Circular polarization of gravitational waves from early-Universe helical turbulence

Tina Kahniashvili [email protected] McWilliams Center for Cosmology and Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA Faculty of Natural Sciences and Medicine, Ilia State University, 0194 Tbilisi, Georgia Abastumani Astrophysical Observatory, Tbilisi, GE-0179, Georgia Department of Physics, Laurentian University, Sudbury, ON P3E 2C, Canada Axel Brandenburg [email protected] Nordita, KTH Royal Institute of Technology and Stockholm University, 10691 Stockholm, Sweden Department of Astronomy, AlbaNova University Center, Stockholm University, 10691 Stockholm, Sweden Faculty of Natural Sciences and Medicine, Ilia State University, 0194 Tbilisi, Georgia McWilliams Center for Cosmology and Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA Grigol Gogoberidze Grigol˙[email protected] Faculty of Natural Sciences and Medicine, Ilia State University, 0194 Tbilisi, Georgia Sayan Mandal [email protected] Physics and Astronomy Department, Stony Brook University, Stony Brook, New York 11794, USA Faculty of Natural Sciences and Medicine, Ilia State University, 0194 Tbilisi, Georgia Alberto Roper Pol [email protected] Université de Paris, CNRS, Astrophysique et Cosmologie, Paris, F-75013, France Faculty of Natural Sciences and Medicine, Ilia State University, 0194 Tbilisi, Georgia

Abstract

We perform direct numerical simulations to compute the net circular polarization of gravitational waves from helical (chiral) turbulent sources in the early Universe for a variety of initial conditions, including driven (stationary) and decaying turbulence. We investigate the resulting gravitational wave signal assuming different turbulent geneses such as magnetically or kinetically driven cases. Under realistic physical conditions in the early Universe we compute numerically the wave number-dependent polarization degree of the gravitational waves. We find that the spectral polarization degree strongly depends on the initial conditions. The peak of the spectral polarization degree occurs at twice the typical wavenumber of the source, as expected, and for fully helical decaying turbulence, it reaches its maximum of nearly 100% only at the peak. We determine the temporal evolution of the turbulent sources as well as the resulting gravitational waves, showing that the dominant contribution to their spectral energy density happens shortly after the activation of the source. Only through an artificially prolonged decay of the turbulence can further increase of the gravitational wave amplitude be achieved. We estimate the detection prospects for the net polarization, arguing that its detection contains clean information (including the generation mechanisms, time, and strength) about the sources of possible parity violations in the early Universe.

pacs:

