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Gravitational Radiation from Primordial Helical MHD Turbulence

We consider gravitational waves (GWs) generated by primordial inverse-cascade helical magneto-hydrodynamical (MHD) turbulence produced by bubble collisions at the electroweak phase transitions (EWPT). Compared to the unmagnetized EWPT case, the spectrum of MHD-turbulence-generated GWs peaks at lower frequency with larger amplitude and can be detected by the proposed Laser Interferometer Space Antenna (LISA).

We consider gravitational waves (GWs) generated by primordial inverse-cascade helical magnetohydrodynamical (MHD) turbulence produced by bubble collisions at the electroweak phase transitions (EWPT). Compared to the unmagnetized EWPT case, the spectrum of MHD-turbulencegenerated GWs peaks at lower frequency with larger amplitude and can be detected by the proposed Laser Interferometer Space Antenna (LISA). PACS numbers: 98.70.Vc, When detected, primordial cosmological GWs will provide a very valuable probe of the very early Universe [1]. Various mechanisms that generate such GWs have been discussed: quantum fluctuations [2]; bubble wall motion and collisions during phase transitions [3,4]; cosmic strings [5]; cosmological magnetic fields [6,7,8]; and plasma turbulence [9,10,11,12]. From the direct detection point of view, GWs generated during the EWPT are promising since their peak frequency lies in or near the LISA [13] frequency band [14], however, to produce a detectable signal the EWPT must be strong enough [15,16,17]. Currently discussed EWPT models do not predict an observable GW signal from bubble collisions [17], nor for GWs produced by unmagnetized turbulence [12].
Here we study the generation of GWs during a firstorder EWPT assuming that bubble collisions produce helical MHD turbulence. 1 In the case of unmagnetized hydrodynamical turbulence the peak frequency of the GW power spectrum is determined by the inverse turn-over time of the largest eddy and the energy-scale when the GW is generated. Recently discussed modifications of the standard EWPT model place the transition at a higher energy-scale [16]. As a result, the GW power spectrum peak frequency is shifted to higher frequency which, since the GW spectrum is sharply peaked, reduces the possibility of detection by LISA. On the other hand, in the case of MHD turbulence the presence of an energy inversecascade leads to an increase in the effective size of the largest eddy (now associated with an helical magnetic field), and can result in the GW power spectrum peaking in the LISA band, with amplitude large enough to be detected by LISA. We adapt the technique developed in Ref. [12] to study this case here. We model MHD turbulence and obtain the GW spectrum by using an analogy with the theory of sound wave production by hydrodynamical turbulence [22,23,24,25].
Since the turbulent fluctuations are stochastic, so are the generated GWs. The GW energy density is [14] ρ Here the times t ′(′′) = t − |x − x ′(′′) |, i and j are spatial indices (repeated indices are summed), the source S ij (x, t) = T ij (x, t) − δ ij T k k (x, t)/3 is the traceless part of the stress-energy tensor T ij , G is the gravitational constant, and we use natural units with = 1 = c. We assume that the turbulence exists for a time short enough to neglect the cosmological expansion during GW production. We consider metric perturbations in the farfield limit (i.e. for x ≫ d, where d is a characteristic length-scale of the source region), where GWs are the only metric perturbations [26], and replace |x − x ′ | by |x| in Eq. (1). If the turbulence is stationary then the GW spectral energy density I(x, ω) (ρ GW (x) = dωI (x, ω) where ω is the angular frequency) is [12] Here H ijij (x ′ , k, ω) (where k is a proper wavevector) is the (double traced) four-dimensional Fourier transform of the two-point time-delayed fourth-order correlation tensor, S ij (x ′ , t)S lm (x ′′ , t + τ ) /w 2 , with respect to x ′′ − x ′ and τ , where w = ρ + p is the enthalpy density and p and ρ the pressure and energy density of the plasma. We assume that primordial MHD turbulence is generated at time t ⋆ at proper length-scale l 0 = 2π/k 0 with characteristic velocity perturbation v 0 . 