Dark energy and neutrino masses from future measurements of the expansion history and growth of structure

We forecast the expected cosmological constraints from a combination of probes of both the universal expansion rate and matter perturbation growth, in the form of weak lensing tomography, galaxy tomography, supernovae, and the cosmic microwave background incorporating all cross-correlations between the observables for an extensive cosmological parameter set. We allow for nonzero curvature and parameterize our ignorance of the early universe by allowing for a non-negligible fraction of dark energy (DE) at high redshifts. We find that early DE density can be constrained to 0.2% of the critical density of the universe with Planck combined with a ground-based LSST-like survey, while curvature can be constrained to 0.06%. However, these additional degrees of freedom degrade our ability to measure late-time dark energy and the sum of neutrino masses. We find that the combination of cosmological probes can break degeneracies and constrain the sum of neutrino masses to 0.04 eV, present DE density also to 0.2% of the critical density, and the equation of state to 0.01—roughly a factor of 2 degradation in the constraints overall compared to the case without allowing for early DE. The constraints for a space-based mission are similar. Even a modest 1% dark energy fraction of the critical density at high redshift, if not accounted for in future analyses, biases the cosmological parameters by up to $2\sigma$. Our analysis suggests that throwing out nonlinear scales (multipoles $>1000$) may not result in significant degradation in future parameter measurements when multiple cosmological probes are combined. We find that including cross-correlations between the different probes can result in improved constraints by up to a factor of 2 for the sum of neutrino masses and early dark energy density.

Figure 1

Energy density (top panel) and equation of state (bottom panel) of early dark energy and a cosmological constant. At low redshifts the EDE mimics a dark energy component with the same density and EOS at present, and decouples after redshifts of a few, the exact redshift depending on the size of the EDE fraction ${\Omega }_e$.

Figure 2

Matter power spectrum $P(k)$ ($(\mathrm{Mpc}/h{)}^3$) against wave number $k(h/\mathrm{Mpc})$ at $z=[0,1,5,10]$ (high to low) in four distinct cosmologies: $\Lambda \mathrm{CDM}$ (dotted, black line), wCDM (dashed, red line, $w_0=-0.9$ and ${\Omega }_e=0$), $\Lambda \mathrm{CDM}$ with massive neutrinos (solid, green line, $\sum _{}^{}{m}_{\nu }=0.3\mathrm{eV}$), and a CDM universe with EDE (dot-dashed, blue line, ${\Omega }_e=0.05$, $w_0=-1$).

Figure 3

Redshift-dependent geometric weight $W_{ij}(z)=(9/16)({\Omega }_m{H}_0^2{)}^{2}H(z){\chi }^{7/2}(z){\zeta }_i^{\kappa }(z){\zeta }_j^{\kappa }(z)$ [see text following Eq. (13)] of CMB weak lensing (CMBL) and tomographic low-redshift weak lensing bins for a flat CDM model with ${\Omega }_e=0.1$ and $w_0=-1.0$. For CMB lensing and the fifth tomographic bin we also plot the kernel for ${\Omega }_e=0$. The lensing kernel captures approximately the redshift dependence of the integrand for the lensing power spectrum (including that from the matter power spectrum). The diminishment of the kernels for ${\Omega }_e>0$ stems from the increase of $H(z)$ with increasing ${\Omega }_e$.

Figure 4

Top, left panel: Convergence power spectra $\ell (\ell +1)C_{ij}^{\kappa \kappa }/2\pi$ for the case of five tomographic bins in the fiducial cosmology. We include the expected noise for LSST as a band about the curve. Middle, left panel: Logarithmic derivative $d\ln C_{ij}^{\kappa \kappa }/d\ln p_k$ of the convergence power spectrum with cosmological parameters $p_k$ for the third tomographic bin ($i=j=3$). The derivatives of the other tomographic bins have similar shapes. Bottom, left panel: The subwindow zooms in on the logarithmic derivatives with sum of neutrino masses (dotted blue line) and EDE density (dot-dot-dashed red line). Right panels: Same as left panels except for galaxy power spectra. The noise contribution (both LSST and JDEM) is at most on sub-percent level and decreases towards smaller scales, as the noise is constant with scale whereas $C_{\ell }^{gg}$ increases with scale. This is to be contrasted with lensing tomography where $C_{\ell }^{\kappa \kappa }$ peaks at around $\ell =10$ and thereafter decreases continuously.