98.70.Vc, 98.80.-k

^†^†preprint: NORDITA-2020-102

I Introduction

A remarkable possible source of stochastic gravitational waves (GWs) from the early Universe is turbulence in the primordial plasma induced either from cosmological first-order (electroweak or QCD) phase transitions Witten:1984rs ; Hogan:1986qda ; Kamionkowski:1993fg , or from the primordial magnetic fields that are coupled to the cosmological plasma Brandenburg:1996fc ; Christensson:2000sp ; Kahniashvili:2010gp ; Brandenburg:2017rnt . The GW signal is potentially detectable by the Laser Interferometer Space Antenna (LISA) in the case of strong enough turbulent motions present at the electroweak scale (assuming that the total energy in the turbulence is up to 1–10% of the total thermal energy at the moment of generation) Kosowsky:2001xp ; Caprini:2006jb ; Gogoberidze:2007an ; Kahniashvili:2008pe ; Caprini:2009yp . Since GWs propagate almost freely from the moment of generation until today¹¹1We discard the GW damping due to neutrino free streaming Durrer:1997ta ; Weinberg:2003ur or from anisotropic stresses Deryagin:1986qq ., the detection of GWs sourced by primordial turbulence will open a new window to understand physical processes in the very early stages of the evolution of the Universe (at the femtoseconds timescale); see Ref. Caprini:2019egz and references therein. Moreover, several theoretical extensions of the standard model (SM) of particle physics and cosmology (which is insufficient to explain the matter-antimatter asymmetry in the Universe) imply parity symmetry²²2 The baryon asymmetry of the Universe can be explained through spontaneous lepton number symmetry breaking at a cosmological phase transition Cohen:1990it ; Cohen:1990py . violation at the electroweak energy scale being possibly manifested through helical (chiral) turbulent motions and/or magnetic fields Long:2013tha ; Dorsch:2016nrg . As expected, such parity-violating turbulent sources will produce circularly polarized GWs Kahniashvili:2005qi ; Kahniashvili:2008er ; Kisslinger:2015hua ; Alexander:2018fjp ; Anand:2018mgf ; Niksa:2018ofa ; Ellis:2020uid analogously to the GWs produced via Chern-Simons coupling Alexander:2004us ; Lyth:2005jf . Furthermore, chiral inflationary GWs might be responsible (through the gravitational anomaly) for gravitational leptogenesis which, in turn, manifests itself in the neutrino sector by being successful in either the Dirac or Majorana neutrino mass scenario Adshead:2017znw . Moreover, the gravitational anomaly effect and, correspondingly, a successful electroweak baryogenesis scenario appear viable in the case when the GWs are generated through helical magnetic fields Abedi:2018top . Additionally, the primordial magnetic fields that are responsible for the circularly polarized GWs and, correspondingly, for the baryon asymmetry, can serve as seeds for large-scale magnetic fields in the Universe; see Ref. Vachaspati:2020blt for a review and references therein. Under this consideration the scheme is as follows: helical magnetic fields act as seeds for large-scale magnetic fields, produce circularly polarized GWs, and enable successful lepto- and baryogenesis through the gravitational anomaly. Obviously, if detected, the GW polarization can be a clean measure of the deviations from the SM and will provide us with a better understanding of the nature of parity symmetry and its violation. In fact, the consideration of the different forms of “driving” is dictated by different possible sources: for scenarios in which the magnetic field is the primary (dominant) source of turbulence (e.g., inflationary or phase transition-generated magnetic fields) we deal with “magnetically driven” turbulence, while turbulence with primary fluid motions (e.g., sound waves) is “kinetically driven”, and could then result in magnetic field generation through dynamo action. This can lead to a growth of the magnetic energy by orders of magnitude even though the kinetic energy of the turbulence decays Brandenburg:2017rnt . However, this is not considered in the present paper.

Each of these sources might be parity symmetry violating ones. As we show below, the polarization spectra are different for different forms of driving, and thus the measurement of these spectra can lead to an identification of the source and the energy-scale at which parity violation happened. Interestingly, our results may help us understand the origin of the cosmic seed magnetic field, i.e., it may help discriminate between astrophysical and cosmological magnetogenesis scenarios. Perhaps most importantly, the investigation of the GW background in combination with its polarization might convincingly establish that observations demand a cosmological magnetic field that cannot be generated by a mechanism operating within the confines of the SM.

The detection of circular polarization of the stochastic GW background is a challenging task Seto:2006hf ; Smith:2016jqs ; Masui:2017fzw , and the planar interferometers cannot measure net polarization in the case of isotropic backgrounds Seto:2007tn ; Seto:2008sr . However, the dipolar anisotropy induced by our proper motion with respect to the cosmic reference frame makes it possible to measure the net circular polarization of the stochastic GW background Seto:2006dz ; Crowder:2012ik , and recently it has been shown that the net polarization of GWs could be detected with a signal-to-noise ratio of order one by LISA if the strength of the signal achieves $h_{0}^{2}\Omega_{\rm GW}\sim 10^{-11}$ (with $\Omega_{\rm GW}$ the fraction between the GW energy density and the critical density today ${\mathcal{E}}_{\rm cr}=3H_{0}^{2}/(8\pi G)$ , with $H_{0}=100h_{0}\,{\rm km}\,{\rm s}^{-1}{\rm Mpc}^{-1}$ the Hubble parameter today and $G$ is the gravitational constant) Domcke:2019zls . These findings make it extremely important to properly compute all characteristics (such as the amplitude, the spectral shape, and the polarization degree) of the GW signal from primordial helical (chiral) sources.

