Homework Set (Week 04) Introduction to Astroparticle Physics
Georg G. Raffelt Max-Planck-Institut fur¨ Physik (Werner-Heisenberg-Institut) F¨ohringer Ring 6, 80805 Munchen¨ Email: raffelt(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 fills the universe where p = wρ. 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 infinity within a finite 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 significantly 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 different 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 √ 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<