Small scale magnetic field evolution in the first objects formed in the universe

Abstract

Large scale magnetic fields in galaxies are thought to be generated, by a mean field dynamo. In order to have generated the fields observed, the dynamo would have had to have operated for a sufficiently long period of time. However, magnetic fields of similar intensities to the one in our galaxy, are observed in high redshift galaxies, where a mean field dynamo would not have had time to produce the observed fields. MHD turbulence produces small scale magnetic fields at a faster rate than it does mean fields, which can diffuse toward larger scales. If the turbulence is helical, magnetic fields generated at small scales can become correlated over large scales. We study the evolution of magnetic field correlations in the first objects formed in the universe, due to the action of a turbulent, helical, stochastic dynamo, for redshifts 5 < z < 10. Ambipolar diffusion can play a significant role in this process due to the low level of ionization of the gas in the first objects. We show that for reasonable values of the parameters that characterize the turbulent plasma in the time interval considered, fields can grow to high intensities (~ muG), with large coherence lengths (~ 2 - 6 kpc).

Small scale magnetic field evolution in the first objects formed in the universe

Alejandra Kandus^I; Reuven Opher^I; Saulo M. R. Barros^II

^IDepartamento de Astronomía, IAG-USP, Rua do Matão 1226, Cidade Universitária, CEP: 05508-900, São Paulo, SP, Brazil

^IIDepartamento de Matemática Aplicada, IME-USP, Rua do Matão 1010, Cidade Universitária, CEP: 05508-900, São Paulo, SP, Brazil

ABSTRACT

Large scale magnetic fields in galaxies are thought to be generated, by a mean field dynamo. In order to have generated the fields observed, the dynamo would have had to have operated for a sufficiently long period of time. However, magnetic fields of similar intensities to the one in our galaxy, are observed in high redshift galaxies, where a mean field dynamo would not have had time to produce the observed fields. MHD turbulence produces small scale magnetic fields at a faster rate than it does mean fields, which can diffuse toward larger scales. If the turbulence is helical, magnetic fields generated at small scales can become correlated over large scales. We study the evolution of magnetic field correlations in the first objects formed in the universe, due to the action of a turbulent, helical, stochastic dynamo, for redshifts 5 < z < 10. Ambipolar diffusion can play a significant role in this process due to the low level of ionization of the gas in the first objects. We show that for reasonable values of the parameters that characterize the turbulent plasma in the time interval considered, fields can grow to high intensities (~ mG), with large coherence lengths (~ 2 – 6 kpc).

1 Introduction

The physical processes proposed to explain the origin and evolution of the magnetic fields detected in galaxies and clusters of galaxies, can be divided into two main classes: 1) cosmological mechanisms and 2) local astrophysical processes [1, 2]. Until now, neither of them has provided a satisfactory explanation for the generation of the magnetic fields observed.

In order to explain the fields observed in our galaxy and in small redshift galaxies, a mean field dynamo is commonly invoked [3]. The dynamo would have had to have operated for a time on the order of the age of the universe to have attained the observed intensities. However, the presence of equally intense and coherent fields in high redshift galaxies [4, 5], where the mean field dynamo would not have had enough time to amplify the field to the observed values, casts doubt on the mean field dynamo paradigm as the preferred generation mechanism.

The incidence of star formation regions of the highest intensity increases monotonically with redshift [6]. Therefore the rate of occurrence of supernovae was also higher in the past than at present. Supernovae shocks disturb the plasma in which they are immersed, producing turbulent motions of the gas. If the occurence of supernovae was much higher in the past than at the present, the plasma of the first formed objects must have been more turbulent than observed galactic plasmas at low redshifts.These first stars that formed in the universe began to reionize the gas. Therefore the turbulence is MHD turbulence, with a degree of ionization far from complete or homogeneous.

It is known that MHD turbulence generates stochastic magnetic fields (magnetic noise) at a faster rate than mean fields. If the turbulence is strongly non-helical, the fields induced are confined to small scales [7, 8 ]. However if it is helical, magnetic field correlations over large scales occurs [9, 8].

