Homework Set (Week 04) Introduction to Astroparticle Physics
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Homework Set (Week 04) Introduction to Astroparticle Physics Georg G. Ra®elt Max-Planck-Institut furÄ Physik (Werner-Heisenberg-Institut) FÄohringer Ring 6, 80805 MunchenÄ Email: ra®elt(at)mppmu.mpg.de http://wwwth.mppmu.mpg.de/members/raffelt ! Teaching 17 November 2009 1 Phantom Energy and the Big Rip In the lectures we have considered the barotropic fluid equation of state for what ¯lls the universe where p = w½. Matter (\dust") has w = 0, vacuum energy w = ¡1, and all considered cases had w ¸ ¡1. The opposite possibility w < ¡1 is called \phantom energy" because it has numerous pathological properties and probably can not exist. Still, for the fun of it consider this hypothesis. It leads to the cosmic scale factor exploding to in¯nity within a ¯nite time in the future (\big rip"). Shortly before that time the cosmic expansion will be so fast that even atoms are ripped apart [R. Caldwell, M. Kamionkowksi & N. Weinberg, Phys. Rev. Lett. 91 (2003) 071301, see the attached copy or http://arxiv.org/abs/astro-ph/0302506]. Of course, today the observational limits on w shown in the lectures are much more restrictive < < (¡1:2 » w » ¡0:8), still allowing for a value of w signi¯cantly smaller than ¡1. (i) Read the paper (even if you will not understand everything) and reproduce the estimate for the time until the big rip. (ii) Estimate roughly the time before the big rip when atoms are torn apart. To this end use our previous result from Exercise 4 of Week 01 where we showed that today the force on an atomic electron by the Hubble expansion is roughly 10¡36 that of the Coulomb force, i.e., 36 atoms are ripped apart when H exceeds roughly 10 times the present-day value H0. (While H shrinks with time for w > ¡1 and remains constant for w = ¡1, it grows for phantom energy w < ¡1.) In the cited paper this estimate is performed in a somewhat di®erent way, but reaching a similar conclusion. 1 2 Galactic rotation curves and the local dark matter density As discussed in the lectures, the rotation curve of a spiral galaxy allows one to determine its matter density as a function of radius, at least assuming a roughly spherical mass distribution (consisting primarily of dark matter). (i) Derive the result that the orbital velocity of planets p around a central mass decreases as 1= r. (ii) If the observed rotation curve of a galaxy is exactly flat and the mass distribution exactly spherical, derive the corresponding matter density as a function of radius. (iii) In our Milky Way, the roughly constant rotation velocity around the galactic center is approximately 220 km s¡1. The solar system is at a galactocentric distance of about 8.5 kpc. How large is the dark matter density in our neighborhood? (iv) How does this compare with the cosmic average matter density? 2 PHYSICAL REVIEW LETTERS week ending VOLUME 91, N UMBER 7 15 AUGUST 2003 Phantom Energy: Dark Energy with w<ÿ1 Causes a Cosmic Doomsday Robert R. Caldwell,1 Marc Kamionkowski,2 and Nevin N. Weinberg2 1Department of Physics & Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, New Hampshire 03755, USA 2Mail Code 130-33, California Institute of Technology, Pasadena, California 91125, USA (Received 20 February 2003; published 13 August 2003) We explore the consequences that follow if the dark energy is phantom energy, in which the sum of the pressure and energy density is negative. The positive phantom-energy density becomes infinite in finite time, overcoming all other forms of matter, such that the gravitational repulsion rapidly brings our brief epoch of cosmic structure to a close. The phantom energy rips apart the Milky Way, solar system, Earth, and ultimately the molecules, atoms, nuclei, and nucleons of which we are composed, before the death of the Universe in a ‘‘big rip.’’ DOI: 10.1103/PhysRevLett.91.071301 PACS numbers: 98.80.Cq Hubble’s discovery of the cosmological expansion, But what about w<ÿ1? Might the convergence to crossed with the mathematical predictions of Friedmann w ÿ1 actually be indicating that w<ÿ1? Why re- and others within Einstein’s general theory of relativity, strict our attention exclusively to w ÿ1? Matter with has long sparked speculation on the ultimate fate of the w<ÿ1, dubbed ‘‘phantom energy’’ [19], has received Universe. In particular, it has been shown that if the increased attention among theorists recently. It certainly matter that fills the Universe can be treated as a pressure- has some strange properties. For example, the energy less fluid, which would be the case for galaxies, then the density of phantom energy increases with time. It also Universe expands forever (if it has a Euclidean or hyper- violates the dominant-energy condition [20,21], a