Zero-frequency magnetic fluctuations in homogeneous cosmic plasma revisited
ZZero-frequency magnetic fluctuationsin homogeneous cosmic plasma revisited
F. Caruso
Centro Brasileiro de Pesquisas F´ısicasRua Dr. Xavier Sigaud 150, 22290-180, Rio de Janeiro, RJ, Brazil [email protected] and V. Oguri
Universidade do Estado do Rio de JaneiroRua S˜ao Francisco Xavier 524, 20550-013, Rio de Janeiro, RJ, Brazil [email protected]
ABSTRACT
Magnetic fluctuations in a non-magnetized gaseous plasma is revisited andcalculated without approximations, based on the fluctuation-dissipation theorem.It is argued that the present results are qualitative and quantitative different formprevious one based on the same theorem. In particular, it is shown that it is notcorrect that the spectral intensity does not vary sensitively with k cut . Also thesimultaneous dependence of this intensity on the plasma and on the collisionalfrequencies are discussed. Subject headings: plasma physics; magnetic fluctuations
1. Introduction
Fluctuations of physical quantities near zero frequency have been investigated by severalauthors since the papers of Johnson (1928) and Nyquist (1928). A general theory on thefluctuation-dissipation theorem, which will be the starting point of this paper, was developedby Kubo, R.(1957). To the best of our knowledge, a concrete expression for the low-frequencyspectrum of fluctuations of magnetic fields in a thermal plasma was obtained for the first timeby Tajima, T.; Cable, S; Shibata, K. & Kulrsrud, R.M. (1992). They found a peak around ω = 0 magnetic fluctuation which was interpreted as the evanescent energy component of a r X i v : . [ a s t r o - ph . C O ] A p r
2. The first predictions
The fluctuation-dissipation theorem developed by Kubo, R. (1957) is able to deal withthe thermal fluctuations inside a plasma in or near thermal equilibrium. The expressionfor the magnetic field fluctuation in an homogeneous isotropic non-magnetized equilibriumplasma was obtained by Tajima, T.; Cable, S; Shibata, K. & Kulrsrud, R.M. (1992) lookingat waves in such a plasma. In an electron-positron plasma, for example, the magneticfluctuations in wave number and frequency space is given by (cid:104) B (cid:105) (cid:126)k,ω π = 2 (cid:126) ωe (cid:126) ω/k B T − ηω p × (1) × k c ( ω + η ) k c + 2 ω ( ω p − ω − η ) k c + [( ω − ω p ) + η ω ] ω where k B is the Boltzmann constant and ω p and η are, respectively, the plasma and thecollisional frequencies. Here ω p = ω p e − + ω p e + , and ω p = (cid:115) n πZe γm being n the particle density inside the plasma, Z is the atomic number of these constituents, e and m are respectively the charge and the mass of the electron, and γ is the Lorentz factorgiven by γ = (1 + kT /mc ). This result can be integrated in d (cid:126)k = 4 πk d k to get (theFourier transform) (cid:104) B (cid:105) ω π = (cid:90) d (cid:126)k (2 π ) (cid:104) B (cid:105) (cid:126)k,ω π (cid:90) ∞−∞ (d ω/ π )( (cid:104) B (cid:105) ω / π ).The integral of eq. (1) over wave numbers shows a high wave number divergence. Ac-cording to Tajima, T.; Cable, S; Shibata, K. & Kulrsrud, R.M. (1992), this is expectedsince the derivation is based on classical fluid equations of motion and the constant collisionfrequency η is independent of k . However, they prefer to carry on their analyzes in thesimpler phenomenological approach. In order to overcome the large k dependence, they firsttake the limit η → k to infinity, which corresponds to thevanishing cross section of collisions as k → ∞ . This is a very delicate point and we will turnback to this point in Section 3. For both the high frequency and high wave number limitsthe authors emphasized that the expression of eq. (1) has a substantial value only where ω − c k − ω p (cid:39)
0. The combined high-frequency and high wave-number limits were get byletting η →
0. The expression for the low-frequency spectrum was obtained by breaking upthe k integral into two intervals, by introducing a cutoff value k cut , with x cut ≡ k cut c/ω pe . Inthe integration from 0 to k cut , η was kept finite while in the integral from k cut it was used theapproximation η →
0. The expressions obtained for the high and low parts of the spectrumwas, respectively: (cid:104) B (cid:105) ω π = T π δ ( ω ) (cid:90) ω p ω p + c k k d k + 12 πc (cid:126) e (cid:126) ω/k B T − ω − ω p ) / (2)and (cid:104) B (cid:105) ω π = 1 π (cid:126) ω (cid:48) e ( (cid:126) ω (cid:48) pe /k B T ) ω (cid:48) − η (cid:48) (cid:16) ω pe c (cid:17) × (cid:90) x ( ω (cid:48) + η (cid:48) ) x + · · · d x + (3)+ (cid:126) ( ω (cid:48) − ω (cid:48) p ) / πe ( (cid:126) ω pe /k B T ) ω (cid:48) − (cid:16) ω pe c (cid:17) × Θ( ω − (cid:113) c k + ω p )where Θ is the Heaviside step function, η (cid:48) ≡ η/ω pe , ω (cid:48) ≡ ω/ω pe , and ω (cid:48) p ≡ ω p /ω pe .Finally the zero frequency limit of the magnetic fluctuations is give bylim ω → (cid:104) B (cid:105) ω π = (cid:126) ω (cid:48) π ( e (cid:126) ω pe ω (cid:48) /k B T −
1) 2 (cid:16) ω pe c (cid:17) η (cid:48) (cid:90) x cut0 d x (4)At this point the frequency spectral intensity was plotted for a temperature T = 10 Kby requiring that the value of k cut (or x cut ) provide a smooth behavior at the joint betweenthe low-frequency spectrum and the black-body spectrum. The choice was k cut ∼ ω pe /c or( x cut ∼ k cut and thatnear ω = 0 the spectrum goes like ω − . Let us now show our results.
