Eur. Phys. J. C (2019) 79:119 https://doi.org/10.1140/epjc/s10052-019-6619-5

Regular Article - Theoretical Physics

Chiral plasma instability and primordial gravitational waves

Sampurn Anand1,2,a, Jitesh R. Bhatt1,b, Arun Kumar Pandey1,c 1 Theoretical Physics Division, Physical Research Laboratory, Ahmedabad 380009, India 2 St. Stephen’s College, University Enclave, New Delhi 110007, India

Received: 6 July 2018 / Accepted: 21 January 2019 © The Author(s) 2019

Abstract It is known that cosmic magnetic field, if present, [8–13], cosmological magnetic fields [14–17] and turbulence can generate anisotropic stress in the plasma and hence, can in the plasma [16,18,19]. act as a source of gravitational waves. These cosmic magnetic In the early universe, before the electroweak phase tran- fields can be generated at very high temperature, much above sition, many interesting phenomena have taken place. For electroweak scale, due to the gravitational anomaly in pres- instance, it has been shown by several authors [20–25] that ence of the chiral asymmetry. The chiral asymmetry leads to in presence of asymmetry in the left-handed and the right- instability in the plasma which ultimately leads to the gener- handed particles in the early Universe, there will be an insta- ation of magnetic fields. In this article, we discuss the gener- bility which leads to the generation of turbulence in the ation of gravitational waves, during the period of instability, plasma as well as (hyper-charge) magnetic fields. In refer- in the chiral plasma sourced by the magnetic field created ence [26], it has been shown that these magnetic fields can due to the gravitational anomaly. We have shown that such be generated even in absence of net chiral charge but due gravitational wave will have a unique spectrum. Moreover, to the gravitational anomaly. The magnetic field generated depending on the temperature of the universe at the time of its via this mechanism are helical in nature. However, helicity H = 1 3 · generation, such gravitational waves can have wide range of ( B V d x A B) of these magnetic fields are not com- frequencies. We also estimate the amplitude and frequency pletely conserved due to the fact that large but finite conduc- of the gravitational waves and delineate the possibility of its tivity gives a slight time variation of the helicity density. In detection by the future experiments like eLISA. presence of chiral imbalance, the chiral charge conservation equation, which is valid at temperature T > TR ≈ 80 TeV α [20], is given as: ∂η(Δμ+ π HB) = 0 where Δμ = μR−μL ,  1 Introduction α and η are the asymmetry in the chiral chemical poten- tials and U(1)Y fine structure constant and conformal time Gravitational wave (GW) once generated, propagates almost respectively. At the onset of the instability, HB ≈ 0 and at unhindered through the space-time. This property makes GW subsequent time, helicity will grow at the expense of chiral a very powerful probe of the source which produces it as chemical potential. In this regime, Δμ can be regarded as well as the medium through which it propagates (see [1– constant. On the other hand, at temperature T < 80 TeV the 3] and references therein). From the cosmological point of above conservation equation is not valid as chiral flipping Γ f view, the most interesting gravitational radiation is that of rate is non-vanishing and the hence, the conservation equa- the stochastic gravitational wave (SGW) background. Such tion is given by [22] gravitational radiations are produced by events in the early   d α stages of the Universe and hence, may decipher the physics of Δμ + HB =−Γ f Δμ. (1) those epochs. Several attempts have already been made in this dη π regard and various sources of SGW have been considered.   1/2 Γ = TR = 90 List of SGW source includes quantum fluctuations during Here f T and M∗ π3 Mpl , where geff M√∗ 8 geff inflation [4–7], bubble wall collision during phase transition and Mpl = 1/ G are relativistic degree of freedom and the Planck mass respectively [20]. In this regime, non-linearity a e-mail: [email protected] sets in and the magnetic fields are generated. The generated b e-mail: [email protected] magnetic fields show inverse cascade behaviour, where mag- c e-mail: [email protected] netic energy is transfered from small scale to large-scale.

