Cosmology · Complementary probes of cosmic history

Cosmic clocks and gravitational-wave standard sirens

Compare a finite differential-age calculation with a separate inspiral-amplitude example, retaining dating, inclination and redshift assumptions.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

Back to the chapter

After reading, you can

Explain the connection between differential ages and an expansion rate.
Identify finite-bin error in a supplied clock calculation.
Show why siren amplitude needs orientation and redshift information.

Compare time differences under stated dating assumptions

A cosmic-clock method uses changes of suitable stellar-population ages with redshift to constrain elapsed cosmic time. A galaxy’s stellar age is not automatically the age of the Universe.

The relation is H(z) = −1/[(1+z)dt/dz], where t is cosmic proper time in the chosen homogeneous model. Relating stellar-age differences to dt requires population selection, dating and formation-history assumptions.

Supply a flat coasting history t(z) = T₀/(1+z), with arbitrary positive T₀, and assume the compared passive populations formed at the same time t_form = 0.1T₀. Their stellar ages are τ = t−0.1T₀.

The common formation time cancels from age differences only because it was assumed common. A formation time that drifts with redshift would change the derivative. The chosen flat coasting model is not the negatively curved Milne spacetime.

Compute the finite-bin clock bias

Choose a redshift-bin half-width h = 0.1 and estimate D_h = [τ(z+h)−τ(z−h)]/(2h). For the supplied law, D_h = −T₀/[(1+z)²−h²].

Putting this finite difference into the clock relation gives H_FD T₀ = (1+z)−h²/(1+z). The exact local derivative instead gives H_exact T₀ = 1+z.

The fractional difference is −h²/(1+z)². Thus the finite-bin calculation has a known approximation error even with perfectly specified ages and no measurement noise.

The numbers below are outputs of the selected law. They are not galaxy ages in years, measured H values or precision forecasts for a clock survey.

Chosen flat coasting law with common formation time and bin half-width h = 0.1
Bin center z	H_exact T₀	H_FD T₀	Fractional error (H_FD/H_exact)−1
0.5	1.5	1.493333	−0.004444
1	2	1.995	−0.0025
2	3	2.996667	−0.001111

Orientation and distance both affect amplitude

A gravitational-wave standard siren uses a modeled source waveform and received strain to infer luminosity distance. In an inspiral model, phase evolution constrains combinations such as the redshifted chirp mass.

For a separate leading quasicircular-inspiral example, supply a positive waveform length scale L_* and v = cos i, where i is inclination. The polarization envelopes are h_plus = L_*(1+v²)/(2D_L) and h_cross = L_*v/D_L.

These are envelopes of polarizations in phase quadrature. They are not simultaneous waveform values or the strain of a detector, which also depends on its sky and polarization response.

Define d = D_L/(10¹¹L_*). An edge-on source at d = 1 and a face-on source at d = 2 have the same scaled plus envelope, 0.5, while their cross envelopes differ. One amplitude alone therefore leaves an inclination–distance ambiguity in this example.

Separate supplied leading-inspiral envelopes; the two polarizations are in phase quadrature
d = D_L/(10¹¹L_*)	v = cos i	10¹¹h_plus	10¹¹h_cross
1	0	0.5	0
1	0.5	0.625	0.5
1	1	1	1
2	0	0.25	0
2	0.5	0.3125	0.25
2	1	0.5	0.5

Optional mathematical path: restore the length and regime

For observed cyclic gravitational-wave frequency f and redshifted chirp mass M_c,z = (1+z)M_c, take L_* = 4(GM_c,z)⁵ᐟ³(πf)²ᐟ³/c⁴. This has length units and makes both strain envelopes dimensionless.

Supply GM_c,zf/c³ = 10⁻⁵ and equal-mass symmetric mass ratio η = 1/4. The leading relative orbital-speed estimate is [π(GM_c,zf/c³)η⁻³ᐟ⁵]¹ᐟ³c, about 0.041635c.

At the smaller supplied distance, D_L/(c/f) is about 3982.537799. These chosen scales place the harmonic in a weak-inspiral far-zone regime; no actual source mass, frequency or event is measured by assigning these ratios.

The normalization uses a redshifted mass, so waveform scale does not separately determine intrinsic mass and cosmological redshift. A luminosity distance alone therefore does not provide an independent source redshift or H₀.

Additional assumptions are needed before combining probes

The clock and siren examples are separate constructions, not a joint cosmological fit. One illustrates differential dating and finite bins; the other illustrates orientation, waveform normalization and distance.

