--- title: "Read the LISA sound-wave template and its limits" description: "Read section 2.2.1 of arXiv:2403.03723v2 with two lifetime papers: what the template assumes about sound lifetime and walls, and what LISA could identify." canonical_url: "https://sandboxphysics.com/stories/phase-transition-gravitational-waves/read-the-lisa-sound-wave-template-and-its-limits/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/phase-transition-gravitational-waves/read-the-lisa-sound-wave-template-and-its-limits/" last_modified: "2026-11-12" --- [Phase-transition gravitational waves](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/) # Read the LISA sound-wave template and its limits Sandbox Physics · 2026-11-12 Open arXiv:2403.03723v2, "Gravitational waves from first-order phase transitions in LISA: reconstruction pipeline and physics interpretation" (Caprini et al., published in JCAP 10 (2024) 020). This guide reads its section 2.2.1 on sound waves, the sound-wave panel of its parameter-dependence figure, and its section 5 on reconstructing thermodynamic parameters. Two companion papers on the lifetime of the sound source, arXiv:2003.07360v4 and arXiv:2007.08537v3, show where the template is a choice rather than a result. ## Before you begin Read the chapters on wall dynamics and on the spectrum today. You should be comfortable with the symbols {{K}}, {{H\_\*R\_\*}}, {{\\xi\_w}} and {{H\_\*\\tau\_{\\rm sw}}}. ## What kind of claim each source makes All five papers used in this course are published in journals: arXiv:2403.03723 in JCAP 2024, arXiv:2003.07360 in JCAP 2020, arXiv:2007.08537 in JCAP 2021, arXiv:1910.13125 in JCAP 2020 and arXiv:1004.4187 in JCAP 2010. We read the arXiv versions named here, not the journal typesetting. None of them reports an observation: no phase-transition background has been detected. Their content is of three kinds. Fits to fluid simulations, such as the slopes and the constant $A_{\rm sw}$, are numerical results within the simulated range. The template, the lifetime rule and the LISA forecasts are model predictions. Questions the papers raise but do not settle, such as the spectrum near the Jouguet speed or after shocks form, are open problems. Our own Monte Carlo record and profile solutions are synthetic calculations made for this course. Reading: Caprini et al., Gravitational waves from first-order phase transitions in LISA: reconstruction pipeline and physics interpretation, arXiv:2403.03723v2 / JCAP 10 (2024) 020 · Ellis, Lewicki & No, Gravitational waves from first-order cosmological phase transitions: lifetime of the sound wave source, arXiv:2003.07360v4 / JCAP 07 (2020) 050 · Guo, Sinha, Vagie & White, Phase transitions in an expanding universe: stochastic gravitational waves in standard and non-standard histories, arXiv:2007.08537v3 / JCAP 01 (2021) 001 · Caprini et al., Detecting gravitational waves from cosmological phase transitions with LISA: an update, arXiv:1910.13125v2 / JCAP (2020), doi:10.1088/1475-7516/2020/03/024 ## Section 2.2.1: read each symbol before the equations The section opens with the physical picture: bulk fluid motion is mostly compressional at first, and after collisions the compression waves source gravitational waves. It then cites lattice simulations and the "Higgsless" approach, which replaces the scalar field by a prescribed, expanding true-vacuum background so that the fluid and its shocks can be solved accurately. Equations sw\_shape, sw\_amplitude and Rstar define the template. Read $\Delta_w=\xi_{\rm shell}/\max(\xi_w,c_s)$ carefully: the text says $\xi_{\rm shell}=|\xi_w-c_s|$ holds for subsonic deflagrations and detonations, "the cases we focus on in this work, as we set $\xi_w$ close to unity". A reader who plugs a hybrid wall into the same formula is outside what the authors state. The kinetic fraction is $K\simeq0.6\,\kappa\alpha/(1+\alpha)$. The factor 0.6 comes from many-bubble simulations and is called a conservative estimate compared with the 2020 review, which used $K=\kappa\alpha/(1+\alpha)$. Common misreading: comparing amplitudes from the two papers at the same $\alpha$ and $\xi_w$ without noticing that $K^2$ differs by a factor 0.36. Reading: Caprini et al., Gravitational waves from first-order phase transitions in LISA: reconstruction pipeline and physics interpretation, arXiv:2403.03723v2 / JCAP 10 (2024) 020 · Espinosa, Konstandin, No & Servant, Energy budget of cosmological first-order phase transitions, arXiv:1004.4187v1 / JCAP 1006:028 (2010) ## The sound-wave panel of the parameter figure Find the panel labelled fig:pars\_PT\_sw in the source, the second panel of the parameter-dependence figure. Its caption