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In reference [27], it is shown that the currently observed stated otherwise, we will work in terms of comoving vari- −10 baryon asymmetry (ηB ∼ 10 ) can be generated if the ables defined as, magnetic fields produced above electroweak scale undergoes = 2(η) (η), μ = (η)μ, = , the inverse cascade and the strength of the magnetic field is Bc a B c a kc ak (3) ( −14 − −12) of the order 10 10 G at the galactic scale. The gen- where, B, μ and k represents the physical magnetic fields, erated magnetic fields induce a anisotropic stress so that their chemical potential, and the wave number respectively. It is 2/ π energy density B 8 must be a small perturbation, in order clear from the convention used here that all the comoving to preserve the isotropy of the Friedmann-Robertson-Walker quantities are dimensionless. In terms of these comoving background. This condition allows us to use the linear pertur- variables, the evolution equations of fluid and electromag- bation theory. In this perturbation scheme, peculiar velocity netic fields are form invariant [29–31]. Therefore, we will and magnetic fields are considered to be first order in per- work with the above defined comoving quantities and omit turbations. At sufficiently large length scale, the effect of the the subscript “c” in our further discussion. fluid on the evolution of magnetic fields can be neglected. However, at small length scales the interaction between the fluid and the magnetic field become very crucial. At the inter- 2 Gravitational anomaly and magnetic fields in the mediate length scale, plasma undergoes Alfven oscillations early universe and on a very small scale (viscous scale), these fields undergo exponential damping due to the shear viscosity [28]. Thus, Although the origin of large-scale magnetic field is still an the large-scale magnetic fields are important for the physics unsolved issue in cosmology, several attempts have been at the cosmic scale. made to address the issue. It has been discussed in the lit- We have already mentioned that the seed magnetic field erature that there are processes in the early universe, much can be generated even in absence of net chiral charge but above electroweak scale, which can lead to more number of due to gravitational anomaly [26] and the magnetic field right-handed particle than the left-handed ones and remains thus generated can produce instability in the plasma. These in thermal equilibrium via its coupling with the hypercharge magnetic fields contribute a anisotropic stress to the energy- gauge bosons [32,33]. Furthermore, if the plasma has rota- momentum tensor and hence can act as a source for the gener- tional flow or external gauge field present, there could be a ation of the GWs. The underlying physics of GW generation current in the direction parallel to the vorticity due to rota- is completely different from previously considered scenar- tional flow or parallel to the external field. The current par- ios. Therefore, it is important to investigate the generation allel to the vorticity is known as chiral vortical current and and evolution of GW in this context. In this article, we com- the phenomenon is called as chiral vortical effects (CVE) pute the metric tensor perturbation due to the chiral magnetic [34–38]. Similarly, the current parallel to the external mag- field. Since chiral magnetic field, which sources the tensor netic field is known as chiral magnetic current and the phe- perturbations, has a unique spectrum, the GWs generated is nomenon is called as chiral magnetic effect (CME) [39–43]. expected to have a unique signature in its spectrum as well. CVE and CME are characterized by the transport coefficients Moreover, we compute the amplitude and frequency of the ξ and ξ (B) respectively. The form of these coefficients can be GW and show its dependences on the model parameters. Con- obtained by demanding the consistency with the second law sequently, any detection of SGW in future measurements like μ μ of thermodynamics (∂μ s ≥ 0, with s being the entropy eLISA will constrain or rule out such theoretical constructs. density). Thus, in presence of chiral imbalance and gravita- This paper is organized as follows: in Sect. 2 we outline tional anomaly, which arises due to the coupling of spin with the generation and evolution of magnetic field due to gravita- gravity [44], the coefficients for each right and left particle tional anomaly and chiral imbalance. We discuss the genera- have the following form [37,38,41] tion of SGW in Sect. 3. We present our results in Sect. 4 and     μ 2 μ finally conclude in (5). Throughout this work, we have used 2 2 ni i DT 2 ni i  = = = ξi = C μ 1 − + 1 − , c kB 1 unit. We have also considered Friedman- i 3 (ρ + p) 2 (ρ + p) Robertson-Walker metric for expanding background space- (4)     time 2 ( ) n μ D n T   ξ B = μ − i i − i . i C i 1 (5) 2 2 2 i j 2 (ρ + p) 2 (ρ + p) ds = a (η) −dη + δij dx dx , (2) In above equations ‘i’ stands for each species of the chiral where scale factor a(η) have dimension of length, whereas plasma. The constants C and D are related to those of the conformal time η and conformal coordinate xi are dimen- chiral anomaly and mixed gauge-gravitational anomaly and sionless quantities. In the radiation dominated epoch a = are given as C =±1/4π 2 and D =±1/12 for right and 1/T , we can define conformal time η = M∗/T . Unless left-handed chiral particles respectively. The variables n, ρ 123 Eur. Phys. J. C (2019) 79:119 Page 3 of 9 119

