Constraining neutrinos and dark energy with galaxy clustering in the dark energy survey

We determine the forecast errors on the absolute neutrino mass scale and the equation of state of dark energy by combining synthetic data from the Dark Energy Survey (DES) and the cosmic microwave background Planck surveyor. We use angular clustering of galaxies for DES in seven redshift shells up to $z\sim 1.7$ including cross-correlations between different redshift shells. We study models with massless and massive neutrinos and three different dark energy models: $\mathrm{\Lambda }$ cold dark matter (CDM) ($w=-1$), $w\mathrm{CDM}$ (constant $w$), and $w_a\mathrm{CDM}$ [evolving equation of state parameter $w(a)=w_0+w_a(1-a)$]. We include the impact of uncertainties in modeling galaxy bias using a constant and a redshift-evolving bias model. For the $\mathrm{\Lambda }\mathrm{CDM}$ model we obtain an upper limit for the sum of neutrino masses from $\mathrm{DES}+\text{Planck}$ of $\mathrm{\Sigma }{m}_{\nu }<0.08\mathrm{eV}$ (95% C.L.) for a fiducial mass of $\mathrm{\Sigma }{m}_{\nu }=0.047\mathrm{eV}$, with a $1\sigma$ error of 0.02 eV, assuming perfect knowledge of galaxy bias. For the $w\mathrm{CDM}$ model the limit is $\mathrm{\Sigma }{m}_{\nu }<0.10\mathrm{eV}$. For a $w\mathrm{CDM}$ model where galaxy bias evolves with redshift, the upper limit on the sum of neutrino masses increases to 0.29 eV. DES will be able to place competitive upper limits on the sum of neutrino masses of 0.1–0.3 eV and could therefore strongly constrain the inverted mass hierarchy of neutrinos. In a $w\mathrm{CDM}$ model the $1\sigma$ error on constant $w$ is $\mathrm{\Delta }w=0.03$ from DES galaxy clustering and Planck. Allowing $\mathrm{\Sigma }{m}_{\nu }$ as a free parameter increases the error on $w$ by a factor of 2, with $\mathrm{\Delta }w=0.06$. In a $w_a\mathrm{CDM}$ model, in which the dark energy equation of state varies with time, the errors are $\mathrm{\Delta }{w}_0=0.2$ and $\mathrm{\Delta }{w}_a=0.42$. Including neutrinos and redshift-dependent galaxy bias increases the errors to $\mathrm{\Delta }{w}_0=0.39$ and $\mathrm{\Delta }{w}_a=0.99$.

Figure 1

Top panel: The CMB temperature anisotropy power spectrum $C_l^{TT}$ for different neutrino models: a $\mathrm{\Lambda }\mathrm{CDM}$ model with massless neutrinos (black solid line), a $\mathrm{\Lambda }\mathrm{CDM}$ model with $\sum m_{\nu }=0.047\mathrm{eV}$ (light green dashed line) and a $\mathrm{\Lambda }\mathrm{CDM}$ model with $\sum m_{\nu }=0.235\mathrm{eV}$ (blue dashed line). The physical densities of cold dark matter and baryons are ${\omega }_c=0.1109$ and ${\omega }_b=0.02258$ respectively. Bottom panel: The change in $C_l^{TT}$ for the two neutrino models relative to the case with massless neutrinos.

Figure 2

Top panel: The effect of the dark energy equation of state parameter $w$ on the CMB temperature anisotropy power spectrum $C_l^{TT}$: a $\mathrm{\Lambda }\mathrm{CDM}$ model with $w=-1.0$ (black solid line), a $w\mathrm{CDM}$ model with $w=-0.8$ (light green dashed line) and a $w\mathrm{CDM}$ model (blue dashed line) with $w=-1.2$. Bottom panel: The change in $C_l^{TT}$ relative to the case with $w=-1$.

Figure 3

Top panel: Angular clustering of galaxies in three redshift shells with mean photo-$z$ of $z=0.4$ (black), $z=0.8$ (red) and $z=1.2$ (blue). The solid lines depict models with massless neutrinos and dashed lines are for a $\mathrm{\Lambda }\mathrm{CDM}$ model with massive neutrinos, where ${\mathrm{\Omega }}_{\nu }=0.005$ and $\sum m_{\nu }=0.235\mathrm{eV}$. Bottom panel: The spectrum suppression relative to the case with massless neutrinos.