fixes the black reference spectrum at $K=0.1$, $H_*R_*=0.1$, $\xi_w=0.9$ and $T_*=1\,\mathrm{TeV}$; each coloured line varies one parameter, and the dashed grey line is the LISA noise. We read the caption and the equations, not the plotted pixels. From the equations you can predict what each family of lines must do, and then check it in the figure. Changing $T_*$ shifts the curve sideways without changing its height. Changing $\xi_w$ moves only the second break, through $\Delta_w$. Changing $H_*R_*$ moves both breaks and also changes the height, because both $H_*R_*$ and the lifetime enter the amplitude. Changing $K$ raises the curve as $K^2$ while the lifetime stays at 1, and more slowly once shocks form first. The Lab draws the same template and lets you make these one-parameter changes yourself; the gold curve disappears outside the domain of the template. Reading: Caprini et al., Gravitational waves from first-order phase transitions in LISA: reconstruction pipeline and physics interpretation, arXiv:2403.03723v2 / JCAP 10 (2024) 020 · Original benchmark tutorial materials (standard-library Python) ## How long the sound lasts: three versions Ellis, Lewicki and No (arXiv:2003.07360v4) estimate the shock-formation time as $\tau_{\rm sh}\sim L_f/\bar U_f$, with $\bar U_f^2\simeq\tfrac34\kappa\alpha/(1+\alpha)$. Their abstract argues that in models without a zero-temperature barrier, and in scale-invariant models, sound lasts only a fraction of a Hubble time. The 2020 LISA review reduces the power by $H_*\tau_{\rm sh}=H_*R_*/K^{1/2}$. The 2024 template uses $H_*R_*/\sqrt{K/\Gamma}$ and caps it at 1. For our benchmark, $\alpha=0.2$, $\beta/H_*=50$, $\xi_w=0.9$, these give $0.285$, $0.247$ and $0.368$. The physics is the same; the definitions of fluid speed differ. Guo, Sinha, Vagie and White (arXiv:2007.08537v3) replace the sharp cap by $\Upsilon=1-1/\sqrt{1+2H_*\tau_{\rm sh}}$ in radiation domination, which gives $0.241$. What happens to gravitational-wave production after shocks form is not settled in these papers. The 2024 template stops counting the sound and adds turbulence as a separate, uncertain contribution with its own efficiency parameter. The lifetime factor is therefore one of the largest modelling uncertainties in the amplitude. ![Two source-lifetime factors against the shock-formation time, with the three benchmarks marked](https://sandboxphysics.com/_astro/lifetime.DhHDV3uS.svg) The template factor $\min(H_*\tau_{\mathrm{sh}},1)$ (gold) and the expansion suppression factor $\Upsilon = 1 - 1/\sqrt{1+2H_*\tau_{\mathrm{sh}}}$ of Guo et al. (2021) (purple). Filled and open points place the three benchmarks on each curve. The two agree when $H_*\tau_{\mathrm{sh}} \ll 1$; they differ most at $H_*\tau_{\mathrm{sh}} = 1$, where the template factor is about $2.4$ times $\Upsilon$. [Open the full figure](https://sandboxphysics.com/_astro/lifetime.DhHDV3uS.svg) Reading: Ellis, Lewicki & No, Gravitational waves from first-order cosmological phase transitions: lifetime of the sound wave source, arXiv:2003.07360v4 / JCAP 07 (2020) 050 · Guo, Sinha, Vagie & White, Phase transitions in an expanding universe: stochastic gravitational waves in standard and non-standard histories, arXiv:2007.08537v3 / JCAP 01 (2021) 001 · Caprini et al., Detecting gravitational waves from cosmological phase transitions with LISA: an update, arXiv:1910.13125v2 / JCAP (2020), doi:10.1088/1475-7516/2020/03/024 · Caprini et al., Gravitational waves from first-order phase transitions in LISA: reconstruction pipeline and physics interpretation, arXiv:2403.03723v2 / JCAP 10 (2024) 020 ## Wall speed is an input, not an output In every formula here $\xi_w$ is a free parameter. None of the papers we read computes it from the friction between wall and plasma. The 2020 review states that its ansatz is most reliable when the wall speed is far from the Chapman–Jouguet speed: for deflagrations well below $c_s$, for detonations well above it. Near that speed the shells are thin, with thickness about $R_*|\xi_w-c_s|/c_s$, and the spectral shape fails. This matters for identifiability. If the wall speed of a real transition sits near the sound speed, the template would mismodel the second break, which is exactly the feature that carries $\xi_w$. Reading: Caprini et al., Detecting gravitational waves from cosmological phase transitions with LISA: an update, arXiv:1910.13125v2 / JCAP (2020), doi:10.1088/1475-7516/2020/03/024 · Caprini et al., Gravitational waves from first-order phase transitions in LISA: reconstruction pipeline and physics interpretation, arXiv:2403.03723v2 / JCAP 10 (2024) 020 ## Section 5: what LISA could identify