When computing the GW signal from early-Universe turbulent sources, previous studies (with the exception of Ref. Pol:2019yex ) assumed stationary hydrodynamic turbulence and a turbulence duration set by a fraction of the Hubble time at the moment of generation ( $H_{\star}^{-1}$ ), making it possible to use the simplified GW equation with a discarded term that describes the expansion of the Universe ( $\sim H=a^{-1}{da/dt_{\rm ph}}$ , where $t_{\rm ph}$ denotes the physical time, and $a$ is the scale factor). Moreover, the magnetic field and primordial plasma coupling, and correspondingly, the turbulence decay have been neglected in those analytic studies, making it impossible to study the GW source dynamics and the temporal dependence of the GW amplitude and spectral characteristics beyond the dilution due to the expansion of the Universe. These shortcomings will be avoided by numerically simulating nonstationary turbulence with all terms included, allowing for a full coupling between magnetic fields and plasma motions. We apply hydrodynamic and magnetic forcing terms that have a realistic representation of the time dependent generation of turbulence.

Refer to caption — Figure 1: Evolution of magnetic energy (left) and growth of GW energy density (right) for simulations where the driving is turned off at $t=1.1$ (black dotted line), or the strength of the driving is reduced linearly in time over the duration $\tau=0.2$ (green), $0.5$ (blue), $1$ (red), or $2$ (black). Time is in units of the Hubble time at the moment of source activation. The magnetic and GW energy densities are in units of the radiation energy density.

II Numerical Modeling

Recently we have performed for the first time direct numerical simulations of magnetohydrodynamic (MHD) turbulence in the early Universe accounting for the expansion of the Universe using the Pencil Code PC , and numerically computed the resulting stochastic GW background and relic magnetic fields Pol:2019yex . To scale out expansion effects, we use appropriately scaled comoving variables and conformal time. The full set of MHD equations is then similar to the usual MHD equations (Brandenburg:1996fc, ), and the GW equation is written for the scaled strains and the comoving total (magnetic and kinetic) traceless-transverse stress tensor $T_{ij}^{\rm TT}$ . These simulations allowed us to conclude that the proper inclusion of coupling effects results in the extension of the GW spectrum at lower frequencies due to the power transfer at large scales, until the causal horizon determined by the comoving Hubble frequency, $f_{H}=1.65\cdot 10^{-5}{\rm Hz}\,(g_{\star}/100)^{1/6}(T_{\star}/100{\rm GeV})$ with $g_{\star}$ and $T_{\star}$ being the relativistic degrees of freedom and the temperature at the moment of the GW source activation. In terms of the GW energy density per logarithmic frequency interval as a fraction of critical density today, $h_{0}^{2}\Omega_{\rm GW}(f)$ , the spectrum in the frequency range $f_{\rm H}\leq f\leq f_{\rm S}$ (where the typical frequency of the source is $f_{\rm S}=2Nf_{\rm H}$ ³³3The factor “2” is due to the quadratic nature of the turbulent source., and $N$ determines the number of turbulent eddies per linear Hubble length scale)⁴⁴4The parameter $N$ is determined by the physical stirring wave number of the source, $k_{0}^{\rm phys}$ , via $H_{\star}/N=2\pi/k_{0}^{\rm phys}$ , and to the comoving peak wave number through $k_{0}=k_{0}^{\rm phys}(a_{\star}/a_{0})$ . Here $a_{\star}$ and $a_{0}$ are scale factors corresponding to the moment of generation and today, and $a_{\star}/a_{0}=8.0\times 10^{-16}(100/g_{\star})^{1/3}(100\,{\rm GeV}/T_{\star})$ . is $\propto f$ , as opposed to $f^{3}$ for the causal low-frequency ( $f<f_{H}$ ) tail obtained analytically Kahniashvili:2008pe , while in terms of the comoving dimensionless strain amplitude $h_{c}$ , conventionally written as $h_{c}(f)=1.263\cdot 10^{-18}(f/1{\rm\,Hz})^{-1}\left[h_{0}\Omega_{\rm GW}(f)\right]^{1/2}$ Romano:2016dpx , we observe the scaling $h_{c}\propto f^{-1/2}$ in the frequency range $f_{H}<f<f_{S}$ . At this point we distinguish three parts of $\Omega_{\rm GW}(f)$ : the low-frequency region below the causal horizon $f<f_{H}$ , the intermediate region $f_{H}<f<f_{S}$ , and the high-frequency region $f>f_{S}$ . However, due to computational limitations, we are unable to reproduce the entire spectrum in our numerical simulations. At high frequencies, $f>f_{\rm S}$ , numerical simulations agree well with the analytical estimate. In particular, for Kolmogorov turbulence with a slope $-5/3$ , the high frequency tail $h_{c}(f)$ ( $\Omega_{\rm GW}(f)$ ) scales as $\propto f^{-7/3}$ ( $\propto f^{-8/3}$ ); see Ref. Pol:2019yex for more details. Earlier work Niksa:2018ofa showed that the high-frequency scaling of the GW spectrum depends strongly on the assumptions about the turbulence and the modeling of the time-decorrelation function. Since the spectrum of the stochastic GW background determines the detectability of GWs in a given experiment, realistic turbulence simulations are essential for establishing the sensitivity of upcoming GW experiments to early-Universe physics Romano:2016dpx .

