Axion constraints from white dwarf cooling times

Volume 166B, n u m b e r 4

PHYSICS LETTERS

23 January 1986

AXION CONSTRAINTS FROM WHITE DWARF COOLING TIMES ~

Georg G. R A F F E L T Max- Planck- lnstitut ff4r Physik und Astrophysik - Werner- Heisenberg- lnstitut ff~r Physik Postfach 401212, D-8000 Munich 40, Fed. Rep. Germany Received 15 October 1985

Hypothetical, pseudoscalar particles would be abundantly emitted from the interior of white dwarfs through bremsstrahlung processes. These stars would then rapidly cool. F r o m the observed number of hot degenerates we find a new b o u n d on the Yukawa coupling to electrons of g < 4 × 10-13. For "invisible axions" this translates into a new b o u n d on the Peccei-Quinn scale of v > 1 × 109 GeV, corresponding to m a < 3 × 10 - 2 eV.

1. Introduction. Ever since axions where recognized [1] as a necessary consequence of an attempt to solve the CP problem of strong interactio~a [2] there have been numerous astrophysical [3-6] and cosmological [7,8] arguments concerning their possible existence. This applies, in particular, to so-called "invisible axion" models [9] ,1 which contain the energy scale v as a free parameter at which the Peccei-Quinn symmetry is broken. This parameter determines the mass and interaction strength of the axions. From a combination of astrophysical [4] and cosmological [7] arguments this parameter has been constrained to the interval 4 × 107 GeV ~ v <~ 1012 GeV, corresponding to axion masses of about 1 eV ~ m a 10 -5 eV. If axions would exist and if they were near the "right side" of this window they would presumably constitute the "dark matter" which is believed to dominate the universe [8]. If they were near the "left side" of this interval their interaction with photons, electrons and nucleons would be strong enough that their emission from hot plasmas would provide an efficient cooling mechanism for stars [ 4 - 6 ] . It was recognized for a long time that for values of v up to 109 GeV axion emission would be the dominant cooling mechanism for red giants, i.e., helium Based on work to be submitted as a doctoral thesis to the Ludwig-Maximihans-Universifat, Munich.

*t For a recent review see ref. [10]. 402

burning stars [4,5]. Recently Iwamoto [6] has demonstrated that this would also be the case for neutron stars in their late stages of cooling where usually photon emission from the surface is thought to determine the thermal history of these stars. It was impossible, however, to derive reliable bounds on o from such considerations because there was no unambiguous observational evidence excluding the possibility that these stars would emit much more energy in the form of axions than in the form of photons. This situation appears to be different for white dwarfs. Cooling theories in combination with observational evidence on white dwarf birth rates predict galactic number densities of these stars per luminosity range in good agreement ,2 with observations [12,13]. Therefore the speed of white dwarf cooling appears to be well known. Any hypothetical new cooling mechanism must then respect this observationally established time scale. We shall therefore presently calculate the energy emission rate from the interior of degenerate stars due to the process

•2 This does n o t apply to very dark degenerate stars, L < 10"4Le, where a deficiency o f observed dwarfs exists. This apparent drop in t h e luminosity f u n c t i o n h a s been used b y Freese [11 ] to e o n s ~ a i n patameters o f hypothetical

magnetic monopoles. We are only concerned, howeveL with the high luminosity range, L > lO-3Le.

0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-HoUand Physics Publishing Division)

Volume 166B, number 4

PHYSICS LETTERS

(1)

e- + ( Z , A ) - * e - + ( Z , A ) + a

where (Z, A) represents a nucleus of charge Ze and mass number A. We shall not consider photo production processes of the sort 3' + e - ~ e - + a because the flux of incident "blackbody photons" is suppressed by plasma effects [5]. This bremsstrahlung rate would also apply to other hypothetical, pseudoscalar particles if they are light enough to be emitted from white dwarfs, i.e., m ~ 1 keV. Translating this cooling rate into the number of white dwarfs observable at high luminosities we shall derive a new upper bound on the Yukawa coupling of pseudoscalar particles to electrons. According to the DFS-model of "invisible axions" [9] this result implies a new lower bound on the Peccei-Quinn scale and a new upper bound on the axion mass.