Figure 4

Top panel: Angular clustering of galaxies in three redshift shells with mean photo-$z$ of $z=0.4$ (black), $z=0.8$ (red) and $z=1.2$ (blue). The solid lines represent a model with $w=-0.8$, dotted lines show a $\mathrm{\Lambda }\mathrm{CDM}$ model with $w=-1$, whereas the dashed lines are for a model with $w=-1.2$. Bottom panel: The relative difference in the spectra with respect to the case with $w=-1$. The scale dependence is due to transfer function $T(k)$ effects.

Figure 5

Angular power spectrum of galaxies with cosmic variance (orange) and shot noise (black) contributions at $z=0.4$ (solid blue) and $z=1.6$ (dot dashed red), for a $\mathrm{\Lambda }\mathrm{CDM}$ model with massless neutrinos, ${\mathrm{\Omega }}_{\nu }=0$. The error due to shot noise in the shell at $z=0.4$ is negligible compared to cosmic variance. Galaxy bias is $b=1$.

Figure 6

Marginalized constraints when combining Planck and DES synthetic data for three dark energy models with massless neutrinos. The solid black line is the base $\mathrm{\Lambda }\mathrm{CDM}$ model, while the red line shows the $w\mathrm{CDM}$ model with a constant EOS parameter $w$. The blue dashed line is a $w_a\mathrm{CDM}$ model with a time-variable equation of state in the form of $w(a)=w_0+w_a(1-a)$. Here and throughout, $w$ stands for either $w_0$ in a $w_a\mathrm{CDM}$ model or for $w$ in a $w\mathrm{CDM}$ model. The y-axis is the normalized posterior probability and the vertical bar indicates the fiducial parameter value. Galaxy bias is set to $b=1$.

Figure 7

Marginalized 68% and 95% likelihood contours when combining Planck and DES synthetic data for three dark energy models with massless neutrinos (${\mathrm{\Omega }}_{\nu }=0$). The solid black line is the base $\mathrm{\Lambda }\mathrm{CDM}$ model, while the red solid line is for a $w\mathrm{CDM}$ model with a constant EOS parameter $w$. The blue dashed line is a $w_a\mathrm{CDM}$ model with a time-variable EOS $w(a)$. Galaxy bias is set to $b=1$. Plotting ${\sigma }_8$ vs ${\mathrm{\Omega }}_{de}$ and $H_0$ vs ${\sigma }_8$ shows the change in the direction of degeneracies, although the ellipses remain centered on the fiducial parameter values marked by the symbol $\times$.

Figure 8

Same as Fig. 6 but for a model with massive neutrinos, where ${\mathrm{\Omega }}_{\nu }=0.001$ or $\sum m_{\nu }=0.047\mathrm{eV}$. The parameter estimates in the $w_a\mathrm{CDM}\nu$ model are highly biased, due to degeneracies between the sum of neutrino masses and other parameters (shown in Fig. 10). Estimates for the $w\mathrm{CDM}\nu$ model (solid red line) are only slightly affected.

Figure 9

Same as Fig. 8 but for a model with ${\mathrm{\Omega }}_{\nu }=0.005$ or $\sum m_{\nu }=0.235\mathrm{eV}$. Since in this model we can recover the input neutrino mass, degeneracies between the sum of neutrino masses and other parameters are less problematic, and parameter estimates are therefore less biased.

Figure 10

Same as Fig. 7, but for a model with massive neutrinos, where ${\mathrm{\Omega }}_{\nu }=0.001$ or $\sum m_{\nu }=0.047\mathrm{eV}$. The contours in the $w_a\mathrm{CDM}\nu$ model are now shifted away from the fiducial values due to parameter degeneracies. The $\sum m_{\nu }\text{−}{\sigma }_8$ contours show the rotation of the degeneracies for models with massive neutrinos and constant $w$, relative to a model with massive neutrinos only.

Figure 11

Same as Fig. 10, but for a model with massive neutrinos, where ${\mathrm{\Omega }}_{\nu }=0.005$ or $\sum m_{\nu }=0.235\mathrm{eV}$. Parameter degeneracies are now less severe, and so the $w_a\mathrm{CDM}\nu$ contours shift more towards the fiducial parameter values, compared to contours in Fig. 10.

