Cosmic String Loop Gravitational Wave Power Calculator

Estimate how fast a cosmic string loop radiates away its energy

Cosmic strings are hypothetical one-dimensional defects that may have formed in the early universe. If they exist, loops of string would oscillate, develop cusps and kinks, and radiate energy. In many standard treatments, the dominant energy loss channel is gravitational radiation. That is the specific question this calculator addresses: given a loop tension, a loop length, and an emission efficiency factor, how much power would the loop emit as gravitational waves, and how long would it take the loop to decay?

This is a calculator about scale as much as it is about arithmetic. The numbers involved in cosmic string work are usually extreme. A modest-looking change in the dimensionless tension can move the power by orders of magnitude, while a larger loop can survive much longer even if its instantaneous power is unchanged. Because of that, it helps to read the result not as a single absolute prediction for nature, but as a compact way to compare scenarios, test intuition, and understand the tradeoff between loop size and emission strength.

The page uses the dimensionless quantity Gμ/c² for tension because that is how cosmic string constraints are commonly quoted in cosmology and gravitational-wave literature. You do not need to enter the string mass per unit length μ directly. Instead, you enter the dimensionless combination, which the calculator converts internally into μ in kilograms per meter. That keeps the inputs closer to the language used in papers and lecture notes.

What each input means in physical terms

Dimensionless tension (Gμ/c²) is the most important driver in the model. Here, μ is the string mass per unit length, G is Newton's gravitational constant, and c is the speed of light. Because the combination is dimensionless, it is easy to compare across models. Larger values mean a heavier, more energetic string per unit length. In the formulas below, tension affects both the loop's stored energy and its radiation rate, but it enters the power more strongly than the lifetime. If you increase this value by a factor of ten, the gravitational-wave power rises by a factor of one hundred, while the lifetime shrinks by a factor of ten.

Loop length (m) is the total length of the closed cosmic string loop in meters. In this simplified model, the loop energy is proportional to length, so longer loops contain more energy. Length also controls lifetime very directly: if two loops have the same tension and the same Γ value, the longer loop lives longer because there is more string to radiate away. Unlike tension, loop length does not appear in the standard total power formula used here, so changing length alters the stored energy and the decay time without changing the instantaneous power output.

Emission efficiency Γ is a dimensionless coefficient that summarizes how effectively the loop emits gravitational waves. In many rough estimates, Γ is taken to be of order 50, though different loop shapes and modeling choices can shift it. A larger Γ increases power linearly and reduces lifetime linearly. In other words, if you double Γ while keeping everything else fixed, the loop emits twice as much gravitational-wave power and decays in half the time.

Those three inputs capture a clean idealized picture: a loop with fixed length at the moment you examine it, a given tension, and a gravitational-wave loss rate summarized by Γ. The calculator is intentionally simple. It does not try to model every feature of loop evolution, detector response, network statistics, redshift, or non-gravitational emission channels. Its value is clarity: you can see exactly which assumption moves which result.

Formulas used by the calculator

The starting point is the dimensionless tension variable, which we can call x. It is defined by the familiar combination of constants and string tension:

From that, the physical mass per unit length is recovered as

The total energy stored in a loop of length L is then

For the gravitational-wave loss model used here, the power is

and the decay timescale is simply energy divided by power:

That last expression is especially useful for intuition. It shows immediately that lifetime grows with loop length and shrinks with both Γ and dimensionless tension. It also explains why the result panel can be sanity-checked very quickly. If you multiply length by ten, the lifetime should become ten times larger. If you multiply tension by ten, the lifetime should become ten times smaller. If your result does not move in that direction, the most likely issue is an input misunderstanding rather than a coding problem.

The power relation hides another useful simplification. If you substitute μ = x c² / G into the power formula, then power scales as P ∝ Γx², with the constants collecting into a fixed factor c⁵/G. That means length does not control the instantaneous power in this simple model. Length matters because it determines how much total energy sits in the loop to begin with. Tension matters more sharply because it changes the string energy density and the emission strength at the same time.

How this page fits the general idea of a calculator

Even though this page is about a very specific piece of gravitational physics, it still follows the same broad structure as many scientific calculators: inputs go in, a mathematical model maps them to outputs, and the results are shown in units that let you compare scenarios. The next two MathML blocks express that generic idea. They are not the main cosmic-string formulas, but they are still useful as an abstract summary of how calculators turn variables into results.

On this page, the specific function f is the gravitational-wave model above, and the weights and combinations are set by fundamental constants and by Γ. The point is not that cosmic string physics is merely a weighted sum. Rather, it is that a transparent calculator lets you inspect the mapping from input to output and understand which parameter is dominating your result.

