Mapping Cosmological Observables to the Dark Kinetics

TL;DR

The paper develops a systematic framework to map the inhomogeneous kinetics of dark sectors to observable CMB and LSS features, emphasizing horizon-entry physics as the key conduit for dark perturbations to imprint on visible matter. Using a canonical perturbation formalism, it shows how anisotropic stress, effective stiffness, propagation speed, and clustering of dark components translate into distinct signatures in $\Phi$, $\Psi$, and their combinations, providing clear guidance on when CMB or LSS data are most informative. It also clarifies the relationship between dark dynamics and modified gravity, arguing that many MG effects can be mimicked by GR-coupled dark sectors, but distinct tests—such as equivalence-principle violations, superluminal flows, and gravitational-wave phenomenology—can differentiate them. The results highlight the complementary roles of CMB and LSS, the importance of horizon-entry physics, and the potential for future high-precision data (e.g., Planck/ACT) to tighten constraints on dark radiation, early quintessence, and MG scenarios, thereby advancing our understanding of cosmic acceleration and the nature of gravity.

Abstract

We study systematically which features in the cosmic microwave background (CMB) and large-scale structure (LSS) probe various inhomogeneous properties of the dark sectors (including neutrinos, dark matter, and dark energy). We stress, and quantify by simple formulas, that the primary CMB anisotropies are very susceptible to the gravitational potentials during horizon entry, less at recombination. The CMB thus allows us to scan Φ+Ψand the underlying dark kinetics for all redshifts z~1-10^5. LSS, on the other hand, responds strongest to Φat low redshifts. Dark perturbations are often parameterized by the anisotropic stress and effective sound speed (stiffness). We find that the dark anisotropic stress and stiffness influence the visible species at the correspondingly early and late stages of horizon entry, and affect stronger respectively the CMB and LSS. The CMB yet remains essential to probing the stiff perturbations of light neutrinos and dark energy, detectable only during horizon entry. The clustering of dark species and large propagation speed of their inhomogeneities also map to distinctive features in the CMB and LSS. -Any parameterization of the signatures of dark kinetics that assumes general relativity can effectively accommodate any modified gravity (MG) that retains the equivalence principle for the visible sectors. This implies that formally the nonstandard structure growth or Φ/Ψratio, while indicative, are not definitive MG signatures. The definitive signatures of MG may include the strong dependence of the apparent dark dynamics on visible species, its superluminality, and the nonstandard phenomenology of gravitational waves.

Figures (6)

• Figure 1: The evolution of CMB overdensity $d_{ }( ,k)$ (oscillating, red curves) and the corresponding gravitational driving term $D=-3(\Phi+\Psi)$ (monotonically falling, brown curves) in the radiation era for $R_b\ll 1$ and negligible photon diffusion. The gravitational driving of the CMB [eq. (\ref{['dot_gk']})] is illustrated with a spring that connects the $d_$ and $D$ curves. The panels demonstrate the impact of perturbations in decoupled neutrinos (left) and quintessence (right). --- Left panel shows $d_$ and $D$ for: tightly-coupled photons only (dashed), the standard model with three neutrinos (wider solid), and the same model after switching off neutrino anisotropic stress as a source of $\Phi-\Psi$ (thinner solid). --- Right panel displays: the earlier photon-only model (dashed) and a fictitious model where, as in the standard model, $59\%$ of energy density is in photons but the remaining $41\%$ is now in a tracking quintessence $$, with $w_ =1/3$ (solid).
• Figure 2: Growth of CDM overdensity $d_c( ,k)$ (rising black) during radiation domination (left) and matter domination (right). Brown curves show the value of $\Phi$, responsible for the ultimate growth of $_c/ _c$ by a future time of its observation, eq. (\ref{['growth_grf']}). --- Left: coupled photons only (dashed), photons plus 3 standard neutrinos (wide solid), and photons plus a tracking ($w_ =1/3$) scalar field $$ replacing the neutrino density of the previous model (thin solid). --- Right: pressureless matter only (solid), and a model with equal densities of matter and a scalar field that tracks it, $w_ =0$ (dashed).
• Figure 3: An accurate mechanical analogy for general-relativistic evolution (\ref{['dot_gk']}) of an acoustic CMB mode. The CMB overdensity $d_{ }$ equals the denoted distance to the tip of a pendulum with an internal frequency $=kc_s$ and its suspension point driven as specified by the gravitational driving term $D$, eq. (\ref{['D_def']}). For adiabatic initial conditions the evolution starts with $d_{\rm in}=3 _{\rm in}$, where $_{\rm in}$ is the superhorizon value of the Bardeen curvature. In the radiation era, initially, $D_{\rm in}=\frac{4}{3}(1+\frac{1}{5}R_ )/(1+\frac{4}{15}R_ )d_{\rm in}$ and in the matter era $D_{\rm in}=\frac{6}{5}d_{\rm in}$BS04, c.f. Figs. \ref{['fig_CMB_rad']} and \ref{['fig_CMB_mat']}. The advantages of this approach over considering $T^{(\rm Newt)}\!/T$ or $\Theta_{\rm eff}= T^{(\rm Newt)}\!/T+\Phi=\frac{1}{3}d_{ }+\Phi+\Psi$ as independent dynamical variables include: (A) Independence of the gravitational force that drives $\ddot d_{ }$ from $\ddot d_{ }$ itself. (B) Epoch- and scale-independence of the relation between the superhorizon perturbation $d_{\rm in}$ and the inflation-generated conserved curvature $_{\rm in}$ ($d_{\rm in}=3 _{\rm in}$). (C) Direct cause-effect connection between local physical dynamics and the apparent changes of the variable that describes CMB perturbations.
• Figure 4: Gravitational suppression of CMB temperature anisotropy by growing cosmic structure in the matter era (5-fold for $\Delta T/T$, 25-fold for power $C_l$). --- Left top: Evolution of $d_{ }$ and the driving term $D$ during matter domination with $R_b$ set negligible (solid). Evolution of $d_{ }$ from the same primordial perturbation if the metric becomes homogeneous before the horizon entry (dashed). --- Right: Suppression of the CMB temperature power $C_l$ for $\lesssim 100$ by CDM inhomogeneities. The solid curve shows $C_l$ in the concordance $\Lambda$CDM model with adiabatic initial conditions. The dashed curve describes the same model with changed initial CDM perturbations: $d_{\rm CDM}$ is artificially set to zero on superhorizon scales (the superhorizon $d_{ }$, $d_{ }$, and $d_b$ are unchanged). Then CDM inhomogeneities and the associated potential in the matter era are reduced. The smoother metric suppresses the CMB power at $\lesssim 100$ less, as the $C_l$ plots (obtained with CMBFAST) clearly show. --- Left bottom: The plots of $\Phi$ and $\Psi$ transfer functions confirm that the last model has smaller $\dot\Phi+\dot\Psi$, hence, the enhancement of its power at low $$ cannot be attributed to the ISW effect.
• Figure 5: The isocontours of the kernel $p_ (x_1,x_2)\equiv j_ (x_1)\,j_ (x_2)$ of projecting perturbation $k$-modes to $C_l$, eqs. (\ref{['C_l_expand']}, \ref{['pl_def']}). The figure is for $=10$. The labels on the contours show the values of $p_{10}/10^{-3}$. Note that $p_ (x_1,x_2)\approx 0$ whenever either $x_1<$ or $x_2<$. Also note that $p_ \ge0$ along the line $x_1=x_2$ (diagonal dashed line), but $p_$ oscillates through positive and negative values along any line $x_1=cx_2$ with $c\not=1$ (e.g., lower dashed line).