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Astronomy & Astrophysics manuscript no. magic5 (DOI: will be inserted by hand later)

February 1, 2008

arXiv:0705.1906v1 [astro-ph] 14 May 2007

The tension of cosmological magnetic ?elds as a contribution to dark energy

Ioannis Contopoulos and Spyros Basilakos

Research Center for Astronomy, Academy of Athens, GR-11527 Athens, Greece, e-mail: icontop@academyofathens.gr; svasil@academyofathens.gr Received / Accepted

Abstract. We propose that cosmological magnetic ?elds generated in regions of ?nite spatial dimensions may manifest themselves in the global dynamics of the Universe as ‘dark energy’. We test our model in the context of spatially ?at cosmological models by assuming that the Universe contains non-relativistic matter ρm ∝ α?3 , dark energy ρQ ∝ α?3(1+w) , and an extra ?uid with ρB ∝ αn?3 that corresponds to the magnetic ?eld. We place constraints on the main cosmological parameters of our model by combining the recent supernovae type Ia data and the di?erential ages of passively evolving galaxies. In particular, we ?nd that the model which best reproduces the observational data when ?m = 0.26 is one with ? B ? 0.03, n ? 7.68, ?Q ? 0.71 and w ? ?0.8. Key words. Cosmology, magnetic ?elds

1. Introduction

Recent advances in observational cosmology based on the analysis of a multitude of high quality observational data (type Ia supernovae, hereafter SNIa, cosmic microwave background, large scale structure, age of globular clusters, high redshift galaxies), strongly indicate that we are living in a ?at (? = 1) accelerating Universe containing a small baryonic component, non-baryonic cold dark matter needed to explain the clustering of extragalactic sources, and an extra component with negative pressure, usually called ‘dark energy’, needed to explain the present accelerated expansion of the Universe (eg. Riess et al. 1998; Perlmutter et al. 1999; Efstathiou et al. 2002; Caldwell 2002; Percival et al. 2002; Spergel et al. 2003; Tonry et al. 2003; Riess et al. 2004; Tegmark et al. 2004; Corasaniti et al. 2004). From a theoretical point of view, various candidates for the exotic dark energy have been proposed, most of them characterized by an equation of state pQ = wρQ with w < ?1/3 (see Caldwell 2002; Peebles & Ratra 2003; Corasaniti et al. 2004 and references therein). A particular case of dark energy is the traditional Λ-model which corresponds to w = ?1. Note that a redshift dependence of w is also possible but present measurements are not precise enough to allow meaningful constraints (eg. Dicus & Repko 2004; Wang & Mukherjee 2006). From the observational point of view and for a ?at geometry, a variety of studies, and especially the SNIa data, indicate that w < ?1 (eg. Riess et al. 2004; Basilakos & Plionis 2005; Sanchez et al. 2006; Spergel et al. 2006; Wood-Vasey et al. 2007 and references therein). For such a ?uid the condition w < ?1 is problematic since it presents instabilities and causality problems (de

la Macorra 2007). This may be considered as an indication that the dark energy interacts with another ?uid, for example magnetic ?elds. Tsagas (2001) gave an interesting perspective to the problem by claiming that the e?ect of a primordial magnetic ?eld may resemble that of dark energy through the coupling between the magnetic ?eld and space time. In the present paper, we would like to investigate the potential of present day large scale magnetic ?elds to account for the e?ect of dark energy in the observed acceleration of the expansion of the Universe. As we show in § 2, if the magnetic ?eld is highly tangled, it cannot account for the cosmic tension implied by the presence of dark energy. On the other hand, we argue that, if the cosmic magnetic ?eld is generated in sources whose overall dimensions remain unchanged during the expansion of the Universe, the stretching of this ?eld by the expansion generates a tension that behaves as dark energy. In order to test our model, we introduce in § 3 an extra energy density term in the cosmological equations which we associate with the magnetic ?eld. In § 4 we place constraints on the main parameters of our model by performing a join likelihood analysis utilizing the ‘gold sample’ of SNIa data (Riess et al. 2007) as well as the ages of the passively evolving galaxies (Simon et al. 2005). We conclude with a discussion on the possible values of cosmological magnetic ?elds in § 5.