2 The dynamics of MHD turbulence is dominated by Alfvén waves for which the magnetic and kinetic energy densities are in approximate equipartition [27]. In this case 4πw is the characteristic magnetic field perturbation expressed in velocity units. While MHD turbulence is isotropic on large scales, it is locally anisotropic on small scales [28], resulting in small-scale anisotropy in the generated GW background. However, GWs are generated mainly by the largest eddies [12] so we adopt an isotropic turbulence model, and thus the magnetic field two-point correlation function is Here are the magnetic field energy and helicity densities. The is an autocorrelation function that determines the characteristic function f (η(k), τ ) that describes the temporal decorrelation of turbulent fluctuations. In the following we use [29]. After generation primordial turbulence freely decays. We adopt the decaying MHD turbulence model of Refs. [30,31]. For non-zero initial magnetic helicity turbulence decay is a two stage process. First decay stage dynamics is governed by a direct cascade of energy density lasting for a time τ s0 = s 0 τ 0 , several times (s 0 ∼ 3 − 5) longer then the characteristic largest-eddy turn-over time τ 0 = l 0 /v 0 = 2π/k 0 v 0 . During the first stage energy density flows from large to small scales and finally dissipates on scales ∼ l d = 2π/k d (k d ≫ k 0 ) where one of the Reynolds numbers becomes of order unity. Due to the selective decay effect [27] magnetic helicity is nearly conserved during this stage [31]. To compute the GWs generated by decaying MHD turbulence, we assume that decaying turbulence lasting for time τ s0 is equivalent to stationary turbulence lasting for time τ s0 /2. This can be justified using the Proudman [22,25] argument for (unmagnetized) hydrodynamical turbulence. Consequently, when computing the emitted GWs we ignore the time dependence of E M (k, t) and H M (k, t). We also assume small initial magnetic helicity, 2 We assume that the usual and magnetic Reynolds numbers are much greater than unity on scales ∼ l 0 , otherwise there is no turbulence. Throughout this paper the symbol ∼ represents equality to the accuracy of a dimensionless multiplicative constant of order unity.
For E M (k, t) and η(k) we use the Kolmogorov model, for k 0 < k < k d . Here ε ∼ k 0 v 3 0 is the energy dissipation rate per unit enthalpy.
At the end of the first stage turbulence relaxes to a maximally helical state, α s0 ∼ 1 [31,32]. Accounting for conservation of magnetic helicity, the characteristic velocity and magnetic field perturbations at this stage are Second stage dynamics is governed by a magnetic helicity inverse cascade. If both Reynolds numbers are large at the end of the first stage, magnetic helicity is conserved during the second stage. The magnetic eddy correlation length evolves as ξ M (t) ∼ l 0 1 + t/τ 1 [30,31] where τ 1 ∼ l 0 /v 1 = τ 0 / √ α ⋆ is the characteristic energy containing eddy turn-over time at the beginning of the second stage. The magnetic E M (t) and kinetic E K (t) energy densities evolve as [30,31] These imply that the characteristic turn-over (τ to ) and cascade (τ cas ) timescales evolve as To compute the GWs emitted during the second stage we use the stationary turbulence model that has the same GW output. Introducing the characteristic wavenumber k ξ (t) = 2π/ξ M (t) and using Eqs. (5) we find E M ∼ wv 2 1 k ξ (t)/k 0 and E K ∼ wv 2 1 [k ξ (t)/k 0 ] 2 since b 1 ∼ v 1 . The time when turbulence is present on scale ξ M (t) is determined by Eq. (6) which can be rewritten as τ cas ∼ τ 1 [k 0 /k ξ (t)] 2 . So instead of considering decaying turbulence, we consider stationary turbulence with a scale-dependent duration time (time during which the magnetic energy is present at the scale), τ s1 ∼ τ 1 [k 0 /k] 2 (for k = k ξ this coincides with τ cas ).
