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Home , Black body, Gravitational redshift

Three Distinctive Pitfall • • Almost everybody in the public (including news papers, – the amount by which the wavelength of is stretched scientific magazines, books, etc.) confuses the “expansion – c.f., – wavelength of light is squeezed redshift” with the “Doppler redshift”. They are different. – Confusion may arise from the use of the term, “recession • Doppler Redshift velocity”. – Wavelength is stretched by relative motion – z = (relative velocity)/c – However, galaxies are not moving in comoving coordinates, • This formula is correct only when velocity is much smaller than c except for motion due to mutual gravitational attraction between • Gravitational Redshift galaxies. Space between galaxies is expanding. – Wavelength is stretched by strong – It is the peculiar velocity that causes the Doppler redshift, not – z = (1/2)[()/c]^2 the expansion of the . • This formula is correct only when velocity is much smaller than c – Keep in mind the difference. • Expansion Redshift • Observed velocity of a galaxy – Wavelength is stretched by expansion of space = recession velocity : due to expansion of space – z = (scale factor at reception)/(scale factor at emission) – 1 + peculiar velocity : due to galaxy’s motion

…light-years away Expanding Cosmic Sphere Model • Newspapers often say that “astronomers have discovered the most • Let’s pick a small spherical region (filled with distant galaxy at 12 billion light years away”. What do they really mean? matter) in the universe. – As we have already learned, what astronomers measure is redshift. One has V to use the measured redshift for calculating “how many years ago it was – Radius: L when light was emitted”. – : M • Redshift How many billion years ago V L • 0.01 0.14 – Density: ! • 0.1 1.3 V • 0.5 5.0 – Expansion Velocity: V ! • 1 7.7 • 2 10.2 • 3 11.4 V • 4 12.0 • 5 12.3 To do this conversion, we have • Will this region expand forever? • 10 13.0 to know how R has changed • 100 13.4 – It depends on the expansion velocity, V. with time. Escape Velocity Friedmann Equation V • Let’s write the expansion velocity using the • Escape velocity is given by escape velocity as follows: 2 – V2 = (V )2 + C – (Vescape) =2GM/L V escape L – The constant, “C”, determines the fate of the • The mass is M=(4"G/3)!L3 V region: 2 2 ! – (Vescape) = (8"G/3)!L • C<0: recollapse • C=0, C>0: expand forever V • Recalling the form of the escape velocity, one • Is V larger or smaller than Vescape? obtains the following “Friedmann equation”: – V2 = (8"G/3)!L2 + C – V< Vescape: the region will recollapse in the future – The velocity-distance relation, V=HL, gives the – V= Vescape: the region will expand forever most popular form of the Friedmann equation: 2 2 – V> Vescape: the region will expand forever – H = (8"G/3)! + C/L • This equation determines how the universe expands and what the fate of the universe is --- extremely important in cosmology!

Acceleration The fate of the universe • In addition to the Friedmann equation, • We have now two fundamental cosmological – H2 = (8 G/3) + C/L2 " ! equations: which describes the expansion velocity, there is the second equation which describes the acceleration of the expansion: – H2 = (8"G/3)! + C/L2 + #/3 – a = -(4"G/3)!L – a = -(4"G/3)!L + #L/3 • Notice the negative sign in front: the presence of matter always decelerates (a<0) the expansion. • These equations determine how the scale factor, • Einstein did not like this. R, changes with time. (remember L=Rl) – Einstein wanted the universe to be static – neither expanding or – We have three quantities to specify contracting – however, this equation does not permit it. • !, C, # – He added the “cosmological constant”, #, to this equation in order – Here is the bottom line: we need to determine these to cancel the effect of matter: quantities by observations. • a = -(4"G/3)!L + #L/3 = 0 – This # is what Einstein called later “the biggest blunder”. • These are called the “cosmological parameters” and determination of the cosmological parameters has been the • More importantly, # can accelerate the expansion, unlike most important task in cosmology as they determine the the ordinary matter. evolution of R in the past and in the future. Where is relativity? Expanding Cosmic Sphere • The derivations of the Friemann equation so far did not use relativity – it was totally Newtonian. • Matter inside of the sphere does not V • Where is relativity? go outside of the sphere. (Matter – In Newtonian picture, it is not clear what C really means or what doesn’t escape) V determines C. L – In relativistic picture, C is actually related to the geometry of space: – Therefore, mass inside of the sphere is ! V • C<0: spherical geometry conserved. • C=0: flat geometry – of matter in the sphere is given • C>0: hyperbolic geometry by Einstein’s relation: E=Mc2 V – In other words, the geometry of the universe determines the fate of the universe!! – So, energy of matter is also conserved. • Spherical geometry: recollapse • Flat or hyperbolic geometry: expand forever • Since mass energy is conserved, energy density decreases as 1/L3 as the sphere expands.

Radiation in the sphere Important consequences • Energy of matter is constant. • Next, let’s consider (or light or ) • Energy of radiation decreases as 1/L. – Would energy of radiation also be • Currently, the matter energy dominates over conserved? the radiation energy, but… – Radiation continues to lose its energy as the sphere expands, due to the – As we go back in time, the radiation energy expansion redshift! steadily increases and eventually wins! • Remember the expansion redshift stretches • Hot fire-ball universe dominated by radiation wavelength and energy of radiation is inversely proportional to wavelength. – Energy of radiation decreases as 1/L L ! goes down as 1/L L Time • Since energy goes as 1/L, energy density decreases as 1/L4 as the Temperature Temperature sphere expands. high low Cosmic Microwave Background Spectrum of CMB • In the past, when the energy was dominated by radiation, • The cosmic microwave background (CMB) has a the universe was filled with hot radiation. -body spectrum. • This radiation still fills the universe today. – also have a nearly black-body spectrum • Why don’t we see it by eyes at night then? – CMB has a perfect black-body: the most beautiful • Temperature of this radiation (called the cosmic black-body in the universe microwave background radiation) has gone down so much that we don’t see it. • A black-body spectrum is determined by – Temperature is only 2.73 degrees above . (2.73 K) temperature only. – Compare this extremely low temperature with temperature of the – There is a peak in a black-body spectrum which shifts surface of the : 5800K – When the size of the universe was about 1/2000 of the present to longer wavelength as temperature goes down. size, temperature of the universe was about the same as that of the brightness T high Sun.

T low wavelength

• The spectrum of CMB has a peak at 1.1mm. • Let’s compare it with… – Microwave oven: 12cm – Cellular phone: 20cm – UHF Television: 39-64cm – FM radio: 3m – AM radio: 300m You can “see” CMB by TV (not by cable TV of course!), and you can “hear” CMB by cell phone!!