In this study, we explore the hypothesis that the magnetic fields observed in high redshift galaxies are created by small scale, stochastic, turbulent helical dynamos, rather than mean field dynamos. Following Ref. [8], we take into account the backreaction of the growing magnetic fields on the turbulent plasma, in the form of ambipolar diffusion. The corresponding equations for the stochastic dynamo are therefore nonlinear in the magnetic field, producing a scale of coherence larger than those in linear theory.

2 Our Model

We study a gas cloud that is assumed to have collapsed at a high redshift, z > 10. At z ~ 10, the cloud would have had a low magnetization level and a high level of turbulence. Thus, it would have been similar to the turbulent, low ionization molecular clouds observed in our galaxy, albeit with a much smaller initial magnetic field and a higher level of turbulence.

In all galaxies, the supernova rate is a direct measure of the cosmic ray intensity [10]. Thus, we can infer that cosmic rays were already present in considerable intensities in high redshift galaxies, modifying the plasma turbulence as they do in our galaxy. We take into account phenomenologically, the effect of cosmic rays, supernova shocks, and powerful stellar winds from massive stars by varying the turbulent parameters over a broad range.

2.1 Magnetic field evolution equations

The evolution equation for the magnetic field is given by the induction equation

where B is the magnetic field, v the velocity of the fluid and h is the Ohmic resistivity. v( = v_T + v_D) is the sum of an external stochastic field component v_T, and an ambipolar drift component v_D, which describes the non-linear back-reaction of the Lorentz force. This back-reaction is due to the force that the ionized gas exerts on the neutral gas through collisions of the ions with the neutral atoms.

We take B to be a homogeneous, isotropic, Gaussian random field with a negligible mean value. Therefore, we take the equal time, two point correlation of the magnetic field as áBⁱ(x, t)B^j(y, t)ñ = M^ij(r, t) , where

and r = |x - y| . The symbol á ñ denotes a double ensemble average over both the stochastic velocity and stochastic B fields. M_L(r, t) and M_N(r, t) are the longitudinal and transverse correlation functions respectively, of the magnetic field. H(r, t) is the helical term of the correlations. The induction equation can be converted into evolution equations for M_L and H [8]:

where

and

The velocity v_T is assumed to be an isotropic, homogeneous, Gaussian random field, with a zero mean value and a delta function correlation in time (Markovian approximation). Its two point correlation function is formlly identical to the one for the magnetic field (3), with M_L replaced by T_LL, M_N by T_NN and H by C. The velocity v_D = a[ (Ñ × B) × B] , where a = t/4pr_i, t is the characteristic response time, and r_i is the ion density. h is the microscopic diffusion coefficient. T_LL( 0) - T_LL(r) are the scale dependent turbulent diffusion coefficients. 2aM_L(0, t) is the correction due to ambipolar diffusion. 2C(0) - 2C(r) is the scale-dependent a effect (responsible for inducing magnetic fields correlated on scales larger than L_c). 4aH(0, t) is the nonlinear decrement of the a effect due to ambipolar diffusion. Finally, G(r) is a term which allows for the rapid generation of magnetic fluctuations due to velocity shear and the existence of the small-scale dynamo.

We are interested in the evolution of M_L, since this function provides information about the coherence of the induced large scale field. A positive value of this function over a given length indicates that the magnetic field is coherent in this region. Therefore this length will be taken as the coherence scale of the induced field. The maximum scale of coherence attained in each case can be estimated as the region about r = 0, within which M_L is positive. M_L is the tensor product of parallel field vectors, evaluated at two points separated by a distance r. We can estimate the induced magnetic field intensity at all points where M_L > 0 as B ~ M_L(r) /M_L^1/2(0).

