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On the detection of the tensor-to-scalar ratio r
using the CMB B-modes
Agnès Ferté
In collaboration with: Julien Peloton, Julien Grain, Radek Stompor
Based on: arxiv:2506.06409
CMB B-modes caused by tensor perturbations and lensing of E-modes
(Q,U) maps
E and B modes power spectra
Tensor-to-scalar ratio r
CMB B-modes at low-l probe the primordial Universe
Lensing B modes:
1st direct detection by
Polarbear (arxiv:1403.2369)
Primordial B modes:
r<0.12 (Bicep2/Keck + Planck)
Estimation of B modes power spectrum on a masked sky: the E-to-B leakage

Function of the mask
1 % of the sky
Standard estimation of B-modes
Mode counting variance
From Grain et al, 2009
Pure pseudo spectrum method (x2pure code)
+ optimisation of the window function
1 % of the sky
Other methods
pure
mode counting
Smith, 2006
Grain et al, 2009
Ferté et al, 2013
From Ferté et al, 2013
Pure pseudo spectrum method (x2pure code)
+ optimisation of the window function
71 % of the sky
Other methods
pure
mode counting
Smith, 2006
Grain et al, 2009
Ferté et al, 2013
Impact of leakage on the forecasts of CMB B modes detection?
• 1st investigation: ideal experiments
• 2nd investigation: 3 selected realistic experiments
Signal-to-noise ratio computed using Fisher matrix
Error on r
Covariance matrix computed using:
- Mode counting variance =
-
Pure method =
pure estimation + MC simulations
1st approach: ideal small scale experiments
1%
r=0.1
Mode-counting variance for optimal sky coverage
r = 0.1
Optimal sky fraction
2nd approach: 3 realistic experiments
Spec.
Sky fraction
Noise (muK-arcmin)
Beam (arcmin)
Small scale
Array of telescope
Satellite
1%
36%
71%
inhomogenesous
(average of 5)
1
2
8
3
8
Uncertainties on the power spectrum
r = 0.1
Accessible r at 3σ detection
Example of satellite xp
mode counting
pure
Small scale xp
Telescopes array
Satellite mission
Mode counting
r > 0.038
r > 0.0011
r > 0.0009
Pure method
r > 0.11
r > 0.0051
r > 0.0026
Foreground impact on the detection of r
Remove the first bin: 2 < l < 20
Example of satellite xp
Small scale xp
Telescopes array
Satellite mission
Mode counting
r > 0.04
r > 0.0016
r > 0.0013
Pure method
r > 0.11
r > 0.0053
r > 0.0028
TO TAKE AWAY
• The pure estimation of B modes is awesome
• The mode-counting approach is enough to state on optimal sky coverage
• The pure method is mandatory to perform realistic forecasts on r detection
• Potential future satellite mission: r ≥ 10-3
Stage 4 array of telescope: r ≥ 10-2
Current ground-based or balloon borne: r ≥ 10-1
• Side ‘personal’ note: the CMB TB and EB correlations hold information on
fundamental physics (e.g. : parity breaking, birefringence, primordial
magnetic field…)
From NASA
Effect of binning
Optimised window functions
SNR on r
Small scale
Array of telescopes