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1、arXiv:0806.3496v1 astro-ph 23 Jun 2008On Modified Dispersion Relations and the Chandrasekhar MassLimitMichael Gregg1 and Seth A. Major1,1Department of Physics, Hamilton College, Clinton NY 13323 USA(Dated: June 2008)AbstractModified dispersion relations from effective field theory are shown to alter
2、 the Chandrasekharmass limit. At exceptionally high densities, the modifications affect the pressure of a degenerateelectron gas and can increase or decrease the mass limit, depending on the sign of the modifica-tions. These changes to the mass limit are unlikely to be relevant for the astrophysics
3、of whitedwarf or neutron stars due to well-known dynamical instabilities that occur at lower densities.Generalizations to frameworks other than effective field theory are discussed.Electronic address: smajorhamilton.edu1I. INTRODUCTIONThe principle of Lorentz Invariance is at the heart of the contem
4、porary formulation ofphysical theory, in particular the Standard Model and general relativity. Given the central-ity of Lorentz Invariance (LI) it is wise to explore as many avenues as possible that testthis principle. One avenue investigated in recent years is the physics of modified dispersionrela
5、tions (MDR). A remarkable result of these studies is that astrophysical data significantlylimit Planck scale - suppressed modifications (see e.g. 1, 2 for reviews). The successfullimits on the modifications are due to the high degree of sensitivity of particle process thresh-olds to Lorentz Violatio
6、n (LV). Given the delicate interplay between the modifications of theparticles involved, it is helpful to look for systems in which the effects of the modificationon a single particle type are isolated. An apparently ideal system is the physics of a degen-erate electron gas since the pressure of suc
7、h a gas supports the gravitational attraction ofwhite dwarfs. Further, the Chandrasekhar mass limit, about 1.4M, of white dwarf stars isobtained in the ultra-relativistic limit, precisely where modifications of dispersion relationsare expected to be large.Modified dispersion relations often take the
8、 form of an expansion in LV termsE2 = p2 +m2 +3 p3MP +4p4M2P +. (1)where the parameters i may differ for different particle species and MP is the Planckmass, which we take to be MP = radicalbigplanckover2pi1/(4piG) 3.45 1027 eV (c = 1). In such modelsthere is clearly a preferred frame, which we take
9、 to be the one where the cosmic microwavebackground is isotropic.14 While these modified dispersion relations could be viewed simplyas a phenomenological expansion to test LV, this form has been suggested in a variety ofsettings including string theory tensor vacuum expectation values, heuristic cal
10、culations ofthe semiclassical limit of loop quantum gravity, spacetime foam, non-commutative geometry,analogs of emergent gravity, and some braneworld models 1, 2.Given the energy scale of the modifications, it might seem that testing such modificationsmight simply be impossible, However even the ea
11、rly work 4, 5 demonstrated that particleprocess thresholds are highly sensitive to these modifications. Using several particle pro-cesses and observed energies, much of the parameter space is ruled out 4, 5. For instancethe recent work of Macione et. al. achieves a limit on the parameter space of el
12、ectrons ofless than 105 3.2In this paper we report on numerical solutions to the exact equations for the Chan-drasekhar mass limit with modified dispersion relations. We extend the analysis of 6,allowing the density to vary as a function of radius, and find some significant differenceswith the resul
13、ts reported in that paper. In particular we find that the mass limit maybe raised or lowered depending on the sign of the modifications in the electron dispersionrelation. Further, unlike the solutions in 6, we find that physical radii for the star inequilibrium exist for both signs of the MDR param
14、eter . As interesting as this new masslimit is, however, there would be no effect on white dwarf astrophysics. The effects are onlyimportant in the relativistic regime far above astrophysically accessible densities.This paper is organized as follows. The next two sub-sections are devoted to discussi
15、onsof MDR and the mass limit. In section II we derive the corrections to the mass limit due toMDR. In the final section III we summarize the results and comment on the applicability ofthe calculation to other MDR frameworks.A. Modified Dispersion RelationsTo achieve precise limits on the parameters
16、it is necessary to have some additional knowl-edge of the dynamics of the field theories. This may be achieved in the context of effectivefield theory, where effective field theory is used to determine the (non-renormalizable) massdimension five (or higher) LV operators.Myers and Pospelov found that there are essentially only three operators that simulta-neously break local LI and preserve gauge and rotation invariance. Introducing a preferredframe fi
