Recompute a benchmark sound-wave spectrum

What is in the package

The ZIP contains four files. inputs/benchmarks.json holds the three benchmarks, four deliberately invalid cases and a seven-row table of the largest strength with a deflagration profile, from our own bag-model solver. recompute_benchmark.py does the calculation with the Python standard library only. expected.json is the output we obtained with Python 3.9.6. README.md lists units, conventions and known limits.

Units: $\beta/H_*$ is dimensionless, the wall speed is in units of $c$, the temperature in GeV, frequencies in Hz, and every energy density is the dimensionless $h^2\Omega_{\rm GW}$. The number of relativistic degrees of freedom is fixed at $g_*=100$ and assumed equal to the entropy count.

The output is a parametric template evaluated at fixed inputs. Nothing in the package solves the Einstein equations or the fluid equations in time.

Reading: Original benchmark tutorial materials (standard-library Python) · Caprini et al., Gravitational waves from first-order phase transitions in LISA: reconstruction pipeline and physics interpretation, arXiv:2403.03723v2 / JCAP 10 (2024) 020

Run it and compare

The first command writes your result. The second compares every number with expected.json and prints MATCH when all agree to a relative tolerance of $10^{-9}$. A mismatch lists the path of each differing field, which tells you where your environment or an edit changed the calculation.

python3 recompute_benchmark.py inputs/benchmarks.json > my-output.json
python3 recompute_benchmark.py inputs/benchmarks.json --compare expected.json

Reading: Original benchmark tutorial materials (standard-library Python) · Caprini et al., Gravitational waves from first-order phase transitions in LISA: reconstruction pipeline and physics interpretation, arXiv:2403.03723v2 / JCAP 10 (2024) 020

Follow one benchmark through every step

Open the entry detonation-lisa-band: $\alpha=0.2$, $\beta/H_*=50$, $\xi_w=0.9$, $T_*=200\,\mathrm{GeV}$. The script first decides whether the wall is a deflagration, a hybrid or a detonation. The Jouguet speed is $0.828$, so this is a detonation. The efficiency fit then gives $\kappa=0.2742$, and $K=0.02742$.

Next come the bubble size $H_*R_*=0.05273$, the fluid speed $\bar v_f=0.1434$ and the shock time $H_*\tau_{\rm sh}=0.3676$. The field lifetimeLimitedBy reads shock formation. The shell thickness $\Delta_w=0.3585$ and $H_{*,0}=3.30\times10^{-5}\,\mathrm{Hz}$ give $f_1=0.1252\,\mathrm{mHz}$ and $f_2=0.8729\,\mathrm{mHz}$.

The amplitude follows: $h^2\Omega_{\rm int}=2.630\times10^{-11}$, $h^2\Omega_2=1.419\times10^{-11}$, and a numerical peak of $1.595\times10^{-11}$ at $0.677\,\mathrm{mHz}$. The peak lies between the two breaks because the middle segment still rises as $f$. Finally the four-year SNR with the A and E channels is 634.

Most common mistake: using the wall speed instead of $\max(\xi_w,c_s)$ in $R_*$. For a detonation they agree, so the error only shows in the slow-deflagration benchmark, where it would shrink $H_*R_*$ from $0.0169$ to $0.0117$.

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)

Checks that do not reuse the template

Each result carries a redshiftCheck block. It recomputes $H_{*,0}$ from entropy conservation, the Friedmann equation, the CMB temperature $2.7255\,\mathrm{K}$ and today's entropy count $3.909$, then compares it with the template constant. At 200 GeV it gives $3.2934\times10^{-5}\,\mathrm{Hz}$ against $3.3\times10^{-5}$; the dilution factor gives $1.6404\times10^{-5}$ against $1.64\times10^{-5}$.

The field expansionSuppressionUpsilon gives the factor $\Upsilon$ of Guo et al. In our tests a fourth-order Runge–Kutta integral of the source weight $(a_s/a)^3$ over radiation-dominated expansion reproduces their closed form, which checks it without trusting the printed equation. For the benchmark $\Upsilon=0.241$, so their rule would lower the amplitude by a third relative to the minimum rule.

In the repository tests, the efficiency fits are compared with our own profile solutions (largest relative difference 7.4 percent away from the Jouguet speed), and our noise curve reproduces the low-frequency ratio $20/9$ to the independent sensitivity curve of Robson, Cornish and Liu.

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 · 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

Four cases the script refuses

The failureCases list is part of the test. A hybrid wall at $\xi_w=0.7$ with $\alpha=0.2$ is refused because the shell rule $|\xi_w-c_s|$ is stated only for subsonic deflagrations and detonations. A wall at exactly the sound speed is refused for the same reason. The 2020 LISA review warns that near the Chapman–Jouguet speed the shells become thin and the spectral shape fails.

A strong, slow deflagration with $\alpha=0.5$ and $\xi_w=0.25$ is refused because no deflagration profile exists; our bound there is $\alpha=0.429$. A transition with $\beta/H_*=2.5$ is refused because the bubbles would reach the Hubble radius and the flat-space template no longer applies. If your edited script returns a spectrum for any of these, the edit has removed a physical check.

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 · Original benchmark tutorial materials (standard-library Python)

Errors you should carry with the numbers

The efficiency fit is quoted to better than 15 percent, and that error enters the amplitude squared through $K^2$. The amplitude constant $A_{\rm sw}=0.11$ is an average over simulations with different $\alpha$ and $\xi_w$; we did not find a quoted spread in the section we read. The three published shock-time definitions differ by up to 50 percent for this benchmark.

The optical-metrology noise in arXiv:2403.03723v2 is printed with a corner of $2\times10^{-3}\,\mathrm{mHz}$. We use 2 mHz, the value in the earlier LISA noise model of Robson, Cornish and Liu. The literal printed value would raise the benchmark SNR from 634 to 716, a 13 percent change. Treat the SNR as a noise-only estimate: the 2024 paper expects the astrophysical foregrounds to reduce the reach of LISA significantly.

Checklist for your next template paper: write down units and $g_*$; find which wall regimes the shell rule covers; recompute the redshift constants yourself; find the lifetime rule and its definition of fluid speed; list the cases the formula should refuse.

Reading: Espinosa, Konstandin, No & Servant, Energy budget of cosmological first-order phase transitions, arXiv:1004.4187v1 / JCAP 1006:028 (2010) · Caprini et al., Gravitational waves from first-order phase transitions in LISA: reconstruction pipeline and physics interpretation, arXiv:2403.03723v2 / JCAP 10 (2024) 020