Observational effects of the early episodically dominating dark energy

We investigate the observational consequences of the early episodically dominating dark energy on the evolution of cosmological structures. For this aim, we introduce the minimally coupled scalar-field dark energy model with the Albrecht-Skordis potential, which allows a sudden ephemeral domination of a dark energy component during the radiation or early matter era. The conventional cosmological parameters in the presence of such an early dark energy are constrained with WMAP and Planck cosmic microwave background radiation data including other external data sets. It is shown that in the presence of such an early dark energy, the estimated cosmological parameters can deviate substantially from the currently known $\mathrm{\Lambda }$ cold dark matter ($\mathrm{\Lambda }\mathrm{CDM}$)-based parameters, with best-fit values differing by several percent for WMAP and by a percent level for Planck data. For the latter case, only a limited amount of dark energy with episodic nature is allowed since the Planck data strongly favor the $\mathrm{\Lambda }\mathrm{CDM}$ model. Compared with the conventional dark energy model, the early dark energy dominating near the radiation-matter equality or at the early matter era results in the shorter cosmic age or the presence of tensor-type perturbation, respectively. Our analysis demonstrates that the alternative cosmological parameter estimation is allowed based on the same observations even in Einstein’s gravity.

Figure 1

Top left: A closeup picture of the scalar-field potential [Eq. (1) in units of $H_0^2$] in an EDE model with ${\varphi }_0=-4$, $A=0.01$, and $\lambda =10$ (black solid curve). The cases for $A=0.005$ and 0.02 are shown as dotted curves, while that of the SDE model ($A=100$, ${\varphi }_0=-3$) as a gray curve with the potential amplitude suppressed by a factor of 300 for ease of display. Top right: Evolution of $\varphi$ and its time derivative ${\varphi }^{\prime }=d\varphi /d\ln a$ in the EDE (black) and the SDE (gray curve) models as a function of the scale factor $a(t)$ normalized to unity at present. Bottom: Evolution of background density parameters (${\mathrm{\Omega }}_i$) and energy densities (${\mu }_i$) of radiation ($i=r$, red), matter ($m$, brown), and scalar field ($\varphi$, blue curves) in the EDE model. The gray and black curves indicate the evolutions for the SDE and fiducial $\mathrm{\Lambda }\mathrm{CDM}$ models, respectively.

Figure 2

Evolution of density perturbations of baryon (${\delta }_b$) and dark energy (${\delta }_{\varphi }$) components (left) and that of scalar-field perturbation variables ($\delta \varphi$ and $\delta {\varphi }^{\prime }=d\delta \varphi /d\ln a$; right column) in the EDE model considered in Fig. 1 (red curves), for comoving wave numbers $k=0.01$ (top) and $0.1{\mathrm{Mpc}}^{-1}$ (bottom panels). As the temporal gauge condition, the CDM-comoving gauge (where the velocity of the cold dark matter vanishes) has been chosen. The gray and black curves indicate the evolutions for the SDE and fiducial $\mathrm{\Lambda }\mathrm{CDM}$ models, respectively.

Figure 3

Evolution of density parameters (${\mathrm{\Omega }}_i$; $i=r$, $m$, $\varphi$) and dark energy equation-of-state parameters ($w_{\varphi }$) (top), and power spectra of baryonic matter density (middle left), and of CMB temperature anisotropy (middle right) and polarization (bottom panels) in our EDE models with ${\varphi }_0=-5$ (red), $-4$ (yellow), $-3$ (green), $-2$ (blue curves). For each value of ${\varphi }_0$, $A$ has been adjusted to give ${\mathrm{\Omega }}_{\varphi }=0.3$ at the peak of early episodic domination of the dark energy component. In all EDE models, we set $\lambda =10$. Gray curves represent the results of the SDE model and black curves those of the fiducial $\mathrm{\Lambda }\mathrm{CDM}$ model. The matter power spectra are normalized to the $\mathrm{\Lambda }\mathrm{CDM}$ model prediction at $k=0.1h{\mathrm{Mpc}}^{-1}$, while the CMB anisotropy power spectra at $\ell =10$. For the matter and CMB temperature anisotropy power spectra, the ratio of the EDE model power spectrum to the $\mathrm{\Lambda }\mathrm{CDM}$ prediction is also shown based on such a normalization.

