More accurate gravitational wave backgrounds from cosmic strings

This paper presents a general procedure for calculating gravitational wave backgrounds from cosmic string loops with evolving shapes due to gravitational backreaction, revealing that such backgrounds are 3% to 30% lower than prior predictions and providing updated comparisons for current and future detectors.

The Gist

Imagine the universe is filled with invisible, ultra-thin "fishing lines" called cosmic strings. These aren't made of nylon; they are defects in the fabric of space-time itself, left over from the very beginning of time. As these strings wiggle and vibrate, they create ripples in space-time known as gravitational waves.

For a long time, scientists have been trying to predict exactly what the "sound" of these waves would look like if we could listen to them with our detectors. This background hum is called the Gravitational Wave Background (GWB).

This paper is like a major software update for the scientists' prediction models. Here is the story of what they found, explained simply:

1. The Old Way: The "Smooth Rock" Model

Previously, cosmic strings were modeled like smooth, perfect rubber bands — including in earlier work by two of this paper's own authors (Olum and Blanco-Pillado). The assumption was that as a string vibrates and loses energy, it does so at a steady, predictable pace. You could picture it like this: the string starts big, shrinks a little bit, and eventually disappears, radiating waves the whole time.

That earlier approach used a simplified math trick — a "toy model" — that smoothed out all the rough edges. It was a good first pass, but it was like trying to predict the sound of a guitar by only looking at a smooth, featureless cylinder instead of the actual instrument with its strings and frets. This paper is the authors refining their own earlier work with a much more detailed simulation.

2. The New Way: The "Rough, Wiggly String" Reality

The authors of this paper decided to stop guessing and start simulating. They used supercomputers to watch cosmic strings evolve in a virtual universe, paying close attention to gravitational backreaction.

What is backreaction?
Think of a cosmic string as a person running on a treadmill.

• The Old View: The person runs at a steady speed, burning calories at a constant rate until they stop.
• The New View: The act of running changes the runner. As they run, they get tired, their form changes, and they might trip over their own feet. The energy they burn isn't constant; it spikes when they stumble and slows down when they recover.

In the case of cosmic strings, as they vibrate, they emit gravitational waves. This emission actually changes the shape of the string itself. The string gets "smoother" over time, losing its tiny, jagged kinks.

3. The Big Discovery: They Die Faster

The computer simulations revealed a surprising truth: Cosmic strings don't live as long as we thought.

Because these strings are "wiggly" and rough when they are young, they radiate energy (gravitational waves) much more aggressively at the beginning of their lives. They burn through their energy quickly.

• The Analogy: Imagine a campfire. The old model thought the fire burned steadily for an hour. The new model shows that the fire has a huge, roaring burst of flames in the first 10 minutes, burning through its fuel much faster, and then it dies out sooner than expected.

4. What This Means for the "Sound"

Because the strings die faster, there are fewer of them around at any given time to make noise.

• The Result: The predicted "hum" of the universe (the Gravitational Wave Background) is quieter than previously thought.
• The Magnitude: Depending on the frequency, the signal is about 3% to 30% quieter than the old models predicted.

5. Why Should We Care?

You might ask, "If it's just a little quieter, does it matter?"

Yes, for two reasons:

1. The Hunt for the Signal: We have detectors like LIGO (on Earth), LISA (a future space telescope), and Pulsar Timing Arrays (listening to spinning stars). These detectors are incredibly sensitive. If we are hunting for a whisper, knowing exactly how loud that whisper should be is crucial. If we thought the whisper was a shout, we might have missed it or misinterpreted the data. Now that we have a more accurate "volume knob," we can tune our detectors better.
2. Proving New Physics: If we do detect this background hum, it will confirm that cosmic strings exist. This would be a massive breakthrough, proving theories about the Big Bang and even String Theory. But to be sure we found it, our predictions must be precise. This paper gives us that precision.

Summary

The authors took a complex, messy, real-world simulation of cosmic strings and realized that the old, simplified math was slightly wrong. The strings are "rougher" and "shorter-lived" than we thought.

The takeaway: The universe's background hum from cosmic strings is likely a bit quieter than we hoped, but now we know exactly how quiet it is. This helps us listen more carefully for the faint, cosmic music that could unlock the secrets of the universe's birth.

Technical Summary

1. Problem Statement

Cosmic strings are topological defects predicted by various extensions of the Standard Model of particle physics and string theory. As they evolve, they form a network of long strings and loops that emit gravitational waves (GWs), creating a stochastic Gravitational Wave Background (GWB).

Previous calculations of this GWB relied on "toy models" to account for gravitational backreaction (the effect of GW emission on the string's own shape). These models typically assumed:

• The power radiated by a loop ($\Gamma$) is constant over its entire lifetime (often fixed at a canonical value of $\Gamma \approx 50$).
• Loops are "smoothed" via a Lorentzian convolution, effectively treating all loops as "middle-aged" regardless of their actual evolutionary stage.

