Cosmological tests of general relativity: A principal component analysis

The next generation of weak lensing surveys will trace the evolution of matter perturbations and gravitational potentials from the matter dominated epoch until today. Along with constraining the dynamics of dark energy, they will probe the relations between matter overdensities, local curvature, and the Newtonian potential. We work with two functions of time and scale to account for any modifications of these relations in the linear regime from those in the $\Lambda \mathrm{CDM}$ model. We perform a principal component analysis (PCA) to find the eigenmodes and eigenvalues of these functions for surveys like the Dark Energy Survey and Large Synoptic Survey Telescope. This paper builds on and significantly extends the PCA analysis of Zhao et al. [Phys. Rev. Lett. 103, 241301 (2009)] in several ways. In particular, we consider the impact of some of the systematic effects expected in weak lensing surveys. We also present the PCA in terms of other choices of the two functions needed to parametrize modified growth on linear scales, and discuss their merits. We analyze the degeneracy between the modified growth functions and other cosmological parameters, paying special attention to the effective equation of state $w(z)$. Finally, we demonstrate the utility of the PCA as an efficient data compression stage which enables one to easily derive constraints on parameters of specific models without recalculating Fisher matrices from scratch.

Figure 1

The eigenmodes of $\mu$ and $\gamma$ for $\mathrm{LSST}(+\mathrm{SN}+\mathrm{CMB})$, with all other parameters fixed to fiducial values.

Figure 2

The combined eigenmodes of $(\mu ,\gamma )$ for $\mathrm{LSST}(+\mathrm{SN}+\mathrm{CMB})$, with all other parameters fixed to fiducial values.

Figure 3

The eigenmodes of $\Sigma$ for $\mathrm{LSST}(+\mathrm{SN}+\mathrm{CMB})$, with all other parameters fixed to fiducial values.

Figure 4

The uncertainties (square roots of covariance eigenvalues) associated with the eigenmodes of $\mu$, $\gamma$, $\Sigma$, and the combined $(\mu ,\gamma )$ case for $\mathrm{LSST}(+\mathrm{SN}+\mathrm{CMB})$, with all other parameters fixed to fiducial values.

Figure 5

The eigenmodes of $\mu$ and $\gamma$ for $\mathrm{LSST}(+\mathrm{SN}+\mathrm{CMB})$ after marginalizing over the high-$z$ bins, with all other parameters fixed to fiducial values.

Figure 6

The combined eigenmodes of $(\mu ,\gamma )$ for $\mathrm{LSST}(+\mathrm{SN}+\mathrm{CMB})$ after marginalizing over the high-$z$ bins, with all other parameters fixed to fiducial values.

Figure 7

The eigenmodes of $\Sigma$ for $\mathrm{LSST}(+\mathrm{SN}+\mathrm{CMB})$ after marginalizing over the high-$z$ bins, with all other parameters fixed to fiducial values.

Figure 8

The uncertainties associated with the eigenmodes of $\mu$, $\gamma$, $\Sigma$, and the combined $(\mu ,\gamma )$ modes for $\mathrm{LSST}(+\mathrm{SN}+\mathrm{CMB})$, after marginalizing over the high-$z$ bins only (M.o. high-$z$), and after marginalization over the high-$z$ bins and the galaxy bias parameters (M.o. high-$z+\mathrm{\text{bias}}$), with all other parameters fixed to fiducial values.

Figure 9

The eigenmodes of $\mu$ and $\gamma$ for $\mathrm{LSST}(+\mathrm{SN}+\mathrm{CMB})$, after marginalizing over the high-$z$ bins and the galaxy bias parameters, with all other parameters fixed to fiducial values.

Figure 10

The eigenmodes of $\mu$ and $\gamma$ for $\mathrm{LSST}(+\mathrm{SN}+\mathrm{CMB})$ after marginalizing over all other parameters.

Figure 11

The combined eigenmodes of $(\mu ,\gamma )$ for $\mathrm{LSST}(+\mathrm{SN}+\mathrm{CMB})$ after marginalizing over all other parameters.