ˆ and p are respectively the number density, energy density where (e1, e2, e3 = k) form a right-handed orthonormal ˆ and pressure density. basis with e2 = k × e1. We choose e1 to remain invari- Using the effective Lagrangian for the standard model, ant under the transformation k →−k while e2 flip its sign. one can derive the generalized Maxwell’s Eq. [45], In this basis, the magnetic field can be decomposed as

(η, ) = +(η, )ε+( ) + −(η, )ε−( ). ∇ × B = j , (6) Bi k B k i k B k i k (13)

= + In this basis, the evolution equation for the |B±|2 can be where j is defined as j jv j5 with jv being the vector obtained from Eq. (11) current and j5 is the axial current. Vector and axial currents     respectively takes the following form: ∂|B˜ ±|2 2 2 2 = −k2 ± ξ (B)k |B˜ ±|2 + ±nk + ξk2 μ = μ + σ μ + ξ ωμ + ξ (B) μ , ∂η σ σ 2 jv nv u E v v B (7) μ ( ) ± 2 = μ + ξ ωμ + ξ B μ . ×|v | Fs (14) j5 n5 u 5 5 B (8) where Fs = η − η0 for η − η0 ≤ 2π/(kv) and zero for In the above equations, any quantity xv,(5) denotes the sum η − η ≥ 2π/(kv). Also magnitude of the wave vector k is (difference) of the quantities pertaining to right and left 0 μ μν μ μνσδ represented as k, i.e. |k|=k.FromEq.(14), it is clear that, handed particles. Also E = uν F , B = 1/2ε uν Fσδ, μ μνσδ when first term is dominant over the second term, magnetic and ω = 1/2ε uν∂σ uδ are the electric, magnetic and the modes will grow exponentially with time as vorticity four vectors respectively. We have ignored the dis- placement current in Eq. (6). Taking uμ = (1, v) and using     2η 2η |B±|2 =|B |2 k2 − (k ∓ k )2 , Eqs. (7)– (8), one can show that 0 exp σ ins exp σ ins j0 = n = n + n (9) v 5  ( ) where k = α Δμ/π is the value of the k at which magnetic j = nv + σ(E + v × B) + ξ ω + ξ B B , (10) ins modes have maximum growth rate. The exponential growth ξ = ξ + ξ ξ (B) = ξ (B) + ξ (B) with v 5 and v 5 . Assuming the of the magnetic modes is true only in the linear regime. In velocity field to be divergence free field, i.e. ∇ · v = 0 and this regime, chiral chemical potential remain constant and taking curl of Eq. (6) along with the expression for current magnetic fields are generated at the cost of chiral imbalance. from Eq. (10) we obtain, However, when generated magnetic field is sufficiently large, the non-linear effects become prominent. In this case, we ∂B n 1 = ω + ∇2B + ∇ × (v × B) need to consider the evolution of the chemical potential [20, ∂η σ σ ( ) 23], which is given by Eq. (1). ξ ξ B > + ∇ × ω + ∇ × B. (11) At temperature T 80 TeV, perturbative processes that σ σ lead to the flipping of the chirality are small compared to the In our previous work [26], we discussed that the seed mag- expansion rate of the Universe and hence we can consider netic field (for which B in the right hand side of Eq. (11)is the chiral symmetry to be an exact symmetry of the theory. 2 μ zero) can be generated even if n = 0. The T term in ξi [see Therefore, a chemical potential R,L for each species can be Eq. (4)], which arises due to the gravitational anomaly, acts defined. However, at T < 80 TeV, the chiral symmetry is not as a source for the generation of seed field. On the other hand, exact due to the chiral flipping of the right-handed particles presence of finite chiral imbalance such that μ/T 1, T 2 to the left-handed particles and vice versa. As a consequence, term in ξ still acts as source of seed magnetic field but non- the number densities of these particle are not conserved and zero ξ (B) triggers instability in the system. This result is in therefore, we can not define the chemical potentials for right agreement with the previous studies where it was shown that and left handed particles [22,47]. In order to obtain the veloc- in the presence of a chiral imbalance in the plasma, much ity profile, we used the scaling symmetry [48,49] rather than above the Electroweak scale (T > 100 GeV), there can be solving the Navier-Stokes equation. In our earlier work [26], + − − / instability known as chiral plasma instability [46]. we have obtained |v |=|v |=πk 3 2v. Here v is given The production and evolution of the magnetic field can by be seen through the evolution equation given in Eq. (11).  ( + )/  ( − )/ k 1 m 2 η m 2 6 In order to do so, we decompose the divergence-free vector v(k,η)= v . (15) i k (η ) η fields, e.g. magnetic field, in the orthonormal helicity basis, i 0 0 ε± In above equation, m is a positive integer and v , k are arbi- i , defined as i i trary function encoding the boundary conditions. We also −i showed that the scaling law allows more power in the mag- ε±( ) = √ ( ) ± ( ) ( · ), k [e1 k i e2 k ] exp i k x (12) = ∝ 7 2 netic field. For n 0, EB k at larger length scale whereas 123 119 Page 4 of 9 Eur. Phys. J. C (2019) 79:119