These simplifications fail to capture the time-dependent behavior of the power spectrum, particularly how small-scale structure (kinks) is shed early in a loop's life, leading to inaccuracies in predicted GWB amplitudes and spectral shapes.

2. Methodology

The authors developed a new, rigorous framework combining large-scale numerical simulations with a generalized analytical formalism:

• Numerical Backreaction Simulations:

  • Utilized a piecewise-linear model to simulate Nambu-Goto cosmic string loops interacting only via gravity.
  • Evolved a specific sub-population of 71 loops ("70%-evaporated no-majors") until they lost 70% of their initial length.
  • Tracked the evolution of the power spectrum ($P_n$) and the radiation coefficient ($\Gamma$) as a function of the loop's normalized age ($\zeta$).
  • Found that $\Gamma$ is not constant; it starts high (due to initial small-scale structure) and decreases over time, approaching an asymptotic value of $\approx 47$.

• New Analytical Formalism:

  • Derived a general procedure to calculate the GWB density ($\Omega_{gw}$) where the radiated power $P_n$ and the loop lifetime depend on the loop's age ($\zeta$).
  • Introduced the normalized loop age $\zeta = \Gamma_* G\mu (t - t') / L'$ and the evaporation fraction $\chi = 1 - L/L'$.
  • Established a relationship between $\zeta$ and $\chi$ based on the numerical data, allowing for the integration of loop populations over cosmic history.
  • Derived the loop number density $n(L, \zeta, t)$ accounting for the changing expansion rate of the universe (radiation vs. matter eras) and the non-constant $\Gamma$.

• Extrapolation:

  • Since simulations only reached 70% evaporation, the authors extrapolated the results to $\chi = 1$ (total evaporation) by assuming the asymptotic $\Gamma \approx 47$ holds for the final 30% of the loop's life.

3. Key Contributions

• Time-Dependent Power Spectrum: Replaced the assumption of constant $\Gamma$ with a dynamic model where $\Gamma$ evolves as the loop sheds small-scale structure.
• Accurate Loop Lifetime Calculation: Demonstrated that because young loops radiate more efficiently (higher $\Gamma$), they lose energy faster and have shorter lifetimes than predicted by constant-$\Gamma$ models.
• Updated GWB Prediction: Provided the first GWB calculation for cosmic strings that fully incorporates numerical backreaction results rather than phenomenological smoothing.
• Public Data Release: Made tabular data for the new GWB spectra available for tensions $G\mu \in [10^{-22}, 10^{-8}]$ and frequencies $f \in [10^{-12}, 10^5]$ Hz.

4. Key Results

• Reduced Amplitude: The new GWB predictions are lower than prior predictions by 3% to 30%, depending on the frequency and string tension.
  • The reduction is most significant (up to ~30%) at frequencies just below the spectral "bump" (associated with the radiation-to-matter transition).
  • At high frequencies (radiation-era plateau), the reduction approaches a factor of $\approx 0.87$ (consistent with the ratio of the new effective lifetime to the old one).
• Spectral Shifts: The peak of the "bump" in the GWB spectrum is shifted to slightly higher frequencies compared to previous models. This is because younger loops (which dominate the high-frequency emission) have higher $\Gamma$ and thus evolve differently than the "average" loops in toy models.
• Detector Sensitivity:
  • The reduction in amplitude implies that current upper limits on string tension ($G\mu$) derived from non-detections (e.g., by NANOGrav) are slightly weakened (i.e., the allowed range for $G\mu$ extends slightly higher).
  • However, the change is not large enough to invalidate existing bounds significantly.
  • Future detectors like LISA, DECIGO, BBO, Einstein Telescope (ET), and Cosmic Explorer (CE) will be able to distinguish between the old and new models due to the specific spectral features (steps and bumps) and the frequency-dependent reduction.

5. Significance

• Precision Cosmology: This work provides the most accurate template to date for cosmic string GWBs. As GW astronomy moves from detection to precision characterization, using incorrect templates can lead to misinterpretation of signals or missed detections.
• Model Validation: The results confirm that gravitational backreaction significantly alters the spectral shape of the GWB, invalidating the "constant $\Gamma$" approximation for high-precision studies.
• Future Prospects: The authors note that while the amplitude reduction is moderate, the specific spectral distortions mean that a detection pipeline using the old model might misidentify a signal or estimate the wrong tension. The new templates are essential for the next generation of GW observatories to correctly constrain or confirm the existence of cosmic strings and the physics of the early universe.

In conclusion, this paper establishes that cosmic string loops live shorter lives and radiate more efficiently in their youth than previously thought, leading to a systematically lower and spectrally shifted gravitational wave background.