# Chiral plasma instability and primordial gravitational waves

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## Abstract

It is known that cosmic magnetic field, if present, can generate anisotropic stress in the plasma and hence, can act as a source of gravitational waves. These cosmic magnetic fields can be generated at very high temperature, much above electroweak scale, due to the gravitational anomaly in presence of the chiral asymmetry. The chiral asymmetry leads to instability in the plasma which ultimately leads to the generation of magnetic fields. In this article, we discuss the generation of gravitational waves, during the period of instability, in the chiral plasma sourced by the magnetic field created due to the gravitational anomaly. We have shown that such gravitational wave will have a unique spectrum. Moreover, depending on the temperature of the universe at the time of its generation, such gravitational waves can have wide range of frequencies. We also estimate the amplitude and frequency of the gravitational waves and delineate the possibility of its detection by the future experiments like eLISA.

## 1 Introduction

Gravitational wave (GW) once generated, propagates almost unhindered through the space-time. This property makes GW a very powerful probe of the source which produces it as well as the medium through which it propagates (see [1, 2, 3] and references therein). From the cosmological point of view, the most interesting gravitational radiation is that of the stochastic gravitational wave (SGW) background. Such gravitational radiations are produced by events in the early stages of the Universe and hence, may decipher the physics of those epochs. Several attempts have already been made in this regard and various sources of SGW have been considered. List of SGW source includes quantum fluctuations during inflation [4, 5, 6, 7], bubble wall collision during phase transition [8, 9, 10, 11, 12, 13], cosmological magnetic fields [14, 15, 16, 17] and turbulence in the plasma [16, 18, 19].

We have already mentioned that the seed magnetic field can be generated even in absence of net chiral charge but due to gravitational anomaly [26] and the magnetic field thus generated can produce instability in the plasma. These magnetic fields contribute a anisotropic stress to the energy-momentum tensor and hence can act as a source for the generation of the GWs. The underlying physics of GW generation is completely different from previously considered scenarios. Therefore, it is important to investigate the generation and evolution of GW in this context. In this article, we compute the metric tensor perturbation due to the chiral magnetic field. Since chiral magnetic field, which sources the tensor perturbations, has a unique spectrum, the GWs generated is expected to have a unique signature in its spectrum as well. Moreover, we compute the amplitude and frequency of the GW and show its dependences on the model parameters. Consequently, any detection of SGW in future measurements like eLISA will constrain or rule out such theoretical constructs.

*B*, \(\mu \) and

*k*represents the physical magnetic fields, chemical potential, and the wave number respectively. It is clear from the convention used here that all the comoving quantities are dimensionless. In terms of these comoving variables, the evolution equations of fluid and electromagnetic fields are form invariant [29, 30, 31]. Therefore, we will work with the above defined comoving quantities and omit the subscript “c” in our further discussion.

## 2 Gravitational anomaly and magnetic fields in the early universe

*i*’ stands for each species of the chiral plasma. The constants

*C*and

*D*are related to those of the chiral anomaly and mixed gauge-gravitational anomaly and are given as \(C=\pm 1/4\pi ^2\) and \(D=\pm 1/12\) for right and left-handed chiral particles respectively. The variables

*n*, \(\rho \) and

*p*are respectively the number density, energy density and pressure density.

*i.e.*\(\mathbf {\nabla }\cdot \mathbf {v} = 0\) and taking curl of Eq. (6) along with the expression for current from Eq. (10) we obtain,

*e.g.*magnetic field, in the orthonormal helicity basis, \(\varepsilon ^{\pm }_i\), defined as

*k*, i.e. \(|\mathbf{k}|=k\). From Eq. (14), it is clear that, when first term is dominant over the second term, magnetic modes will grow exponentially with time as

*k*at which magnetic modes have maximum growth rate. The exponential growth of the magnetic modes is true only in the linear regime. In this regime, chiral chemical potential remain constant and magnetic fields are generated at the cost of chiral imbalance. However, when generated magnetic field is sufficiently large, the non-linear effects become prominent. In this case, we need to consider the evolution of the chemical potential [20, 23], which is given by Eq. (1).

*m*is a positive integer and \(\mathrm{v}_i\), \(k_i\) are arbitrary function encoding the boundary conditions. We also showed that the scaling law allows more power in the magnetic field. For \(n = 0\), \(E_B\propto k^7\) at larger length scale whereas for \(n\ne 0\), \(E_B\propto k^5\) instead of \(k^7\) [23]. However, in the both scenarios \(E_B\) is more than that of the case without considering the scaling symmetry [23]. This aspect is important for this analysis as more power in the helical magnetic field can generate larger anisotropic stress.

*S*(

*k*) is the symmetric and \({\mathscr {A}}(k)\) is the helical part of the magnetic field power spectrum. \(P_{ij} = \delta _{ij} - {\hat{k}}_i\,{\hat{k}}_j\) is the transverse plane projector which satisfies: \(P_{ij}{\hat{k}}_j = 0 \), \(P_{ij}P_{jk} = P_{ik}\) with \({\hat{k}}_i = k_i/k\) and \(\epsilon _{ijk}\) is the totally antisymmetric tensor. Using Eq. (16) and the reality condition \(B^{\pm *}({\varvec{k}})=-B({\varvec{k}})^{\pm }\), one can show that

### 2.1 Anisotropic stress

*f*(

*k*) and

*g*(

*k*) are defined as follows:

*f*(

*k*) and

*g*(

*k*) also depends on time because of the time dependence of the magnetic fields. The integral form of these two functions are

## 3 Gravitational waves from chiral magnetic fields

*k*for different temperature at fix number density and for different number density at fix temperature.