2. Tension in a cosmological magnetic ?eld

It is well known that a positive pressure in the expanding cosmic ?uid contributes to the deceleration of the expansion. It is easy to understand this when we realize that every expanding part of the Universe is pushed against the expansion by

2

Ioannis Contopoulos and Spyros Basilakos: Cosmological Magnetic Fields

its neighboring parts. As a result, each expanding part performs work against its surroundings, and thus loses kinetic energy. The exact opposite takes place when the expanding ?uid feels a tension force. In that case, each expanding part is pulled outwards by its neighbors, and the work done on it by its neighbors contributes to the acceleration of its expansion (e.g. Harwit 1982). Dark energy is, therefore, the cosmic tension that accounts for the observed present accelerated expansion of the Universe. When one talks about tension, one immediately comes to think about magnetic ?elds. Magnetic ?eld lines may be considered as strained ropes, with a highly anisotropic pattern of tension and pressure. We do observe magnetic ?elds up to several tens of ?G in galaxy clusters (see Carilli & Taylor 2002 for a review), but their origin remains a mystery. There is a general belief that cosmic magnetic ?elds are produced through some type of dynamo process that ampli?ed a weak protogalactic seed magnetic ?eld of the order of 10?20 G (e.g. Ruzmaikin, Shukurov & Sokolo? 1988; Kulsrud et al. 1997). In this picture, the magnetic ?eld permeates the cosmic ?uid which is assumed highly conductive. In other words, the sources of the cosmic magnetic ?eld are electric currents distributed more or less isotropically throughout the expanding cosmic ?uid. Furthermore, because of ?ux conservation, the magnetic ?eld scales with the expansion of the Universe as α?2 , where α is the Universe scale factor, and therefore, the magnetic energy contained in any expanding volume of the Universe scales as α?1 . Such a magnetic ?eld generates a positive isotropic (on average) pressure pB = ρB /3, and ρB ≡ B2 /8π ∝ α?4 . The same result may be obtained after averaging out the magnetic pressure and tension terms. In other words, the equation of state of a highly tangled magnetic ?eld is the same as that obtained for a ?uid of highly relativistic particles, and therefore, it cannot account for the cosmic tension implied by the presence of dark energy. The above led the community to conclude that isotropic tension, or equivalently negative pressure, is peculiar to a scalar ?eld (Zeldovich 1986). Here, we would like to investigate the potential of a di?erent scenario in which the magnetic tension does manifest itself in the expansion of the Universe as dark energy. We thus propose that the sources of cosmological magnetic ?elds are of ?nite dimensions, and that these dimensions remain unchanged during the expansion of the Universe. It is interesting to note that, galaxy clusters are the largest gravitationally bound structures in the Universe and as such, they have decoupled from the background expansion. What we have in mind here is some mechanism that results in the generation of magnetic ?elds of order B0 around ‘sources’ of spatial dimensions r0 (such as galaxies, or clusters of galaxies). Such a scenario may not be unreasonable. In fact, we have already proposed a physical picture where magnetic ?elds are generated without the need for a dynamo mechanism, through the Poynting-Robertson effect on electrons in a highly conducting plasma around bright gravitating sources such as active galactic nuclei, black holes, neutron stars, and protostars (Contopoulos & Kazanas 1998; Contopoulos, Kazanas & Christodoulou 2006). If we assume the existence of such sources of cosmic magnetic ?elds, it is natural to further assume that the magnetic

?eld around each source has a dipolar structure. In that case, the magnetic ?eld drops with distance r as B(r) ≈ B0 r r0

?3

(1)

in the region in?uenced by a source of size r0 at its center. We argue that the Universe may be ?lled with several such expanding regions of comoving size Rα(t), which contain cosmic magnetic sources of size r0 ? Rα. The magnetic ?eld energy in each such region is equal to

Rα

EB =

r=r0

B2 r 3 B2 (r) 4πr2 dr ≈ 0 0 . 8π 6

(2)