2.2 Characterizing the high redshift plasma

We considered a cloud at z ~ 10 and followed the evolution of the magnetic correlations until z ~ 5 (~ 10⁹ years). The value taken for the cut-off scale of the turbulence, l_c ~ 1 AU, is similar to that for present objects [2]. Assuming L_c >> l_c, we studied the range of values 10 pc L_c 100 pc. We assumed that the height h of the turbulent eddies of the high redshift object is of the same order of magnitude as L_c. In order to estimate the correlation velocity V_c on the scale L_c, we used the expression (V_c/L_c) ~ e, where e is the turbulent energy dissipated per unit mass per unit time. This expression assumes that the energy is dissipated on the order of a single rotation of the eddies of size L_c at the angular frequency W ~ V_c /L_c. We then have V_c ~ (eL_c)^{1/ 3}. Supernova explosions are a major contributor to the galactic turbulent energy. The energy associated with a supernova remnant in our galaxy is about 3 × 10⁵⁰ erg, with about one third transformed into kinetic energy of the ambient gas. Larger values for the supernova remnant energy and the mass of the gas involved in the explosions, will produce higher turbulent velocities. We assumed that at redshifts 5-10, f explosions occurred every 5 years and that the mass of the gas involved was 10¹⁰ [2]. As noted above, the star formation and supernova rates were very high in the past. The indicated star formation rate from observations increased by a factor of ~ 50, in going from z ~ 0 to z ~ 8 (see e.g., fig. 4 in Lanzetta et al. [6]). The expected values for f are then 1 < f 10. A value of f ~ 0.1 corresponds to the present supernova rate in our galaxy. We, thus, have e 0.3 × f cm² s^-3. For the considered values of L_c, the expected range of values for V_c is 9.59 km s^-1

V_c 96.5 km s^-1. These values are 3 - 10 times larger than those in our galaxy [2]. Assuming that the largest velocity corresponds to the largest eddy, we have W ~ 10^-13 s^-1. We estimated that the baryon density is r_n(z) = r_n(0) (1 + z)³b, where r_n(0) is the present baryon density and b is a compression factor, which can be much greater than ~ 200 (virial collapse). In our galaxy, the particle density is ~ 1 cm^-3 or r_n ~ 10^-24 g cm^-3. The average baryon density in the universe today is ~ 10^-30 g cm^-3. Thus, for our galaxy, the compression factor is b ~ 10⁶. We assumed that the cloud that we are studying in the interval 5 < z < 10, collapsed virially at a high redshift, creating a large b. Reasonable values for b are, then, in the range 200 < b < 10⁷. Taking r_n(0) ~ 0.05 r_c(0), where r_c(0) 0.9 × 10^-29 g cm^-3 is the present critical density (assuming a fiducial factor, h ~ 0.7, for the Hubble constant), we obtain 4 × 10^-26 g cm^-3r_n (z = 10) 2.3 × 10^-21 g cm^-3 for the baryon density in our high redshift cloud. We estimated the ion mass density as r_i ~ gr_n, with 0.001 g 1, which gives an ion density in the range 4 × 10^-29g cm^-3r_i 2.3 × 10^-21 g cm^-3. At z ~ 10, the cosmic microwave radiation temperature was (1 + z)T₀ ~ 30 K. For 5 z 10, we considered plasma cloud temperatures in the interval 30 K T 10³ K. Using these values and estimating the thermal velocity of the ions as v_n = (3k_BT/m_p)^1/2, we obtained 10⁴ cm s^-1

v_n 10⁵ cm s^-1. Comparing these values with V_c, we see that we are dealing with mildly supersonic turbulence. Due to the the relatively low temperatures of the plasma, the ion-neutral collision cross section is s_in 10^-15 cm ² [13]. The ion-neutral collision frequency is n_in = s_in n_n v_th, giving 10^-16s^-1n_in 10^-10s^-1. The electrical resistivity can be estimated as h = (c^2/4p) (m_e n_en/e²n_e), where n_e is the electron number density, m_e the electron mass, and n_en = á s_en v_e ñ n_n is the electron-neutral collision frequency. Taking n_e = n_i (charge neutrality), T_e ~ T_i, and using v_e ~ ( 3k_BT_e/m_e)^1/2, we obtain h ~ 5 × 10³ cm ² s^-1, which is extremely small. The magnetic Reynolds number is R_m = L_cV_c /h ~ 10 ²³ - 10 ²⁴, which means that at high redshifts, plasma turbulence was the main mechanism for diffusion and dissipation. Thus the first term in equation (5) can be neglected. Since the ion-neutral collision was the dominant interaction in the plasmas considered, we took the characteristic response time as t ~ . The coefficient ''a" in the non-linear terms in equations (3) and (4) can then assume values in the interval 4.3 ×10 ³⁰ g^-1 cm³s a 2.5 × 10 ⁴⁴ g^-1 cm³ s.