3. General Result
We have integrated eq. (1) over k analytically, without any approximation, by the partialfractions technique. The exact result for the indefinite integral over the wave number is: S ( ω ) ≡ (cid:104) B (cid:105) ω π = u ω by + u ω r / (cid:26)
12 sin 3 θ (cid:12)(cid:12)(cid:12)(cid:12) R − ( y ) R + ( y ) (cid:12)(cid:12)(cid:12)(cid:12) + (5)+ cos 3 θ (cid:20) arctan (cid:18) y − a (cid:48) b (cid:48) (cid:19) + arctan (cid:18) y + a (cid:48) b (cid:48) (cid:19)(cid:21)(cid:27) with the following definitions: (cid:96) D ≡ c/ω p is the Debye length, x ≡ ω/ω p , y ≡ k/k = kc/ω , η (cid:48) = ηT /ω p , a ≡ − ( x + η (cid:48) ) − , b ≡ ( η (cid:48) /x )( x + η (cid:48) ) − , R ± ( y ) = y ± a (cid:48) y + r (cid:48) , a (cid:48) = (cid:112) ( r + a ) / b (cid:48) = (cid:112) ( r − a ) / r = ( a + b ) / , sin(3 θ/
2) = ( ba (cid:48) + ab (cid:48) ) r − / , cos(3 θ/
2) = ( aa (cid:48) − bb (cid:48) ) r − / ,and u ω ≡ (cid:126) /π e (cid:126) ω/k B T − ω c = (cid:126) /π e x/x p − x (cid:96) D (6)where x p ≡ k B T / (cid:126) ω p .Note that the general result shows only a linear divergence in k restricted to the firstterm of eq. (5). This term, however, cannot be simply discarded by doing the limit η → k , as did by Tajima, T.; Cable, S; Shibata, K. & Kulrsrud, R.M.(1992), since it plays a very important role when the strict limit ω/ω p → ω (cid:28) ω p ) is to be considered, even when k is large. Indeed, ifwe discard it for all large values of k it can be shown that the limit x (cid:28) S ( ω ) will benegative. Therefore, our result for the definite integral can be put in the form S ( ω ) = S + S + S (7)where S = u ω b (cid:90) ∞ d y ; S = πu ω a (cid:48) b + ab (cid:48) ); S = πu ω aa (cid:48) − bb (cid:48) ) (8) 5 –The term S will be taken as S = u ω b y cut with y cut as large as we want. This will renderthe confrontation with the previous result easier. We note that for k > cm − , S → aa (cid:48) = 1 √ ω + η − ω p ω / ( ω + η ) / (cid:104)(cid:113) ( ω + η )[( ω − ω p ) + η ω ] + ω ( ω + η − ω p ) (cid:105) / and bb (cid:48) = 1 √ ηω p ω / ( ω + η ) / (cid:104)(cid:113) ( ω + η )[( ω − ω p ) + η ω ] − ω ( ω + η − ω p ) (cid:105) / Compared to eqs. (2)-(4) it is immediately evident how our result is different from those ofequations, showing a much more complicated dependence of the frequency spectrum on thevariable ω , and on the parameters ω p and η , which dependencies on plasma temperature areshown, respectively, in Figures 1. T(K) ) - ( s p ω T(K) η -5 -3 -1 Fig. 1.— (a) Plasma frequency and (b) plasma collison frequency, both as a function ofplasma temperature.Our full result S ( ω ) is shown in Fig. 2, where both the normalized (gray curve) and nonnormalized (green curve) spectral intensities are given, for k cut = 10 cm − e T = 10 K.In this figure we have also plotted each one of the terms that contribute to S ( ω ) areshown in different colors. The deep we see in this figure near x = 1 tends to disappear as k cut goes to high values; as k cut decreases up to (cid:39) cm − , the ordinate of the deep tendsto zero.The detail of the non normalized spectral intensity near ω (cid:39) ω p is shown in Fig. 3. Itshows naturally a smooth behavior between the low-frequency spectrum and the blackbody 6 – x -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1
10 1 10 Sp ec t r a l i n t e n s it y ( S w = S1 + S2 + S3 ) -27 -22 -17 -12 -7 -2 SwS1(x,1e20)S2(x,1e20)S3(x,1e20)Planck(x)Sw/So
Fig. 2.— Non normalized S ( ω ) and normalized ( S ( ω ) /S ◦ ) spectral intensities showing thedifferent components of the spectrum, S , S and S , as given by eq. (8).spectrum, which is constructed by hand in Tajima, T.; Cable, S; Shibata, K. & Kulrsrud,R.M. (1992).If we compare this graph to the correspondent one shown in Fig. 1(a) of the paperof Tajima, T.; Cable, S; Shibata, K. & Kulrsrud, R.M. (1992), we see that our result, for x >