Obviously, the expansion of the Universe will gradually stretch each dipolar con?guration described by eq. (1) into a monopolar con?guration described by B(r) → B0 r r0

?2

(3)

in the region around each source1 . The magnetic ?eld energy in the expanding region will approach the value

Rα

EB =

r=r0

B2 r 3 B2 (r) 4πr2 dr → 0 0 , 8π 2

(4)

which is 3 times larger than the expression in eq. (2)! We showed here that the magnetic energy E B contained inside a region of size Rα(t) increases with increasing α. This may be parametrized around the present epoch with a simple power law

3 E B = f B2 r0 αn , 0

(5)

where, f ? 1/3, and n is a positive parameter to be determined from observations (see § 4). As a result, ρB = 3 f 2 r0 B 4π 0 R

3

αn?3 , and

(6) (7)

n p B = ? ρB . 3

Note that in our picture, the magnetic ?eld B0 is not primordial, because if that were the case, by the time the Universe doubles its size, the magnetic ?eld in each region of magnetic in?uence will have e?ectively completely transformed itself from purely dipolar to purely (split) monopolar. When that happens, the magnetic ?eld contribution to the cosmological tension dies out. In our scenario, we expect that the physical mechanism responsible for the generation of the cosmic magnetic ?eld (e.g. Contopoulos & Kazanas 1998) requires a certain number of years τ in order to build a value of the order of B0 .

Actually, this is a split-monopole con?guration where space is separated by an equatorial current sheet discontinuity across which the radial magnetic ?eld changes direction. Such is the case in stellar magnetospheres.

1

Ioannis Contopoulos and Spyros Basilakos: Cosmological Magnetic Fields

3

3. Cosmological evolution

We test our hypothesis by introducing an extra term that accounts for the magnetic ?eld in the standard cosmological equations. In particular, we assume that the Universe is homogeneous, isotropic, ?at, and consists of the following three components denoted by subscripts ‘m’, ‘Q’ and ‘B’ respectively: non-relativistic matter (with zero pressure), an exotic ?uid (dark energy), and a magnetic ?uid with an equation of state given by eq. (7). The corresponding equation of state parameters w ≡ pQ /ρQ and n ≡ ?3pB /ρB are assumed here for simplicity to be constant. Therefore, the evolution of the ?uid densities ρm , ρQ and ρB is given by ρm = ?3Hρm , ρQ = ˙ ˙ ?3(w + 1)HρQ and ρB = (n ? 3)HρB. The scale factor of the ˙ Universe α(t) evolves according to the Friedmann equation: H2 ≡ α ˙ α

2

or ?m + ?Q (3w + 1)α?3w + ?B (?n + 1)αn = 0 . I I (12)

Therefore, in order for the latter equation to contain roots in the interval of α ∈ [0, 1], we obtain a theoretical boundary for the possible (n, wQ ) values, namely 3w?Q ? n?B < ?1 with w < 0 and n > 0 . (13)

It is obvious that when ?B → 0, the above constraint tends to the Quintessence case w < ?1/3, as it should.

4. Likelihood Analysis

In order to constrain the cosmological parameters in our model, we use the ‘gold’ sample of 182 supernovae of Riess et al. 2007. In particular, we de?ne the likelihood estimator2 as: LSNIa (c) ∝ exp[?χ2 (c)/2] with: SNIa

182

=

8πG (ρm + ρQ + ρB ) . 3

(8)

Di?erentiating the Friedman equation and using at the same time the above formalism we obtain 4πG α ¨ =? [ρm + (3w + 1)ρQ + (?n + 1)ρB ] . α 3 (9)

χ2 (c) = SNIa

i=1

Mth (zi , c) ? Mobs (zi ) σi

2

.