Real clock analyses need stellar-population modeling and control of formation histories. Real siren analyses need a waveform, orientation and sky response, strain calibration and a treatment of lensing and other relevant uncertainties.

Ideally resolving both polarization responses can supply information absent from one envelope, but a detector network’s actual ability to do this requires its response and noise model. The table supplies no instrumental sensitivity or forecast.

Compare probes by asking what primitive signal is measured, how physics maps it to a parameter and which calibration or modeling assumptions remain. Independent-looking quantities do not become unconditional clocks or distances merely by receiving different names.

The two panels are separate supplied constructions, not one joint cosmology or a precision forecast. Panel A assumes a spatially flat coasting history t/T₀ = 1/(1+z), arbitrary positive time T₀, common stellar-population formation time 0.1T₀ and centered redshift half-width h = 0.1. This flat history is not the negatively curved Milne spacetime. Stellar age τ = t−0.1T₀ is not cosmic age. The common offset cancels from the stipulated finite age difference D_h = [τ(z+h)−τ(z−h)]/(2h). Exact differentiation gives H_exact T₀ = 1+z, while this finite-bin calculation gives H_FD T₀ = (1+z)−0.01/(1+z). Panel A has a linear bin-center z axis from 0.5 to 2 and a linear fractional-error ordinate 100(H_FD/H_exact−1) from −0.5 to 0 percent. A dotted violet curve with diamonds follows −1/(1+z)² percent. Values at z = 0.5, 1, 2 are approximately −0.444444, −0.25, −0.111111 percent. The native table gives rows (z,H_exact T₀,H_FD T₀) = (0.5,1.5,1.493333), (1,2,1.995), (2,3,2.996667). Every sampled age is positive, but no age in years or measured H is supplied. Panel B is a separate leading quasicircular-inspiral example with positive waveform length L_* = 4(GM_c,z)^(5/3)(πf)^(2/3)/c⁴, observed cyclic frequency f and redshifted chirp mass M_c,z. Define d = D_L/(10¹¹L_*) and v = cos i. Its linear v axis runs from 0 to 1, and zero-based scaled dimensionless envelope 10¹¹h from 0 to 1.1. Solid blue circles show plus at d = 1: (1+v²)/2. Dashed orange squares show cross at d = 1: v. Dotted green diamonds show plus at d = 2: (1+v²)/4. Dash-dot violet squares show cross at d = 2: v/2. At v = 0, 0.5, 1 the four series respectively give (0.5,0.625,1), (0,0.5,1), (0.25,0.3125,0.5), (0,0.25,0.5). Same endpoints may intentionally coincide. The plus envelope for edge-on d = 1 equals that for face-on d = 2, both 0.5; the corresponding cross envelopes are 0 and 0.5. These are coefficients of polarizations in phase quadrature, not simultaneous waveform samples or detector strain. Instrument response, orientation, dating, formation histories, lensing and calibration remain required in real analyses. A distance alone supplies neither source redshift nor H₀. The source mass/frequency and all response values are supplied, not observations. Shapes, distinct line patterns, explicit scales and the clock table provide noncolor interpretation. Original Ark teaching model.

Worked reasoning

Compare two orientations and distances

Can the edge-on d = 1 and face-on d = 2 cases share the same plus envelope?

For edge-on v = 0, 10¹¹h_plus = 1/2 and 10¹¹h_cross = 0.
For face-on v = 1 at d = 2, 10¹¹h_plus = 2/(2×2) = 1/2.
Its scaled cross envelope is 1/2, different from the edge-on case.

Yes: plus amplitude alone is degenerate here, while an ideal additional polarization response can distinguish these specified cases.

Try the reasoning

At z = 1, does the finite clock calculation exactly equal H_exact T₀ = 2?
Show the answer and reasoning
No: H_FD T₀ = 1.995.
The finite half-width produces a fractional error −0.0025 under the supplied law.
Will a redshift-dependent stellar formation time necessarily cancel from differential ages?
Show the answer and reasoning
No.
Only the explicitly common formation-time offset cancels in this example.
Does a siren luminosity distance alone supply source redshift or H₀?
Show the answer and reasoning
No.
The waveform uses redshifted source parameters, and a distance-to-expansion inference needs additional information and assumptions.

What to carry forward

Differential dating constrains expansion through population and derivative assumptions.
Siren amplitude contains distance together with orientation and waveform response.
The two probes offer different information, with distinct calibration and model requirements.

References

Sources supporting the explanations, physical distinctions and examples in this lesson.