PRL 96, 161302 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending28 APRIL 2006

Production of Axions by Cosmic Magnetic Helicity

L. Campanelli1,2,* and M. Giannotti1,2,†

1Dipartimento di Fisica, Universita di Ferrara, I-44100 Ferrara, Italy2INFN–Sezione di Ferrara, I-44100 Ferrara, Italy

(Received 13 January 2006; revised manuscript received 8 March 2006; published 26 April 2006)

0031-9007=

We investigate the effects of an external magnetic helicity production on the evolution of the cosmicaxion field. It is shown that a helicity larger than �few� 10�15 G�2 Mpc, if produced at temperaturesabove a few GeV, is in contradiction with the existence of the axion, since it would produce too much of anaxion relic abundance.

DOI: 10.1103/PhysRevLett.96.161302 PACS numbers: 98.80.Cq, 14.80.Mz, 98.62.En

Recently, the topic of cosmic magnetic helicity genera-tion and phenomenology has been widely discussed (see,e.g., [1–8]). An in-depth study of this very peculiar quan-tity could help in understanding the nature of the cosmicmagnetic field itself and its generation. Magnetic helicity isdefined as

HB�t� �1

V

ZVd3xA � r �A; (1)

where A� � �A0;A� is the electromagnetic field. In a flatuniverse described by a Robertson-Walker metric, ds2 �dt2 � R2dx2, where R�t� is the expansion parameter nor-malized so that, at the present time t0, R�t0� � 1, theelectric and magnetic fields are defined as E � �R�1 _Aand B � R�2r�A, where a dot indicates the derivativewith respect to the cosmic time t. Hence, the helicity can bewritten as HB�t� � �1=V�R

2RV d

3xA � B. It should benoted that in the literature the definition of HB is usuallygiven without the factor R2. In any case, the two definitionscoincide at the present time. However, we note that, sincethe early Universe is a very good conductor, magneticfields are frozen into the plasma and then B scales intime as B / R�2 [3]. Therefore, magnetic helicity, as de-fined in Eq. (1), remains constant (after its generation).

The magnetic helicity is related to the topological prop-erties of the magnetic field; it is a CP-odd pseudoscalarquantity and, if different from zero, would reveal a macro-scopic P andCP violation in the Universe. Let us introducethe magnetic energy and magnetic helicity spectraEB�k; t� � �2�=V�k2B�k� � B��k� and H B�k; t� ��4�=V�R2k2A�k� � B��k�, respectively, where B�k� andA�k� are the magnetic field and the vector potential inFourier space and k � jkj. In terms of the spectra, themagnetic energy EB�t� � �1=2V�

RV d

3xB2�x; t� and themagnetic helicity are EB �

RdkEB and HB �

RdkH B.

From the above definitions, it follows that any magneticfield configuration satisfies the inequality jH B�k; t�j �2R4k�1EB�k; t� [6]. A straightforward integration in k leadsto the so-called ‘‘realizability’’ condition jHBj � Hmax

B �2R3�BEB, which fixes the maximal helicity associated witha given magnetic field configuration. Here �B�t� �2�R

Rdkk�1EB�k; t�=EB is the magnetic field correlation

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length. Therefore, bounds on the magnetic field can bedirectly translated into bounds on magnetic helicity.Before discussing this point more deeply, it is useful tointroduce the following parametrization for the helicityproduced at the temperature T�:

HB�T�� � 3:3� 10�19rg��T��

1=2

g�S�T��GeV

T�G2 Mpc; (2)

where g� and g�S count the total numbers of effectivelymassless degrees of freedom referring to the energy andentropy density of the Universe, respectively. (From nowon, g� and g�S will be considered equal.) The choice r � 1gives the maximal helicity associated with a magnetic fieldcorrelated on the Hubble scale, �B � H�1 (H is the Hubbleparameter), and whose energy density equals the totalradiation energy density at T�. Thus, the parameter r � 1measures the fraction of helicity with respect to the maxi-mum possible at a certain time.

It is known from the big bang nucleosynthesis (BBN)argument that a magnetic field B at the temperature T ’0:1 MeV, correlated on the Hubble scale at that time,cannot be more intense than 1011 G [4]. This yields theupper limit r& rBBN’82g��T��

1=2�T�=GeV�. On the otherhand, the analysis of the observed anisotropy of the cosmicmicrowave background (CMB) radiation requires thebound 10�9 G on the scales of 1 Mpc [4] for the magneticfield (today), which leads to r & rCMB ’ 0:2g��T��1=2��T�=GeV�. Observe that for magnetic helicity producedbefore 1 GeV neither of these limits give any interestingconstraints, since r is always expected to be less than 1.

