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Vorlesung 6+7

Vorlesung 6+7. Roter Faden: Cosmic Microwave Background radiation (CMB) Akustische Peaks Universum ist flach. Zum Mitnehmen. Pfeiler der Urknalltheorie: Hubble Expansion CMB Kernsynthese 1) beweist dass es Urknall gab und 2,3) beweisen,dass Univ. am Anfang heiss war. Bisher: Ausdehnung

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Vorlesung 6+7

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  1. Vorlesung 6+7 • Roter Faden: • Cosmic Microwave Background radiation (CMB) • Akustische Peaks • Universum ist flach

  2. Zum Mitnehmen • Pfeiler der Urknalltheorie: • Hubble Expansion • CMB • Kernsynthese • 1) beweist dass es Urknall gab und 2,3) beweisen,dass Univ. am Anfang heiss war

  3. Bisher: Ausdehnung und Alter des Universums berechnet. Wie ist die Tempe- raturentwicklung? Am Anfang ist die Energiedichte dominiert durch Strahlung.

  4. Nach Rekombination ‘FREE STREAMING’ der Photonen

  5. Last Scattering Surface (LSS)

  6. Temperaturentwicklung des Universums

  7. Observing the Microwave Background Bell Labs (1963) (highlights, there are many others) COBE satellite (1992) WMAP satellite (2003)

  8. 45 times sensitivity WMAP vs COBE WMAP

  9. Lagrange Punkt 2

  10. Himmelsabdeckung

  11. Cosmology and the Cosmic Microwave Background The Universe is approximately about 13.7 billion years old, according to the standard cosmological Big Bang model. At this time, it was a state of high uniformity, was extremely hot and dense was filled with elementary particles and was expanding very rapidly. About 380,000 years after the Big Bang, the energy of the photons had decreased and was not sufficient to ionise hydrogen atoms. Thereafter the photons “decoupled” from the other particles and could move through the Universe essentially unimpeded. The Universe has expanded and cooled ever since, leaving behind a remnant of its hot past, the Cosmic Microwave Background radiation (CMB). We observe this today as a 2.7 K thermal blackbody radiation filling the entire Universe. Observations of the CMB give a unique and detailed information about the early Universe, thereby promoting cosmology to a precision science. Indeed, as will be discussed in more detail below, the CMB is probably the best recorded blackbody spectrum that exists. Removing a dipole anisotropy, most probably due our motion through the Universe, the CMB is isotropic to about one part in 100,000. The 2006 Nobel Prize in physics highlights detailed observations of the CMB performed with the COBE (COsmic Background Explorer) satellite. From Nobel prize 2006 announcement

  12. Early work The discovery of the cosmic microwave background radiation has an unusual and interesting history. The basic theories as well as the necessary experimental techniques were available long before the experimental discovery in 1964. The theory of an expanding Universe was first given by Friedmann (1922) and Lemaître (1927). An excellent account is given by Nobel laureate Steven Weinberg (1993). Around 1960, a few years before the discovery, two scenarios for the Universe were discussed. Was it expanding according to the Big Bang model, or was it in a steady state? Both models had their supporters and among the scientists advocating the latter were Hannes Alfvén (Nobel prize in physics 1970), Fred Hoyle and Dennis Sciama. If the Big Bang model was the correct one, an imprint of the radiation dominated early Universe must still exist, and several groups were looking for it. This radiation must be thermal, i.e. of blackbody form, and isotropic.

  13. First observations of CMB The discovery of the cosmic microwave background by Penzias and Wilson in 1964 (Penzias and Wilson 1965, Penzias 1979, Wilson 1979, Dicke et al. 1965) came as a complete surprise to them while they were trying to understand the source of unexpected noise in their radio-receiver (they shared the 1978 Nobel prize in physics for the discovery). The radiation produced unexpected noise in their radio receivers. Some 16 years earlier Alpher, Gamow and Herman (Alpher and Herman 1949, Gamow 1946), had predicted that there should be a relic radiation field penetrating the Universe. It had been shown already in 1934 by Tolman (Tolman 1934) that the cooling blackbody radiation in an expanding Universe retains its blackbody form. It seems that neither Alpher, Gamow nor Herman succeeded in convincing experimentalists to use the characteristic blackbody form of the radiation to find it. In 1964, however, Doroshkevich and Novikov (Doroshkevich and Novikov 1964) published an article where they explicitly suggested a search for the radiation focusing on its blackbody characteristics. One can note that some measurements as early as 1940 had found that a radiation field was necessary to explain energy level transitions in interstellar molecules (McKellar 1941). Following the 1964 discovery of the CMB, many, but not all, of the steady state proponents gave up, accepting the hot Big Bang model. The early theoretical work is discussed by Alpher, Herman and Gamow 1967, Penzias 1979, Wilkinson and Peebles 1983, Weinberg 1993, and Herman 1997. CN=Cyan