2. Bremsstrahlung emission rates. The process eq. (1) may occur through the bremsstrahlung amplitudes fig. la in analogy with the usual photo bremsstrahlung. Furthermore there exists the electro-Primakoff amplitude fig. lb due to a two-photon coupling of axiom which is analogous a similar vertex of neutral pions. The contribution of this latter amplitude, however, turns out to be very much smaller than that of the former ones and may thus be neglected. Concentrating on the amplitudes fig. la we begin our calculation of the energy emission rate e per unit mass (to be measured, e.g., in erg g-1 s - l ) by a guess of the major dependence on the relevant parameters. The lagrangian density for the interaction of elec-

~

e

~(z,A) ~(Z,A)

e

e

la

11

e

(a)

(b)

Fig. 1. Feynman amplitudes contributing to the production of axiom in electron-nucleus collisions. (a) Bremsstraldung amplitudes in analogy with photon emission. (b) ElectroPdmakoff amplitude.

23 January 1986

trons with axions is Z?aee = ig~e75 @eea where @e is the electron field, Ca is the pseudoscalar axion field and g is a dimensionless coupling constant given in terms of the Peccei-Quinn scale as g = me/O. We note that axion emission by an electron is analogous to "magnetic emission" of a photon through a spin-fiip amplitude where a = e2/4n ~ 1/137 is replaced by a a -g2/4rr ~ 2.1 X 10-26(109 GeV/u)2. Noting that the usual photo bremsstrahlung cross section is inversely proportional to the photon frequency co [14] we expect that the axionic analogue is directly proportional to the axion energy E a due to an extra factor ofEa2 from the spin-flip nature of the process. Since we are interested in an energy emission rate a further factor E a occurs. Axion energies will be on the order of the temperature of the plasma, hence we expect an overall factor T 2 in the emission rate. For conditions of degenerate matter the phase space available to electrons in scattering events is considerably reduced. For the process eq. (1) this means that the electron must initially and finally lie within a shell of approximate thickness T near the Fermi surface. Therefore we expect a further factor (T[EF)2 from the electron phase-space integrals where one integral is from the cross section, and one from averaging over the initial electron momenta. For axion emission from electron-electron scattering, e - + e - ~ e + e - + a, this factor would be (T[EF)3. Therefore this latter process is suppressed relative to electronnucleus scattering by a factor TIE F and may thus be neglected. Considering a plasma of only one species of nuclei we then expect for the energy emission rate per unit mass the following expression (in natural units where / i = c = k B = 1):

e = (Z2a2%/A)(T4/m2mu)~(pF),

(2)

where m u = 1.66 X 10 -24 g is the atomic mass unit and ~ is a numerical factor depending only on the Fermi momentum PF of the degenerate electron gas. Such a factorization can indeed be approximately obtained in the limit T ~ E F where the initial and final electron momenta can be taken as [Pl I ~ IP21 ~ PF. Then the axion energy and the difference between the initial and final electron energies enter only in the phase-space integrals which take into account the Fermi-Dirac distribution of the initial electrons and the Pauli-blocking of the outgoing electrons. Taking 403

serves as a basis for the following white dwarf cooling calculation.

the axion to be approximately massless we find

= 17r2(me/EF)2 d~22

Xf ~

d~ 3

f ~

[2(1--c12)--(c13--e23)2]

X [(1 - c12 + 2ot/lr~F)2(1 --/~F c13)(1 -- flFC23)] - 1 , (3) where flF = PF/EF is the electron velocity at the Fermi surface, c12 is the cosine of the angle between the directions of motion of the incident and outgoing electrons etc., where the index 1 refers to the incident, 2 to the outgoing electron and 3 to the axion. The first term in the denominator arises from the Fourier transform of the screened static Coulomb potential and is equal to [(IAI2 + K2)/2p 2 ] 2 where the momentum transfer has been approximated as IAI2 2p2F(1 -- c12 ). The term K2 accounts for the screening of a static Coulomb field in an environment of freely moving degenerate electrons. Under such conditions the Coulomb potential Ze[r must be effectively replaced by a Yukawa potential Ze e- Kr/r. In the Thomas-Fermi approximation we fred * a r2 = (4~/ rr)PFE F such that ~2/2p2 = 2a/rt~ F. For white dwarfs the relevant Fermi momenta are on the order of 1 MeV. In this range the expression eq. (3) may neither be evaluated in a non-relativistic approximation nor may the result of ref. [6] be used which applies only in the ultra-relativistic limit. Therefore we have integrated the full expression eq. (3) which amounts to one trivial and one non-trivial analyric and two numerical quadratures. We find ~ = 1.78 for PF = 0.2 MeV, 1.99 for 0.4 MeV, 2.05 for 0.6 MeV, 2.04 for 0.8 MeV, and 2.00 for 1.0 MeV. We conclude that for the relevant range of PF we may use the fLxed value ~ = 2.0 as a fair approximation. Then the axion emission rate of white dwarfs depends only on the internal temperature of these stars. Numerically we find e = 3.30 X 10 - 3 erg g-1 s-1