5 7 for n = 0, EB ∝ k instead of k [23]. However, in the both stress energy tensor given in Eq. (20). This can be done by scenarios EB is more than that of the case without consider- using the projection operator: Piljm(k) =[Pil(k) Pjm(k)− 1 ( ) ( )] ing the scaling symmetry [23]. This aspect is important for 2 Pij k Plm k which leads to [50] this analysis as more power in the helical magnetic field can Π (k) = P (k) T (k). (21) generate larger anisotropic stress. ij iljm lm The two point correlation function for the helical magnetic At this stage, we will evaluate the two-point correlation func- fields is given as: tion of the energy-momentum tensor which will be used in 3 the later part of the calculation. The two point correlation of ∗  (2π)  Bi (k)B (k )= δ(k − k ) PijS(k) the stress-energy tensor takes the form j 2

ˆ +i  k A (k) (16) ∗  1 1 ijl l T (k)T (k )= d3 p d3q ij lm ( π)6 ( π)2 4 2 4 a where S(k) is the symmetric and A (k) is the helical part ˆ ˆ ×  ( ) ∗( − ) ∗( ) ( − ) of the magnetic field power spectrum. Pij = δij − ki k j is Bi p B j p k Bl q Bm q k ˆ  the transverse plane projector which satisfies: Pijk j = 0, ˆ PijPjk = Pik with ki = ki /k and ijk is the totally anti- + (..)δij + (..) δlm + (..)δijδlm symmetric tensor. Using Eq. (16) and the reality condition ±∗ ± (22) B (k) =−B(k) , one can show that + +∗  − −∗  B (k)B (k ) + B (k)B (k ) It was shown in Ref. [51] that only first term in the angular  =−(2 π)3 S(k)δ(k − k ) (17) bracket will have a non-vanishing contribution in the two- ∗ + +∗  − −∗  point correlation function of the anisotropic stress Π Π . B (k)B (k ) − B (k)B (k ) ij lm Therefore, we will not evaluate other terms. Moreover, = ( π)3 A ( )δ( − ). 2 k k k (18) around the chiral instability, the magnetic field profile is Note that, A (k) represents the difference in the power of left- Gaussian and the major contribution to the anisotropic stress handed and right-handed magnetic fields, however a maxi- come from this regime only. Therefore, we can safely assume mally helical magnetic fields configuration can be achieved that the magnetic fields are Gaussian and hence four-point when A (k) = S(k). correlator in the integrand can be expressed, using Wick’s theorem, in terms of two-point correlators as 2.1 Anisotropic stress Bi (ki )B j (k j )Bl (kl )Bm(km) Tensor component of metric perturbation, which results in the =Bi (ki )B j (k j )Bl (kl )Bm(km) (23) gravitational waves, are sourced by the transverse-traceless +Bi (ki )Bl (kl )B j (k j )Bm(km) part of the stress-energy tensor. In this work, we assume that +Bi (ki )Bm(km)B j (k j )Bl (kl ) . the anisotropic stress is generated by the magnetic stress- energy tensor which is given by After a bit of lengthy but straightforward calculation, we   obtain the two point correlations of the energy momentum 1 1 T = B B − δ B2 , tensor and the transverse-traceless part of the energy momen- ij 2 i j ij (19) a 2 tum tensor is