Figure 12

The sum of neutrino masses plotted as a function of the lightest state $m_0$ for the inverted ($m_0=m_3$) and normal ($m_0=m_1$) hierarchy. The filled contours are the $3\sigma$ regions allowed by the results of solar and atmospheric neutrino flavor oscillations. Overplotted are the upper limits from our $\mathrm{DES}+\text{Planck analysis}$, showing that in $\mathrm{\Lambda }\mathrm{CDM}$ [$\mathrm{\Sigma }{m}_{\nu }<0.08\mathrm{eV}$ (95% C.L.)] and $w\mathrm{CDM}$ [$\mathrm{\Sigma }{m}_{\nu }<0.10\mathrm{eV}$ (95% C.L.)], the combination of DES and CMB data could distinguish between the normal and inverted hierarchy provided that we have full knowledge of galaxy bias.

Figure 13

Marginalized 68% and 95% likelihood contours from DES and Planck for a $\mathrm{\Lambda }\mathrm{CDM}\nu$ model with massive neutrinos and ${\mathrm{\Omega }}_{\nu }=0.005$ or $\sum m_{\nu }=0.235\mathrm{eV}$, taking into account uncertainty in galaxy bias modeling. The solid black line is the base $\mathrm{\Lambda }\mathrm{CDM}\nu$ model assuming a full knowledge of galaxy bias by setting $b=1$. The green thin dashed line is a $\mathrm{\Lambda }\mathrm{CDM}\nu +b$ model with a single free bias parameter $b$. The magenta thick dashed line is a $\mathrm{\Lambda }\mathrm{CDM}\nu +b_i$ model with a redshift-evolving bias, parametrized with seven free bias parameters $b_i$, one per redshift bin. The fiducial cosmology is marked by the symbol $\times$.

Figure 14

Same as Fig. 13 but for a $w\mathrm{CDM}\nu$ model with massive neutrinos and ${\mathrm{\Omega }}_{\nu }=0.005$ or $\sum m_{\nu }=0.235\mathrm{eV}$ and a constant EOS parameter $w$, where $w\ne -1$.

Figure 15

Same as Fig. 14 but for a $w_a\mathrm{CDM}\nu$ model with massive neutrinos and ${\mathrm{\Omega }}_{\nu }=0.005$ or $\sum m_{\nu }=0.235\mathrm{eV}$ and a time-variable EOS parameter $w(a)$.

Figure 16

Marginalized posterior distributions for $\sum m_{\nu }$ and ${\sigma }_8$ where $\sum m_{\nu }=0.235\mathrm{eV}$. Shown are six models with different assumptions about the galaxy bias amplitude. The four solid curves in black, red, blue and green are for the single bias model with $b=\{1,2,2.5,3\}$ respectively, and the two dashed curves are for a seven-parameter model: a redshift-evolving model which has all $b_i$ set to 1 (dashed black lines) and a model with the $b_i$ values from Table 4 (dashed blue lines).

Figure 17

Marginalized posterior distributions with $\sum m_{\nu }=0.047\mathrm{eV}$ (left panel) and $\sum m_{\nu }=0.235\mathrm{eV}$ (right panel) for dark energy models with $w=-1$, $w\ne -1$ and time-varying equation of state $w(a)$ for models with $b=1$ and a redshift-evolving bias model.

Figure 18

1D marginalized limits on the equation of state parameters $w$ and $w_a$ for massless and massive neutrino models. Models marked with a * are for a neutrino mass of $\sum m_{\nu }=0.047\mathrm{eV}$. The constraints on $w$ in the left panel are only from constant $w$ models. Models with a redshift-evolving bias are plotted in red, whereas models that assume perfect knowledge of bias with $b=1$ are plotted in black.

Figure 19

Likelihood contours and 1D marginalized constraints for $\mathrm{\Lambda }\mathrm{CDM}\nu +b$ and $w\mathrm{CDM}\nu +b$ with ${\mathrm{\Omega }}_{\nu }=0.005$ including a single free bias parameter $b$. The input values of the parameters are marked with $\times$.

Figure 20

1D and 2D marginalized constraints for $\mathrm{\Lambda }\mathrm{CDM}$ $\nu +b_i$ with ${\mathrm{\Omega }}_{\nu }=0.005$ including a redshift-evolving bias. Shown are degeneracies in a model with seven bias parameters. We only show four of the seven bias parameters.

Figure 21

1D and 2D marginalized constraints for $w\mathrm{CDM}\nu +b_i$ with ${\mathrm{\Omega }}_{\nu }=0.005$ including a redshift-evolving bias and EOS parameter $w$.