Cosmological effect of coherent oscillation of ultralight scalar fields in a multicomponent universe

TL;DR

The paper extends the Turner–Ratra equivalence between coherently oscillating ultralight scalar fields and cold dark matter from a single-component, flat FLRW setting to a multicomponent, curved cosmology with possible interactions and dynamic dark energy. It constructs a general averaging framework using $\phi_\pm$ to track the fast scalar-field oscillations against slower cosmological evolution, deriving modified Friedmann equations that incorporate curvature, additional fluids, and decay into radiation. A key result is that, with a specific initial-condition constraint, the scalar-field sector continues to mimic dust-like CDM across cosmological epochs and even during gravitational collapse, while outlining the limits of validity, notably $m \gg H$ and redshift bounds. The work also demonstrates that small DM-to-radiation decays can be cosmologically viable and provides a foundation for studying perturbations and structure formation within this generalized equivalence.

Abstract

The idea that coherent oscillations of a scalar field, oscillating over a time period that is much shorter than the cosmological timescale, can exhibit cold dark matter (CDM) like behavior was previously established. In our work we first show that this equivalence between the oscillating scalar field model and the CDM sector is exact only in a flat Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime in the absence of cosmological constant and any other possible matter components in the universe when the mass of the scalar field is very large compared to the Hubble parameter. Then we show how to generalize the equivalence between the coherently oscillating scalar field model and the CDM sector in a spatially curved universe with multiple matter components. Using our general method, we will show how a coherently oscillating scalar field model can represent the CDM sector in the presence of non-minimal coupling of the CDM sector with radiation. Our method is powerful enough to work out the dynamics of gravitational collapse in a closed FLRW spacetime where the coherently oscillating scalar field model represents the CDM sector. We have, for the first time, presented a consistent method which specifies how a coherently oscillating scalar field model, where the scalar field is ultralight, acts like the CDM sector in a multicomponent universe.

Figures (5)