III Results

The spectral form of the GW spectrum at low frequencies is independent of the initial conditions and the turbulence model while the amplitude of the signal strongly depends on the model chosen. The universal form of the GW spectrum does not allow us to discriminate between helical and nonhelical sources and thus limits our ability to determine the parity violation in the early Universe. This leads us to the present study with its main focus on GW circular polarization estimates and the question whether the detection of polarization can help in the identification of distinct properties of the source.

Table 1: Characteristic parameters of the runs of Fig. 1.

Run ${\cal P}$ ${\cal E}_{\rm M}^{\max}$ ${\cal E}_{\rm K}^{\max}$ ${\cal E}_{\rm GW}^{\rm sat}$ ${h}_{\rm rms}^{\rm sat}$ $\lambda=0.1$ $-13.5$ $0.0367$ $0.0134$ $6.3\times 10^{-8}$ $2.5\times 10^{-6}$ $\lambda=0.2$ $-9.37$ $0.0368$ $0.0135$ $7.6\times 10^{-8}$ $2.8\times 10^{-6}$ $\lambda=0.5$ $-3.32$ $0.0372$ $0.0137$ $1.1\times 10^{-7}$ $3.4\times 10^{-6}$ $\lambda=1$ $-1.43$ $0.0378$ $0.0140$ $1.6\times 10^{-7}$ $4.1\times 10^{-6}$ $\lambda=2$ $-0.63$ $0.0381$ $0.0141$ $2.4\times 10^{-7}$ $5.1\times 10^{-6}$

III.1 Decay with Decreased Driving

We first address the temporal evolution of the GW spectrum. As it was shown in Ref. Pol:2019yex , the GW spectrum becomes stationary shortly after the driving of the source ends (i.e., when the free decay stage of the source starts), while the energy density of the source is still present. To demonstrate this, we drive magnetic fields with an electromotive force, ${\cal F}_{i}=(\delta_{ij}+\sigma\epsilon_{ijl}\hat{k}_{l}){\cal F}_{j}^{(0)}$ , consisting of plane waves that are delta-correlated in time. Here, $-1\leq\sigma\leq 1$ quantifies the fractional helicity, and ${\cal F}_{j}^{(0)}$ is a nonhelical plane-wave forcing term. Plasma motions are selfconsistently driven by the Lorentz force. Purely hydrodynamic motions are driven by a ponderomotive force analogous to ${\cal F}_{i}$ .

In Fig. 1, we show the temporal evolution of the source and the growth of the GW energy density for the driven ( $1<t<1.1$ ) and decaying stages ( $t>1.1$ ), where the driving decreases linearly for a duration $\tau=0.1$ – $2$ , although $\tau>0.5$ may be unrealistic. As in Ref. Pol:2019yex , we use $1152^{3}$ meshpoints for the runs in Fig. 1 and put $\sigma=0$ ; see Table LABEL:Tsummary_ramp for a summary relevant quantities. During the statistically stationary stage, the GW energy density growth rate is proportional to the duration of turbulence, as was estimated through analytical modeling of Ref. Gogoberidze:2007an . In reality, the driving stage is short compared to the Hubble time-scale, and consists of the few largest eddy turnover times.

