CMB Power Spectrum Explorer

Interactive comparison of RFT vs ΛCDM predictions against Planck 2018 observations

What the CMB tells us

The cosmic microwave background (CMB) power spectrum encodes precise information about the universe's composition, geometry, and early evolution. The angular power spectrum \(C_\ell\) represents temperature and polarization fluctuations as a function of multipole moment \(\ell\), with peaks and valleys that reveal fundamental cosmological parameters. Here we compare RFT's scalaron-driven background evolution against the Planck 2018 measurements.

RFT Parameters

Baryon density parameter
Cold dark matter density
Scalar spectral index
Optical depth to reionization
Amplitude of scalar fluctuations
Non-minimal scalaron coupling
χ² = --

Inputs from RFT background

The CMB spectrum calculation uses the modified Hubble parameter \(H(z)\) and scalaron field evolution from RFT's background cosmology. The key modification comes from the running gravitational coupling \(G_{\text{eff}}(z) = G/(1 + 2\alpha\phi^2/M_P^2)\) which affects the growth of perturbations and the acoustic oscillation pattern in the early universe.

Current parameter values:
  • H₀ = 67.4 km/s/Mpc
  • Ω_m = 0.315
  • Age = 13.8 Gyr
  • Sound horizon = 147 Mpc

Goodness-of-fit vs Planck

The χ² value automatically updates as you adjust parameters, showing how well RFT predictions match Planck 2018 observations. Values χ²/ν ≲ 1.2 indicate excellent agreement, while χ²/ν > 2 suggests significant tension with observations.

What to tweak

Try adjusting the α (RFT coupling) parameter to see how the non-minimal scalaron coupling affects the CMB spectrum. Larger values of |α| modify the effective gravitational strength during recombination, shifting peak positions and heights. The other parameters follow standard ΛCDM interpretations but with RFT's modified background evolution.

Advanced: Parameter values can also be imported from the Λ derivation widget using browser localStorage.

How the Boltzmann code was modified for scalaron mass running

The CMB spectrum calculation incorporates RFT modifications through several key changes to the standard Boltzmann equation solver:

1. Modified Background Evolution

The Hubble parameter includes scalaron energy density:

\[ H^2(z) = H_0^2 \left[ \Omega_m (1+z)^3 + \Omega_r (1+z)^4 + \Omega_\Lambda + \Omega_\phi(z) \right] \]

2. Running Gravitational Coupling

The effective Newton's constant evolves with the scalaron field:

\[ G_{\text{eff}}(z) = \frac{G}{1 + 2\alpha\langle\phi^2\rangle(z)/M_P^2} \]

3. Modified Sound Speed

The photon-baryon sound speed is affected by the varying gravitational strength:

\[ c_s^2(z) = \frac{1}{3(1 + R_b(z))} \quad \text{where} \quad R_b(z) = \frac{3\rho_b}{4\rho_\gamma} \]

These modifications are implemented as hooks in the Boltzmann solver, preserving the standard ΛCDM limit when α → 0 while capturing RFT's unique signatures in the acoustic oscillation pattern.