Section 5 of arXiv:2403.03723v2 states the key counting problem. A double broken power law has three geometric parameters, the amplitude and two breaks, while sound waves depend on four thermodynamic ones, $K$, $H_*R_*$, $\xi_w$ and $T_*$. The inversion is therefore degenerate. The authors show the degeneracies can be lifted when more than one contribution is reconstructed, and they reconstruct geometric parameters first and convert afterwards. Read the limits stated in the abstract and introduction before quoting the forecast of better than about 10 percent accuracy. The simulated data use the same templates for signal, noise and foregrounds as the reconstruction, so template error is absent by construction. General-relativistic effects for bubbles comparable to the Hubble size are neglected, and the noise model has two parameters. The forecast holds "if certain experimental and theoretical uncertainties are solved by the time LISA flies". Checklist for the next paper in this field: which wall regimes the template covers; which definition of $K$ and of fluid speed it uses; how the lifetime is capped; whether foregrounds are included; whether the injected and fitted templates are the same; which numbers are simulation fits, which are forecasts, and which questions remain open. Reading: Caprini et al., Gravitational waves from first-order phase transitions in LISA: reconstruction pipeline and physics interpretation, arXiv:2403.03723v2 / JCAP 10 (2024) 020 · Robson, Cornish & Liu, The construction and use of LISA sensitivity curves, arXiv:1803.01944v2 / Class. Quantum Grav. (2019), doi:10.1088/1361-6382/ab1101 ## Exercises and explained answers 1. A paper quotes $h^2\Omega_{\rm peak}=10^{-10}$ for $\alpha=0.3$, $\xi_w=0.95$ using $K=\kappa\alpha/(1+\alpha)$. Roughly what would the 2024 template give for the same inputs and lifetime? Read the explanation The 2024 template multiplies $K$ by 0.6, so $K^2$ falls by 0.36. If the lifetime is capped at 1 in both, the peak falls to about $3.6\times10^{-11}$. If shocks set the lifetime, $H_*\tau_{\rm sh}\propto K^{-1/2}$ rises and partly compensates, giving a factor near $0.36/\sqrt{0.6}\approx0.46$, but the different fluid-speed definitions then also matter. The point is to check definitions before comparing numbers. 2. LISA measures $f_1$, $f_2$ and $\Omega_2$ perfectly. Which thermodynamic combination can you still not separate without extra input? Read the explanation Both breaks scale as $H_{*,0}/(H_*R_*)\propto T_*/(H_*R_*)$, so $f_1$ fixes only the ratio $T_*/(H_*R_*)$. The ratio $f_2/f_1$ fixes $\Delta_w$ and hence $\xi_w$. The amplitude then fixes $K^2(H_*\tau_{\rm sw})(H_*R_*)$. One relation is missing: you need $T_*$ from a particle-physics model, or a second source such as turbulence. ## Try the prediction ### [Change one template parameter at a time](https://sandboxphysics.com/labs/phase-transition-gravitational-waves/) 1. With the default preset, move only $\xi_w$ from 0.85 to 0.95, then only $\beta/H_*$. The Lab sliders are $\alpha$, $\beta/H_*$ and $\xi_w$, so the wall speed also changes $R_*$ and $\kappa$; still, the second break moves more than the first. 2. Set $\alpha=0.5$, $\xi_w=0.95$ and compare the two lifetime rules in the lifetime plot. ## Benchmark tutorial materials [Download the offline tutorial ZIP](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/materials/phase-transition-benchmark-v1.zip) · [Read units, versions and limits](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/materials/README.md) Standard-library Python recomputes the source lifetime, the two break frequencies, the template spectrum today and the LISA signal-to-noise ratio for three fixed benchmarks, and shows how four failure cases are rejected. It evaluates a published parametric template; it does not solve the Einstein or fluid equations. Files, sizes and SHA-256 - [README.md](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/materials/README.md) (6252 B) `296087f7e0c72f7c9cb613563064b85abfbaf3318d4b66feef069f35bb7685e6` - [recompute\_benchmark.py](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/materials/recompute_benchmark.py) (12375 B) `f2c7c8359bff16583431c8457e4c1f1cf1c995a1c5fc65b597ca2729a4ddf098` - [benchmarks.json](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/materials/benchmarks.json) (2329 B) `daf8124eedb3f23491f9964205f7d32bcd632fd12a762f63b89f832fb4258dde` - [expected.json](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/materials/expected.json) (3733 B) `1cf760ff6941739c0a13a77e9d2946d05d85c8e229c367a955130d2a934115d8` - [phase-transition-benchmark-v1.zip](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/materials/phase-transition-benchmark-v1.zip) (9851 