.Z2A-1T4a26,

(4)

where T 7 -= T/107 K and tx26 = % / 1 0 -26. This result • a In the non-relativistic limit this is r 2 = (4a/Tr)mePF, as derived, e.g., by Shapiro and Teukolsky [12] or in textbooks on solid state physics. Iwamoto [6] has used the relativistic limit K2 = (4~/rr)p~ in the context of axion emission from neutron stars. 404

23 January 1986

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Volume 166B, number 4

3. Axion coolingo/white dwarfs.We are now in a position to calculate the speed of white dwarf cooling due to axion emission. To this end we concentrate on degenerates of relatively high luminosity, L > 10-3Le, For less luminous stars, corresponding to internal temperatures below approximately 10 7 K, cooling calculations are rather complicated due to the onset of crystallization in the core of the degenerate star. In such a crystal the axion emission rate would drop far below the value given by eq. (4) because electrons move quasi freely in a periodic potential. Instead of Coulomb scattering from nuclei one would then have to consider the interaction of electrons with lattice imperfections and phonons. In the luminosity range 1L® > L > 1 0 - 3 L e the classical cooling theory of Mestel [15] ,4 yields resuits in good agreement with more sophisticated, modern calculations [17]. Following this relatively simple approach we take the bulk of the energy resources of the white dwarf as the thermal kinetic energy o f the non-degenerate, heavy components of the plasma. Considering nuclei of mass number A this energy content per unit mass is u = ~T/Amu. If we assume that this energy is carried away mainly by axion emission, its decrease per unit time is du/dt = - e where e is given by eq. (4). This equation may be viewed as a differential equation for the internal temperature which is taken to be constant throughout the entire star due to the large thermal conductivity of the degenerate electron gas. This differential equation may be easily integrated to yield a relation between age and internal temperature of the star. The internal temperature is related to the surface photon luminosity. This relation is mainly determined by the opacity of the non-degenerate white dwarf surface layers. It is approximately given as [151 L/M~.

1.7 X 10-3.T77/2 ,

(5)

where the white dwarf luminosity L and mass M are understood in units of the solar luminosity L o and solar mass Me, respectively. If axion emission is assumed to be the dominant *4 For a modern review see re/. [16].

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Table 1 Number of white dwarfs per 1000 pc 3 in the solar neighborhood for two ranges of absolute visual magnitudes Mv. Npred refers to our prediction for the case when axion cooling is dominant. Nob s refers to observational numbers. My

Npred

Nobs

10.0-11.0 11.0-12.0

0.03/a26 0.08/a26

0.08 a) 0.30 a)

a) Ref. [16].

0.12 b) 0.24 b)

yr.a~lz-2M6/7 (6)

F o r the number o f white dwarfs presently observable at luminosities between L 1 and L 2 we must simply multiply this result with the white dwarf birthrate.

4. Comparison with observations. In order to make contact with observational data we use the present white dwarf birthrate in the solar neighborhood [ 18] o f about 2 X 10 -12 yr - 1 pc - 3 and assume that it has remained constant during the recent galactic history, i.e., for approximately the past 109 years. We furthermore use the typical case o f a carbon white dwarf (Z = 6) with m a s s M = 0.6M o. Then we predict N ~ e d ~-6.1 × 10-4"ot~-l(L~ 6/7 - L i -6/7)

g<4X

10 -13

v > 1 X 109 G e V ,

cooling mechanism we can then derive the following expression for the time interval it takes to cool from the surface luminosity L 1 at time t I to L 2 at the later time t 2 :

X (L~ 6/7 - L 1 6 / 7 ) .

t~26 < 1 is a conservative constraint. It corresponds to the upper bound (8)

on the Yukawa coupling o f any light, pseudoscalar particle to electrons. This appears to be the most stringent bound presently available. F o r "invisible a x i o m " this result translates into a new lower bound on the P e c c e i - Q u i n n scale o f

b) Ref. [12].

