Cosmic String Loop Gravitational Wave Power Calculator
Calculate gravitational-wave losses from a cosmic string loop
This cosmic string loop calculator estimates the energy budget of an oscillating loop under the standard gravitational-radiation loss model. Supply its dimensionless tension, physical length, and gravitational-wave emission coefficient to obtain the loop's mass per unit length, stored energy, radiated power, and characteristic decay time.
Cosmic strings are hypothetical one-dimensional defects that may have formed in the early universe. An oscillating closed loop can develop cusps and kinks and lose energy through gravitational waves. The calculation is an idealized instantaneous estimate, not a prediction that any particular loop exists or that its radiation would be observable at Earth.
Cosmic-string tensions are commonly reported as Gμ/c², rather than as μ in kilograms per meter. The form therefore accepts that dimensionless quantity and converts it internally to μ. This lets the input match the notation widely used in cosmology while retaining SI units for energy, power, and time.
Cosmic string loop inputs and their physical roles
Dimensionless tension (Gμ/c²) sets the string's mass-energy per unit length. Here μ is mass per unit length, G is Newton's gravitational constant, and c is the speed of light. Raising this input increases stored energy and increases gravitational-wave power more sharply: at fixed Γ, power is proportional to the square of the dimensionless tension, whereas lifetime is inversely proportional to it.
Loop length (m) is the total closed-string length at the instant being described. A longer cosmic string loop contains proportionally more energy. In this model, length does not enter the total gravitational-wave power formula, but it does enter the decay time, so longer loops last longer when tension and Γ are unchanged.
Emission efficiency Γ is the dimensionless coefficient used to summarize gravitational-wave emission by the loop. It packages details of the loop's oscillation and shape into one parameter. Increasing Γ raises emitted power linearly and shortens the energy-loss time by the same factor.
These inputs describe a deliberately narrow model of cosmic string loop evolution. The page does not calculate a loop network population, a burst waveform, redshift effects, a detector strain, or particle-emission losses. Its purpose is to make the direct connection between tension, length, gravitational-wave power, and lifetime easy to inspect.
Equations for cosmic string loop energy, power, and lifetime
The cosmic string tension input is represented by x, the dimensionless ratio
For a chosen value of x, the calculator recovers the mass per unit length as
The stored energy of a cosmic string loop with total length L is
Its gravitational-wave power in the model used by this calculator is
The corresponding cosmic string loop decay time is energy divided by gravitational-wave power:
The lifetime expression makes the parameter trends explicit. Multiplying loop length by ten multiplies the lifetime by ten. Multiplying tension or Γ by ten divides the lifetime by ten. These directional checks are useful when comparing runs, especially because cosmic string quantities often span many orders of magnitude.
Substituting μ = x c² / G into the power relation gives the proportionality P ∝ Γx², with c⁵/G as the fixed dimensional factor. This is why changing length affects the energy and lifetime but not the instantaneous power shown by this simplified model.
Reading the scaling of a cosmic string loop calculation
A cosmic string loop result is most informative when the four outputs are read together. Tension controls both the energy density and the radiation strength, Γ controls the efficiency of the energy loss, and length sets the available energy reservoir. A loop can therefore have immense energy yet a brief lifetime if its gravitational-wave power is comparably immense.
For a quick qualitative check, change only one input at a time. Increasing Gμ/c² should increase μ, E, and P while decreasing τ. Increasing length should increase E and τ while leaving P unchanged. Increasing Γ should increase P and decrease τ without changing μ or E. If a run does not follow those relationships, verify that the intended quantities and units were entered.
The most consequential entry to double-check is the tension field. It requires Gμ/c², not μ itself and not Gμ in a different convention. The length must be in meters, while Γ has no units. Keeping those definitions straight matters because the output changes rapidly with tension.
Worked cosmic string loop estimate with the default inputs
With the displayed defaults, Gμ/c² = 1×10-7, L = 1000 m, and Γ = 50, the conversion gives approximately μ ≈ 1.35×1020 kg/m. This reflects the exceptionally large mass-energy per length associated with the chosen hypothetical string tension.