## 4 Results and Discussion

Before discussing the results obtained, we would like to explain various important length scales useful for the magnetohydrodynamic discussion of the generation of GWs due to the chiral instability in presence of the external magnetic fields. Firstly, magnetic modes grow exponentially for \(k= k_{\mathrm{ins}}= \xi ^{(B)}/2 \approx g^2 (\frac{\mu }{T})T\) [25, 26]. Secondly, dissipation due to the finite resistivity of the plasma works at wave numbers \(k< k_{\mathrm{diss}}\sim \sqrt{\frac{g_{\mathrm{eff}}}{6}}gT\) [55]. Therefore, near instability, the wavenumber corresponding to the length scales of interest \(k \ll k_{\mathrm{diss}}\). In the present analysis, we have not considered any dissipation in the plasma and restricted ourself to the scales where there is maximally growing modes of the magnetic fields are available. Therefore, for *k* values larger than the instability scale, this analysis may not be reliable. In Figs. 1 and 2, we have found that GWs peak occurs at \(k_{\mathrm{ins}}\). It is evident that at higher values of *k* i.e. at small length scales, power increases after instability. This is related to the rise in the magnetic energy at large *k*. The rise in the magnetic energy is unphysical as we know that in turbulent system, energy accumulates at smallest scales. This effect in principle can be restricted by going to hyper diffusion scale (instead of \(\nabla ^2\) operator, one needs to introduce \(\nabla ^4\) operator) [56].

We would like to emphasize, in Figs. 1 and 2 that variable \(x=k\eta \) is a dimensionless quantity. In order to interpret the results, we convert *x* in frequency of the GW. Since, \(x=a\, k\eta = \frac{k}{T}\eta ~= \frac{2\pi \nu }{T}\eta \). From this we can get \(\nu \) in terms of *x* as \(\nu = \frac{T}{2\pi \eta } x\). Moreover, peak of the power spectrum of the GW occurs when growth of the instability is maximum which is given by \(\nu _{\mathrm{max}}\approx \frac{16}{9\pi }\left( \frac{\delta ^2}{\sigma /T}\right) T\) [25, 55]. Here \(\delta \) is defined as \(\delta = \alpha ( \mu _R-\mu _L) /T\). The red-shifted value of the frequency can be obtained using the relation: \(\nu _0=\frac{T_0}{T_*}\nu _{\mathrm{max}}\), where \(T_*\) is the temperature at which instability occurs. Hence, we can obtain the frequency at which maximum power is transferred from the magnetic field to GWs. The obtained formula of the frequency in simplified form is \(\nu \approx 10^{9} \, \delta ^2\) Hz, where we have used \(T_0=2.73\) K \(\approx 10^{11}\) Hz in our units and \(\sigma /T=100\). For temperature \(T\sim 10^6\) GeV, \(( \mu _R-\mu _L)/T \sim 10^{-3}\) [23] and thus, \(\delta ^2= 10^{-10}\) (with \(\alpha \approx 10^{-2}\)). Hence frequency where maximum power of GW occurs, is around \(10^{-1}\) Hz. Thus they may be detected in eLISA experiment [57]. Further, the strength of magnetic field changes when chiral charge density *n* change. Figure 2 shows the effect of *n* on GW spectrum. It is apparent that the \(k_{\mathrm{ins}}\) 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 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].

## 5 Conclusion

In the present work, we have extended our earlier works on the generation of primordial magnetic fields in a chiral plasma [25, 26] to the generation of GWs. This kind of source may exist much above electroweak scale. We have shown that the gravitational anomaly generates the seed magnetic field which evolves and create instability in the system. This instability acts as a source of anisotropic stress which leads to the production of gravitational waves. The production and evolution of the magnetic field has been studied using Eq. (11). In order to obtain the velocity profile, we have used scaling properties [48, 49] rather than solving the Navier-Stokes equation. This scaling property results in more power in the magnetic field at smaller *k* as compared to that of the case without scaling symmetries (see [26]). We have calculated power spectrum of the produced GWs and shown that the spectrum has a distinct peak at \(k_{\mathrm{ins}}\) and hence correspond to the dominant frequency of GW. The GW generated at high temperature \(T \ge 10^{6}\) GeV via aforementioned method is potentially detectable in eLISA.

In this work, we have considered massless electrons much above electroweak scale and discussed the production of gravitational waves due to chiral instabilities in presence of Abelian fields belonging to \(U(1)_Y\) group. However, a similar situation can arise in the case of Quark-Gluon Plasma (QGP) at \(T\gtrsim 100\) MeV where quarks are not confined and interact with gluons which may result in instabilities. Thus, GW can be produced in QGP as well.

To conclude, the study of relic GWs can open the door to explore energy scales beyond our current accessibility and give insight into exotic physics.

## Notes

### Acknowledgements

We would like to acknowledge Late Prof. P. K. Kaw for his insight and motivation towards the problem. We also thank Abhishek Atreya for discussions.

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