• Figure 1: Evolution of the effective and dark matter equation-of-state (EoS) parameters with respect to cosmic time (t) across different cosmic regimes, where dark matter is modeled as an oscillating scalar field $\phi$ and dark energy is represented by a cosmological constant with $\Lambda = 1.08\,u^{2}$. The initial conditions deviate from those specified in Eq. (\ref{['initcn']}). The initial conditions correspond to distinct redshifts: $z = 3600$ for the radiation-dominated era, $z = 5$ for the matter-dominated era, and $z = 0.2$ for the dark energy-dominated era. The mass of the dark matter scalar field is fixed at $m = 1.93\times10^{11}\,u$, and the initial conditions for the scalar fields satisfy the relation $\frac{\dot{\phi}_+(t_i)}{\phi_+(t_i)} = \frac{\dot{\phi}_-(t_i)}{\phi_-(t_i)} = -\tfrac{3}{2} \times 10^{3} H(t_i)$ with the initial dark matter energy densities chosen according to the corresponding redshifts for each era. All other cosmological parameters follow the Planck 2018 results Planck:2018vyg, and all quantities are expressed in geometrized units, $u = 10^{-28}\,\mathrm{cm}^{-1}$.
• Figure 2: Evolution of cosmological variables as a function of redshift (z) in a universe composed of radiation (R), baryons (B), dark matter ($\phi$), and dark energy. The horizontal axis shows $\log_{10}(1+z)$, plotted in decreasing order to represent the forward progression of cosmic time. Dark energy is modeled in three ways: (1) a cosmological constant with $\Lambda = 1.08\, u^2$; (2) a quintessence scalar field ($\psi$) with exponential potential $V(\psi) = V_m e^{-\lambda \psi}$, $V_m = 1.38\, u^2$, $\lambda=1$, and initial conditions $\psi(t_i) = 0.0001$, $\dot{\psi}(t_i) = 0.0001~u$; (3) a phantom scalar field with the same potential form and same initial and parameter values except $V_m = 0.91\, u^2$. Dark matter is an oscillating scalar field $\phi$ with mass $m = 1.93\times 10^{11}\, u$ and initial conditions $\phi_\pm(t_i) = 10^{-6}$, $\dot{\phi}_\pm(t_i) = -\frac{3H_i}{2} \times 10^{-6}~u$, with $a_i = 2.24\times10^{-4}$ and $H_i = 1.82\times10^5~u$. All other cosmological parameters follow Planck 2018 Planck:2018vyg, and quantities are expressed in geometrized units $u = 10^{-28}~\mathrm{cm}^{-1}$.
• Figure 3: Time evolution of system variables for an overdense region treated as a closed system with positive curvature ($k = 1$). Radiation $(R)$ is sourced by the decay of an oscillating ultralight scalar field ($\phi$), representing cold dark matter, with mean energy density $\langle \rho \rangle$, and a decay rate $\Gamma = 10^{-4}~u$. Dark energy is incorporated through three distinct models: (1) a cosmological constant with $\Lambda = 1.08~u^2$; (2) a quintessence-like scalar field ($\psi$) with potential $V(\psi) = V_{m} e^{-\lambda \psi}$, where $V_{m} = 1.38~u^2$; and (3) a phantom-like scalar field ($\psi$) with the same potential form but $V_{m} = 0.91~u^2$. For both scalar field models, the parameter $\lambda = 1$ and initial conditions $\psi(t_i) = 0.0001$, $\dot{\psi}(t_i) = 0.0001~u$. All variables are evaluated up to the virialization time $t_\text{v}$. Dark matter $\phi$ has mass $m = 1.93\times 10^{11}\, u$ with initial conditions $\phi_\pm(t_i) = 10^{-6}$, $\dot{\phi}_\pm(t_i) = -\frac{3H_i}{2} \times 10^{-6}~u$, where $a_i = 2.24\times10^{-4}$ and $H_i = 1.82\times10^5~u$. The initial radiation energy density $\rho_{(R)_{i}} = 0.00001~u^{2}$. All quantities are expressed in geometrized units with $u = 10^{-28}~\mathrm{cm}^{-1}$.
• Figure 4: Evolution of cosmological variables as a function of redshift (z) in a universe composed of radiation (R), baryons (B), dark matter ($\phi$), and dark energy, where dark matter decays into radiation with decay constant $\Gamma=10^{-4}~u$. The horizontal axis shows $\log_{10}(1+z)$, plotted in decreasing order to represent the forward progression of cosmic time. Dark energy is modeled in three ways: (1) a cosmological constant with $\Lambda = 1.08\, u^2$; (2) a quintessence scalar field ($\psi$) with exponential potential $V(\psi) = V_m e^{-\lambda \psi}$, $V_m = 1.38\, u^2$, $\lambda=1$, and initial conditions $\psi(t_i) = 0.0001$, $\dot{\psi}(t_i) = 0.0001~u$; (3) a phantom scalar field with the same potential form and same initial and parameter values except $V_m = 0.91\, u^2$. Dark matter is an oscillating scalar field $\phi$ with mass $m = 1.93\times 10^{11}\, u$ and initial conditions $\phi_\pm(t_i) = 10^{-6}$, $\dot{\phi}_\pm(t_i) = -\frac{3H_i}{2} \times 10^{-6}~u$, with $a_i = 2.24\times10^{-4}$ and $H_i = 1.82\times10^5~u$. All other cosmological parameters follow Planck 2018 Planck:2018vyg, and quantities are expressed in geometrized units $u = 10^{-28}~\mathrm{cm}^{-1}$.
• Figure 5: Evolution of the residual of the radiation continuity equation ($\mathcal{R}$) in $u^{3}$ units and its normalized form expressed as a percentage relative to $H\rho_{R}$, shown as functions of redshift in a universe composed of radiation (R), baryonic matter (B), dark matter ($\phi$), and dark energy. The dark matter component decays into radiation with a decay constant $\Gamma = 10^{-4}\,u$. The horizontal axis shows $\log_{10}(1 + z)$, plotted in decreasing order to represent the forward progression of cosmic time. The dark energy sector is modeled using three approaches: (1) a cosmological constant (black curve) with $\Lambda = 1.08\,u^2$; (2) a quintessence-like scalar field (red curve, $\psi$) with an exponential potential $V(\psi) = V_{m} e^{-\lambda \psi}$ and $V_{m} = 1.38\,u^2$; and (3) a phantom-like scalar field (green curve, $\psi$) with the same potential form but $V_{m} = 0.91\,u^2$. In both scalar field models, the parameter $\lambda = 1$ and initial conditions $\psi(t_i) = 0.0001$, $\dot{\psi}(t_i) = 0.0001~u$;. The dark matter field $\phi$ is modeled as an oscillating scalar field with mass $m = 1.93 \times 10^{11}\,u$ and initial conditions $\phi_\pm(t_i) = 10^{-6}$, $\dot{\phi}_\pm(t_i) = -\frac{3H_i}{2} \times 10^{-6}~u$, with $a_i = 2.24\times10^{-4}$ and $H_i = 1.82\times10^5~u$. All other cosmological parameters follow Planck 2018 Planck:2018vyg, and quantities are expressed in geometrized units $u = 10^{-28}~\mathrm{cm}^{-1}$.