Constructing the Universe
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Constructing the universe Subir Sarkar None of us can understand why there is a Universe at all, why anything should exist; that’s the ultimate question. But while we cannot answer this question, we can at least make progress with the next simpler one of what the Universe as a whole is like. Dennis Sciama (1978) 15th Central European Seminar on Particle Physics & Quantum Field Theory, Vienna, 28-29 Nov 2019 This is a ‘popular’ talk but I will include discussion of a recent result that has fundamental implications for cosmology In the Ptolemic/Aristotlean standard cosmology (350 BC➛1600 AD) the universe was static and finite and centred on the Earth (1321) Alligheri The Divine Comedy, Dante Dante The Divine Comedy, This was a simple model and fitted all the observational data … but the underlying principle was unphysical Today we have a ‘standard LCDM model’ of the universe … dominated by dark energy and undergoing accelerated expansion It too is ‘simple’ (if we count L as just one parameter) and fits all the observational data … but lacks a physical foundation The universe appears complex & structured on many scales ... How can we possibly describe it by a simple mathematical model? Although the universe is lumpy, it seems to become smoother and smoother when averaged over larger and larger scales Tegmark Courtesy: Max Max Courtesy: “Data from the Planck satellite show the universe to be highly isotropic” (Wikipedia) We observe a statistically isotropic Gaussian random field of small temperature fluctuations (fully quantified by the 2-point correlations angular power spectrum) isotropy does … unless not there is necessarily isotropy imply around every homogeneity point in space But all we can ever learn about the universe is contained within our past light cone We cannot move over cosmological distances and check … so there are limits if the universe looks to what we can know the same from ‘over (‘cosmic variance’) there’ as it does from here … the Universe must appear to be the same to all observers wherever they are This ‘cosmological principle’ … Edward Arthur Milne (1896-1950) Rouse Ball Professor of Mathematics & Fellow of Wadham College, Oxford, 1928- Until about a hundred years ago all we saw in the sky was our Milky Way … and other stars like our Sun (whose distances could be measured by parallax) Diameter of the Earth’s orbit is ~1000 light-seconds. so if the ‘parallax’ of a star is 1’’, then its distance is 1 parsec ⇒ ~3.3 light-years First measured in 1838 by Wilhelm Bessel for 61 Cygni (0.3”) To measure distances in the universe we need ‘standard candles’ – astronomical sources whose absolute luminosity is known from its correlation with some other property E.g. pulsation period in the case of Cepheid variable stars … and thus we build up the ‘cosmic distance ladder’ to the furthest galaxies In 1923 Edwin Hubble used the 100” Mt Wilson telescope to determine the distance to the Andromeda Nebula He was searching for ‘novae’ … instead he found a ‘Cepheid variable’, which Henrietta Leavitt had shown (in 1912) to be a distance indicator Hubble discovered that Andromeda (M31) is not a cloud of stars and gas in our Milky Way, but a galaxy similar to our own at a very substantial distance … 2.5 million light years (⇒ 0.8 Mpc)! The Universe had suddenly became a lot bigger … and thus began modern cosmology The Hubble Space Telescope (1990-) can resolve Cepheids in galaxies much further away M100 is a galaxy in the Virgo cluster at a distance of 54 million light years (16.4 Mpc) Cepheids can be used to ‘calibrate’ other sources such as supernovae – exploding stars which are bright enough to be seen even further away … using supernovae we can now measure distances of billions of light-years Looking far away is the same as looking back into our past We see the We see the nearest star Sun as it was Proxima Centauri, 8 minutes ago as it was 4 years ago We see the Galactic We see our nearest centre as it was galaxy