  14. Further observations of CMB Following the 1964 discovery, several independent measurements of the radiation were made by Wilkinson and others, using mostly balloon-borne, rocket-borne or ground based instruments. The intensity of the radiation has its maximum for a wavelength of about 2 mm where the absorption in the atmosphere is strong. Although most results gave support to the blackbody form, few measurements were available on the high frequency (low wavelength) side of the peak. Some measurements gave results that showed significant deviations from the blackbody form (Matsumoto et al. 1988). The CMB was expected to be largely isotropic. However, in order to explain the large scale structures in the form of galaxies and clusters of galaxies observed today, small anisotropies should exist. Gravitation can make small density fluctuations that are present in the early Universe grow and make galaxy formation possible. A very important and detailed general relativistic calculation by Sachs and Wolfe showed how three-dimensional density fluctuations can give rise to two-dimensional large angle (> 1°) temperature anisotropies in the cosmic microwave background radiation (Sachs and Wolfe 1967).

  15. Dipol Anisotropy Because the earth moves relative to the CMB, a dipole temperature anisotropy of the level of ΔT/T = 10-3 is expected. This was observed in the 1970’s (Conklin 1969, Henry 1971, Corey and Wilkinson 1976 and Smoot, Gorenstein and Muller 1977). During the 1970-ties the anisotropies were expected to be of the order of 10-2 – 10-4, but were not observed experimentally. When dark matter was taken into account in the 1980-ties, the predicted level of the fluctuations was lowered to about 10-5, thereby posing a great experimental challenge.

  16. The COBE mission • Because of e.g. atmospheric absorption, it was long realized that measurements of the high frequency part of the CMB spectrum (wavelengths shorter than about 1 mm) should be performed from space. A satellite instrument also gives full sky coverage and a long observation time. The latter point is important for reducing systematic errors in the radiation measurements. A detailed account of measurements of the CMB is given in a review by Weiss (1980). • The COBE story begins in 1974 when NASA made an announcement of opportunity for small experiments in astronomy. Following lengthy discussions with NASA Headquarters the COBE project was born and finally, on 18 November 1989, the COBE satellite was successfully launched into orbit. More than 1,000 scientists, engineers and administrators were involved in the mission. COBE carried three instruments covering the wavelength range 1 μm to 1 cm to measure the anisotropy and spectrum of the CMB as well as the diffuse infrared background radiation: DIRBE (Diffuse InfraRed Background Experiment), DMR (Differential Microwave Radiometer) and FIRAS (Far InfraRed Absolute Spectrophotometer). COBE’s mission was to measure the CMB over the entire sky, which was possible with the chosen satellite orbit. All previous measurements from ground were done with limited sky coverage. John Mather was the COBE Principal Investigator and the project leader from the start. He was also responsible for the FIRAS instrument. George Smoot was the DMR principal investigator and Mike Hauser was the DIRBE principal investigator.

  17. The COBE mission • For DMR the objective was to search for anisotropies at three wavelengths, 3 mm, 6 mm, and 10 mm in the CMB with an angular resolution of about 7°. The anisotropies postulated to explain the large scale structures in the Universe should be present between regions covering large angles. For FIRAS the objective was to measure the spectral distribution of the CMB in the range 0.1 – 10 mm and compare it with the blackbody form expected in the Big Bang model, which is different from, e.g., the forms expected from starlight or bremsstrahlung. For DIRBE, the objective was to measure the infrared background radiation. The mission, spacecraft and instruments are described in detail by Boggess et al. 1992. Figures 1 and 2 show the COBE orbit and the satellite, respectively.

