19 Lecture 19: Cosmic Microwave Background Radiation

Transcription

1 PHYS 652: Astrophysis Leture 19: Cosmi Mirowave Bakground Radiation Observe the void its emptiness emits a pure light. Chuang-tzu The Big Piture: Today we are disussing the osmi mirowave bakground (CMB) radiation, the snapshot of the Universe at its infany when it was only about a few hundred thousand years old. We present the spetrum of the radiation and analyze its main features. Importane of the CMB Radiation The CMB radiation is a predition of Big Bang theory. Aording to the Big Bang theory, the early Universe was made up of a hot plasma of photons, eletrons and baryons. The photons were onstantly interating with the plasma through Thomson sattering. As the Universe expanded, adiabati ooling aused the plasma to ool until it beame favorable for eletrons to ombine with protons and form hydrogen atoms. This happened at around 3,000 K or when the Universe was approximately 380,000 years old (z 1100). At this point, the photons sattered off the now neutral atoms and began to travel freely through spae. This proess is alled reombination or deoupling (referring to eletrons ombining with nulei and to the deoupling of matter and radiation respetively). The photons have ontinued ooling ever sine; they have now reahed K and their temperature will ontinue to drop as long as the Universe ontinues expanding (T γ a 1 ). Aordingly, the radiation from the sky that we measure today omes from a spherial surfae, alled the surfae of last sattering. This represents the olletion of points in spae (urrently around 46 billion light years from the Earth) at whih the deoupling event happened long enough ago (less than 400,000 years after the Big Bang, 13.7 billion years ago) that the light from that part of spae is just reahing observers. The Big Bang theory suggests that the CMB radiation fills all of observable spae, and that most of the radiation energy in the Universe is in the osmi mirowave bakground, whih makes up a fration of roughly of the total density of the Universe. Two of the greatest suesses of the Big Bang theory are its predition of its almost perfet blak-body spetrum and its detailed predition of the anisotropies in the CMB radiation. The reent Wilkinson Mirowave Anisotropy Probe (WMAP) has preisely measured these anisotropies over the whole sky down to angular sales of 0.2 degrees. These an be used to estimate the parameters of the standard ΛCDM model of the Big Bang (reall Artile 3). Some information, suh as the shape of the Universe, an be obtained diretly from the CMB radiation, while others, suh as the Hubble onstant, are not onstrained and must be inferred from other measurements. Blakbody spetrum. The funtion desribing the distribution of photons radiated by a blakbody is simply given by the equilibrium BE equilibrium statistis, after taking E = p = ν = ν: f(ν) = 1 e ν/t 1 and the orresponding intensity of the blak-body spetrum is given by the Poisson distribution I(ν) = (370) 4πν3 e ν/t 1. (371) The exellent agreement between theoretial spetrum in eq. (371) is shown in Fig

2 PHYS 652: Astrophysis 98 Figure 31: Intensity of CMB radiation as a funtion of a wavenumber from FIRAS instrument on COBE satellite. The distintion between the theoretial predition and the measured values are all smaller than the thikness of the line. Systemati Bias: The Dipole Anisotropy If CMB radiation looks like a perfet blak-body radiation to one observer, it should not look like a perfet blak-body to other observers who are moving relative to the first observer. The radiation should be Döppler shifted beause of the observer s motion. The observed radiation should appear somewhat bluer (hotter) in the diretion in whih the observer is moving, and somewhat redder (ooler) in the opposite diretion. The relativisti Döppler effets due to the motion of our frame of referene in relation to the frame of referene in whih the CMB radiation is a perfet blak-body need to be aounted for before one an suessfully analyze the CMB spetrum. Relativisti Döppler shift. Assume the observer is moving away from eah other with a relative veloity v. Let us derive the SR relation onneting the frequenies of light emitted in one (denoted with subsript 1) and reeived in another referene system (subsript 2), moving away at speed v. Suppose one wavefront arrives at the observer. The next wavefront is then a distane = /ν 1 away from him/her (where is the wavelength, ν 1 the frequeny of the wave emitted, and is the speed of light). Sine the wavefront moves with veloity and the observer esapes with veloity v, the time observed between rests is t = v = ( v ) = 1 v 1 = ( ) 1 v ν 1 = ν1. (372) However, due to the relativisti time dilation, the observer will measure this time to be t 2 = t γ = 1 γ ( 1 v ), (373) ν1 98

3 PHYS 652: Astrophysis 99 where γ = 1/ 1 v 2 / 2, so the observed frequeny is ν 2 = 1 t 2 = γ and the orresponding relativisti Döppler shift ν ( 2 = γ 1 v ) ν 1 ( 1 v ) ν 1, (374) = 1 v. (375) 2 In a more general ase, when the motion of the two referene frames is given by a vetor ˆn, suh that vˆn = v os θ, the equation for the relativisti Döppler shift beomes ν 2 = 1 vˆn ν 1 2 = 1 v os θ. (376) 2 However, we are moving in relation to the referene frame at rest, so we are ν 1 ν o and observing light whih in the referene frame at rest has frequeny ν 2 ν e, so ν o = 2 ν e 1 v (377) os θ. This means that the temperature observed in the diretion θ, T(θ), is given in terms of the average temperature T as T(θ) T = 2 1 v os θ = ( 1 1 v v os θ + v2 2 ( 2 ) 1/2 ( 1 v os θ ) 1 ) ) v v2 (1 + os θ + 2 os2 θ +... ( os 2 θ 1 ) +... (378) 2 The motion of the observer (us) gives rise to both a dipole and other, higher order orretions. The observed dipole anisotropy, first deteted in 1960 s, implies that v v CMB = 370 ± 10 km/se towards φ = ± 0.8 o, θ = 48.2 ± 0.5 o, (379) where θ is the olatitude (polar angle) and it is in the range 0 θ π and φ is the longitude (azimuth) and it is in the range 0 φ 2π. Therefore θ = 0 at the North Pole, θ = π/2 at the Equator and θ = π at the South Pole. Allowing for the Sun s motion in the Galaxy and the motion of the Galaxy within the Loal Group, this implies that the Loal Group is moving with v LG v CMB 600 km/se towards φ = 268 o, θ = 27 o. (380) This peuliar motion is subtrated from the measured CMB radiation, after whih the intrinsi anisotropy is isolated (Fig. 32), and revealed to be about few parts in Even though minisule, these primordial perturbations provided seeds for the struture of the Universe. 99