The expression for E M yields the time-independent Here C 1 is a constant of order unity, k S is the smallest wavenumber where the inverse cascade stops, and the second equation follows from saturating the causality condition. For the second stage autocorrelation function, which is inversely proportional to the turn-over time (6), we assume η(k) = (k/k 0 ) 2 / √ 2πτ 1 . At the largest scales there is no efficient dissipation mechanism, so the inverse cascade will be stopped at scale l S (t) = 2π/k S where either the cascade timescale τ cas reaches the expansion timescale H −1 easy to see that the first condition is fulfilled first and consequently k 0 /k S ≤ (v 0 /γ) 1/2 α 1/4 ⋆ . To have an inverse cascade requires k 0 /k S ≥ 1, leading to a constraint on initial helicity, γ ≤ M α The magnetic field perturbation stress-energy tensor For the first decay stage we compute for this magnetic part and then double the result to account for approximate magnetic and kinetic energy equipartition for Alfvén waves. During the second stage, according to Eqs. (5), kinetic energy can be neglected compared to magnetic energy. To compute H ijij (k, ω) we assume Millionshchikov quasi-normality [22] and use the convolution theorem (for details see Sec. III of Ref. [12]). Using the (k → 0) aero-acoustic approximation, which is accurate for low Mach number (M ≤ 1, (and slightly overestimates GWs amplitude for the Mach number approaching unity (M → 1) [12]), we find The integral is dominated by the contribution of large scale (k ≃ k S ) perturbations and is maximal at ω For the first-stage direct-cascade turbulence the peak frequency is ω 12]. To determine the peak frequency at the current epoch we need to account for the cosmological expansion which decreases the GW amplitude and frequency by the factor a ⋆ /a 0 , where a ⋆ and a 0 are the values of the cosmological scale factor at the GW generation and current epochs The total GW energy spectrum at a given spacetime event is obtained by integrating over all source regions with a light-like separation from that event, and includes contributions from GW generated during the first and second stages. For the first stage (with duration time τ (ω) is given by Eqs. (21) and (A3) of Ref. [12]. For the second stage contribution we must account for the scale dependence of the cascade time. The total GW fractional energy density parameter at the moment of emission Ω GW,⋆ is 105 12]. Here the index m runs over I and II for the first and second decay stages, H 0 ω ⋆ is an angular frequency at the moment of emission. The current GW amplitude is related to the current fractional energy density parameter through h C (f ) = 1.26 × 10 −18 (1 Hz/f ) h 2 0 Ω GW (f ) 1/2 (where h 0 is the current Hubble parameter H 0 in units of 100 km sec −1 Mpc −1 ) [14], and Here the linear frequency f = (a ⋆ /a 0 )f ⋆ with f ⋆ = ω ⋆ /2π, T ⋆ and g ⋆ are the temperature and effective number of relativistic (all fields) degrees of freedom at scale factor a ⋆ . Figure 1 shows h C (f ) for a few initial magnetic helicity values. GWs emitted during direct-cascade unmagnetized turbulence peak at current f (I) max ≃ M ν ⋆ [12]. We find that the MHD-inverse-cascade generated GW (current epoch) peak frequency is determined by cosmology parameters, f (II) max = H ⋆ a 0 /a ⋆ = 1.6 × 10 −5 Hz (g * /100) 1/6 (T * /100 GeV) and is independent of turbulence parameters. On the other hand, f (II) max = γf (I) max /M is shifted to lower frequency compared to the unmagnetized case. From Eq. (9), the amplitude of MHD-turbulence-generated GWs at the peak is a factor ∼ α 9/8 ⋆ γ −3/4 M 3/4 larger than that in the unmagnetized case.
When modeling turbulence we used the Biskamp and Muller model, [30,31]. If we adopt the helical MHD turbulence model of Banerjee and Jedamzik [32] (also see Refs. [33]) the GW peak frequency remains the same while the amplitude of the signal doubles. Figure 1 shows that even for small values of magnetic helicity the main contribution to the GW energy density is from the second, inverse-cascade stage. The GWs will be strongly polarized since magnetic helicity is maximal at the end of the first stage [11]. LISA should be able to detect such GW polarization [20]. Unlike the unmagnetized case due to the second (inverse-cascade) stage contribution the GW amplitude is large enough at 10 −4 Hz to be detectable by LISA. If the EWPT occurs at higher energies (T ⋆ > 100 Gev) the peak is shifted to higher frequency, closer to LISA sensitivity peak, which leads to a stronger signal. Our formalism is applicable for GW production at an earlier QCD phase transition, assuming the presence of colored magnetic fields [34], or for any other phase transitions [3]; the peak frequency will be shifted according to the changes in T ⋆ and g ⋆ . The GW signal estimated here exceeds that from bubble collisions [4,15,16] or from hydrodynamical, unmagnetized turbulence [9,10,12]. Of course, this strong signal assumes initial non-zero (although small) magnetic helicity, so detection of polarized GWs by LISA will indicate parity violation during the EWPT as proposed in Refs. [18].
We greatly appreciate useful comments from A. Gruzinov and the referees. We acknowledge helpful discussions with R. Durrer, G. Gabadadze, and A. Kosowsky. T. K. acknowledges the hospitality of the Abdus Salam International Center for Theoretical Physics. G. G. and T. K. acknowledge partial support from INTAS 061000017-9258 and Georgian NSF ST06/4-096 grants. B. R. acknowledges US DOE grant DE-FG03-99EP4103.