  18. The COBE success COBE was a success. All instruments worked very well and the results, in particular those from DMR and FIRAS, contributed significantly to make cosmology a precision science. Predictions of the Big Bang model were confirmed: temperature fluctuations of the order of 10-5 were found and the background radiation with a temperature of 2.725 K followed very precisely a blackbody spectrum. DIRBE made important observations of the infrared background. The announcement of the discovery of the anisotropies was met with great enthusiasm worldwide.

  19. CMB Anisotropies • The DMR instrument (Smoot et al. 1990) measured temperature fluctuations of the order of 10-5 for three CMB frequencies, 90, 53 and 31.5 GHz (wavelengths 3.3, 5.7 and 9.5 mm), chosen near the CMB intensity maximum and where the galactic background was low. The angular resolution was about 7°. After a careful elimination of instrumental background, the data showed a background contribution from the Milky Way, the known dipole amplitude ΔT/T = 10-3 probably caused by the Earth’s motion in the CMB, and a significant long sought after quadrupole amplitude, predicted in 1965 by Sachs and Wolfe. The first results were published in 1992.The data showed scale invariance for large angles, in agreement with predictions from inflation models. • Figure 5 shows the measured temperature fluctuations in galactic coordinates, a figure that has appeared in slightly different forms in many journals. The RMS cosmic quadrupole amplitude was estimated at 13 ± 4 μK (ΔT/T = 5×10-6) with a systematic error of at most 3 μK (Smoot et al. 1992). The DMR anisotropies were compared and found to agree with models of structure formation by Wright et al. 1992. The full 4 year DMR observations were published in 1996 (see Bennett et al. 1996). COBE’s results were soon confirmed by a number of balloon-borne experiments, and, more recently, by the 1° resolution WMAP (Wilkinson Microwave Anisotropy Probe) satellite, launched in 2001 (Bennett et al. 2003).

  20. Outlook • The 1964 discovery of the cosmic microwave background had a large impact on cosmology. The COBE results of 1992, giving strong support to the Big Bang model, gave a much more detailed view, and cosmology turned into a precision science. New ambitious experiments were started and the rate of publishing papers increased by an order of magnitude. • Our understanding of the evolution of the Universe rests on a number of observations, including (before COBE) the darkness of the night sky, the dominance of hydrogen and helium over heavier elements, the Hubble expansion and the existence of the CMB. COBE’s observation of the blackbody form of the CMB and the associated small temperature fluctuations gave very strong support to the Big Bang model in proving the cosmological origin of the CMB and finding the primordial seeds of the large structures observed today. • However, while the basic notion of an expanding Universe is well established, fundamental questions remain, especially about very early times, where a nearly exponential expansion, inflation, is proposed. This elegantly explains many cosmological questions. However, there are other competing theories. Inflation may have generated gravitational waves that in some cases could be detected indirectly by measuring the CMB polarization. Figure 8 shows the different stages in the evolution of the Universe according to the standard cosmological model. The first stages after the Big Bang are still speculations.

  21. The colour of the universe • The young Universe was fantastically bright. Why? Because everywhere it was hot, and hot things glow brightly. Before we learned why this was: collisions between charged particles create photons of light. As long as the particles and photons can thoroughly interact then a thermal spectrum is produced: a broad range with a peak. • The thermal spectrum’s shape depends only on temperature: Hotter objects appear bluer: the peak shifts to shorter wavelengths, with: pk = 0.0029/TK m = 2.9106/T nm. At 10,000K we have peak = 290 nm (blue), while at 3000K we have peak = 1000 nm (deep orange/red). • Let’s now follow through the color of the Universe during its first million years. As the Universe cools, the thermal spectrum shifts from blue to red, spending ~80,000 years in each rainbow color. • At 50 kyr, the sky is blue! At 120 kyr it’s green; at 400 kyr it’s orange; and by 1 Myr it’s crimson. This is a wonderful quality of the young Universe: it paints its sky with a human palette. • Quantitatively: since peak ~ 3106/T nm, and T ~ 3/S K, then peak ~ 106 / S nm. Notice that today, S = 1 and so peak = 106 nm = 1 mm, which is, of course, the peak of the CMB microwave spectrum.