4 PHYS 652: Astrophysis 100 Figure 32: The CMB radiation temperature flutuations from the 5-year WMAP data seen over the full sky. The average temperature is 2.725K, and the olors represents small temperature flutuations. Red regions are warmer, and blue older by about K. Angular Power Spetrum We now desribe the tehnique whih allows quantifiation of small-sale flutuations in the CMB radiation field. First, define the normalized temperature Θ in diretion ˆn on the elestial sphere by the deviation from the average: Θ(ˆn) = T T, (381) Seond, we onsider multipole deomposition of Θ(ˆn) in terms of spherial harmonis Y lm : Θ(ˆn) = Θ(θ,φ) = l Θ lm Y lm (θ,φ) (382) l=0 m= l with Θ lm = Θ(ˆn)Y lm (ˆn)dΩ. (383) Integral above is over the entire sphere and Y lm (ˆn) = Y lm (θ,φ) = (2l + 1) (l m)! 4π (l + m)! P l m (os θ)e imφ, (384) with P m l (x) the assoiated Legendre funtions: The basis funtions are orthonormal: π Pl m (x) (1 x2 ) m/2 d m+l ( x 2 2 l l! dx m+l 1 ) l. (385) θ=0 2π φ=0 Y lm Yl m dω = δ ll δ mm, (386) 100

5 PHYS 652: Astrophysis 101 Figure 33: Power spetrum of CMB radiation. where δ nn is the Kroneker delta funtion (=1 when n = n, =0 otherwise), and dω = sinθdφdθ. The field of Gaussian random flutuations is fully haraterized by its power spetrum Θ lm Θ l m. The order m desribes the angular orientation of a flutuation mode, and the degree (multipole) l determines its harateristi angular size. Therefore, in a Universe with no preferred diretion (isotropi), we expet that the power spetrum to be independent of m. Also, in a Universe whih is the same from point to point (homogeneous), we expet that the power spetrum to be independent of l. Finally, we define the angular power spetrum C l to be C l = Θ lm Θ l m = δ ll δ mm C l. (387) The brakets denote the average over the skies with the same osmology. The best estimate of C l is then from the average over m. Cosmi variane. From eq. (382), we an see that eah of the multipoles l is determined by harmonis with m [ l,l], a total of (2l + 1). This poses a fundamental limit in determining the power. This is alled the osmi variane: C l C l = 2 2l + 1. (388) The osmi variane states that it is only possible to observe part of the Universe at one partiular time, so it is diffiult to make statistial statements about osmology on the sale of the entire Universe. The standard Big Bang model features an epoh of osmi inflation. In inflationary models, the observer only sees a tiny fration of the whole Universe. So the observable Universe (the soalled partile horizon of the Universe) is the result of proesses that follow some general physial 101

6 PHYS 652: Astrophysis 102 laws, inluding quantum mehanis and GR. Some of these proesses are random: for example, the distribution of galaxies throughout the Universe an only be desribed statistially and annot be derived from first priniples. This raises philosophial problems: suppose that random physial proesses happen on length sales both smaller than and bigger than the horizon. A physial proess (suh as an amplitude of a primordial perturbation in density) that happens on the horizon sale only gives us one observable realization. A physial proess on a larger sale gives us zero observable realizations. A physial proess on a slightly smaller sale gives us a small number of realizations. Therefore, even if the bit of the Universe observed is the result of a statistial proess, the observer an only view one realization of that proess, so our observation is statistially insignifiant for saying muh about the model, unless the observer is areful to inlude the variane. On small setions of the sky where its urvature an be negleted, the spherial harmoni analysis beomes ordinary Fourier analysis in two dimensions. In this limit l beomes the Fourier wavenumber. Sine the angular wavelength θ = 2π/l, large multipole moments orresponds to small angular sales with l 10 2 representing degree sale separations. The power spetrum is traditionally displayed in literature as (the power per logarithmi interval in l) T 2 l(l + 1) C l T 2 2π CMB, (389) where T CMB is the blak-body temperature of the CMB radiation. Figure 33 shows the measurements of this quantity by several experiments. The power spetrum shown in Fig. 33 begin at l = 2 and exhibit large errors at low multipoles. The reason is that the predited power spetrum is the average power in the multipole moment l an observer would see in an ensemble of Universes. However a real observer is limited to one Universe and one sky with its one set of Θ lm s, 2l + 1 numbers for eah l. This is partiularly problemati for the monopole and dipole (l = 0,1). If the monopole were larger in our viinity than its average value, we would have no way of knowing it. Likewise for the dipole, we have no way of distinguishing a osmologial dipole from our own peuliar motion with respet to the CMB rest frame. Nonetheless, the monopole and dipole are of the utmost signifiane in the early Universe. It is preisely the spatial and temporal variation of these quantities, espeially the monopole, whih determines the pattern of anisotropies we observe today. 102