cher- bolic spatial geometry) or eventually recollapses (if its ished notion that helps prohibit time machines and worm- spatial geometry is that of a 3-sphere). Evidence from holes. However, it is hard to see how time machines and supernova searches [1,2] and the stunning cosmic micro- wormholes would arise with phantom energy. Although wave background (CMB) results from balloon and ground experiments [3–8] and now from Wilkinson Microwave Anisotropy Probe (WMAP) [9,10] that indicate an accel- erating cosmological expansion show that this simple picture is not enough; the Universe additionally consists of some sort of negative-pressure dark energy. The dark energy is usually described by an ‘‘equation- of-state’’ parameter w p=, the ratio of the spatially homogeneous dark-energy pressure p to its energy den- w<ÿ1=3 sity . A value is required for cosmic accel- m eration. The simplest explanation for dark energy is a 1 cosmological constant, for which w ÿ1. However, this cosmological constant is 120 orders of magnitude smaller than expected from quantum gravity. Thus, although we can add this term to Einstein’s equation, it is really only a placeholder until a better understanding of this negative pressure arises. Another widely explored possibility is quintessence [11–16], a cosmic scalar field that is displaced from, but slowly rolling to, the minimum of its potential. In such models, the equation-of-state parameter is ÿ1 <w<ÿ1=3, and the dark-energy den- FIG. 1 (color). Current constraints to the w- m parame- ÿ3 1w ter space. The red solid curves show the age (in Gyr) of sity decreases with scale factor a t as Q / a . w the Universe today (assuming a Hubble parameter H0 Figure 1 shows constraints to the - m parameter 70 km sÿ1 Mpcÿ1 ). The light shaded regions are those allowed space (where m is the pressureless-matter density in (at 2) by the observed cluster abundance and by current super- units of the critical density) from the cluster abundance, nova measurements of the expansion history. The dark orange supernovae, quasar-lensing statistics (see Refs. [17,18] shaded region shows the intersection of the cluster abundance and references therein), and the first acoustic peak in the and supernova curves, additionally restricted (at 2)bythe CMB power spectrum (values taken from Ref. [10]). As location of the first acoustic peak in the cosmic-microwave- the figure shows, w seems to be converging to w ÿ1. background power spectrum and quasar-lensing statistics. 071301-1 0031-9007=03=91(7)=071301(4)$20.00 © 2003 The American Physical Society 071301-1 PHYSICAL REVIEW LETTERS week ending VOLUME 91, N UMBER 7 15 AUGUST 2003 sound waves in quintessence travel at the speed of light, it pansion is accelerating, as a consequence of dark energy does not automatically follow that disturbances in phan- with ÿ1 w<ÿ1=3, then the expansion again contin- tom energy must propagate faster than the speed of light; ues forever. However, in this case, the scale factor grows in fact, there are already several scalar-field models more rapidly than the horizon. As time progresses, gal- for phantom energy in which the sound speed is sublumi- axies disappear beyond the horizon, and the Universe nal [19,21–25]. It is true that these models feature un- becomes increasingly dark. Still, structures that are cur- usual kinetic terms in their Lagrangians, but such terms rently gravitationally bound, such as the Milky Way and may arise in supergravity [26] or higher-derivative- perhaps the local group, remain unaffected. Thus, gravity theories [27]. Theorists have also discussed although extragalactic astronomy becomes less interest- stringy phantom energy [28] and brane-world phantom ing, galactic astronomy can continue to thrive [33]. energy [29]. Connections with the deSitter–conformal With phantom energy, the Friedmann equation govern- field theory correspondence have also been made [30]. ing the time t evolution of the scale factor a t becomes 2 2 2 3 ÿ3 1w To be sure, phantom energy is not something that any H aa=a_ H0 m=a 1 ÿ ma , where theorist would have expected; on the other hand, not too H0 is the Hubble parameter, and the dot denotes a time many more theorists anticipated a cosmological constant. derivative. If m ’ 0:3, then the Universe is already dark- Given the limitations of our theoretical understanding, it energy–dominated, and for w<ÿ1 it will become in- is certainly reasonable to ask what empirical results have creasingly dark-energy–dominated in the future. We thus to say. approximate the subsequent evolution of the scale fac- In Fig. 2 we generalize the analysis of cosmologi- tor by neglecting the first term on the right-hand side. cal constraints to a parameter space that extends to w< Doing so, we find that the scale factor blows up in a time ÿ1 ÿ1 ÿ1=2 ÿ1.