  22. Light Intensity • Hotter objects appear brighter. There are two reasons for this: • More violent particle collisions make more energetic photons. Converting pk ~ 0.003/T m to the equivalent energy units, it turns out that in a thermal spectrum, the average photon energy is ~ kT. So, for systems in thermal equilibrium, the mean energy per particle or per photon is ~kT.Faster particles collide more frequently, so make more photons. In fact the number density of photons, nph  T3. Combining these, we find that the intensity of thermal radiation increases dramatically with temperature Itot = 2.210-7 T4 Watt /m2 inside a gas at temperature T. • At high temperatures, thermal radiation has awesome power – the multitude of particle collisions is incredibly efficient at creating photons. To help feel this, consider the light falling on you from a noontime sun – 1400 Watt/m2 – enough to feel sunburned quite quickly. Let’s write this as Isun. • Float in outer space, exposed only to the CMB, and you experience a radiation field of I3K = 2.210-72.74 = 10 W/m2 = 10-8 Isun – not much!Here on Earth at 300K we have I300K ~ 1.8 kW/m2 (fortunately, our body temperature is 309K so you radiate 2.0 kW/m2, and don’t quickly boil!).A blast furnace at 1500 C (~1800K) has I1800K = 2.3 MW/m2 = 1600 Isun (you boil away in ~1 minute). • At the time of the CMB (380 kyr), the radiation intensity was I3000K = 17 MW/m2 = 12,000 Isun – you evaporate in 10 seconds. • In the Sun’s atmosphere, we have I5800K = 250 MW/m2 = 210,000 Isun. That’s a major city’s power usage, falling on each square meter. • Radiation in the Sun’s 14 million K core has: I = 81021 W/m2 ~ 1019 Isun (you boil away in much less than a nano-second).

  23. Warum ist die CMB so wichtig in der Kosmologie? • Die CMB beweist, dass das Universum früher heiß war • und das die Temperaturentwicklung verstanden ist b) Alle Wellenlängen ab einer bestimmten Länge (=oberhalb den akustischen Wellenlängen) kommen alle gleich stark vor, wie von der Inflation vorhergesagt. c) Kleine Wellenlängen (akustische Wellen) zeigen ein sehr spezifisches Leistungsspektrum der akustischen Wellen im frühen Universum, woraus man schließen kann, dass das Universum FLACH ist und die baryonische Dichte nur 4-5% der Gesamtdichte ausmacht.

  24. Warum akustische Wellen im frühen Universum? P Definiere: δ=Δρ/ρ F=ma FG Newton: F=ma δ``+ (Druck-Gravitation) δ=0 Lösung: Druck gering: δ=aebt, d.h. exponentielle Zunahme von δ (->Gravitationskollaps) Druck groß: δ=aeibt , d.h. Oszillation von δ (akustische Welle) Rücktreibende Kraft: Gravitation Antreibende Kraft: Photonendruck

  25. Photonen, Elektronen, Baryonen wegen der starken Kopplung wie eine Flüssigkeit behandelt → ρ, v, p Dunkle Materie dominiert das durch die Dichtefluktuationen hervorgerufene Gravitationspotential Φ δρ/δt+(ρv)=0 (Kontinuitätsgleichung = Masse-Erhaltung)) v+(v∙)v = -(Φ+p/ρ) (Euler Gleichung = Impulserhaltung) ² Φ = 4πGρ (Poissongleichung = klassische Gravitation) erst nach Überholen durch den akustischen Horizont Hs= csH-1 , (cs =Schallgeschwindigkeit) können die ersten beiden Gleichungen verwendet werden Lösung kann numerisch oder mit Vereinfachungen analytisch bestimmt werden und entspricht grob einem gedämpftem harmonischen Oszillator mit einer antreibenden Kraft Mathematisches Modell Tiefe des Potentialtopfs be- stimmt durch dunkle Materie

  26. Entwicklung der Dichtefluktuationen im Universum -DT / T ~ Dr / r Man kann die Dichtefluktuationen im frühen Univ. als Temp.-Fluktuationen der CMB beobachten!

  27. dim dim compression rarefaction rarefaction bright The first sound waves • gas falls into valleys, gets compressed, & glows brighter b) it overshoots, then rebounds out, is rarefied, & gets dimmer bright bright rarefaction compression compression dim c) it then falls back in again to make a second compression  the oscillation continues sound waves are created • Gravity drives the growth of sound in the early Universe. • The gas must also feel pressure, so it rebounds out of the valleys. • We see the bright/dim regions as patchiness on the CMB.

