NUCLEAR ASTROPHYSIC PHYS 554
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This 10 page Class Notes was uploaded by Dr. Simeon Wiza on Wednesday September 9, 2015. The Class Notes belongs to PHYS 554 at University of Washington taught by Staff in Fall. Since its upload, it has received 14 views. For similar materials see /class/192443/phys-554-university-of-washington in Physics 2 at University of Washington.
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Date Created: 09/09/15
Chapter 2 Baryons Cosrnology Dark Matter and Energy 21 Hubble expansion We are all aware that at the present time the universe is expanding However what will be its ultimate fate Will it continue to expand forever or will the expansion slow and nally reverse In order to see what role the constituent matter and energy 7 baryons photons neutrinos and other stuff not yet identi ed 7 of our universe may play in answering this question we explore their effects in an expanding homogeneous and isotropic universe Consider a small test mass m which sits on the surface of a spherical chunk of this universe having radius R If the mean energy density of the universe is p then the mass contained inside the spherical volume is 4 MR gma The potential energy of the test mass as seen by an observer at the center of the sphere is MRm U 7G7 R while its kinetic energy is 2 2 dt By Hubble7s Law the expansion velocity is given by 1 1 dB 2 T 7mv2 7m 1HR where H is the Hubble constant Although the While the size of H has been debated in the past recent deterrninations give a rather precise value of 71 i 4 krnsMpc One parsec 3262 light years The total energy of the test particle is then 1 2 2 8 Etot T U EWR H 7 gan and the fate of the universe depends on the sign of this number or equivalently with the relation of the density to a critical value 3H2 pm 188 gtlt 10 29h2gcrn3 where h N 071 i 004 is today7s Hubble constant in units of 100 krnsMpc This means p g pmt continued cipansion p g pmt ultimate contraction 22 Photon baryon and neutrino contributions to rnassenergy density So how does the measured rnassenergy density of the universe match up to pent We can 1 certainly do one immediate calculation for photons You are probably aware that pho tons remained in thermal equilibrium with the matter as long as there were free protons and electrons But just as we calculated the n p lt gt d y equilibrium we can evaluate the p 6 lt gt H y equilibrium where H denotes the hydrogen atom Given the ioniza tion potential of H of 136 eV one can calculate when the photons cool to the point that photocapture can no longer ef ciently break up newly formed atoms One can show this corresponds to a temperature of about 1 eV and to a time about 380000 years after the Big Bang After this point the photons decouple from the matter as they no longer see free charges to scatter off This decoupled background of photons is now redshifted to microwave energies For the photon number density dgq 1 3 2 3 n 4 2WW 4 23T7r 408cm where 3 120206 is the Riemann zeta function and T7 the today7s cosmic microwave background temperature measured with great accuracy to be about 273 K Similarly for the energy density in photons 1 p 2 2703 expoTn 1 It follows that photons contribute only 00000485 of the closure density sz w N 46 gtlt 10 34gcm3 Now what we did in BBN allows us to estimate the baryonic or nucleonic contribution to the p as well The baryon to photon number density is 77 which either BBN or cosmic microwave background studies nds to be 773 59 i 08 gtlt10 10 770MB 614 i 025 gtlt 1010 So these values are in great agreement Using the CMB value we then nd nmdems nap13717 251 gtlt 10 7cm3 and thus multiplying by the average nucleon mass a detail 7 but we know the np ratio is 17 for doing this average p1 419 gtlt 10 31gcm3 00442th That is baryons provide only 44 of the closure mass Clearly the electron contribution to p p5 6m57mNpb is then neglible about 2 gtlt10 5 of pmt comparable to the photon contribution One can count the visible77 nucleons7 by integrating over all of the luminous matter in stars and gas clouds Interestingly this yields pvis N 03902pc39rit Thus one concludes roughly half of the baryons are not visible Presumably these nucleons are some place 7 perhaps nonluminous gas clouds 7 because we believe BBN7 and because the BBN prediction for 77 is now con rmed by CMB results This problem is sometimes called the dark baryons problem 7 though there are even more intriguing dark77 problems A second dark problem has to do with large scale gravitational interactions of galaxies7 galaxy clusters7 etc For some time it has been clear that the total p is much larger than that coming from photons and baryons and electrons For example7 Doppler studies of the rotation rates of spiral galaxies indicate that these systems are much more massive than their luminosities seem to suggest prot N 2Opvis This is too large a discrepancy to attribute just to the dark baryons The origin of the dark matter77 responsible for this discrepancy is a matter of current study there are several pos sibilities But regardless of the origin of the dark matter7 it appears that the matterenergy density of our universe is a lot closer to pmt than one would guess from our calculations of pb and p7 Just as we have a OMB7 there will be a relic neutrino spectrum left over from the big bang These neutrinos would have decoupled when temperatures were slightly above 1 MeV Since that rst second of the big bang7 no further interactions have occurred If we had some means to detect these neutrinos7 they would tell us about conditions at that very early time7 eg7 their temperature uctuations probably exceeding tiny over the sky would tell us about the structure of the universe at 1 sec We do the calculation of the neutrino contribution to p making two assumptions First is the assumption that we have three avors thus 6 neutrinos in all7 as the standard model tells us7 all of which are light We will see that this is know