Figure 4

Probability distributions of potential parameters ${\varphi }_0$ and $A$ favored by (a) the WMAP seven-year data and (b) the SDSS DR7 LRG power spectrum. The gray region with $A\lesssim 0.01$ corresponds to the forbidden region where the permanent acceleration occurs. The relative logarithmic probability ($\mathrm{\Delta }\ln P=-\mathrm{\Delta }{\chi }^2/2$) decreases from red to dark blue. The dotted curves represent the epoch of EDE domination at maximum ($a=10^{-6}$, $10^{-5}$, $10^{-4}$, $10^{-3}$ from left to right) while the solid curves indicate the strength of the dark energy at maximum (${\mathrm{\Omega }}_{\varphi }=0.06$, 0.1, 0.2, 0.3, 0.4, 0.5 from bottom to top). Triangles indicate the potential parameters of three EDE models (EDE 1–3) chosen in the Markov chain Monte Carlo analysis (see Fig. 5 and Table 1).

Figure 5

Top: Two-dimensional likelihood contours favored by $\mathrm{WMAP}7+\mathrm{LRG}+H_0$ data sets for EDE 1 (${\varphi }_0=-4$, $A=0.01$; red), EDE 2 (${\varphi }_0=-3$, $A=0.02$; yellow), and EDE 3 (${\varphi }_0=-2$, $A=0.04$; green contours) models. For all models, we set $\lambda =10$. The 68.3% and 95.4% confidence limits are indicated by the thick and thin solid curves, respectively. The results for the SDE (${\varphi }_0=-3$, $A=100$; gray) and $\mathrm{\Lambda }\mathrm{CDM}$ (black contours) models are shown for comparison. Middle and bottom: Marginalized one-dimensional likelihood distributions for each cosmological parameter.

Figure 6

Evolution of background density parameters (top left) and baryon density perturbations (${\delta }_b\equiv \delta {\mu }_b/{\mu }_b$ in the CDM-comoving gauge) at $k=0.01{\mathrm{Mpc}}^{-1}$ relative to the $\mathrm{\Lambda }\mathrm{CDM}$ model (top right), and baryonic matter and CMB anisotropy power spectra (sum of contributions from scalar- and tensor-type perturbations; bottom panels) for three best-fit EDE, $\mathrm{\Lambda }\mathrm{CDM}$, and SDE models, with the same color codes as in Fig. 5. For matter and CMB power spectra, recent measurements from SDSS DR7 LRG [24] and WMAP seven-year [1] data have been added (gray dots with error bars). In the ratio panels, we present the power ratio relative to the $\mathrm{\Lambda }\mathrm{CDM}$ prediction together with the fractional error bars of the observational data including SPT data (at $\ell \gtrsim 600$; blue bars) [29].

Figure 7

Top: Two-dimensional likelihood contours favored by the recent observations ($\text{Planck}+\mathrm{WP}+\mathrm{BAO}$) for the EDE 4 (with ${\varphi }_0$, $A$, $\lambda$ as free parameters; green dashed) and EDE 5 (${\varphi }_0=-5$, $A=0.01$, $\lambda =20$; red contours) models. The 68.3% and 95.4% confidence limits are indicated by the thick and thin solid curves, respectively. The results for the SDE (gray) and $\mathrm{\Lambda }\mathrm{CDM}$ (black contours) models are shown for comparison. Middle and bottom: Marginalized one-dimensional likelihood distributions for each cosmological parameter.

Figure 8

Evolution of background density parameters (top left) and baryon density perturbations (${\delta }_b\equiv \delta {\mu }_b/{\mu }_b$ in the CDM-comoving gauge) at $k=0.01{\mathrm{Mpc}}^{-1}$ relative to the $\mathrm{\Lambda }\mathrm{CDM}$ model (top right), and baryonic matter and CMB anisotropy power spectra (bottom panels) for the EDE 5, SDE, and $\mathrm{\Lambda }\mathrm{CDM}$ models best-fitting $\text{Planck}+\mathrm{WP}+\mathrm{BAO}$ data sets, with the same color codes as in Fig. 7. Note that the CMB temperature power spectrum is the sum of contributions from the scalar- and tensor-type perturbations. For the CMB temperature power spectrum, the recent measurement from Planck data [2] has been added (gray and blue dots with error bars). In the ratio panels, we present the power ratio relative to the $\mathrm{\Lambda }\mathrm{CDM}$ prediction together with the fractional error bars of the observational data (only for the Planck CMB angular power spectrum at small angular scales).