In this Letter, we consider the effects of magnetic helic-ity generation on the cosmic axion field, showing how suchgeneration could considerably amplify the axion’s ex-pected relic abundance �a. In particular, we will showthat the request �a � �matter sets, in most cases, a boundon r of many orders of magnitude smaller that the aboverBBN and rCMB.

The axion field a emerges in the so-called Pecce-Quinn(PQ) mechanism [9] for the solution of the strong CPproblem (for a review see, e.g., [10]). In most models, itis identified with the phase � of a complex scalar field a �fa�, where fa is a phenomenological scale usually re-

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9 10 11 12 13 14 15 16Log10 fa GeV

20

15

10

5

0

Log

10r

TG

eV

FIG. 1. Allowed region for the magnetic helicity and the PQconstant in the case of helicity production before the axioncoherent oscillations. The area under the curve represents theparameter region �a <�matter ’ 0:3 [21].

PRL 96, 161302 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending28 APRIL 2006

ferred to as the PQ or axion constant, and presently con-strained in the very narrow region 109 & fa & 1012 GeVby astrophysical and cosmological considerations [11].(For the so-called hadronic axion models, whose interac-tion with electrons is suppressed, a small region aroundfa 106 GeV is also allowed [12,13]. For other possibil-ities of reconsidering the limits on the PQ constant, see, forexample, Ref. [14].)

For temperatures above the PQ scale, the totalLagrangian is required to be invariant under a constantshift of the phase � (PQ symmetry). However, for lowertemperatures, the symmetry is (spontaneously) broken andthe angle � is fixed on a precise value. We indicate thisinitial value with �i. This phase evolves following theequation [10]:

��� 3H _��@V@�� 0; (3)

where V��; T� � m2a�T��1� cos�� is the instanton in-

duced potential. The temperature-dependent axion massis [15,16] ma�T� ’ 0:1m��=T�3:7, for T � �, andma�T� � m, for T � with m ’ 6:2� 10�6 eV=f12

�f12 � fa=�1012 GeV��, and � 200 MeV is the QCDscale.

As the curvature (mass) term in Eq. (3) becomes domi-nant over the friction (Hubble) term, � begins to oscillatewith the frequency ma�T�. This happens at about thetemperature T � T1 defined by the equation ma�T1� �3H�T1�. Approximately T1 ’ 0:9f�0:175

12 �0:65200 GeV, with

�200 � �=�200 MeV�. During this period of coherent os-cillations, the number of axions in a comoving volumeremains constant, naR3 � const, where na�T� is the axionnumber density. Thus, the axion relic abundance today is

�a ’ 0:2��0:65200 ��

21 � �

_�1=3H1�2�f1:175

12 ; (4)

where �1, _�1, and H1 are, respectively, ��T1�, _��T1�, andH�T1�. If the axion field is not interacting in the region T >T1, and T1 fa, then the axion kinetic energy at T � T1

is reliably negligible, _�1 / �T1=fa�3 1, and conse-

quently �1 ’ �i. Thus, Eq. (4) reduces to �a � �a0 ’0:2��0:65

200 �2i f

1:17512 . Assuming the natural value �i 1,

this gives �a0 ’ 0:3 (the expected value for the dark matterabundance) for f12 1. Much larger values of f12 wouldcause too much axion production and are, therefore,excluded.

However, if the field evolution took place in the presenceof an external magnetic field, the previous analysis needs tobe modified. It is easy to show the importance of themagnetic helicity in this case. In fact, in the presence ofa magnetic field, the right-hand side of Eq. (3) is modifiedas ��=2�f2

a�E � B. Assuming a homogeneous axion field,and observing that �1=V�

RV d

3xE �B � � _HB=�2R3�,Eq. (3) becomes

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��� 3H _��@V@�� �

�

4�f2a

_HB

R3 : (5)

The slightness of the effects of a nonhelical magnetic fieldon the cosmic evolution of the axion field discussed inliterature (see, e.g., [17]) can easily be understood: Eventhough the axion itself produces helicity in the presence ofan external magnetic field [2,6,18] and, therefore, its evo-lution is modified, the effect is negligible since the helicityproduced by the axion itself is small. However, if anothermechanism were responsible for the production of a sig-nificant amount of helicity, the consequences on the axionrelic abundance today could be enormous.

