We present details of numerical simulations of the gravitational radiationproduced by a first order thermal phase transition in the early universe. Weconfirm that the dominant source of gravitational waves is sound wavesgenerated by the expanding bubbles of the low-temperature phase. We demonstratethat the sound waves have a power spectrum with a power-law form between thescales set by the average bubble separation (which sets the length scale of thefluid flow $L_\text{f}$) and the bubble wall width. The sound waves generategravitational waves whose power spectrum also has a power-law form, at a rateproportional to $L_\text{f}$ and the square of the fluid kinetic energydensity. We identify a dimensionless parameter $\tilde\Omega_\text{GW}$characterising the efficiency of this "acoustic" gravitational wave productionwhose value is $8\pi\tilde\Omega_\text{GW} \simeq 0.8 \pm 0.1$ across all oursimulations. We compare the acoustic gravitational waves with the standardprediction from the envelope approximation. Not only is the power spectrumsteeper (apart from an initial transient) but the gravitational wave energydensity is generically larger by the ratio of the Hubble time to the phasetransition duration, which can be 2 orders of magnitude or more in a typicalfirst order electroweak phase transition.
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机译：我们介绍了由早期宇宙中的一阶热相变产生的引力辐射的数值模拟的细节。我们确认引力波的主要来源是由低温相的膨胀气泡产生的声波。我们证明了声波的功率谱在平均气泡间距（确定流体流动的长度比例$ L_ \ text {f} $）和气泡壁宽度之间的比例之间具有幂律形式。声波产生引力波，其功率谱也具有幂律形式，其速率与$ L_ \ text {f} $和流体动能密度的平方成比例。我们确定了一个无量纲参数$ \ tilde \ Omega_ \ text {GW} $，表征了此“声学”重力波产生的效率，其值是$ 8 \ pi \ tilde \ Omega_ \ text {GW} \ simeq 0.8 \ pm 0.1 $我们的模拟。我们将声引力波与根据包络近似得到的标准预测值进行比较。哈勃时间与相变持续时间的比值不仅使功率谱变得更强（除了初始瞬变），而且引力波的能量密度通常也更大，在典型的一阶电弱相变中，它可以为2个数量级或更多。
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