1, is a classical blackbody radiation spectrum which goes to zero at x (cid:39) while theirblackbody spectrum has a much greater width (more than two orders of magnitude).Another important qualitative difference between the non approximate and the approx-imate results is that we found a very peculiar oscillations in S ( ω ) for the x (cid:39) − region ofthe spectrum, as can be seen from Figure 4. These oscillations occur in an x region wherethe classical blackbody spectrum still have a significant value; there is however, in this case,a strong interference in the total spectrum S ( ω ), eq. (7), due to a change of sign of thefunction S ( ω ). Such a kind of behavior was found just for cut-off values of the order of k cut = 10 cm − . For values of k cut greater than this one such fluctuations disappear. Inany case, this feature confirm our statement that the result can vary sensitively with k cut ,contrary to what was sustain by Tajima, T.; Cable, S; Shibata, K. & Kulrsrud, R.M. (1992).Finally, we have studied the behavior of S ( ω, η ) by varying ω e η . The result is shown 7 – x -1
10 1 10 Sp ec t r a l i n t e n s it y ( S w = S1 + S2 + S3 ) SwS1(x,1e20)S2(x,1e20)S3(x,1e20)Planck(x)
Fig. 3.— Detail of the spectral intensity near ω (cid:39) ω p . x -1
10 1 10 Sp ec t r a l i n t e n s it y ( S w = S1 + S2 + S3 ) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 SwS1(x,1e10)S2(x,1e10)S3(x,1e10)Planck(x)
Fig. 4.— Detail of the spectral intensity fluctuations for k cut = 10 cm − . 8 –in Figure 5. Notice that just the peak of the zero-frequency plasma spectrum depends on η (the blackbody part remains unchanged). Indeed, the spectral intensity varies two orders ofmagnitude by varying η by two orders too, namely, it goes from S ( ω, η ) (cid:39) , for η (cid:39) − ,to (cid:39) , for η (cid:39) − . Thus, our result indicates that, when one goes backwards in time,temperature grows, dynamo action is enhanced (since η goes down), and the resonance peakof the zero-frequency plasma peak goes down. x η -8 -7 -6 ) ω S ( Fig. 5.— Dependence of the spectral intensity on η and ω .
4. Discussions
In this paper we have computed the spectrum of magnetic fluctuations of an homoge-neous cosmic plasma avoiding any approximations. Several different behaviors between ourresults and the previous one obtained by Tajima, T.; Cable, S; Shibata, K. & Kulrsrud,R.M. (1992), mainly in the low-frequency part of the spectrum, are found and discussed.It is important to stress that the exact result indicates that the peak of the zero-frequencyspectrum can indeed vary sensitively with the cut-off value ( k cut ).In the light of this new result, and following the papers of Tajima, T.; Cable, S; Shibata,K. & Kulrsrud, R.M. (1992) and Caruso, F. & Oguri, V. (2005), the problem of establishingan upper limit for fractal space dimensionality from COBE data can be revisited.Our results can still be improved towards cosmological applications by computing theFourier transformed volume element d (cid:126)k in terms of curved Riemannian space embedded 9 –into general relativistic spacetime. This will allow us to address the problem of dynamoaction in Einstein cosmology. The dynamo effect in plasmas is a competitive effect betweenconvection of the cosmological fluid and the plasma resistivity. This is the reason why is sointeresting to consider the relation between the dynamo action γ and the plasma resistivityand its frequency. Recently, some new investigations on this directions were addressed byRubashnyi, A.S. & Sokoloff, D.D. (2010), by using the magnetic field correlation tensor inspace of negative curvature, and by de Souza, R. & Opher, R. (2010) in the case of positivecurvature. Also recently, Garcia de Andrade, L.C. (2010) has obtained a constraint ondynamo action from COBE data, using two-dimensional spatial sections of negative curvatureof Friedmann universe, based on the general relativistic magnetohydrodynamic equationderived by Marklund, M. & Clarkson, C.A. (2005).In near future, by using the formula of Kulsrud, R.M; Cen, R.; Ostriker, J.P. & Ryu, D.(1996), d (cid:15) M (cid:15) M = γ where γ is the magnetic field growth rate (dynamo action), we expect to compute γ in termsof the magnetic plasma resistivity ( η ) and plasma frequency ( ω p ). In this way we shallestimate the variation of dynamo action in terms of η and ω p .
5. Acknowledgment
The authors would like to thank Garcia de Andrade for useful comments.
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