In Table LABEL:Tsummary_ramp, we have quoted the values of ${\cal E}_{\rm GW}^{\rm sat}$ and ${h}_{\rm rms}^{\rm sat}$ obtained at the end of the simulation at $t=t_{\rm end}$ . We emphasize that ${\cal E}_{\rm GW}$ is the comoving GW energy density normalized by the critical energy density, which is the same as the radiation energy density during the simulation, and ${h}_{\rm rms}$ corresponds to the scaled strain. To compute the relic observable $h_{0}^{2}{\Omega}_{\rm GW}$ at the present time, we have to multiply ${\cal E}_{\rm GW}^{\rm sat}$ by a factor $(H_{\ast}/H_{0})^{2}a_{0}^{-4}$ ; see Refs. Pol:2019yex ; Pol:2018pao for details, although there we used the symbol ${\Omega}_{\rm GW}$ also for the latter. For the aforementioned fiducial parameters of $g_{*}=100$ and $T_{*}=100\,{\rm GeV}$ , this factor is $1.64\times 10^{-5}$ . The largest value of ${\cal E}_{\rm GW}^{\rm sat}$ quoted in Table LABEL:Tsummary_ramp is $2.4\times 10^{-7}$ and corresponds therefore to $h_{0}^{2}{\Omega}_{\rm GW}=4\times 10^{-12}$ . Likewise, the values of ${h}_{\rm rms}^{\rm sat}$ in Table LABEL:Tsummary_ramp have to be multiplied by $a_{0}^{-1}=8.0\times 10^{-16}$ to obtain the observable $h_{c}$ at the present time. Again, the largest value of ${h}_{\rm rms}^{\rm sat}=5.1\times 10^{-6}$ corresponds therefore to the observable $h_{c}=4\times 10^{-21}$ .

Table 2: Characteristic parameters of the runs of Fig. 1.

Run ${\cal P}$ ${\cal E}_{\rm M}^{\max}$ ${\cal E}_{\rm K}^{\max}$ ${\cal E}_{\rm GW}^{\rm sat}$ ${h}_{\rm rms}^{\rm sat}$ K0 $0.01$ $0$ $0.0080$ $4.5\times 10^{-9}$ $7.0\times 10^{-7}$ K01 $0.31$ $0$ $0.0082$ $4.4\times 10^{-9}$ $6.9\times 10^{-7}$ K03 $0.73$ $0$ $0.0085$ $4.8\times 10^{-9}$ $6.4\times 10^{-7}$ K05 $0.88$ $0$ $0.0091$ $5.3\times 10^{-9}$ $5.7\times 10^{-7}$ K1 $0.95$ $0$ $0.0114$ $6.4\times 10^{-9}$ $5.5\times 10^{-7}$ M0 $-0.01$ $0.0059$ $0.0020$ $1.6\times 10^{-9}$ $4.1\times 10^{-7}$ M01 $0.57$ $0.0078$ $0.0021$ $2.0\times 10^{-9}$ $4.6\times 10^{-7}$ M03 $0.86$ $0.0119$ $0.0024$ $3.5\times 10^{-9}$ $6.9\times 10^{-7}$ M05 $0.94$ $0.0148$ $0.0026$ $4.5\times 10^{-9}$ $8.6\times 10^{-7}$ M1 $0.97$ $0.0168$ $0.0025$ $5.3\times 10^{-9}$ $9.0\times 10^{-7}$

III.2 Approach to a Stationary State

The GW generation can be split into three cases (see Fig. 2 for kinetically and magnetically driven turbulence): non-helical, partially helical ( $|\sigma|<1$ ), and fully helical. Here, we used $512^{3}$ meshpoints and arranged the forcing amplitude such that ${\cal E}_{\rm K}$ and ${\cal E}_{\rm M}$ are around $10^{-2}$ . The viscosity and magnetic diffusivity have equal values and are chosen such that the dissipative subrange is resolved; see Table LABEL:Tsummary_pol for a summary and Ref. DATA for further details. Surprisingly, the kinetically driven turbulence is more efficient in producing GW energy; see Figs. 2(c) and (f). However, in this case the presence of kinetic helicity does not affect the source amplitude – contrary to the magnetically driven case where the amplitude of the source increases substantially with increasing fractional helicity; see Figs. 2(a) and (d). We also present the kinetic and magnetic energy dissipation rates, $\epsilon_{\rm K}$ and $\epsilon_{\rm M}$ , respectively. The dissipation rates remain almost unchanged during the stationary stage as we can expect. In addition, we see that they are almost unaffected by the presence of helicity. One may have expected a correlation between $\epsilon_{\rm K}$ (or $\epsilon_{\rm M}$ ) and ${\cal E}_{\rm GW}$ , but in the magnetically driven case, larger values of $\sigma$ produce even slightly less dissipation at early times. Nevertheless, ${\cal E}_{\rm GW}$ clearly increases with $\sigma$ .