t 2 - t 1 ~ 1.7 X 107

23 January 1986

(7)

for the present number o f white dwarfs in a volume o f 1000 pc 3 and with luminosities between L 1 and L 2. They can be translated into absolute visual magnit u d e s M v. Using eq. (7) o f ref. [13] we f'md l o g L - 0 . 8 forM~ -- 10.0, - 2 . 0 for 11.0 and - 2 . 7 for 12.0. Then the entries for Npred" in table 1 can be easily determined. These expected values can now be compared with the relevant empirical numbers o f Green [ 13] and o f Weidemann [18] which are based on different sets o f observational data (table 1). We conclude that conflicts with these observational data can be avoided only if a26 is smaller than about 0.3. Allowing for the various uncertainties involved in our theoretical arguments and in the observational data we believe that

(9)

and into a new upper bound on the axion mass of m a < 3 X 10 - 2 e V .

(10)

I f these bounds were nearly saturated axion emission would be the dominant cooling mechanism for white dwarfs. The "ecological niche" available to invisible axions has then been further reduced. Their possible existence, however, remains an open question to be hopefully answered in the future. The author thanks Leo Stodolsky for helpful discussions and for useful comments on the manuscript.

References [1] S. Weinberg, Phys. Rev. Lett. 40 (1978) 223; F. Wilczek, Phys. Rev. Lett. 40 (1978) 279. [2] R.D. Peceei and H. Quinn, Phys. Rev. Lett. 38 (1977) 1440. [3] D~A.Dieus, E.W. Kolb, V.L. Teplitz and R.V. Wagoner, Phys. Rev. D18 (1978) 1829; K.O. Mikaelian, Phys. Rev. D18 (1978) 3605; G.G. Rafter and L. Stodolsky, Phys. Lett. l19B (1982) 323; K. Sate, Prog. Thcor. Phys. 60 (1978) 1942; M.I. Vysotsskii, Ya.B. Zeldovieh, M.Yu. Khlopov and V.M. Chechetkin, JETP Lett. 27 (1978) 502. [4] A. Barroso and G.C. Branco, Phys. Lett. l16B (1982) 247; D.A. Dieus, E.W. Kolb, V.L. Teplitz and R.V. Wagoner, Phys. Roy. D22 (1980) 839; J. Ellis and K.A. Olive, Nuel. Phys. B223 (1983) 252; L.M. Krauss, J.E. Moody and F. Wflezek, Phys. Lett. 144B (1984) 391 ; G.G. Raffelt, Astrophysical axion bounds diminished by screening effects, MPI-PAE/PTh 51/85 (August 1985). [5 ] M. Fukugita, S. Watamura and M. Yoshimuxa, Phys. Rev. Lett. 48 (!982) 1522, Phys. Rev. D26 (1982) 1840. [6] N. Iwamoto, Phys. Rev. Lett. 53 (1984) 1198. 405

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[7] J. PreskilL M.B. Wise and F. Wilczek, Phys. Lett. 120B (1983) 127; L.F. Abbott and P. Sikivie, Phys. Lett. 130B (1983) 133; M. Dine and W. Fischler, Phys. Lett. 120B (1983) 137; W.G. Umuh and R.M.Wald, Phys. Rev. D32 (1985) 831; M.S. Turner, Phys. Rev. D32 (1985) 843. [8] R.H. Brandenberger, Phys. Rev. D32 (1985) 501; M. Fukugita and M. Yoshimma, Phys. Lett. 127B (1983) 181; J. Ipser and P. Sikivie, Phys. Rev. Lett. 50 (1983) 925; T. Moody, Phys. Lett. 149B (1984) 328; S.Y. Pi, Phys. Rev. Lett. 52 (1984) 1725; P. Sikivie, Phys. Rev. Lett. 48 (1982) 1156; F.W. Steeker and Q. Shaft, Phys. Rev. Lett. 50 (1983) 928; P.J. Steinhardt and M.S. Turner, Phys. Lett. 129B (1983) 51; M.S. Turner, F. Wilczek and A. Zee, Phys. Lett. 125B (1983) 35. [9] J.E. Kim, Phys. Rev. Lett. 43 (1979) 103; M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, NucL Phys. B166 (1980) 493;

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