  28. Akustische Wellen im frühen Universum Überdichten am Anfang: Inflation

  29. Druck der akust. Welle und Gravitation verstärken die Temperaturschwankungen in der Grundwelle (im ersten Peak) http://astron.berkeley.edu/~mwhite/sciam03_short.pdf

  30. Druck der akust. Welle und Gravitation wirken gegeneinander in der Oberwelle ( im zweiten Peak)

  31. Mark Whittle University of Virginia Viele Plots und sounds von Whittles Webseite http://www.astro.virginia.edu/~dmw8f See also: “full presentation”

  32. Flute power spectra Joe Wolfe (UNSW) Akustische Wellen im frühen Universum Bь Clarinet piano range Modern Flute Überdichten am Anfang: Inflation

  33. peak trough Lineweaver 1997 Sky Maps  Power Spectra We “see” the CMB sound as waves on the sky. Use special methods to measure the strength of each wavelength. Shorter wavelengths are smaller frequencies are higher pitches

  34. Sound waves in the sky This slide illustrates the situation. Imagine looking down on the ocean from a plane and seeing far below, surface waves. The patches on the microwave background are peaks and troughs of distant sound waves. Water waves : high/low level of water surface many waves of different sizes, directions & phases all “superimposed” Sound waves : red/blue = high/low gas & light pressure

  35. Power (Leistung) pro Wellenlänge This distribution has a lot of long wavelength power And a little short wavelength power

  36. Three all-sky maps of the CMB The CMB is highlyuniform, as illustrated here. This means the young Universe is extremely smooth. The oval shapes show a sphericalsurface, as in a global map. The whole sky can be thought of as the inside of a sphere. But not completely: COBE’s 1992 map showed patchiness for the first time. red blue = tiny differences in brightness. Resolution ~7o. Patches in the brightness are about 1 part in 100,000 = a bacterium on a bowling ball = 60 meter waves on the surface of the Earth. WMAP’s now famous 2003 map of CMB patchiness (anisotropy). Resolution ~ ¼o.

  37. Sound in space !?! • Surely, the vacuum of “space” must be silent ? •  Not for the young Universe: • Shortly after the big bang (eg @ CMB: 380,000 yrs) • all matter is spread out evenly(no stars or galaxies yet) • Universe issmaller everything closer together (by ×1000) • the density is much higher (by ×109 = a billion) • 7 trillion photons & 7000 protons/electrons per cubic inch • all at 5400ºF with pressure 10-7 (ten millionth) Earth’s atm. • There is a hot thin atmosphere for sound waves • unusual fluid  intimate mix of gas & light • sound waves propagate at ~50% speed of light

  38. Big Bang Akustik http://astsun.astro.virginia.edu/~dmw8f/teachco/ While the universe was still foggy, atomic matter was trapped by light's pressure and prevented from clumping up. In fact, this high-pressure gas of light and atomic matter responds to the pull of gravity like air responds in an organ pipe – it bounces in and out to make sound waves. This half-million year acoustic era is a truly remarkable and useful period of cosmic history. To understand it better, we'll discuss the sound's pitch, volume, and spectral form, and explain how these sound waves are visible as faint patches on the Cosmic Microwave Background. Perhaps most bizarre: analyzing the CMB patchiness reveals in the primordial sound a fundamental and harmonics – the young Universe behaves like a musical instrument! We will, of course, hear acoustic versions, suitably modified for human ears.

  39. Akustik Ära • Since it is light which provides the pressure, the speed of pressure waves (sound) is incredibly fast: vs ~ 0.6c! This makes sense: the gas is incredibly lightweight compared to its pressure, so the pressure force moves the gas very easily. Equivalently, the photon speeds are, of course, c – hence vs ~ c. • In summary: we have an extremely lightweight foggy gas of brilliant light and a trace of particles, all behaving as a single fluid with modest pressure and very high sound speed. With light dominating the pressure, the primordial sound waves can also be thought of as great surges in light’s brilliance. • After recombination, photons and particles decouple; the pressure drops by 10-9 and sound ceases. The acoustic era only lasts 400 kyr, and is then over.