III.3 Polarization Degree

To estimate the polarization degree, we follow the procedure described in Ref. Pol:2018pao . We use the usual circular polarization basis tensors $e^{\pm}_{ij}=-({\bf e}_{1}\pm i{\bf e}_{2})_{i}\times({\bf e}_{1}\pm i{\bf e}_{2})_{j}/\sqrt{2}$ , we decompose the Fourier transform of the GW strains $h_{ij}({\bf k})=\int d^{3}x\,e^{i{\bf k}\cdot{\bf x}}h_{ij}({\bf x})$ into two states – right- ( $h_{+}$ ) and left-handed ( $h_{-}$ ) circularly polarized GWs $h_{ij}=h_{+}e^{+}_{ij}+h_{-}e^{-}_{ij}$ . The GW circular polarization degree is given by Kahniashvili:2005qi ⁵⁵5As an alternative we can use the decomposition using the linear polarization basis tensors $e^{+}_{ij}({\bf\hat{k}})=e_{i}^{1}e_{j}^{1}-e_{i}^{2}e_{j}^{2}$ and $e^{\times}_{ij}({\bf\hat{k}})=e_{i}^{1}e_{j}^{2}+e_{i}^{2}e_{j}^{1}$ as $h_{ij}({\bf k})=h_{+}({\bf k})e_{ij}^{+}({\bf\hat{k}})+h_{\times}({\bf k})e_{ij}^{\times}({\bf\hat{k}})$ where $h_{+}$ and $h_{\times}$ are gauge independent components corresponding to two polarization modes. In this case the polarization degree will be equal to ${\mathcal{P}}(k)={\langle h^{\star}_{\times}({\mathbf{k}})h_{+}({\mathbf{k}^{\prime}})-h^{\star}_{+}({\mathbf{k}})h_{\times}({\mathbf{k}^{\prime}})\rangle}/{\langle h^{\star}_{+}({\mathbf{k}})h_{+}({\mathbf{k}^{\prime}})+h^{\star}_{\times}({\mathbf{k}})h_{\times}({\mathbf{k}^{\prime}})\rangle}$

{\mathcal{P}}(k)=\frac{\langle h^{\star}_{+}({\mathbf{k}})h_{+}({\mathbf{k}^{\prime}})-h^{\star}_{-}({\mathbf{k}})h_{-}({\mathbf{k}^{\prime}})\rangle}{\langle h^{\star}_{+}({\mathbf{k}})h_{+}({\mathbf{k}^{\prime}})+h^{\star}_{-}({\mathbf{k}})h_{-}({\mathbf{k}^{\prime}})\rangle}=\frac{{\mathcal{H}}(k)}{H(k)},~{}

(1)

where $H({k})$ and ${\mathcal{H}}({k})$ characterize the GW amplitude and polarization (chirality) defined through the Gaussian-distributed GWs wave number-space two-point function (for simplicity of notations we omit the time-dependence):

\frac{\langle h^{\star}_{ij}({\mathbf{k}})h_{lm}({\mathbf{k}^{\prime}})\rangle}{(2\pi)^{3}}=\delta^{(3)}({\bf k}-{\bf k^{\prime}})\left[{\mathcal{M}}_{ijlm}H(k)+i{\mathcal{A}}_{ijlm}{\mathcal{H}}(k)\right],

where $4{\mathcal{M}}_{ijlm}({\mathbf{\hat{k}}})\equiv P_{il}P_{jm}+P_{im}P_{jl}-P_{ij}P_{lm}$ , and $8{\mathcal{A}}_{ijlm}({\mathbf{\hat{k}}})\equiv{\hat{\bf k}}_{q}(P_{jm}\epsilon_{ilq}+P_{il}\epsilon_{jmq}+P_{im}\epsilon_{jlq}+P_{jl}\epsilon_{imq})$ are tensors, $P_{ij}({\bf\hat{k}})\equiv\delta_{ij}-{\hat{k}}_{i}{\hat{k}}_{j}$ is the projection operator, $\delta_{ij}$ and $\epsilon_{ijl}$ are the Kronecker delta and the fully antisymmetric tensor, respectively.

