFRX8Wo8T4sdowGlEMdJG6ttlTw4FPINeJBHOvEuTDFGeHXkJzft2xam6J8dOrQ5/E7fqTFEBczT79psXCpzGhGvMkFI910m0nyGpKWYkL3mpIgnCIzQgPUM5ionys+kROdw3SggjIc3jGk7Vr4kMxUqN48BMxkgP1U9vIv7l9VId1f2M8iTVhOPZoihlUAs4aQSGVBKs2dgQhCU1f4V4iEwd2vRWMiV8Xgr/J+3DqmuauTqqNE7ndRRBGeyBA+CCGmiAC9AELYDBLbgHj+DJurMerGfrZTZasOaZXfAN1usHrkmYIA==</latexit> 4⇡ ❖ gives a¨<latexit sha1_base64="W3z9p+BU+DY0DNjSmBTdURw1nlA=">AAAB/3icdVDLSgMxFL1TX7W+RgU3boJFcFVmRGwFhYIblxXsA9qhZDKZNjSTGZKMUMYu/BU3LhRx62+4829MH4rPA4HDOfdwb46fcKa047xZubn5hcWl/HJhZXVtfcPe3GqoOJWE1knMY9nysaKcCVrXTHPaSiTFkc9p0x+cj/3mNZWKxeJKDxPqRbgnWMgI1kbq2judIIh1hkfoDHU+mdO1i07JPTl2yhX0m7glZ4IizFDr2q8mTdKICk04VqrtOon2Miw1I5yOCp1U0QSTAe7RtqECR1R52eT+Edo3SoDCWJonNJqoXxMZjpQaRr6ZjLDuq5/eWPzLa6c6rHgZE0mqqSDTRWHKkY7RuAwUMEmJ5kNDMJHM3IpIH0tMtKmsYEr4+Cn6nzQOS65p5vKoWD2d1ZGHXdiDA3ChDFW4gBrUgcAN3MEDPFq31r31ZD1PR3PWLLMN32C9vAPzsJVk</latexit> =˙ a =0 and therefore represents a static Universe with size (and mass) determined by the Cosmological constant a =1 / p ⇤ . <latexit sha1_base64="HxnsEqdEV3p7xbyoyFRXNdau+oM=">AAAB/XicdVDJSgNBFHwTtxi3cbl5aQyCpzgjYiIoBLx48BDBLJAZQk9PT9KkZ7G7R4hD8Fe8eFDEq//hzb+xswiuBQ1F1Sve6/ISzqSyrHcjNzM7N7+QXywsLa+srpnrGw0Zp4LQOol5LFoelpSziNYVU5y2EkFx6HHa9PpnI795Q4VkcXSlBgl1Q9yNWMAIVlrqmFsYnSJ735HXQmXOhQ76eNgxi1bJPj6yyhX0m9gla4wiTFHrmG+OH5M0pJEiHEvZtq1EuRkWihFOhwUnlTTBpI+7tK1phEMq3Wx8/RDtasVHQSz0ixQaq18TGQ6lHISengyx6smf3kj8y2unKqi4GYuSVNGITBYFKUcqRqMqkM8EJYoPNMFEMH0rIj0sMFG6sIIu4fOn6H/SOCjZupnLw2L1ZFpHHrZhB/bAhjJU4RxqUAcCt3APj/Bk3BkPxrPxMhnNGdPMJnyD8foBuDmUww==</latexit> ❖ This is the Einstein static Universe. Einstein favoured this model as it gives a Universe that is eternal and finite. ❖ Hubble’s observation of the recession velocity of spiral nebulae in 1929 cast doubt on the static Universe model. .668E 670 Prof. A. S. Eddington, xc. 7, .90. universe expands the individual motions of the galaxies tend to decrease, not only relatively to the scale of the system, but absolutely. In fact OMNRAS. the average random velocity changes proportionately to i/a. Thus, 193 if the expansion during past history has been considerable, we may expect the spiral nebulæ to be nearly “ at rest,” so that the regular scattering apart will not be unduly masked by individual motions. 4. Conservation of Mass.—It simplifies the tracing of the course of expansion of the universe if we may assume that the total mass of the universe remains constant. This is not rigorously true. We may Eddington andconsider either Cosmology the proper mass or the relative mass {i.e. mass rela- tive to axes “at rest”). Unfortunately neither of these is strictly conserved :— (1) Apart from radiation the proper mass is conserved ; the relative mass diminishes owing to the decrease of kinetic energy of random motion mentioned in § 3. ❖ In the 1930, Eddington published(2) a In paper the conversion “On of the matter instability into radiation of relative Einstein’s mass is conserved ; the proper mass diminishes since radiation has no proper mass. spherical world”. Thus both masses may diminish a little in the course of time. However, .668E 668 Prof. A. S. Eddington, xc. 7, these are rather insignificant complications. In what follows we shall .90. generally take £> = o, so that proper mass and relative mass are the same, and both will be conserved to our order of approximation.* OMNRAS. On the Instability of Einstein*s Spherical World. 5. Instability of Einstein’s Universe.—Setting jp = om (4) we have 193 By A. S. Eddington, F.K.S. dPa , % , 77 ) 1. Working in conjunction with Mr. G. C. McVittie, I began some 3^ = a(A - 4 7 )- months ago to examine whether Einstein’s spherical universe is stable. Before our investigation was complete we learnt of a paper by Abbé G. Lemaître * which gives a remarkably complete solution of the various For equilibrium (Einstein’s solution) we must accordingly have 2 2 questions connected with the Einstein and de Sitter cosmogonies. p = A/477. If now there is a slight disturbance so that p < À/477, d ajdt Although not expressly stated, it is at once apparent from his formulae is positive and the universe accordingly expands. The expansion will that the Einstein world is unstable—an important fact which, I think, 2 has not hitherto been appreciated in cosmogonical discussions. Astro- decrease the density ; the deficit thus becomes worse, and dHjdt nomers are deeply interested in these recondite problems owing to their increases. Similarly if there is a slight excess of mass a contrac- connection with the behaviour of spiral nebulæ ; and I desire to review tion occurs which continually increases. Evidently Einstein’s world is the situation from an astronomical standpoint, although my original hope of contributing some definitely new result has been forestalled unstable. by Lemaître’s brilliant solution. The initial small disturbance can happen without supernatural Finitude of space depends on a “ cosmical constant ” À, which occurs interference. If we start with a uniformly diffused nebula which (by in Einstein’s gravitational equations G^ = Xg^ for empty space. On general philosophical grounds f there can be little doubt that this form ordinary gravitational instability) gradually condenses into galaxies, of the equations is correct rather than his earlier form G^,, = o ; but A the actual mass may not alter but the equivalent mass to be used in is so small as to be negligible in all but very large scale applications. ❖ applying the equations for a strictly uniform distribution must be ExceptIn in so fact,far as a value Lemaitre may be suggested by (Eddington’s astronomical survey of student) had analysed cosmological the extragalactie universe, A is unknown ; or it would be better to say slightly altered. It seems quite possible that this evolutionary process that we do not know the lengths of the objects and standards of our started off the expansion of the universe. Once started, it must ordinary scale of experience in terms of the natural cosmical unit of solutions in 1927 implying similarcontinue conclusions, to expand at an increasing but Eddingtonrate. I have not, however,had not been length i /VA. Besides involving A, the shape and size of space depend on the amount of matter contained in the universe and the way it is able to decide theoretically whether the condensation ought to start distributed.read Naturally the space paper will only be atof a perfectlythe sphericaltime! form an expansion rather than a contraction. if the matter