Let us assume, first, that a large amount of helicity isproduced at the temperature T� � T1 and that the timescale of its production is short compared to the time scalet1. (This is not a very strong assumption. For example, inthe �-dynamo mechanism, this time scale is expected to be�t� & t� t1 [5].) Moreover, taking into account thatafter its generation the helicity can be considered a (qua-si)conserved quantity (see, e.g., [19]), we can safely ap-proximate it as a step function, HB�t� � HB�t����t� t��.

When the potential energy is small compared to thekinetic energy and/or the friction, Eq. (5) can be solved,giving _��T� � �HB�T�=�4�f

2aR

3�, that is, the kinetic en-ergy is a decreasing function of time. Let us indicate withTeq the temperature at which the kinetic energy equals thepotential: � _�2=2�jTeq

� hV���ijTeq, where for the sake of

simplicity we are considering the root mean square valueof the potential hV���i � m2

a�T��1� 1=���2p�. It is clear that

only if Teq � T1 will the oscillations start at the tempera-ture T1, where the friction becomes small. In this case, thestandard analysis applies and we find

�a ’ 0:1f1:17512 ��0:65

200 �1� �r=ra�2�; if r � ra; (6)

where r� ra’1:5�10�11g1=2� �T���

�0:65200 f2:175

12 �T�=GeV�corresponds to Teq � T1. However, if Teq < T1, the oscil-

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9 10 11 12 13 14 15 16Log10 fa GeV

10

8

6

4

2

0

Log

10r

T1

2

T 0.1 MeV

T 3 keV

T 100 eV

FIG. 2. Allowed region for the magnetic helicity and the PQconstant in the case of fast helicity production during the axioncoherent oscillations. The areas enclosed by the dashed andcontinuous curves represent the parameter region �a <�matter ’ 0:3 for the plus and minus sign in Eq. (9), respectively.The areas under the horizontal lines represent the region allowedby the CMB analysis for three different temperatures.

PRL 96, 161302 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending28 APRIL 2006

lations cannot start at T � T1, the kinetic energy being stilltoo large, but will be delayed and start at T � Teq. Theresulting axion relic abundance is

�a ’ 0:2f1:17512 ��0:65

200 �r=ra�; if r > ra: (7)

In Fig. 1, we show the cosmologically allowed region�a <�matter ’ 0:3, for the PQ constant fa and the rparameter. Observe that, since we are now consideringhelicity to be generated before the axion oscillations, T� >T1 1 GeV, the BBN and CMB arguments do not set anyconsistent limit on the magnetic helicity. On the otherhand, the limit on r from the axion abundance is extremelysignificant: The curve in Fig. 1 is well approximated byr � ra (for f12 & 1), and this implies [see Eq. (2)] that amagnetic helicity generated at the temperature T� cannotbe larger than

HB ’ 5� 10�30��0:65200 f2:175

12 G2 Mpc: (8)

In the most conservative case, f12 ’ 1, we exclude anyHB * �few� 10�15 G�2 Mpc. We note that a full numeri-cal treatment of the axion evolution equation would revealthe existence of a small region with fa above the cosmo-logical bound. We omit the discussion of this case to avoiduseless complications. Whatever the case, the regionemerges as a fine-tuned compromise between helicityand the � angle and is, therefore, very small.

We will now briefly discuss the phenomenon of helicityproduction during the axion oscillations and show that theeffects on the axion relic abundance are, in this case, lessevident. The peculiarity of this second case is the appear-ance of an additional energy scale, the axion mass, whichwas neglected in the above discussion. In the radiation era,Eq. (5) can be restated as �’� �3H2=4�m2

a�’ � F�t�,where ’ � �R3=2, F�t� � ���=4�f2

a�� _HB=R3=2�, andwe assumed j�j< 1 (see below) so that V��� ’m2a�2=2. For temperatures T �, the axion mass is

constant, ma�T� � m, and much larger than the Hubbleparameter H=m 10�6�2

200f12g��T��T=��2. Hence, fac-tors of order H=ma can be neglected in the previousequation, which therefore describes a forced harmonicoscillator with frequency m and forcing term F�t�. Its so-lution is ’ � ’0 � �1=m�

RttiduF�u� sinm�t� u�, where

’0 is the solution for the free harmonic oscillator. With asuitable choice of initial conditions, we can write ’0 �A sin�mt�, A being related to the actual axion energy den-sity in the case of null magnetic helicity, �a�T0� �12 f

2am

2A2. Let us suppose that the helicity is producedduring the finite interval of time (tA; tB). Since before tAand after tB the evolution of ’ is that of a free oscillator,and since for r & rCMB it results j�j< 1 if T� & 1 MeV,we can apply the standard procedure to find the axion relicabundance today. This reads �a � �a0 � 2�1=2

a0 Rez�jzj2, where z � �fa=

��������2�cp

�RtBtA dtF�t�e

imt, �c is the presentcritical density, and �a0 is the expected axion relic abun-

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dance (today) in the absence of magnetic helicity [see thediscussion below Eq. (4)]. The time profile of the magnetichelicity in the expanding Universe is conveniently parame-trized by the following ‘‘rapidity’’ function: f�t� ��R�=R�3=2t� _HB=HB�t��, where t� is a reference time be-tween tA and tB. Therefore, the forcing term can be writtenas F�t� � ���=4�f2

a��HB�T��=R3=2� �f�t� and, using

Eq. (2), z becomes

z ’ 3� 107�1=2200f

�112 r�T�=��1=2f�m�; (9)

where f�!� �R�1�1 dtf�t�e

i!t is the Fourier transform off�t�. The axion relic abundance can be exactly calculatedonce the rapidity function f�t� is known. In the case of slow(�t� m�1) helicity production, it results f�m� 0 and,consequently, �a ’ �a0. On the other hand, if helicityproduction is fast (�t m�1), we have jf�m�j Re�f�m�� 1. To give a simple estimate of �a, we observethat for f12 < 1 it results that �a0 �a and, therefore,�a jzj2. This yields the upper bound r &

10�8�1=2200f12��=T��1=2, which translates into a bound on

the present magnetic helicity

HB & 10�27�1=2200g��T��

�1=2f12

�GeV

T�

�3=2

G2 Mpc: (10)

This limit is always stronger than the one from the BBNand, for T� * fewf2=3

12 keV, it also results that r < rCMB

(see Fig. 2). In this figure, we have included the regionabove the cosmological limit fa > 1012 GeV (see the con-tinuous line), since the analysis is straightforward in thiscase. Simply note that the minus sign in Eq. (9) corre-sponds to a reduction of the axion relic abundance.

To conclude, we have pointed out that a strong connec-tion between magnetic helicity and axion cosmology ex-

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PRL 96, 161302 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending28 APRIL 2006

ists. In particular, we have studied the influence of primor-dial magnetic helicity generation on the present axion relicabundance. The result is that, allowing for possible helicitygeneration in the early Universe, the knowledge of the PQconstant would not be sufficient to determine the axionrelic abundance today. Indeed, suppose axions were de-tected with the PQ constant strictly under 1012 GeV. Thiswould not exclude the possibility that they represent thedark matter component of our Universe, if an appropriateamount of helicity has been generated (see Fig. 1). Wefocused our analysis on two different cases correspondingto helicity production before or during the axion coherentoscillations. Let us consider the former, and more interest-ing, case first. We look at two different possibilities withinthis case: (i) First, if axions were detected as cold darkmatter particles, this would severely constrain the allowedamount of magnetic helicity in our Universe, according toEq. (8). On the other hand, (ii) if a helicity greater than�few� 10�15 G�2 Mpc were revealed at the present time,this would be in contradiction with the existence of theinvisible axion itself, unless it is assumed that helicityproduction took place at a temperature above the PQ scale.It should be noted that the bounds found are considerablystronger than the ones deduced from the BBN or CMBanalysis.

Finally, if helicity production took place during theprimordial axion oscillations, the constraints on the helic-ity depend on the time scale of its production and areindeed relevant only if this is much smaller than the typicalaxion oscillation time 1=ma. Also in this case, the boundsfound, Fig. 2 and Eq. (10), are relevant compared to theones from the BBN and CMB.

We consider these results of interest in view of nextgeneration experiments devoted to the axion search andfuture experiments on CMB, which are expected to besensitive enough for the detection of a possible helicalmagnetic field [7,20].

We thank A. D. Dolgov for his careful reading of themanuscript and M. Giovannini and A. Mirizzi for helpfuldiscussions.

*Electronic address: campanelli@fe.infn.it†Electronic address: giannotti@fe.infn.it

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