Note that the spatial indices are raised, lowered and con-  ( ) ∗ ( ) Tij k Tlm k tracted by the Kronecker delta such that B2 = δijB B .The  i j 1 magnetic field component B then coincide with the comov- = δ(k − k) d3 p S(p)S(k − p) i 2 4 2 4(4π) a ing magnetic field which in our notation is Bc = a B.In   Fourier space, the stress-energy tensor for the magnetic field Pil( pˆ ) Pjm(k − p) + Pim( pˆ ) Pjl(k − p) take the following form  − H ( ) H ( − )   ˆ (− )  p k p ila jmb pa k p b a−2  (24) T (k) = d3q B (q)B∗(q − k) +   ˆ (− ) + H ( ) ( − ) ij ( π)4 i j imc jld pc k p d i p S k p 2 2     P (k− p) pˆ +  P (k− p) pˆ −1 ( ) ∗( − )δ . ila jm a imc jl c Bl q Bl q k ij (20)  2 + i H (k − p)S(p)  P ( pˆ )(k− p) jmb il b We are interested in the generation of GW and hence, we  +  ( ˆ )(− ) need to extract the transverse traceless component of the jld Pim p k p d 123 Eur. Phys. J. C (2019) 79:119 Page 5 of 9 119 and 3 Gravitational waves from chiral magnetic fields

Π ( )Π∗ ( )=P ( ) P ( ) ( ) ∗ ( ). ab k cd k aibj k cldm k Tij k Tlm k We have seen that the chiral magnetic field generated at very (25) high temperature can produce anisotropic stress which leads to tensor perturbation in the metric. To linear order, the small tensor perturbation in the FLRW background can be written Above equation can also be written in terms of a most general as: isotropic transverse traceless fourth rank tensor which obeys P = P = P = P abcd bacd abdc cdab as [28,52] 2 2 2 i j ds = a (η)[−dη + (δij + 2hij)dx dx ], (32) 1 Π (k)Π ∗ (k)= [M f (k)+iA g(k)]δ(k−k), where the tensor perturbation satisfies the following condi- ab cd 4 abcd abcd 4a tions hi = 0 and ∂i h j = 0. In this gauge, these tensor per- (26) i i turbations describe the GW whose evolution equation can be obtained by solving Einstein’s equation which, to the linear with a definition of M and A as abcd abcd order in hij, is given as: M = + −  +  + 2 = π Π , abcd Pac Pbd Pad Pbc Pab Pcd hij 2 Hhij k hij 8 G ij (33) 1 ˆ Aabcd = ke(Pbdace + Pacbde + Padbce + Pbcade) 2 where prime denotes the derivative with respect to the con- η = 1 ∂a(η) (27) formal time and H a(η) ∂η . The time dependence in the right hand side of the Eq. (33) comes from the fact that the which follows following properties: magnetic field is frozen in the plasma. Therefore, Πij(k,η)is a coherent source, in the sense that each mode undergoes the M = M = M = M abcd cdab abdc bacd same time evolution. Assuming that the tensor perturbations Aabcd = Acdab =−Aabdc =−Abacd has a Gaussian distribution function, the statistical properties can be described by the two-point correlation function given Mabab = 4 as, Maacd = Mabcc = 0