2.3 Characterizing the turbulence

In studying turbulence, it is usually assumed that the fluid is incompressible (Ñ.v = 0). The functions T_NN and T_LL are, then, related in the way described by Subramanian [8]. For compressible fluids, (Ñ × v_T = 0) these functions are related by T_LL = T_NN + rdT_NN/dr [11]. Adopting the above relation between T_NN and T_LL, since astrophysical plasmas are compressible, the fluid flow correlation functions can be written as

with A_N = V_cL_c/3 [12]. (In our study, l_c is much smaller than the numerical resolution used. We therefore considered M_L(0) = M_L(l_c)).

3 Results and conclusions

The magnetic correlations that result from the evolution of the turbulent kinematical dynamo are found to be independent of the initial field correlations. The resulting intensity of the magnetic field is very sensitive to the value of both V_c and the degree of ionization of the plasma, while the final correlation length L_M of the magnetic field is found to be very sensitive to the values of V_c and W. Larger values of p and/or of q produce larger values of M_L.

To illustrate the dependence of varying the parameters, we investigated the following turbulent cases:

• L_c = 33 pc, V_c = 45 km s^-1 and W = 4.5 × 10^-13 s ^-1. We obtained an average magnetic field B ~ 1.1 × 10^-6 G and a final correlation length, L_M 1.7 kpc.

• L_c = 80 pc, V_c = 96 km s^-1 and W = 9.6 × 10^-13 s ^-1. We obtained an average magnetic field B ~ 1.4 × 10^-6 G and a final correlation length, L_M ~ 5.4 kpc.

In both cases we took p = 4/3, q = 2, h ~ 10³ cm² s^-1, and a 9.76 × 10³⁹ cm³ s g^-1.

In Fig. 1 we plot the final M_L, for different values of a. We show the final M_L for different values of V_c in Fig.2. The growth of M_L as a function of time for various initial conditions is shown in Fig. 3.

Figure 3

The age of the universe at z = 10 is 4.7 × 10⁸ years. A galaxy at this redshift has already been observed [5]. The star formation rate in this galaxy is observed to be extremely high. A high level of turbulence produced by supernovae and star formation can then be assumed to exist already at z ~ 10 and, thus, throughout the redshift interval 5 < z < 10.

In summary, we found that for a reasonable set of turbulent, astrophysical parameters, magnetic fields on the order of 10^-6 G, as are observed in high redshift objects are generated in less than about 10⁹ years (the time elapsed between z ~ 10 and z ~ 5). They are coherent on scales of L_M 3.5 - 5.4 kpc.

Our model for the generation and evolution of magnetic correlations is relatively simple. Our results, which are preliminary due to the simple evolution equations used, suggest that the reionization process of the universe, involving the formation of the first stars, played an important role in determining the features of the magnetic fields detected in high redshift objects.

Acknowledgments

We thank George Morales for useful comments. We acknowledge E. Opher for careful proof reading of this manuscript. This work was partially supported by the Brazilian financing agency FAPESP (00/06770-2). A.K. acknowledges the FAPESP fellowship (01/07748-3). R. O. acknowledges partial support from the Brazilian financing agency CNPq (300414/82-0).

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Received on 2 February, 2004; revised version on 19 May, 2004

Publication Dates

• Publication in this collection
  01 Mar 2005
• Date of issue
  Dec 2004

History

• Received
  02 Feb 2004
• Accepted
  19 May 2004