The GW polarization degree depends, as expected, on the fractional helicity of the source. On the other hand, the evolution of helical sources is determined by the initial fractional helicity: the coupling between the partially helical magnetic field and the plasma motions leads to a reconfiguration of the magnetic field at large scales through free decay, resulting in the growth of the fractional helicity due to the increase of the correlation length and the corresponding decrease of the magnetic energy until the fully helical stage is developed and inverse cascading starts Tevzadze:2012kk . However, for weakly helical sources, a substantial time-period is needed to reach a fully helical configuration. On the other hand, our simulations show that the dominant contribution to the GW signal occurs shortly after the source has reached its maximum: the subsequent decay of the magnetic field causes a decline of the turbulent driving of GWs. This decline is further enhanced by the expansion of the Universe, although this effect is small if the decay time of the turbulence is short compared with the Hubble time; see Fig. 1. We see that, even with a substantially extended decay phase of the turbulence of twice the Hubble time, the final GW production is enhanced by only a factor of about four. Thus, the GW polarization will retain information about the initial fractional helicity of the source.

As we have highlighted above, previous works Kahniashvili:2005qi ; Kisslinger:2015hua considered stationary turbulence with two different models for helical turbulence realization: (i) Kolmogorov-like helical turbulence with two different spectral slopes, ${-5/3}$ and $-11/3$ for the spectral energy and helicity densities, $E_{\rm M}(k)$ and $H_{\rm M}(k)$ , respectively, and (ii) helicity-transferring turbulence (if helicity transfer and small-scale helicity dissipation dominates Kraichanan:1973 ) with the spectral indices $-7/3$ and $-10/3$ for $E_{\rm M}(k)$ and $H_{\rm M}(k)$ Moiseev:1996 ⁶⁶6Note that Ref. Kahniashvili:2005qi used a different convention for the spectral indices referring to the power spectra of the symmetric $[P_{S}(k)\propto E_{\rm M}(k)/k^{2}]$ and the antisymmetric parts $[P_{A}(k)\propto H_{\rm M}(k)/k]$ .. The former case seems most suitable for describing the usual nonhelical turbulence experiencing forward cascading Borue:1997 . The difference in using these spectral shapes is determined by the effect of helicity on the energy dissipation length. For highly helical turbulence, the helicity dissipation length is larger, so the two region description Kahniashvili:2005qi might be justified using Kolmogorov-like turbulence at large wavenumbers and approximating the low wave number tail with helicity transfer turbulence Ditlevsen:2001 . Following this description, Ref. Kahniashvili:2008er and later also Ref. Ellis:2020uid discussed two stages – first the fractionally helical one and later the fully helical one with inverse transfer to describe more precisely the GW generation by helical MHD turbulence. However, these estimates suffer due to (i) the assumption of stationary turbulence; (ii) neglecting the decay and correspondingly temporal dynamical effects (the GW spectrum becomes stationary shortly after source activation).

III.4 Dependence on the Driving

We now investigate the GW polarization spectra dependence on the nature of driving. We show the polarization degree spectra in Fig. 3 for continuous pumping of kinetic or magnetic energy and helicity at intermediate scales. The parameters of these simulations are the same as those used for Fig. 2; see also Ref. DATA . We see a substantial difference between the kinetic and magnetic initial sources at the low frequency tail due to the inverse cascade for the magnetic sources that is absent in the kinetically driven case: more precisely the energy density spectrum is unchanged after the decay stage starts, while the transfer of magnetic helicity to the large scales results in the increasing of the polarization degree. These results confirm that the polarization degree is scale dependent: $\propto k^{-0.5}$ at large wavenumbers, which is shallower than the $k^{-1}$ expected for Kolmogorov-like helical turbulence with different spectral indices for the magnetic spectral energy density ( $n_{S}=-5/3$ ) and the spectral helicity density ( $n_{H}=-11/3$ ). In our simulations, the actual indices are a bit smaller, which also explains the shallower slope in the polarization degree.