Andromeda as 30,000 years ago it was 2.5 million years ago We see the Virgo We see galaxies in cluster as it was the Hubble Ultra 54 million years ago Deep Field as they were - up to 12 billion years ago But there is something odd about the spectra of distant galaxies … they are shifted towards the red (longer wavelengths) as if they are travelling away from us - Doppler effect? The New Yorker “Every time I see Edwin Hubble, he is moving rapidly away from me!” The ‘expansion’ of the universe Hubble (1931) to De Sitter: “The interpretation, we feel, should be left to you and the very few others who are competent to discuss the matter with authority”. Hubble’s data (1929) (2002) The redshift of distant galaxies is not a Doppler effect … it occurs because the wavelength of light is apparently increased by the stretching of space-time (aka ‘expansion of the universe’) λobserved/λemitted =1 + z = robserved/remitted : Wayne Hu : Wayne Courtesey This picture also makes it clear that the expansion has no ‘centre’ The changing view, going back in back timegoing changingThe view, Theview back,now and later Robert Osserman, Poetry of the Universe: A Mathematical Exploration of the Cosmos,1996 The Gravity of matter should be slowing down the expansion rate … however it was claimed in1998 that distant SNIa appear fainter than expected for “standard candles” in a decelerating universe This was interpreted as Þ accelerated expansion below z ~ 0.5 The observations are made at one instant (the redshift is taken as a proxy for time) so this is not a direct measurement of acceleration, nevertheless it is presently more direct than all other such ‘evidence’ PHYS 652: Astrophysics 19 After straightforward yet tedious calculations (which I relegate to homework), we obtain the com- ponents of the Ricci tensor: 0 a¨ R0 =3, a Standard cosmological model 0 The universe is isotropic + homogeneous (when averaged on ‘large’ scales) Ri =0, ⇒ Maximally1 -symmetric2 space-time + ideal fluid energy-momentum tensor Ri = aa¨ +2˙a +2k δα. j 2 a2 µ ⌫ β ds g!µ⌫ dx dx " 1 ⌘ R(93)µ⌫ Rgµ⌫ + λgµ⌫ = a2(⌘) d⌘2 dx¯2 − 2 The t t component of the Einstein’s equation given in eq. (92) becomes − − Einstein =8⇡GNTµ⌫ 3¨a ⇥ 1 ⇤ =8πG (ρ + P )+ (ρ P ) , (94) a Robertson#− -Walker 2 − $ or 4πG a¨ = (ρ +3P ) a. (95) − 3 ⇢m k ⇤ ⌦m 2 , ⌦k 2 2 , ⌦⇤ 2 The i i component of the Einstein’s(3 equationH /8⇡GN) is (3H a ) (3H − <latexit sha1_base64="51ZpQU/7JoM1nBLtKyfzpNYhoVQ=">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</latexit>sha1_base64="AdU/Mra5g13ogMW+ETWlZL5ZXyo=">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</latexit>sha1_base64="RqT+kXNNr6jO9+ym5MT2++WvKAA=">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</latexit> ⌘ 0 ⌘ 0 0 ⌘ 0 1 2 1 aa¨ So+2˙ thea +2 Friedmannk =8πG-Lemaitre(ρ P equation) , ⇒ ‘cosmic sum rule’:(96) W + W + W = 1 a2 #2 − $ m k L ! " or We observe: 0.8Ωm - 0.6ΩL ≈ -0.2 (Supernovae), Ωk ≈ 0.0 (CMB), Wm ~ 0.3 (Clusters) 2 2 aa¨ +2˙a +2k =4πG(ρ P )a , (97) 2 infer universe is dominated− by dark energy: WΛ = 1 - Wm - Wk ~ 0.7 ⇒ Λ ~ 2H0 The eqs. (95)-(97) are the basic equations connecting the scale factor a to ρ and P .Toobtaina closed system of equations, we onlyTo need drive an accelerated equation ofexpansion state P = requiresP (ρ), which negative relates pressureP and (ρP.< -r/3) so this is 1/4 2 1/4 -12 -1/2 2 The system then reduces to two equationsinterpreted for as two vacuum unknownsenergy:a and (rL)ρ. = (H0 /8pGN) ~ 10 GeV << GF ~ 10 GeV It is, however, beneficial to furtherThis makes massage no physical these basic sense equ …ations exacerbates into a set the that (old) is Cosmological more easily Constant problem! solved. Solving the eq. (97) fora ¨,weobtain 2˙a2 2k a¨ =4πG(ρ P )a + , (98) − − a a which can be combined with eq. (95) to cancel out P dependence and yield 16πGρa 2k 2˙a2 =0, (99) 3 − a − a or 2 8πG 2 a˙ + k = ρa . (100) 3 When combined with the eq. (62) derived in the context of conservation of energy-momentum tensor, and the equation of state, we obtain a closed system of Friedmann