(14)

In this framework, we de?ne the density parameters ?m (α), ?Q (α) and ?B (α) as ?m (α) ≡ ?m α?3 ρm ≡ 2 , ρm + ρQ + ρB E (α)

?3(1+w)

where M is the distance modulus M = 5logDL + 25, DL (z) is the luminosity distance DL (z) = (1 + z)x(z), zi is the observed redshift, σi is the observed uncertainty, and c is a vector containing the cosmological parameters that we want to ?t (Riess et al. 2007). Note, that x is the coordinate distance related to the redshift through x(z) = c H0

z 0

ρQ ?Q α ?Q (α) ≡ ≡ ρm + ρQ + ρB E 2 (α)

?3+n

,

dy . E(y)

(15)

?B (α) ≡ with

ρB ?B α , ≡ ρm + ρQ + ρB E 2 (α)

E(α) = ?m α?3 + ?Q α?3(1+w) + ?Bα?3+n

1/2

.

(10)

Here, the Hubble parameter is given by H(α) = H0 E(α), where H0 is the Hubble constant at the present time. Also, ?m + ?Q + ?B = 1. Note that in the context of our model, Λ-models correspond to (w, ρB ) = (?1, 0) or (n, ρQ ) = (3, 0), while if n = 0 or w = 0, the extra ?uid behaves like pressureless matter. It is interesting to mention that the interplay between the values of w and n could yield ?at cosmological models for which there is not a one-to-one correspondence between the global geometry and the expansion of the Universe. Indeed, in a ?at low-?m model with (w, n) = (?1/3, 1) we have the same dynamics as in an open Universe, despite the fact that these models have a spatially ?at geometry! In this cosmological scenario, there is an epoch which corresponds to a value of α = αI , where αI = 0. This is called ¨ the in?ection point. After that epoch we reach an acceleration phase with α > 0. Eq. (9) thus implies that at the in?ection ¨ point, ρm,I + (3w + 1)ρQ,I + (?n + 1)ρB,I = 0 , (11)

We remind the reader that we work here within the framework of ?at cosmology (?m + ?Q + ?B = 1) with non-zero large scale magnetic ?elds ?B ≥ 0. Furthermore, we use the results of the HST key project (Freedman et al. 2001) and ?x the Hubble parameter to its nominal value H0 ? 72 km/s/Mpc. The matter density ?m remains the most weakly constrained cosmological parameter. In principle, ?m is constrained by the maximum likelihood ?t to the WMAP and SNIa data, but in the spirit of this work, we want to use measures which are completely independent of the dark energy component. An estimate of ?m without conventional priors is not an easy task in observational cosmology. However, many authors using mainly large scale structure studies, have tried to put constraints to the ?m parameter. In particular, from the analysis of the power spectrum, Sanchez et al. (2006 and references therein) obtain a value ?m ? 0.24. Moreover, Feldman et al. 2003 and Mohayee & Tully 2005 analyze the peculiar velocity ?eld in the local Universe and obtain the values ?m ? 0.3 and ? 0.22 respectively. In addition, Andernach et al. 2005, based on the cluster mass-to-light ratio, claim that ?m lies in the interval 0.15?0.26 (see Schindler 2002 for a review). In the present paper, we decided to ?x ?m to the value 0.26. In this case, the vector of unknown cosmological parameters is c ≡ (?Q , w, n). We, therefore, sample the various parameters as follows: the dark energy density ?Q ∈ [0.01, 0.74] in steps of 0.01, the dark energy parameter w ∈ [?4, ?0.1] in steps

2

Likelihoods are normalized to their maximum values.

4

Ioannis Contopoulos and Spyros Basilakos: Cosmological Magnetic Fields

Fig. 2. Left panel: We show the e?ective equation of state parameter as a function of the scale factor of the Universe. Right panel: We present the evolution of the density parameters ?m (solid), ?Q (dot-dashed) and ?B (dashed). The vertical lines corresponds to the present epoch. where H(z) is the Hubble parameter (see section 3), H(z) = H0 E(z). The dashed lines in ?g. 1 represents the 1σ, 2σ, and 3σ, con?dence levels in the (?Q , w) plane. In this case, we ?nd that the best ?t solution is ?Q ? 0.56 (?B ? 0.18), w ? ?0.68, and n ? 8 (with T ? 13.2Gyr). We now join the likelihoods, Ljoint (c) = LS NIa × LH , and the overall function peaks at ?Q = 0.71+0.03 (?B ? 0.03), ?0.26 w = ?0.80+0.14 and n ? 7.68+2.42 (or wB ? ?2.56). Note that, ?0.04 ?4.00 the corresponding age of the Universe is 13.2Gyr, while solving numerically eq. (12) the in?ection point is at αI ? 0.57 (or zI ? 0.76). Finally, due to the fact that the association of the extra term in the Friedman equation with the magnetic ?eld is arbitrary, we may equally well consider the solution ?B ? 0.71 (?Q ? 0.03), w ? ?2.56 and n ? 2.4 (or wB ? ?0.8).