Figure 12

The eigenmodes of $\Sigma$ for $\mathrm{LSST}+\mathrm{SN}+\mathrm{CMB}$ after marginalizing over all other parameters.

Figure 13

The uncertainties associated with the eigenmodes of $\mu$, $\gamma$, $\Sigma$, and the combined $(\mu ,\gamma )$ modes for $\mathrm{LSST}+\mathrm{SN}+\mathrm{CMB}$ after marginalizing over all other parameters. Two cases are shown: with $w$ bins fixed to their fiducial value of $-1$ (dots joined by lines), and with $w$ bins varied and marginalized over (triangles).

Figure 14

The eigenmodes of $\mu$ and $\gamma$ for $\mathrm{DES}+\mathrm{SN}+\mathrm{CMB}$ after marginalizing over all other parameters.

Figure 15

The eigenmodes of the combined $(\mu ,\gamma )$ modes for $\mathrm{DES}+\mathrm{SN}+\mathrm{CMB}$ after marginalizing over all other parameters.

Figure 16

The uncertainties associated with the eigenmodes of $\mu$, $\gamma$, $\Sigma$, and combined $(\mu ,\gamma )$ modes for $\mathrm{DES}(+\mathrm{SN}+\mathrm{CMB})$ and for $\mathrm{LSST}(+\mathrm{SN}+\mathrm{CMB})$ errors, after marginalization over all other parameters.

Figure 17

Top: The uncertainties associated with the combined $(\mu ,\gamma )$ eigenmodes for LSST (left) and DES (right) for four cases: (I) without systematics, but with $w$ fixed; (II) without systematics, but with $w$ varied; (III) with systematics, but with $w$ fixed; and (IV) with systematics, and with $w$ varied. Bottom: The uncertainties for LSST (left) and DES (right) relative to case 1.

Figure 18

Top: The uncertainties associated with eigenmodes of $\mu$ for LSST (left) and DES (right) for four cases: (I) without systematics, but with $w$ fixed; (II) without systematics, but with $w$ varied; (III) with systematics, but with $w$ fixed; and (IV) with systematics, and with $w$ varied. Bottom: The uncertainties for LSST (left) and DES (right) relative to case 1.

Figure 19

The eigenmodes for $\mathrm{LSST}(+\mathrm{SN}+\mathrm{CMB})$. The upper (lower) panel corresponds to the case without (with) systematics. The 10th, 11th, and 12th modes of $\mu$ without systematics are compared to the 8th, 11th, and 12th modes of $\mu$ with systematics, respectively. These modes are chosen since they correspond to the last $z$-independent and first $z$-dependent mode, respectively, in the two cases (with and without systematics). Analogously, the 4th eigenmode of $\gamma$ without systematics is compared to the 4th mode of $\gamma$ with systematics. It is an illustration of how LSST eigenmodes are distorted as a result of accounting for systematics.

Figure 20

The eigenmodes for $\mathrm{DES}(+\mathrm{SN}+\mathrm{CMB})$. The upper (lower) panel corresponds to the case without (with) systematics. The three modes of $\mu$ and the first mode of $\gamma$ without systematics are compared to the corresponding modes of $\mu$ with systematics. It is an illustration of how DES eigenmodes would be distorted due to systematics.

Figure 21

Top panel: First three eigenmodes of $w(z)$ with (left) and without (right) MG included. Bottom panel: Errors for $w(z)$ eigenmodes with and without MG included (there is a prior of 0.5 on $w$ bins).

Figure 22

Redshift error—distortion: Distortion of the total galaxy distribution due to uncertainties in redshift measurement, plotted here for LSST. $g_i$’s are expansion coefficients of $\Delta \overline{z}$ in terms of Chebyshev polynomials (a3).

Figure 23

Redshift error—shift: Uncertainty in measuring the centroids of the redshift bins (nonvanishing $z$ bias), quantified by $b_i$ coefficients at redshift bin $i$.

Figure 24

Redshift error—scatter: Uncertainties in the photo-$z$ error at redshift $z$ modeled as $\sigma (z)={\sigma }_0(1+z)$, where ${\Delta }_i$ is ${\sigma }_0-{\overline{\sigma }}_0$ at redshift bin $i$.