Figure 5

Top, left panel: Tomographic galaxy-lensing correlations $\ell (\ell +1)C_{ij}^{\kappa g}/2\pi$. Here we only illustrate a subset of cases where galaxy and lensing bins fully overlap. Middle, left panel: Logarithmic derivatives of the power spectra $d\ln C_{ij}^{\kappa g}/d\ln p_k$ with parameters $p_k$ for the third tomographic bin in both galaxy and lensing (similar characteristics for other bins). Bottom, left panel: Zooming in on the derivatives with EDE density and sum of neutrino masses. Right panels: Logarithmic derivatives of the CMB temperature (solid, black line), $E$-mode (dashed, blue line), and lensing potential (dotted, red line) power spectra with cosmology.

Figure 6

Top, left panel: Correlations between weak lensing of the CMB and low-redshift sources $\ell (\ell +1)C_i^{{\kappa }_c\kappa }/2\pi$ for three of the five tomographic bins. Middle, left panel: Logarithmic derivatives of the power spectrum $d\ln C_i^{{\kappa }_c\kappa }/d\ln p_k$ with parameters $p_k$ for the third tomographic bin (similar characteristics for other bins). Bottom, left panel: Zoom on derivatives with EDE density and sum of neutrino masses. Right panels: Same as left panels except for correlations between galaxy tomography and lensing of the CMB.

Figure 7

Top, left panel: ISW-WL correlations $-\ell (\ell +1)C_i^{\kappa T}/2\pi$ for three of the five tomographic bins. Midle, left panel: Logarithmic derivatives of the power spectrum $d\ln C_i^{\kappa T}/d\ln p_k$ with parameters $p_k$ for the third bin (similar characteristics for other bins). Bottom, left panel: Zoom on the EDE and neutrino mass derivatives. Right panels: Same as left panels except for ISW-galaxy correlations.

Figure 8

Weak lensing signal-to-noise $C_{ij}^{\kappa \kappa }(\ell )/\Delta C_{ij}^{\kappa \kappa }(\ell )$ for combinations of the second and fifth tomographic bins in the fiducial cosmology, for LSST (thick line) and JDEM (thin line).

Figure 9

Parameter degeneracies with early dark energy density (${\Omega }_e$) in a flat universe (also see Table ) for Planck measurements of temperature and CMB lensing spectra [TT, EE, TE, ${\kappa }_c{\kappa }_c$, ${\kappa }_c\mathrm{T}$] (dotted-dashed, black curve), along with LSST tomographic weak lensing spectra [$\kappa \kappa$] (dotted, blue curve) and tomographic galaxy spectra [gg] (dashed, turquoise curve). Constraints from SNe are too weak to be visible in the shown parameter regions. The error ellipses from the combination of all these probes (including SNe), incorporating all cross-correlations [see Eq. (48)] is shown as (solid, red) curves.

Figure 10

Error ellipses showing degeneracies between curvature (${\Omega }_k$), early dark energy density (${\Omega }_e$) and sum of neutrino masses ($\sum _{}^{}{m}_{\nu }$). The curves have the same meaning as in Fig. 9. Note the strong degeneracy between curvature and sum of neutrino masses for CMB data (dotted-dashed, black curve) that was pointed out in Ref. 151. This degeneracy is broken when information on curvature from measures of cosmological distances are included, as shown by the solid (red) contour. The constraints in the plane of ${\Omega }_e$ and $\sum _{}^{}{m}_{\nu }$ are shown in Fig. 9.

Figure 11

Bands show the allowed values for the sum of neutrino masses as a function of the lightest neutrino mass eigenstate. The branch with higher values is for the inverted mass hierarchy. We have used the constraints on the neutrino mass squared differences from the Particle Data Group [161] to create bands of the allowed regions. The 3 horizontal lines show the $1\sigma$ Fisher matrix error estimates for the cases $\mathrm{PK}+{\Omega }_k$ (0.22 eV, Table ), $\mathrm{PK}-{\Omega }_e$ (0.15 eV, Table ), and ${\mathrm{PKL}}^{\kappa }{\mathrm{L}}^g{\mathrm{L}}^s+{\Omega }_k$ (0.04 eV, Table ).

Figure 12

Error ellipses showing degeneracies between the sum of neutrino masses ($\sum _{}^{}{m}_{\nu }$) and the effective number of neutrinos ($N_{\mathrm{eff}}$) in a flat universe, where we have marginalized over the early dark energy density. The curves have the same meaning as in Fig. 9. Even for our extended parameter spaces, the CMB temperature as measured by Planck will be able to determine the possible existence of extra relativistic species, while the combination of experimental probes is extremely helpful in pinning down the sum of neutrino masses.