Worked example using the default values

Suppose you keep the default entries: Gμ/c² = 1×10^-7, L = 1000 m, and Γ = 50. The calculator first converts the dimensionless tension into a mass per unit length. With the constants built in, that gives roughly μ ≈ 1.35×10²⁰ kg/m. That number is enormous by everyday standards, which is one reason cosmic strings are such exotic objects in theoretical cosmology.

Next, it computes the total loop energy. Multiplying μ by the loop length and by c² gives about E ≈ 1.21×10⁴⁰ J. Then the gravitational-wave power comes out to roughly P ≈ 1.81×10⁴⁰ W. Finally, dividing energy by power yields a lifetime of about τ ≈ 6.67×10^-1 s, which is about 2.11×10^-8 years.

That combination may look surprising at first: how can the loop contain such huge energy and still live for less than a second? The answer is that the power is also huge. In this simplified estimate, strong tension and efficient emission make the loop radiate energy away at an extraordinary rate. This is why both outputs belong together. Power alone can sound dramatic but incomplete. Lifetime tells you whether the source stays around long enough to matter as a persistent object, while energy tells you how much was available to radiate in total.

A useful quick check is the compact lifetime form τ = L/(Γxc). Plugging in the same numbers gives 1000 divided by 50 × 10^-7 × 299,792,458, which is again about 0.67 seconds. If your manual estimate and the calculator disagree wildly, recheck the tension entry first. The most common mistake is to enter μ itself where the form expects the dimensionless quantity Gμ/c².

How to interpret the result panel

The result panel reports four values: μ in kilograms per meter, total loop energy E in joules, gravitational-wave power P in watts, and the lifetime τ in both seconds and years. Because the scales are huge, the page uses scientific notation. That is not cosmetic. It is the most readable way to compare scenarios without losing track of orders of magnitude.

When you interpret the output, start with direction rather than exactness. Ask whether the power increases when you raise tension or Γ, and whether the lifetime increases when you raise length. If those trends match your expectation, the model is behaving correctly. Then look at the order of magnitude. For example, if you change tension by one decade and power fails to move by roughly two decades, that would be a sign to check the input values or units.

Remember too that this calculator estimates intrinsic loop quantities, not detector strain or event rate. A gigantic power output does not automatically imply an easily observable signal at Earth. Detectability depends on distance, redshift, frequency spectrum, burst structure, and the response of whatever gravitational-wave experiment you care about. This page answers a narrower but still important question: given a loop with specified properties, what are its basic energy-loss scales?

Assumptions and limits of the model

The calculator assumes that gravitational radiation is the main channel by which the loop loses energy. That is a standard toy-model choice and often a good first estimate, but some cosmic string scenarios allow additional particle emission or different small-scale structure effects. If those channels are important, the real lifetime could differ from the estimate shown here.

It also treats Γ as a single constant. In reality, Γ summarizes loop shape, mode structure, cusp and kink activity, and other details. Using one fixed coefficient is useful for comparison work, but it hides the underlying complexity. If you are reading a paper with a different Γ convention, use the paper's value consistently when you compare your hand calculations with the page output.

The model further assumes a single isolated loop at a given instant. It does not include the cosmological history of a loop network, redshift evolution, intercommutation probabilities, fragmentation, or the frequency-dependent spectrum of emitted gravitational waves. It also does not attempt to convert the power into an observed flux or characteristic strain. Those topics require additional assumptions and often a much more elaborate pipeline than a one-page calculator should impose.

Still, these simplifications are a strength when you use the tool for education or scenario testing. Because the equations are short and explicit, you can see the tradeoffs immediately. Longer loops live longer. Higher tension makes loops brighter but shorter-lived. Higher Γ also makes loops brighter but shorter-lived. Those are exactly the relationships a quick calculator should make intuitive.

Practical ways to use this calculator well

If you are comparing theoretical cases, change one variable at a time. Start with a baseline tension, length, and Γ. Then raise only the tension and note the new power and lifetime. Next restore the original tension and alter only the length. This one-change-at-a-time approach makes the scaling behavior obvious and prevents one big input shift from hiding what another input did.

If you are using numbers from literature, check the symbol definitions carefully. Some sources discuss μ directly, others quote Gμ, and many quote the dimensionless ratio Gμ/c². This page expects the dimensionless ratio. Likewise, make sure the loop length is in meters and Γ is dimensionless. Small transcription errors can produce huge output differences because the model spans such a large dynamic range.

Finally, use the calculator as a conversation partner with the formulas, not as a replacement for them. Run a case, then ask why the answer changed. That habit is especially valuable in cosmology and gravitational-wave work, where the meaning of a result often matters more than its raw numerical value.