Bumblebee cosmology: The FLRW solution and the CMB temperature anisotropy

TL;DR

The paper tests whether dark energy can be replaced by a massive vector field in the bumblebee gravity framework. It derives an FLRW background with nonminimal couplings to curvature and implements a dedicated CMB code to compute the temperature power spectrum using scalar perturbations and line-of-sight integration, including careful initialization for the early universe. A key contribution is the explicit background solution with an expansion rate ${\cal H}$ determined by the bumblebee parameters, plus a complete scalar perturbation framework and a publicly released code that reproduces $\Lambda$CDM-like expansion but fails to match the Planck $C_\ell$ at low multipoles due to a suppressed metric potential $\Psi$. The findings place strong constraints on action-based vector-tensor gravity as a dark-energy substitute, while the released code provides a transparent tool for evaluating other modified gravity scenarios via CMB observables.

Abstract

We put into test the idea of replacing dark energy by a vector field against the cosmic microwave background (CMB) observation using the simplest vector-tensor theory, where a massive vector field couples to the Ricci scalar and the Ricci tensor quadratically. First, a remarkable Friedmann-Lemaître-Robertson-Walker (FLRW) metric solution that is completely independent of the matter-energy compositions of the universe is found. Second, based on the FLRW solution as well as the perturbation equations, a numerical code calculating the CMB temperature power spectrum is built. We find that though the FLRW solution can mimic the evolution of the universe in the standard $Λ$CDM model, the calculated CMB temperature power spectrum shows unavoidable discrepancies from the CMB power spectrum measurements.

Figures (8)

• Figure 1: Prior parameter space (the shaded region). The blue lines $\xi_2/\tilde{V}_1 = 1$ and $\xi_2/\tilde{V}_1=0$ correspond to $q_0=0$ and $\alpha=0$, while the dashed black line $\xi_1+2\xi_2=0$ corresponds to $q_0 \rightarrow \infty$ and $\alpha \rightarrow \infty$. The red line is the contour for the age of the universe to be $0.68\, H_0^{-1}$, which is about $9.5 \times 10^9$ years with $H_0 \approx 70\, {\rm km/s/Mpc}$. The dots are 5 representative sets of parameters to use for demonstrating the CMB results later (colors consistent with those in Figs. \ref{['fig2']} and \ref{['fig4']}). The star is the best-fit result for the spatially flat bumblebee model in Ref. tempxu.
• Figure 2: ${\cal H}$ vs. $\ln{a}$. The vertical line at $\ln{a}=-7$ marks the epoch of recombination. The parameters of the bumblebee solutions correspond to the dots in Fig. \ref{['fig1']}. The standard $\Lambda$CDM solution in GR has parameters $H_0=67.4\, {\rm km/s/Mpc}$, $\Omega_{m0}=0.315$, and $T_{\rm CMB}=2.726\, K$Planck:2018vyg.
• Figure 3: Representative results of the CMB temperature power spectrum calculated in the bumblebee theory. Upper panel: Comparing the results when changing the parameter $q_0$. Lower panel: Comparing the results when changing the parameter $\alpha$. The bumblebee cosmological model has the conventional parameters $H_0=70\, {\rm km/s/Mpc}$, $\Omega_{b0}=0.05$, $\Omega_{c0}=0$, $T_{\rm CMB}=2.7\, K$ and $n_s=1$. The standard $\Lambda$CDM result, which is calculated using the parameters $H_0=67.4\, {\rm km/s/Mpc}$, $\Omega_{b0}=0.0493$, $\Omega_{c0}=0.2657$, $T_{\rm CMB}=2.726\, K$, and $n_s=0.965$Planck:2018vyg, is plotted in both panels for comparison. We have normalized the bumblebee results to match the standard $\Lambda$CDM result at $l=200$.
• Figure 4: Examples of $\Theta_l$ calculated using Eq. (\ref{['losint']}) in the bumblebee cosmological model compared with the standard $\Lambda$CDM results. The parameters for the bumblebee cosmological model are set to be $H_0=70\, {\rm km/s/Mpc}$, $\Omega_{b0}=0.05$, $\Omega_{c0}=0$, $T_{\rm CMB}=2.7\, K$, $n_s=1$, $q_0=-0.5$, and $\alpha=-1.1$. The parameters of the standard $\Lambda$CDM model are $H_0=67.4\, {\rm km/s/Mpc}$, $\Omega_{b0}=0.0493$, $\Omega_{c0}=0.2657$, $T_{\rm CMB}=2.726\, K$, and $n_s=0.965$Planck:2018vyg.
• Figure 5: Typical numerical solutions for the representative perturbation variables $\Psi, \ \Theta_0$, and $v_b$. Four Fourier modes with different values of $k$ are shown. The vertical line at $\ln{a}=-7$ marks the epoch of recombination. The parameters for the bumblebee cosmological model are set to be $H_0=70\, {\rm km/s/Mpc}$, $\Omega_{b0}=0.05$, $\Omega_{c0}=0$, $T_{\rm CMB}=2.7\, K$, $n_s=1$, $q_0=-0.5$, and $\alpha=-1.1$. The parameters of the standard $\Lambda$CDM model are $H_0=67.4\, {\rm km/s/Mpc}$, $\Omega_{b0}=0.0493$, $\Omega_{c0}=0.2657$, $T_{\rm CMB}=2.726\, K$, and $n_s=0.965$Planck:2018vyg. Note that the $10^{-5}$ factor is only for $\Psi$ in the bumblebee model.