PeaMabcd = Mebcd   ∗ ( ,η)  ( ,η)=1δ3( − ) M ( ,η) A = A hij k hlm k k k ijlmSGW k Pea abcd ebcd 4  Mabcd Mabcd = Mabcd Mabcd = 8 + i Aijlm HGW(k,η) ,

Aabcd Mabcd = 0 (34)

Aabab = Aaacd = Aabcc = 0. (28) where SGW and AGW characterizes the amplitude and polar- The functions f (k) and g(k) are defined as follows: ization of the GWs. With the above definition, we can write,

 1  ∗   1 ∗  δ(k − k ) S = M h (k) h (k ) (35) δ(k − k ) f (k) = MabcdTab(k) T (k ) GW ( π)3 ijlm ij lm 2 cd 2  i ∗  δ( − ) H = 1 A   ( ) ∗ ( ). δ(k − k ) g(k) =− AabcdTab(k) T (k ). (29) k k GW ijlm hij k hlm k (36) 2 cd (2π)3 We point out that the functions f (k) and g(k) also depends We now choose a coordinate system, for which unit vectors on time because of the time dependence of the magnetic are eˆ1, eˆ2, eˆ3, in which GW propagates in the eˆ3 direction. fields. The integral form of these two functions are Further, we introduce  1 1  f (k) = d3 p (1 + γ 2)(1 + β2) S(p)S(k − p) ± 3 ± ± ( π)2 e =− (ε × ε ) (37) 4 4  ij 2 i j + 4 γβH (p)H (k − p) (30)

1 1 which forms basis for a tensor perturbations and satisfy g(k) = d3 p (1 + γ 2)βS(p) H (k − p) . δ ± = ˆ ± = 2 (4π)2 the following properties: ij eij 0, ki eij 0 and ± ∓ = / (31) eij eij 3 2[53,54]. The right-handed and left handed circularly polarized state of the GWs are represented by + ˆ where p =|p|, (k − p)=|k − p|, γ = k · pˆ and β = and − sign respectively. In this basis, polarization tensor and kˆ · (k − p) = (k − pγ)/ k2 + p2 − 2γ pk. modes of the GWs can be written as follows: 123 119 Page 6 of 9 Eur. Phys. J. C (2019) 79:119

Π ( ) = +Π +( ) + −Π −( ), ( ) ( ) ij k eij k eij k (38) where c1 x and c2 x are undetermined coefficients which + + − − is given as hij(k,η) = h (k,η)e + h (k,η)e . (39) ij ij x ±  ±  s (x )  On using equations in (38), Eq. (29) can be expressed as: c (x) =− dx y (x ) (47) 1 w(x) k2 0 xin x ±  3 ± s (x ) δ( − ) ( ) = Π +( )Π +∗( ) c (x) = dx j (x) k k f k k k 2  2 0 (48) 2 x w(x ) k − −∗  in +Π (k)Π (k ) (40) where w(x) = j (x) y (x) − y (x) j (x) = 1 .Wehave δ( − ) ( ) =−3Π +( )Π +∗( ) 0 0 0 0 x2 k k g k k k ±( ) ±( ) 2 calculated c1 x and c2 x using equations Eqs. (47) and − −∗  −Π (k)Π (k ) . (41) (48) under the limits of x > 1. In this limit, the second term with y0 diverges, therefore, first term dominates over second Adding and subtracting Eqs. (40) and (41) we obtain, one. In this case, in the radiation dominated epoch, the two polarizations of the tensor perturbations can be written as: − −∗ − f (k) + g(k) = 3Π (k)Π (k)≈3|Π |2 (42) + + + +∗ + h (x) = c (1) j (x) f (k) − g(k) = 3Π (k)Π (k)≈3|Π |2 . (43) 1 0  90 f − g =− j (x) (x ) 2 0 log in (49) Similarly, we can write Eq. (36)as: π geff 3 −( ) = −( ) ( ) h x c1 1 j0 x δ( − ) ( ,η) = 3 +( ,η) +∗( ,η)  k k SGW k h k h k 90 f + g 2 =− j (x) (x ), − −∗  2 0 log in (50) +h (k,η)h (k ,η) π geff 3  3 + +∗  δ(k − k )HGW(k,η) =− h (k,η)h (k ,η) here x = 1inc1(1) signifies the value at the time of horizon 2 crossing. After horizon crossing, these gravitational waves −h−(k,η)h−∗(k,η) propagate without any hindrance. However, their energy and polarization stretched by the scale factor, similar to the mag- Therefore, components of the GWs evolve as netic radiation energy. The time derivative of the of the Eqs. (49) and (50)is     ± ( ,η)+ a ± ( ,η)+ 2 ±( ,η)= π Π ±( ), h k 2 h k k h k 8 G k + 90 f − g  a h (x) =− j (x) log(x ), (51) π 2 0 in (44) geff  3 − 90 f + g  h (x) =− j (x) log(x ), (52) π 2 0 in here Πij is the mean square root value of the transverse trace- geff 3 less part of the energy momentum tensor. In terms of dimen- In real space, the energy density of the gravitational waves sionless variable x = kη, the above equation reduces to is defined as