  40. Where the sound comes from? • A too-quick answer might be: “of course there’s sound, it was a “big bang” after all, and the explosion must have been very loud”. This is completely wrong. The big bang was not an explosion into an atmosphere; it was an expansion of space itself. The Hubble law tells us that every point recedes from every other – there is no compression – no sound. Paradoxically, the big bang was totally silent! • How, then, does sound get started? Later we’ll learn that although the Universe was born silent, it was also born very slightly lumpy. On all scales, from tiny to gargantuan, there are slight variations in density, randomly scattered, everywhere – a 3D mottle of slight peaks and troughs in density. • We’ll learn how this roughness grows over time, but for now just accept this framework. The most important component for generating sound is dark matter. Recall that after equality (m = r at 57 kyr) dark matter dominates the density, so it determines the gravitational landscape.

  41. Where the sound comes from? • Everywhere, the photon-baryon gas feels the pull of dark matter. • How does it respond? It begins to “fall” towards the over-dense regions, and away from the under-dense regions. Soon, however, its pressure is higher in the over-dense regions and this halts and reverses the motion; pushing the gas back out. This time it overshoots, only to turn around and fall back in again. The cycle repeats, and we have a sound wave! • The situation resembles a spherical organ pipe: gas bounces in and out of a roughly spherical region. [One caveat: “falling in” and “bouncing out” of the regions is only relative to the overall expansion, which continues throughout the acoustic era.] • Notice there is a quite different behavior between dark matter and the photon-baryon gas. Because the dark matter has no pressure (it interacts with nothing, not even itself), it is free to clump up under its own gravity. In contrast, the photon-baryon gas has pressure, which tries to keep it uniform (like air in a room). However, in the lumpy gravitational field of dark matter, it falls and bounces this way and that in a continuing oscillation.

  42. Bow+string your ears microphone & amplifier & antenna ariel & amplifier speakers radio waves sound sound few 100 miles Listener Concert hall few µsec delay sound waves glow your ears telescope computer speakers light gravity + hills/valleys sound sound microwaves very long way ! Listener Big Bang 14 Gyr delay ! How does sound get to us ? Consider listening to a concert on the radio:

  43. The Big Bang is all around us ! • Since looking in anydirection looks back to the foggy wall • we see the wall in alldirections. • the entire sky glows with microwaves • the flash from the Big Bang is all around us! Big Bang Near Far Now red-shift Then Far Near Big Bang Then red-shift Now Big Bang

  44. Akustische Peaks von WMAP

  45. A 220 Hz Frequency (in Hz) CMB Sound Spectrum Click for sound acoustic non-acoustic Lineweaver 2003

  46. Kugelflächenfunktionen l=4 l=8 l=12 Jede Funktion kann in orthogonale Kugelflächenfkt. entwickelt werden. Große Werte von l beschreiben Korrelationen unter kleinen Winkel.

  47. Vom Bild zum Powerspektrum • Temperaturverteilung ist Funktion auf Sphäre: ΔT(θ,φ) bzw. ΔT(n) = ΔΘ(n) T T n=(sinθcosφ,sinθsinφ,cosθ) • Autokorrelationsfunktion: C(θ)=<ΔΘ(n1)∙ΔΘ(n2)>|n1-n2| =(4π)-1 Σ∞l=0 (2l+1)ClPl(cosθ) • Pl sind die Legendrepolynome: Pl(cosθ) = 2-l∙dl/d(cos θ)l(cos²θ-1)l. • Die Koeffizienten Cl bilden das Powerspektrum von ΔΘ(n). mit cosθ=n1∙n2

  48. Das Leistungsspektrum (power spectrum) ω = vk= v 2/λ

  49. Temperaturschwankungen als Fkt. des Öffnungswinkels Θ 180/l

  50. x Raum-Zeit Inflation t Entkopplung max. T / T unter 10 Position des ersten Peaks Berechnung der Winkel, worunter man die maximale Temperaturschwankungen der Grundwelle beobachtet: Maximale Ausdehnung einer akust. Welle zum Zeitpunkt trec: cs* trec (1+z) Beobachtung nach t0 =13.8 109 yr. Öffnungswinkel θ = cs * trec * (1+z) / c*t0 Mit (1+z)= 3000/2.7 =1100 und trec = 3,8 105 yr und Schallgeschwindigkeit cs=c/3 für ein relativ. Plasma folgt: θ= 0.0175 = 10(plus (kleine)ART Korrekt.) Beachte: cs2≡ dp/d = c2/3, da p= 1/3 c2 nλ/2=cstr

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