from both cosmology7 and from a combination of tritium B decay and recent discoveries of neutrino oscillations The upper bound on the masses of the light neutrinos is about 1 eV With this assumption neutrinos are relativistic when they decoupled It follows that each neutrino avor eg7 V5 and De contributes n 7 g 7 3C3T327r2 V 2703 MaTV 1 It thus follows 3 T J 3 v 4T n7 What about TV In the very early universe electrons positrons neutrinos and photons would all be relativistic and in equilibrium characterized by a single temperature Then there is an epoch around 1 MeV when the neutrinos have decoupled but the electrons are relativistic and in equilibrium with the photons Let the temperature of this epoch be called T Still sometime later the positrons and electrons annihilate into two ys This will clearly heat the photons It can be shown the constancy of the entropy then relates the new T7 after annihilation to TV by 2 7 p 13 T7 95 p p where the superscripts T tell us to evaluate this at temperature T when the electrons were relativistic and in equilibrium with the photons Thus p727d3q q ZP e 27735PQT7 1 8 7 It follows that Tyi 4 13 TIE Thus today7s cosmic neutrino background temperature is about 192 K It follows 34 71 Kin in Now consider today when the temperatures are low Nothing has occurred to change 7 If neutrinos were massless they would contribute very little to the mass energy clearly So lets assume they have a mass We assume that mass is large compared to today7s kinetic energy 7 that7s the only way to make them important On the other hand they cannot be so massive to invalidate our assumptions of relativistic neutrinos on decoupling We know this is true experimentally It follows that their contribution to the massenergy is their number density times their mass Summing over three avors 3 pcn t r pV in imz 00106 imz We will see later that the maximum of the sum over neutrino masses using only laboratory and neutrino oscillations is 66 eV And the minimum from the neutrino mass difference measured with atmospheric neutrinos is 0055 eV Using h071 we nd 00011 g PixPm g 014 So two things are important about this First there is neutrino dark matter Second based on laboratory data only it could be signi cant though never more than 17 the closure density It turns out that the large scale structure of our universe is sensitive to neutrino mass a topic that will likely come up next quarter Current analyses that take into account large scale structure surveys the OMB and other cosmological tests appear to require that the sum of neutrino masses cannot exceed 1 eV Thus while more model dependent one concludes 00011 g JVpm g 0026 That is neutrinos could be about as important as the visible baryons in the universe7s massenergy budget but not more 23 Dark matter We have gone through the calculation of the cosmological density of baryonic matter from the theory of BBN and the measurements of light element abundances in reasonable detail We have also mentioned the CMB constraint on 77 the baryon to photon ratio This comes from an analysis of temperature uctuations in the Cosmic Microwave Background black body spectrum 7 which Eric Agol will likely discuss in detail in his cosmology course but we will also summarize below Temperature anisotropies are found at the level of 1 part in 105 and involve a typical angular size of about one degree One can understand the general physics relatively simply First the observers very carefully measure the temperature of the black body radiation as a function of solid angle plotting the very small variations in this temperature as a function of multipolarity In a Legendre expansion the Z of the multipole maps into distance the higher the Z for a peak in the power spectrum the more rapid the variation with change of solid angle The picture of structure formation is that dark matter seeds 7 areas of higher density 7 form the gravitational potential into which ordinary matter falls This picture presumes that there is some spectrum of density uctuations associated with early cosmology Ordi nary or baryonic matter acts differently from the dark matter because it not only responds to gravity but also interacts with radiation Gravity causes ordinary matter to ow into potential wells radiation pressure increases in regions of higher density and thus acts to resist strong compression of ordinary matter The result are acoustic oscillations of the or dinary matter that re ect the time scale 7 the time matter has had to ow since the Big Bang There are a couple of processes that connect temperature variations in the CMB to density uctuations and thus to the structure ofthe universe at recombination The most important physics at least on smaller scales of most interest to us is the heating and cooling associated with the interactions between ordinary matter and radiation as that matter is acoustically compressed or rare ed lf matter ows into a gravitational potential and achieves a higher density some of that kinetic energy associated with the in ow will be converted into heating of the plasma Thus a hot spot in the CMB at small scales indicates a high density region while a cold spot indicates a rare ed region There are a couple of other effects that can also alter the temperature One7 the Sachs Wolfe7 effect has to do with the gravitational red shift If a photon comes out of a region of high density 7 and thus from deeper in the gravitational well 7 it will loose more energy 7 opposite ofthe effect describe above This effect becomes more effective on larger scales7 as a large scale overdensity generates a stronger gravitational potential and a larger gravitational red shift Thus it has a different signature Such temperature uctuations7 and their connections to density uctuations7 probe the dy namic processes that govern structure formation The kinetic energy of in ow is transfered to the plasma by processes like Compton scattering This provides a radiation pressure that resists matter ow7 and can