α ± ± + ± + ± = s , 1   h 2 h h (45) ρGW = h h . (53) x k2 16 π Ga2 ij ij    π ( ) ∓ ( ) 2 where s±(k,η) = 8 G f k g k and the parameters Note that a factor of a in the denominator comes from the a2 3 fact that h is the derivative with respect to conformal time. α = 1 and α = 2 indicates the radiation dominated and the In Fourier space, the energy density of the gravitational wave matter dominated epoch respectively. In the radiation dom- is given as inated epoch, the homogeneous solution of the Eq. (45)are the spherical Bessel function j0 and y0. In our case, magnetic η dk dSGW field is generated at in in the radiation dominated epoch due SGW(k) = (54) to chiral instability leading to anisotropic stress which in turn k d log k generates the gravitational waves. Thus, the general solution of Eq. (45) can be given as, with 3 dSGW(k) k + − ± ± ± = (|h |2 +|h |2). ( ) = ( ) ( ) + ( ) ( ) 4 2 6 (55) h x c1 x j0 x c2 x y0 x (46) d logk 2 M∗ a (2π) G 123 Eur. Phys. J. C (2019) 79:119 Page 7 of 9 119  With this definition, we can define power spectrum evaluated < ∼ geff numbers k kdiss 6 gT [55]. Therefore, near insta- at the time of generation as bility, the wavenumber corresponding to the length scales of Ω interest k k . In the present analysis, we have not con- d GW,s = 1 dSGW,s diss 4 dlogk (ρ , /M∗ ) dlogk sidered any dissipation in the plasma and restricted ourself c s   3 2   to the scales where there is maximally growing modes of the 16πk 90 f (k)  = j (x)log(x ) 2 . magnetic fields are available. Therefore, for k values larger ( π)6 2 π 2 2 0 in 3 2 as geff Hs than the instability scale, this analysis may not be reliable. In (56) Figs. 1 and 2, we have found that GWs peak occurs at kins.It is evident that at higher values of k i.e. at small length scales, where ρc,s is the critical density of the universe at the time of generation of GW. Once gravitational waves are produced, power increases after instability. This is related to the rise k they are decoupled from the rest of the Universe. This implies in the magnetic energy at large . The rise in the magnetic that the energy density of the gravitational waves will fall as energy is unphysical as we know that in turbulent system, a−4 and frequency redshifts as a−1. Hence, the power spec- energy accumulates at smallest scales. This effect in princi- trum at today’s epoch can be given as ple can be restricted by going to hyper diffusion scale (instead of ∇2 operator, one needs to introduce ∇4 operator) [56].     4 dΩ , dΩ , a ρ , GW 0 ≡ GW s s c s . (57) dlogk dlogk a0 ρc,0