Fig. 1. Likelihood contours in the (?Q , w) plane for ?m = 0.26. The contours are plotted where ?2lnL/Lmax is equal to 2.30, 6.16 and 11.83, corresponding to 1σ, 2σ and 3σ con?dence level. Note that the continuous and the dashed lines correspond to the SNIa and H(z) results respectively, while the shadowed area is ruled out by stellar ages. of 0.02, and the magnetic ?eld scaling parameter n ∈ [0.1, 10] in steps of 0.02. Doing so, the likelihood function peaks at ?Q ? 0.5 (?B ? 0.24) with w ? ?1.32 and n ? 0.44 (or wB ? ?0.15) which corresponds to an age of the Universe of 12.3 Gyr. The latter appears to be ruled out by stellar ages. Indeed, in order to put further constraints on our solutions we use additionally the so called age limit, given by the age of the oldest globular clusters in our Galaxy (? 12.5?13 Gyr; Caputo, Castellani & Quatra 1988; Cayrel et al. 2001; Hansen 2002 and 2004; Krauss 2003 and references therein). Taking into account the above age limit, the resulting best ?t solution is: ?B ? 0.04 (?Q ? 0.7), w ? ?0.8 and n ? 2.52 (or wB ? ?0.84), corresponding to an age of 13.1 Gyr. In ?g.1 (solid lines) we present the 1σ, 2σ and 3σ con?dence levels in the (?Q , w) plane by marginalizing over n. It is evident that ?Q is degenerate with respect to w and that all the values in the interval 0 ≤ ?Q ≤ 0.74 are acceptable within the 1σ uncertainty. Therefore, in order to put further constraints on ?Q we additionally use measures of H(z) (see Simon et al. 2005) from the di?erential ages of passively evolving galaxies (hereafter H(z) data). Note that the sample contains 9 entries. Doing so, the H(z) likelihood function can be written as: LH (c) ∝ exp[?χ2 (c)/2] with: H

9

5. Discussion

The above values of ?B correspond to an average cosmic magnetic ?eld B = (8π?Bρcr c2 )1/2 ≈ 650?1/2h ?G , B (17)

χ2 (c) = H

i=1

H th (zi , c) ? H obs (zi ) σi

2

,

(16)

where, ρcr ? 1.88 × 10?29h2 g cm?3 is the critical density of the Universe; h is the Hubble constant in units of 100 km/s/Mpc; and c is the speed of light. If ?B ? 0.03, B ? 80?G, whereas if ?B ? 0.71, B ? 400?G. In the present work we do not discuss what physical mechanism may account for the generation of such high magnetic ?elds. Magnetic ?elds of the order of several tens of ?G have been observed in the centers of cooling ?ow clusters. We refer the reader to Vogt & En?lin (2003) (and references therein) for a detailed discussion of the techniques used to estimate such high values of cluster magnetic ?elds. Furthermore, a number of authors have investigated the possibility that ? 50?G ?elds may provide magnetic pressure support in cluster atmospheres (e.g. Loeb & Mao 1994; MiraldaEscude & Babul 1995; Dolag & Schindler 2000; but see also Rudnick & Blundell 2003). We argued that dark energy (or