halt that ow Likewise7 if a ow is reversed7 motion of matter outward in a gravitational well must lead to a cooling of the plasma7 by energy conservation The timescale for possible acoustic oscillations 7 compression and rarefaction 7 is governed by the age of the universe at recombination7 380000 years This limits the size scale of uctuations if the scale is too large7 there is insuf cient time for matter at that scale to fully condense It is relatively easy to appreciate intuitively that the largest structures that can be seen must correspond to the largest area that can condense over the lifetime of the universe By condense here we mean reach the density where the radiation pressure just halts the ow It is helpful to think of the process as an oscillator7 with gravity working to compress the spring7 and with radiation resisting the compression and becoming more effective as the spring is compressed When a spring oscillates7 at the points of maximum compression and maximum rarefaction7 the spring is at rest If one were to 77sample77 the spring during its motion7 therefore7 the 77power would collect at these extremes At recombination7 of course7 the sampling time is xed at 380000 What varies are the springs 7 the variety of density uctuations that presumably follow some characteristic spec trum A special spring 7 a special size scale 7 are the uctuations that7 over 380000 years7 allow matter to reach the point of maximum compression Power will collect in this mode Another special mode corresponds to a size scale about half of this There the matter has time to reach the point of maximum compression7 be forced outward by the radiation pres sure7 and then again come to rest as gravity once again overcomes the diminishing radiation pressure These are the 7139 and 27139 peaks in the power spectrum One can continue7 forming a second compression 37139 etc Figure 2 shows 7 with somewhat diminishing clarity 7 the rst compression7 second rarefaction7 and third compression power peaks in the CMB temperature uctuations7 as measure by WMAP and other CMB probes The rst peak corresponds to an Z of about 200 7 an angular scale of about one degree The second cor responds to an angular size of about half a degree Peaks in the power spectrum are not seen for is much smaller than 200 7 there has not been enough time for large scale regions to compress to high density gure 1 The NASAWM pm a the lmpmlure nuke cm swam s vex eumsmengulsxsmlssm ehoulmed yee m sexeanch Mame red s w 1mm Thls mm amny ucmeums at the me 0 1m smattering whmh g m x m umelhe Mme drmnhmeum end elm me energy m a New es mam mm mm ms 0 Mammy 80 Boom98 Q CBI Maxima l DASI 500 1 0 l multipole Figure 2 The measured power spectrum for CMB temperature uctuations From Wayne Hu7s Web page As the mechanism for the rarefaction is the interaction of radiation with ordinary matter interactions the peak structure must be sensitive to the baryon to photon ratio 77 that we introduced in our BBN discussion The baryons act as a source of inertia in the compres sion and rarefaction It should be clear that if one increases the number of baryons then the amplitudes of the oscillations should increase there is more inertia on infall that the radiation has to resist overcome and reverse This is shown in the third gure Low baryon density tends to reduce the radio of the rst two peaks corresponding to compression and rarefaction The result is in good agreement with the BBN determination as we noted earlier favoring just slightly larger values of 77 In terms of a closure density it corresponds to a plumm of about 00442 Both the OMB and the BBN calculations 7 based on radically different physics governing the universe at very different times 7 give similar results We noted before this implies that about half of the baryonic matter is nonluminous Among the possible hiding places are MACHOS 7 massive compact halo objects being probed in gravitational microlensing searches 7 and matter hidden in nonluminous gas clouds There are also a couple of reliable determinations of the total matter density The height of the rst acoustic peak in the CMB spectrum is quite sensitive to the matter density as shown in the fourth gure The position of this peak requires pM N 0268 i 0018 fraction of the critical density Red shift surveys measurements of the shape of the power spectrum for large scale matter inhomogeneities also probe this quantity giving pM N 040 i 006 These results are in reasonable agreement with each other as well as with the values derived by combining the known baryon density with the baryon to total mass density ratio in clus ters Distance Type la supernovae can be used as standard candles 7 even at large distances and thus at past times 7 to probe the Hubble expansion These indicate that in addition to dark matter space the vacuum is characterized by some dark energy This dark energy is a sort of negative pressure working against gravity causing the universe to expand more rapidly than it would due to matter along They nd PA PM W 04 Again this sensitivity is physically very plausible Matter retards expansion dark energy accelerates it Thus the expansion rate should test the difference Actually one can ap preciate that things are actually richer as cold matter and dark energy evolve differently cosmologically Since we know how the former evolves as the universe stretches careful 11 measurements can determine the equation of state of the dark energy Distant supernovae are the tool for probing the condition of the universe at earlier times Combined cosmological analyses also give p N 10 i 004 That is7 the universe is close to critical density Combined with the above results7 one de duces that pM 0277 again with 0044 of this being baryons visible and otherwise and the rest something beyond the standard model like the lightest stable supersymmetric par ticle The remainder is the dark energy7 pA 073 7 whose nature is simply not understood
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