B) `541d13d3b1b14734d368dd74d8bfe360c935c893c74dd5d2c10206f64fce4bbf` [Previous: Recompute a benchmark sound-wave spectrum](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/recompute-a-benchmark-sound-wave-spectrum/) ## Sources and calculation scope 1. [Caprini et al., Gravitational waves from first-order phase transitions in LISA: reconstruction pipeline and physics interpretation, arXiv:2403.03723v2 / JCAP 10 (2024) 020](https://arxiv.org/abs/2403.03723v2) Read in the arXiv v2 LaTeX source: abstract and introduction, section 2.2.1 (sound-wave double broken power law, amplitude, mean bubble separation and redshift factors), section 3 and appendix A (two-parameter LISA noise and response) and the opening of section 5 on reconstruction. The noise equation prints the optical-metrology corner as 2e-3 mHz; our materials use 2 mHz and report both results. Figure 2 was read from its caption only. 2. [Caprini et al., Detecting gravitational waves from cosmological phase transitions with LISA: an update, arXiv:1910.13125v2 / JCAP (2020), doi:10.1088/1475-7516/2020/03/024](https://arxiv.org/abs/1910.13125v2) Read the LaTeX source in two places: the sound-wave signal section (signal.tex, about lines 180 to 260) and the definition of the root-mean-square fluid velocity and kinetic-energy fraction (prelim.tex, lines 171 to 182). Used for the earlier shock-time definition only. 3. [Espinosa, Konstandin, No & Servant, Energy budget of cosmological first-order phase transitions, arXiv:1004.4187v1 / JCAP 1006:028 (2010)](https://arxiv.org/abs/1004.4187v1) Read the fluid equations for a single self-similar bubble, the deflagration, hybrid and detonation profiles, and the appendix-B efficiency fits with their stated precision of better than 15 percent. Our own profile solutions in the bag model check the fits independently. 4. [Ellis, Lewicki & No, Gravitational waves from first-order cosmological phase transitions: lifetime of the sound wave source, arXiv:2003.07360v4 / JCAP 07 (2020) 050](https://arxiv.org/abs/2003.07360v4) Read the LaTeX source for the shock-formation time, the mean fluid velocity written through the efficiency, and the argument that the sound-wave source switches off well before a Hubble time for strong transitions. 5. [Guo, Sinha, Vagie & White, Phase transitions in an expanding universe: stochastic gravitational waves in standard and non-standard histories, arXiv:2007.08537v3 / JCAP 01 (2021) 001](https://arxiv.org/abs/2007.08537v3) Read the main LaTeX file for the expansion suppression factor for a radiation-dominated universe. It is shown beside the template lifetime factor for comparison; the template itself does not use it. 6. [Robson, Cornish & Liu, The construction and use of LISA sensitivity curves, arXiv:1803.01944v2 / Class. Quantum Grav. (2019), doi:10.1088/1361-6382/ab1101](https://arxiv.org/abs/1803.01944v2) Located the optical-metrology and test-mass noise terms and the sensitivity formula in the LaTeX source. Used only as an independent check of the noise curve shape and of the 2 mHz corner; the paper was not read section by section. 7. [Original benchmark tutorial materials (standard-library Python)](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/materials/README.md) Our script, three fixed benchmarks, four failure cases and the expected output. It evaluates the published parametric template, the efficiency fits and the noise model; it does not solve the Einstein or fluid equations and contains no browser Engine code, paper text or LISA data. ## Keep reading Phase-transition gravitational waves · Reading path ### From bubble nucleation to the spectrum LISA could see 1. [Article **What a bubbling early universe would leave in gravitational waves**](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/what-a-bubbling-early-universe-leaves-in-gravitational-waves/) 2. [Article **How bubbles appear and fill space**](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/how-bubbles-appear-and-fill-space/) 3. [Article **How bubble walls push the plasma**](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/how-bubble-walls-push-the-plasma/) 4. [Article **From the sound lifetime to the spectrum today**](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/from-sound-lifetime-to-the-spectrum-today/) 5. [Article **Recompute a benchmark sound-wave spectrum**](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/recompute-a-benchmark-sound-wave-spectrum/) 6. [You are here **Read the LISA sound-wave template and its limits**](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/read-the-lisa-sound-wave-template-and-its-limits/)