Ioannis Contopoulos and Spyros Basilakos: Cosmological Magnetic Fields

5

equivalently cosmic tension) manifests itself as a tension agent between galaxy clusters, and as such, it obviously acts in intracluster space. Moreover, there is a theoretical indication that strong magnetic ?elds lie in regions of signi?cantly reduced plasma density (e.g. Gazzola et al. 2007), which would make observations of cosmic magnetic ?elds through Farady rotation or X-ray emission in intra-cluster cosmic voids very di?cult. We must keep in mind that the study of cosmic magnetic ?elds on scales of clusters of galaxies is a fairly new area of research (see Carilli & Taylor 2002 for a review), and therefore, we cannot preclude future observational surprises. In particular, any observation of magnetic ?elds of the order of 100?G over cosmological scales would give credence to our scenario. We would like to end this section with a short discussion on the equation of state of the dark energy. As we mentioned in the introduction, there is strong indication for an equation of state more complicated than the simple assumption of a constant ratio between the pressure and the energy density. We may thus combine our two dark energy ?uids into one with an e?ective dark energy parameter we? PQ + P B Pe? = . = ρe? ρQ + ρ B (18)

2. Each source is associated with a region of magnetic in?uence around it where the large scale ?eld is due to the central source. The sources are uniformly and isotropically distributed throughout the Universe. 3. As the Universe expands, the magnetic ?eld in each region of in?uence is stretched, and the total magnetic ?eld energy grows. This results in the acceleration of the expansion. The acceleration will decrease unless new sources are continuously generated throughout the Universe. 4. We model the e?ect of the magnetic ?eld with an extra term ρB ∝ αn?3 in the Freedman equations. The model that best reproduces the observational data when ?m = 0.26 is one with ?B ? 0.03, n ? 7.68, ?Q ? 0.71 and w ? ?0.8, which yields an average cosmic magnetic ?eld of the order of ? 80?G. Obviously, we may equally well consider the solution ?B ? 0.71, n ? 2.4, ?Q ? 0.03 and w ? ?2.56. The latter corresponds to an average cosmic magnetic ?eld of the order of ? 400?G.

References

Andernach, H., Plionis, M., Lopez-Cruz, O., Tago, E., Basilakos, S., 2005, Astronomical Society of the Paci?c Conference Series, vol. 329, p. 289-293. Nearby Large-Scale Structures and the Zone of Avoidance, Proceedings of a meeting held in Cape Town, South Africa, March 28 – April 2, 2004, San Francisco: Astronomical Society of the Paci?c, 2005. Basilakos, S. & Plionis, M., 2005, MNRAS, 360, L35 Caldwell, R. R., 2002, Physics Letters B, 545, 23 Caputo, F., Castellani, V., Quarta, M. L, 1988, A Self-Consistent Approach to the Age of Globular Cluster M15, The Early Universe: Reprints Edited by Edward W. Kolb and Michael S. Turner. Frontiers in Physics, Reading: Addison-Wesley, p.263 Carilli, C. L. & Taylor, G. B. 2002, Ann.Rev.A& A, 40, 319 Cayrel, R., et al. 2001, Nature, 409, 691 Contopoulos, I. & Kazanas, D. 1998, ApJ, 508, 859 Contopoulos, I., Kazanas, D. & Christodoulou, D. M. 2006, ApJ, 652, 1451 Corasaniti, P. S., Kunz, M., Parkinson, D., Copeland, E. J., Bassett, B. A., 2004, Phys. Rev. Lett., 80, 3006 de la Macorra, A., 2007, submitted (astro-ph/0701635) Dicus, D. A. & Repko, W.W., 2004, Phys.Rev.D, 70, 3527, Dolag, K. & Schindler, S. 2000, A& A, 364, 491 Efstathiou, G. et al. 2002, MNRAS, 330, L29 Feldman, H. et al. 2003, ApJL, 596, L131 Freedman, W., L., et al., 2001, ApJ, 553, 47 Gazzola, L., King, E. J., Pearce, F. R. & Coles, P. 2007, MNRAS, 375, 657 Hansen B. et al. ApJL, 2002, 574, 155 Hansen B. et al. ApJS, 2004, 155, 551 Harwit, M. 1982, in Astrophysical Concepts (Wiley: New York) Krauss, L. M., 2003, ApJ, 596, L1 Kulsrud, R. M., Cen R., Ostriker, J. P. & Ryu, D. 1997, ApJ, 480, 481 Linder E. V., Phys. Rev. Lett., 2003, 90, 1301 Loeb, A. & Mao, S. 1994, ApJ, 435, 109 Miralda-Escude, J. & Babul, A. 1995, ApJ, 449, 18 Mohayaee, R. & Tully, B., 2005, AJ, 130, 1502 Peebles P. J. E., & Ratra, B., 2003, RvMP, 75, 559 Perlmutter, S. et al. 1999, ApJ, 517, 565 Percival, J. W. et al. 2002, MNRAS,337, 1068 Riess, A. G. et al. 1998, AJ, 116, 1009

