T = m hc 2 k B = K

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Corey Perkins
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1 PHYSICS 43/53: Cosmology Midterm Exam Solution Key (06). [5 points] Short Answers (5 points each) (a) What are the two assumptions underlying the cosmological principle? i. The Universe is homogeneous on large scales (same in all places). ii. The Universe is isotropic on large scales (same in all directions). (b) The Higgs boson has an approximate mass energy of 5 GeV. At approximately what expansion factor was the universe hot enough to create Higgs bosons? (Higgs freeze-out.) The corresponding temperature is: So: T m hc k B K a T 0 T.73 K K E.C. (3 points) Recalling that Ω rel 4.h 0 5 and assuming that your answer in part (a) is during the radiation dominated epoch, how long after the big bang is Higgs freeze-out? Solving the Friedmann equation: Since: so: t H a t Ωrel 9.78h Gyr s t a s 8 0 s (c) What is the comoving horizon distance in a flat universe with w + as a function of a. Note: this is not a universe we ve seen before. Ω X, so: ρ a 6 χ c a 0 a a 6 c a (d) In units of the Hubble time, how old would the w + universe be toy? The age is simple: t a a a 6

2 (e) Please describe in a sentence how (historically) we measure the following steps in the distance ladder. ( gets you full credit, 3 gets you + extra credit). The distances to the nearest stars. Stellar parallax. Distances up to 0 s of pc. For space-based observations, we can get up to kpc. The distances to star clusters in our galaxy. Primarily through measuring the distance modulus of HR diagrams, but if any of you have heard of the moving cluster method (not discussed in class), I ll be both impressed and give full credit. The distances to the nearest galaxies. The period-luminosity relationship of Cepheid variables.. [30 points] Let s discuss our actual universe a little bit. (a) (0 points) To within 0% of the Planck values, give a best estimate to Ω M, Ω DE, Ω k, w (for Dark Energy), and h. The actual values are: Ω M 0.35 ± 0.07 Ω DE ± 0.07 w.3 ± 0.4 Ω K ± (b) What is the approximate accepted value of the deceleration parameter, q 0? More importantly, describe in a sentence or two the observations that constrain this parameter. Hint: For the numerical calculation, you may find your answers in the previous problem helpful. Numerically, it s just: ( ) ΩM q 0 Ω Λ 0.53 Note that I gave 0.55 in class. The deceleration is negative so the universe is accelerating. The major observational constraints (at least as you ve seen so far) are Type a supernova measurements. At modest redshift, z < 0.5, we found: D L dz ( + q ) 0 z While I don t expect you to memorize or re-derive the equation, the upshot is that in an accelerating universe, the luminosity distance is larger than it would be otherwise. This is, in essence, because the expansion rate toy is faster than it would be otherwise. Luminosities in SNa are relatively fixed (stanrd candles), so by measuring their apparent brightness, we get the luminosity distance. (c) At what expansion redshift did/will the universe start to get dominated by Dark Energy? I would like a numerical value. Matter-lamb equality is reached when: Ω M Ω Λ

3 or or a 3 ΩM Ω Λ z a 0.3 (d) Suppose robot cosmologists still exist in a trillion years. Approximately what Hubble constant will they measure? You may express your answer in terms of present-y cosmological parameters. What sort of universe (it has a name) will they essentially live in? At arbitrarily high expansion factor: Their cosmology will be a desitter universe. H ΩΛ (e) Give arguments for rk matter. Again, this should be in a sentence or two. If you wish to argue against rk matter, be my guest, but your argument will have to be very persuasive to get any credit. i. Rotation curves of galaxies suggest mass far outside of stellar/gas regions. ii. Dynamics of cluster members suggest missing matter. iii. Gas+stellar estimates put Ω B 0.04, far less than Ω M below. iv. The Bullet cluster (and other dynamics systems) have mass reconstructions with large structures nowhere near the hot gas. 3. [5 points] Measuring masses (a) The sun orbits the center of the Milky Way at a radius of approximately 8kpc and with a circular speed of 00 km/s. Approximately how much mass is contained interior to the sun s orbit? This is a straightforward application of the rotation curve relation: or M(R) v R G kg M M (b) The luminosity density of the local universe is approximately. 0 8 L / Mpc 3. The Milky Way has an approximate luminosity of. 0 0 L. Assuming all of the light comes from galaxies like the Milky Way, what is the density of these galaxies? This is meant to be relatively easy: n gal j L MW 0.0 Mpc 3 (c) Assume all of these galaxies have a fixed mass/light ratio. What would Ω M be under those circumstances? First compute the density: ρ M MW n gal M / Mpc kg/m 3

4 Thus: Ω M ρ ρ c 0.007h with reasonable values of h 0.7, about (d) Suppose the rotation curve of the Milky Way is flat out to 6 kpc. If that is, indeed, the case, what is the mass interior to 6 kpc? Does this answer shed any light into the smallness (or largeness, I suppose) of your answer in the previous part? In that case, the mass is a linear function of radius. Thus: M M In other words, it s likely we haven t captured all of the mass in the galaxies. Further, as a spiral galaxy, the mass/light ratio of the MW is likely to be below average. (e) Apropos of mass measurements, we can also use gravitational lensing. The Einstein ring SDSS J is shown below: The ring is a lensed quasar (arbitrarily far away), while the lens (in the center) is at a redshift of z 0.08 (you may use the simple version of distance). The radius of the ring is. Based on that, what is the mass of the lens? What sort of object is the lens, based on its mass and by comparison with your result from the previous parts of the problem? The Einstein radius relation may be computed: D cz 64h Mpc M c Dθ E 4G h kg 3 0 h M where θ E rad. Based on the earlier parts of the problem it seems reasonable to suppose that the lens is a galaxy.

5 4. [0 points] We observe a cluster of galaxies at a redshift of z 3. For simplicity, you may assume that we live in an Einstein-deSitter universe. We ll test that assumption shortly. (a) In units of /, How far in the past are we observing the galaxy? First note that at z 3, a 0.5. Recall that lookback time is: t a i a [ ] / i 3 [ ] t H (b) In units of c/, what is the comoving distance to the galaxy? Same basic idea: Well that s convenient! χ c a i a c [ a i ] c (c) It has a physical radius of.5h Mpc. What is the angular radius in an Einstein-deSitter universe? Please express your answer in arcseconds. We re dealing with a flat universe so: Thus: S k χ + z 750h Mpc θ.5h Mpc 750h 0.00 rad 4 Mpc (d) Suppose clusters of galaxies were stanrd rulers (they aren t, but for the purposes of this problem suppose they are) with the fixed proper radius above. Suppose you actually measure an angular radius of about 90. What (qualitatively) does that tell you about the geometry of the universe we live in (in the hypothetical world of this problem)? First note that we re dealing with large redshifts, so the acceleration is less important than the curvature. In closed universes, D A is small, which means that things appear bigger than in flat universes (larger θ). In open universes (which this is), things appear smaller than in flat universes. 90 < 4, so we re in an open universe. For what it s worth, I used the numbers for Ω M 0.3, Ω K 0.7.

World of Particles Big Bang Thomas Gajdosik. Big Bang (model)

World of Particles Big Bang Thomas Gajdosik. Big Bang (model) Big Bang (model) What can be seen / measured? basically only light (and a few particles: e ±, p, p, ν x ) in different wave lengths: microwave to γ-rays in different intensities (measured in magnitudes)