For that 1000-meter loop, the stored energy is approximately E ≈ 1.21×1040 J, while the gravitational-wave power is approximately P ≈ 1.81×1040 W. The resulting energy-over-power timescale is approximately τ ≈ 6.67×10-1 s, or 2.11×10-8 years.
This short lifetime is consistent with the calculation rather than a contradiction of the huge energy value: the assumed loop radiates at an equally huge power. The compact form τ = L/(Γxc) gives the same check directly. For the default values, 1000 divided by 50 × 10-7 × 299,792,458 is about 0.67 seconds.
This example is a model calculation, not an observational claim. Different tensions, loop sizes, or emission efficiencies produce substantially different values, and physical cosmic string scenarios may include effects that this compact treatment leaves out.
Interpreting cosmic string loop power and decay outputs
The cosmic string loop result panel lists μ in kilograms per meter, loop energy E in joules, gravitational-wave power P in watts, and lifetime τ in seconds and years. Scientific notation is used because these quantities can be far outside familiar terrestrial scales.
Start by checking the scaling before focusing on individual digits. A one-decade increase in tension should move power by roughly two decades, while a one-decade increase in loop length should move lifetime by one decade. This approach makes it easier to distinguish a real physical scaling from an accidental unit or transcription error.
The outputs are intrinsic properties of the modeled loop. They are not a prediction of gravitational-wave flux, characteristic strain, frequency content, burst rate, or detector signal. Connecting loop power to an observation requires distance, cosmological evolution, emission spectrum, and detector assumptions that are outside this calculator.
Assumptions behind the cosmic string gravitational-wave model
This cosmic string loop estimate assumes gravitational radiation is the relevant energy-loss channel. That is a common starting point for simple loop calculations, but other scenarios can include particle emission or effects associated with small-scale structure. If such channels matter, the actual decay time need not equal the value calculated here.
The model also uses a single constant Γ. In more detailed treatments, gravitational-wave emission depends on loop shape, mode content, cusps, kinks, and related dynamics. Γ is useful for controlled comparisons, provided the same convention is used throughout a calculation.
The calculator treats one isolated loop at one instant. It does not model network formation, intercommutation, fragmentation, cosmological redshift, or the frequency-dependent gravitational-wave spectrum. Those questions require additional physical assumptions beyond a three-input energy-loss estimate.
Within those limits, the equations remain valuable for building intuition. Higher tension makes a loop more powerful but shorter-lived; a larger Γ has the same qualitative tradeoff; and a longer loop supplies more energy and survives longer without altering the model's instantaneous power.
Using the cosmic string loop calculator carefully
For cosmic string parameter comparisons, vary one quantity at a time from a baseline case. First change tension and observe the squared response of power and inverse response of lifetime. Then restore it and vary loop length, followed by Γ. This isolates the physical role of each parameter.
When transferring values from papers or notes, inspect the definition next to every symbol. Sources may use μ, Gμ, or Gμ/c²; this calculator specifically expects Gμ/c². Confirm that length is expressed in meters and that Γ is dimensionless before comparing numerical outputs.
Use the displayed results alongside the equations rather than independently of them. Asking why a loop became brighter, dimmer, longer-lived, or shorter-lived is the quickest way to turn the calculation into a useful check of cosmic string gravitational-wave scaling.
Cosmic string loop parameter-tuning mini-game
This optional canvas challenge turns cosmic string loop scaling into a quick tuning exercise. Adjust tension, length, and Γ until the modeled power and lifetime fall inside the target windows. Higher tension and Γ raise power, whereas a longer loop supports a longer lifetime.
The game is separate from the calculator result above. It uses the same qualitative scaling laws, but it does not modify the calculator's math or inputs.
Takeaway: in the calculator, power rises roughly with tension squared, while lifetime grows with loop length and falls when either tension or Γ increases.