Using the evolution of ρQ and ρB the e?ective dark energy parameter as a function of time is given by we? (α) = 3w ? nνα3w+n 3(1 + να3w+n ) where ν= ?B . ?Q (19)

Using our best ?t parameters, we? ? ?0.9 for α ? 1, we? ? ?0.8 in the limit α ? 1, and we? ? ?2.6 in the limit α ? 1. In the left panel of ?g. 2, the solid line shows the evolution of the e?ective equation of state parameter as a function of the Universe scale factor. A ?rst order Taylor expansion around the present epoch (Linder 2003) yields we? (α) ? ?0.87 + 0.36(1 ? α) . (20)

In the right panel of ?g. 2 we show the evolution of the density parameters ?m (solid line), ?Q (dot-dashed) and ?B (dashed). It is interesting that, although at the present time the dark energy in dominant, before the in?ection point (αI ? 0.57) the Universe was matter dominated, i.e. ?m (α ? 1) ≈ 1. In fact, we estimate that prior to the in?ection point, ?B ≤ 0.2%?m , which corresponds to an average cosmic magnetic ?eld B ≤ 30?G. This value was well under equipartition, and therefore, one may argue that, at an early enough epoch, matter is able to generate the cosmological magnetic ?elds required in our scenario (for example, through the Poynting-Robertson mechanism described in Contopoulos & Kazanas 1998). We conclude with a summary of the main elements of our scenario: 1. The cosmological magnetic ?eld is generated in sources of characteristic size r0 with characteristic value B0 . The main idea is that the size of these sources does not follow the overall expansion of the Universe. We know that the expansion of the Universe manifests itself over length scales larger than the typical size of clusters of galaxies. This leads us to suggest that the size of our putative magnetic ?eld sources is of the order of a few Mpc.

6

Ioannis Contopoulos and Spyros Basilakos: Cosmological Magnetic Fields

Riess, A. G. et al. 2004, ApJ, 607, 665 Riess, A. G. et al. 2007, ApJ, 659, 98 Rudnick, L. & Blundell, K. M. 2003, ApJ, 588, 143 Ruzmaikin, A. A., Shukurov, A. M. & Sokolo?, D. D. 1988, in Astrophysics and Space Science Library, Magnetic Fields in Galaxies (Dordrecht: Kluwer) Sanchez, A. G., Baugh, C. M., Percival, W. J., Peacock, J. A., Padilla, N. D., Cole, S., Frenk, C. S., Norberg, P., 2006, MNRAS, 366, 189 Schindler S., Space Science Reviews, 2002, 100, 299 Simon, J., Verde, L., Jimenez, R., 2005, Phys. Rev. D., 71, 123001 Spergel, D. N. et al. 2003, ApJS, 148, 175 Spergel D. N. et al. ApJ, 2007, in press (astro-ph/0603449) Tegmark M. et al. 2004, Phys.Rev.D., 69, 3501 Tonry et al. 2003, ApJ, 594, 1 Tsagas, C., 2001, Phys. Rev. Letters, 86, 5421 Vogt, C. & En?lin, T. A. 2003, A& A, 412, 373 Wang, Y. & Mukherjee, P., 2006, ApJ, 650, 1 Wood-Vasey, M. W. et al. 2007, submitted (astro-ph/0701041) Zeldovich, Y. B. 1986, Sov. Sci. Rev. E Astrophys. Space Phys, 5, 1

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