Tensor modes, polarisation and detection limits

Tensor perturbations add a different statistical question

Primordial tensor modes describe gravitational-wave perturbations of the geometry. An early-Universe proposal can predict their statistical spectrum in addition to the scalar fluctuations that seed much of the later density structure.

CMB polarisation supplies one route to testing a primordial tensor contribution through its spatial patterns. E and B label components of that polarisation pattern in a harmonic description. They are not the two local Stokes parameters Q and U at one pixel.

A tensor-to-scalar ratio compares appropriately defined tensor and scalar primordial power at an explicit common pivot. Its numerical interpretation requires stated spectrum conventions and a model connecting those primordial quantities to observed polarisation.

The question is therefore larger than whether an instrument records nonzero polarisation or some apparent B power. Foregrounds, lensing, calibration, noise and the observation model must be accounted for in a primordial-tensor interpretation. This lesson adopts no present numerical tensor bound or detection-status claim.

A fixed instrument error can mix pattern modes

Consider one real flat-sky Fourier test mode, with E_true = 10 thermodynamic μK and B_true = 0. These chosen amplitudes are teaching inputs, not a measured sky or an angular bandpower.

Adopt the following explicit instrument-angle error convention: E_obs = cos(2δ)E_true + sin(2δ)B_true and B_obs = −sin(2δ)E_true + cos(2δ)B_true. A positive δ therefore produces negative B_obs from the positive pure-E input.

Here δ is a constant calibration error relative to a fixed sky convention. This is different from merely rotating a correctly calibrated coordinate basis and consistently transforming the complete description. Trigonometric arguments use radians, so a degree label must be multiplied by π/180.

The example omits lensing, foregrounds, beams, noise and other modes. It isolates one calibration sensitivity. It does not compute the transfer of a primordial tensor spectrum or turn a chosen instrument offset into a tensor-to-scalar ratio.

A signed amplitude has nonnegative squared power

Define one-mode power proxies P_E = E_obs² and P_B = B_obs², in μK². The chosen pure-E input gives P_E = 100cos²(2δ) and P_B = 100sin²(2δ). Their sum remains 100 μK² under this rotation.

At δ = 1 degree, B_obs ≈ −0.348995 μK, but P_B ≈ 0.121797 μK² is positive. Squaring removes the amplitude sign; it does not erase the sign information in the cross-product E_obsB_obs.

For the stated convention, that cross-product is −50sin(4δ) μK². At one degree it is about −3.487824 μK². These are algebraic products for one supplied mode, rather than measured EB or BB angular power-spectrum estimates.

The table’s degree offsets were deliberately chosen to make the sensitivity visible. They are not actual instrument-calibration uncertainties, a probability distribution or a survey sensitivity forecast.

A rotation signature does not identify its cause alone

An apparent B contribution exists in this construction even though the true input has no B component. This is enough to demonstrate that apparent B power alone cannot establish a primordial gravitational-wave origin.

Cross-correlations can help diagnose angle calibration when their expected intrinsic values are specified. However, setting intrinsic EB or TB to zero for a calibration procedure is itself an assumption about the relevant sky and propagation signals.

A uniform physical rotation of cosmic polarisation can be degenerate with an instrument-angle error in such an account. A rotation-like signature therefore needs independent calibration information and a stated physical model before its cause is assigned.

This qualification does not assert that either effect is present in a particular dataset. The example demonstrates one dependence that an actual analysis must test alongside its other relevant sources and uncertainties.

Inference needs the full observation model

A real CMB tensor analysis relates observations to a model including signal spectra, foreground contributions, lensing, noise and covariance, with the relevant calibration and instrumental response. A likelihood and its validation determine what the data support under those assumptions.

An upper bound limits a parameter within a stated inference framework and sensitivity. It does not prove that the parameter is exactly zero. A detected contribution also needs tests of its physical interpretation rather than being identified with an origin mechanism by its label alone.

The chosen E-to-B leakage calculation supplies neither a likelihood nor an uncertainty model, sky coverage, tensor spectrum or pivot-normalized ratio. Its P_B values cannot be read as a numerical primordial tensor constraint.

When comparing a result with an early-Universe proposal, retain the statistic, pivot, spectrum convention, nuisance assumptions and distinction between bound and detection. That keeps a model test connected to what was actually inferred.

Worked reasoning

Try the reasoning

1. Can a negative B_obs give positive P_B?

   Show the answer and reasoning

   Yes.

   P_B is the squared real amplitude in this test, so the sign disappears while EB can retain it.

2. Does an EB rotation signature alone uniquely identify an instrument error?

   Show the answer and reasoning

   No.

   The diagnostic depends on assumptions about intrinsic correlations and possible physical rotation; additional calibration information can be required.

3. Does an upper bound establish exactly zero primordial tensor power?

   Show the answer and reasoning

   No.

   A bound is conditional on sensitivity, likelihood and model assumptions. This example supplies no numerical observational bound.

What to carry forward

• Tensor tests require a connection from primordial statistics to observed polarisation.
• Instrument-angle errors can generate apparent B from pure E under a stated convention.
• Amplitude, squared power, upper bounds and physical-origin claims have different meanings.