Assuming that the Universe has expanded adiabatically 10−10 which implies that the entropy per comoving volume is con- 10−20 served leads to 10−30     1/3 a g , T s = eff 0 0 , (58) 10−40 a0 geff,s Ts 10−50 wherewehaveusedgeff for the effective degrees of free- 10−60 dom that contribute to the entropy density also. This is due T =108GeV to the fact that effective degrees of freedom that contribute T =106GeV T =104GeV to the energy and entropy densities are same at very high temperature. Therefore, Eq. (57), using Eq. (58) reads as 10−6 10−5 10−4 10−3       4/3 4 2 dΩGW,0 geff,0 T0 Hs dΩGW,s = Fig. 1 Gravitational wave spectrum as a function of x = kη.Wehave dlogk g , T H dlogk = −6 eff s  s  0  fixed n 10 and varied temperature π 3 2 4/3 = 16 k × 90 geff,0 6 2 2 3(2π) a π geff geff,s   s 4   T0 f (k)  2 10−10 j (x)log(xin) (59) T H 2 0 s 0 10−20 In Figs. 1 and 2, we have shown the variation of GW spec- 10−30 trum with respect to k for different temperature at fix number density and for different number density at fix temperature. 10−40

10−50

4 Results and Discussion 10−60 T =108GeV n =10−6 Before discussing the results obtained, we would like to n =10−4 explain various important length scales useful for the mag- 10−6 10−5 10−4 10−3 netohydrodynamic discussion of the generation of GWs due to the chiral instability in presence of the external mag- = η netic fields. Firstly, magnetic modes grow exponentially for Fig. 2 Gravitational wave spectrum as a function of x k at differ- ent number densities at temperature T∼ 108 GeV. For a large number = = ξ (B)/ ≈ 2( μ ) k kins 2 g T T [25,26]. Secondly, dissipa- density, the effects at large length scale is more than at the small length tion due to the finite resistivity of the plasma works at wave scale 123 119 Page 8 of 9 Eur. Phys. J. C (2019) 79:119

We would like to emphasize, in Figs. 1 and 2 that variable situation can arise in the case of Quark-Gluon Plasma (QGP) x = kη is a dimensionless quantity. In order to interpret at T  100 MeV where quarks are not confined and interact the results, we convert x in frequency of the GW. Since, with gluons which may result in instabilities. Thus, GW can = η = k η = 2πνη ν x ak T T . From this we can get in terms of be produced in QGP as well. ν = T To conclude, the study of relic GWs can open the door to x as 2πηx. Moreover, peak of the power spectrum of the GW occurs when growth of the instability is maximum which explore energy scales beyond our current accessibility and ν ≈ 16 δ2 δ give insight into exotic physics. is given by max 9π σ/T T [25,55]. Here is defined as δ = α(μR − μL )/T . The red-shifted value of the frequency T Acknowledgements We would like to acknowledge Late Prof. P. K. ν = 0 ν ∗ can be obtained using the relation: 0 T∗ max, where T is Kaw for his insight and motivation towards the problem. We also thank the temperature at which instability occurs. Hence, we can Abhishek Atreya for discussions. obtain the frequency at which maximum power is transferred Data Availability Statement This manuscript has no associated data or from the magnetic field to GWs. The obtained formula of the data will not be deposited. [Authors’ comment: This is a theoretical 9 2 the frequency in simplified form is ν ≈ 10 δ Hz, where study and no external data associated with the manuscript.] 11 we have used T0 = 2.73 K ≈ 10 Hz in our units and 6 Open Access This article is distributed under the terms of the Creative σ/T = 100. For temperature T ∼ 10 GeV, (μR −μL )/T ∼ −3 δ2 = −10 α ≈ −2 Commons Attribution 4.0 International License (http://creativecomm 10 [23] and thus, 10 (with 10 ). Hence ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, frequency where maximum power of GW occurs, is around and reproduction in any medium, provided you give appropriate credit 10−1 Hz. Thus they may be detected in eLISA experiment to the original author(s) and the source, provide a link to the Creative [57]. Further, the strength of magnetic field changes when Commons license, and indicate if changes were made. Funded by SCOAP3. chiral charge density n change. Figure 2 shows the effect of n on GW spectrum. It is apparent that the kins is not affected by the number density and hence, the peak does not shift. However, the power in a particular k mode enhances with an References increase in n. This happens due to the fact that for a larger value of n, magnetic field strength is higher at larger k [26]. 1. M. Maggiore, in Sense of Beauty in Physics, 2006 (2006) 2. A. 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