Phase-transition gravitational waves
If the early universe boiled, bubbles of a new phase grew, collided and set the hot plasma ringing. That sound can leave a faint background of gravitational waves. These six pages follow it from the bubbles to the spectrum LISA could measure, and say where the calculation stops being reliable.
What a bubbling early universe would leave in gravitational waves
If the hot early universe changed state through bubbles, their sound would leave a faint hum of gravitational waves near a millihertz today.
How bubbles appear and fill space
Derive how much old phase survives under an exponentially growing nucleation rate, and why the mean bubble spacing is set by the wall speed over beta.
How bubble walls push the plasma
Classify walls as deflagrations, hybrids or detonations, find the Jouguet speed and compute how much released energy becomes fluid motion.
From the sound lifetime to the spectrum today
Combine bubble size, fluid energy and sound lifetime into the LISA sound-wave template, then redshift its frequencies and energy density to today.
Recompute a benchmark sound-wave spectrum
Download one stdlib Python script and three fixed benchmarks, recompute the sound lifetime, spectrum today and LISA SNR, and see four refused cases.
Read the LISA sound-wave template and its limits
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.
Benchmark tutorial materials
Download the offline tutorial ZIP · Read units, versions and limits
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 (6252 B)
296087f7e0c72f7c9cb613563064b85abfbaf3318d4b66feef069f35bb7685e6 - recompute_benchmark.py (12375 B)
f2c7c8359bff16583431c8457e4c1f1cf1c995a1c5fc65b597ca2729a4ddf098 - benchmarks.json (2329 B)
daf8124eedb3f23491f9964205f7d32bcd632fd12a762f63b89f832fb4258dde - expected.json (3733 B)
1cf760ff6941739c0a13a77e9d2946d05d85c8e229c367a955130d2a934115d8 - phase-transition-benchmark-v1.zip (9851 B)
541d13d3b1b14734d368dd74d8bfe360c935c893c74dd5d2c10206f64fce4bbf
Sources and calculation scope
- Caprini et al., Gravitational waves from first-order phase transitions in LISA: reconstruction pipeline and physics interpretation, arXiv:2403.03723v2 / JCAP 10 (2024) 020
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.
- 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
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.
- Espinosa, Konstandin, No & Servant, Energy budget of cosmological first-order phase transitions, arXiv:1004.4187v1 / JCAP 1006:028 (2010)
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.
- Ellis, Lewicki & No, Gravitational waves from first-order cosmological phase transitions: lifetime of the sound wave source, arXiv:2003.07360v4 / JCAP 07 (2020) 050
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.
- 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
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.
- Robson, Cornish & Liu, The construction and use of LISA sensitivity curves, arXiv:1803.01944v2 / Class. Quantum Grav. (2019), doi:10.1088/1361-6382/ab1101
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.
- Original benchmark tutorial materials (standard-library Python)
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.