III.5 Comparison of the Spectra

In Fig. 4 we show the numerator and denominator of the degree of polarization, $H(k)$ and ${\cal H}(k)$ , respectively. The underlying kinetic and magnetic energy and helicity spectra are shown in Fig. 5. The figure shows that the GW polarization is larger at larger length scales. To see whether this could be related to inverse cascading in the magnetic case, we now show in Fig. 5 the corresponding energy and helicity spectra. They show clear inverse cascading of the magnetic energy and helicity spectra in the magnetically driven case. Inverse cascading is clearly absent in the kinetic energy and helicity spectra in the kinetically driven case.

The departure from the theoretical predictions is due to the assumption of a scale-independent time-decorrelation function for magnetic energy and helicity densities. Interestingly, the spectral shape of the polarization degree is independent of the actual indices for energy and helicity, but depends on the difference between them Kahniashvili:2005qi . Obviously, in a realistic case, the proper consideration of time-decorrelation and its dependence on wave numbers is required. In fact, even in the simplified description, different forms of the time-decorrelation function for different models of turbulence (including both compressible and incompressible cases) and its scale-dependence leads to different scaling of the GW spectrum at high frequencies Niksa:2018ofa .

IV Conclusions

In this paper we present numerical simulations of the circular polarization degree of GWs generated through parity violating (helical) turbulent sources in the early Universe. We present our results for GWs generated at electroweak energy scales, but the formalism is not limited to the specific moment of GW generation, and can be adjusted to primordial turbulence sources at any time after inflation and before recombination epochs. We have confirmed that the GW signal reaches its maximal strength faster then the turbulence decays. We have also shown that the slope of the low frequency tail of the GW spectrum is independent of the nature of the turbulent source (i.e., the nature of initial driving, the presence of helicity, etc). This restricts the discrimination between helical and non-helical sources as well between kinetic and magnetic drivings if the signal will be detected. On the other hand, the polarization spectrum not only retains information about the initial characteristic frequency (as well as the GW signal does) and the strength of parity violation of the source, but also manifests the dependence in the driving mechanisms. We have shown that the previously used assumption of stationary turbulence does not predict the GW polarization spectrum for realistic turbulence (the scaling of the spectrum at low and high frequencies, the peak position, etc), and thus might result in inadequate estimates of the detection prospects. In particular, we have shown that the polarization spectra increase at low frequencies due to the inverse cascade, reflecting redistribution of the helical structures, while the energy density of the GWs sustains the scaling that was established soon after the turbulent source activation. In fact, previous works Kahniashvili:2005qi ; Ellis:2020uid predicted a completely different picture for both helical Kolmogorov turbulence and helicity-transferring turbulence. More precisely, no previous work addressed the dependence of the polarization spectra on the temporal characteristics of the source, nor was the helicity transfer-induced increase of polarization spectra at low frequencies detected. There was never a self-consistent description of GW polarization nor an attempt to disentangle the nature of the underlying source (kinetic vs magnetic driving). Based on our results, we argue that the inverse cascade time scale determines the height of the polarization spectra at low frequencies (a second peak) in the MHD case – even for low fractional helicity. The second (smaller) peak is located at lower frequencies. Fortunately, numerical simulations have now become an affordable tool to address these and other questions of relic GW generation and give a more complete picture for the detection prospects by LISA (as for electroweak phase transitions) and by PTAs (as for QCD phase transitions Caprini:2010xv ; Neronov:2020qrl ), and/or any future planned missions, including atomic interferometry Dimopoulos:2007cj .

Data availability—The source code used for the simulations of this study, the Pencil Code, is freely available from Ref. PC . The simulation setups and the corresponding data are freely available from Ref. DATA .

Acknowledgements.

We thank Arthur Kosowsky and Andrii Neronov for useful discussions. Support through the Swedish Research Council, grant 2019-04234, and Shota Rustaveli GNSF (grants FR/18-1462 and FR/19-8306) are gratefully acknowledged. We acknowledge the allocation of computing resources provided by the Swedish National Allocations Committee at the Center for Parallel Computers at the Royal Institute of Technology in Stockholm.

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