Current constraints on the cosmic growth history

We present constraints on the cosmic growth history with recent cosmological data, allowing for deviations from $\Lambda \mathrm{CDM}$ as might arise if cosmic acceleration is due to modifications to general relativity or inhomogeneous dark energy. We combine measures of the cosmic expansion history, from Type 1a supernovae, baryon acoustic oscillations, and the cosmic microwave background (CMB), with constraints on the growth of structure from recent galaxy, CMB, and weak lensing surveys along with integated Sachs Wolfe-galaxy cross correlations. Deviations from $\Lambda \mathrm{CDM}$ are parameterized by phenomenological modifications to the Poisson equation and the relationship between the two Newtonian potentials. We find modifications that are present at the time the CMB is formed are tightly constrained through their impact on the well-measured CMB acoustic peaks. By contrast, constraints on late-time modifications to the growth history, as might arise if modifications are related to the onset of cosmic acceleration, are far weaker, but remain consistent with $\Lambda \mathrm{CDM}$ at the 95% confidence level. For these late-time modifications we find that differences in the evolution on large and small scales could provide an interesting signature by which to search for modified growth histories with future wide angular coverage, large scale structure surveys.

Figure 1

The effect of changing $Q$ (left panels) and $R$ (right panels) in Model I, defined in (8). Shown are the CDM density fluctuation, ${\Delta }_c$, its growth rate, $f$, and ISW and weak lensing transfer functions, $T_I$ and $T_{\kappa }$, today, relative to their values for $\Lambda \mathrm{CDM}$. Four values of $Q_0$ and $R_0$ are shown for a time-varying but scale-independent modification with $s=3$ and $k_c=\infty$, and all other cosmological parameters fixed. On subhorizon scales, boosting the gravitational potential ($Q>1$) boosts the growth of dark matter density perturbations and the lensing potential. If $Q$ is increased, on subhorizon scales the boost in the gravitational potential opposes the late-time decay induced by accelerated expansion. The ISW signal can be progressively reduced, canceled out (for $Q$ a little larger than unity) and turn negative as $Q$ increases. The modification has a larger and opposite effect on CDM growth on horizon scales, where the time evolution of the modification plays a key role. Increasing $Q$ causes growth in ${\Delta }_c$ to slow, or even decay at late times on the largest scales (with $f=d\ln {\Delta }_c/d\ln a<0$). The effect of increasing $R$ is quantitatively degenerate for subhorizon CDM growth and qualitatively similar otherwise, but with a slightly reduced amplitude.

Figure 2

The effect of changing ${\Delta }_{xH}=0(\mathrm{GR})$, $\pm 5\times {10}^{-5}$ in Model II defined in (11) for two different scale dependences, $n_x=0.5n_s$ (left panels) and $n_x=0.75n_s$ (right panels). For this illustration we fix $s_x=1$ and include a cutoff, $k_x=0.01{\mathrm{Mpc}}^{-1}$, and all other cosmological parameters fixed. Shown are the CDM density fluctuation, ${\Delta }_c$, its growth rate, $f$, the ISW transfer function, $T_I$, and equivalent value of $Q$, today, relative to their values for $\Lambda \mathrm{CDM}$. A positive ${\Delta }_{xH}$ is equivalent to $Q>1$, suppressing cold dark matter growth, that can give rise to $f<0$ on large scales, and boosting the ISW component. For $n_x=0.5n_s$ the modification has roughly the same spectral evolution as the CDM perturbations, leading to a scale-independent effect on large scales. Increasing the spectral tilt, $n_x$, exacerbates the impact of the modification for scales just below the horizon.

Figure 3

The effect of the scale-independent and time-evolving modifications in Fig. 1 on the CMB (upper panel), lensing correlation (middle panel), and ISW-galaxy cross correlation (lower panel). For the lensing and ISW-galaxy correlations example redshift bins are shown. With increasing $Q$, the lensing potential, and galaxy number density are boosted while, by contrast, the late-time ISW is suppressed. Both positive ($Q<1$) and negative ($Q>1$) ISW amplitudes give rise to a boost in the CMB power spectrum at large scales. Increasing $Q$ reduces the ISW-galaxy cross correlation, and can lead to anticorrelation at low redshifts (this is relevant for the 2MASS samples, but not the higher redshift LRG sample shown here). Increasing $R$ has a qualitatively degenerate effect on the spectra.

Figure 4

The effect on the CMB temperature power spectrum of a time-independent modification to the Poisson equation (top panel) and the relationship between the two potentials, $\varphi$ and $\psi$ (lower panel). The principle effect of a larger $Q$ is to suppress the early ISW effect on scales comparable with the horizon scale at last scattering. A larger $R$ generates a late-time ISW amplitude even in the matter dominated era, by introducing a multiplicative difference between $\dot{\varphi }$ and $\dot{\psi }$ and boosts the temperature dipole anisotropy, raising the acoustic peaks. In contrast to the time-varying evolution in Fig. 3, constant $Q$ and $R$ have opposing, rather than degenerate, effects on the CMB.

Figure 5

68% and 95% confidence limits for the time- and scale-independent modifications in Model I ($Q=Q_0$ and $R=R_0$, $s=0$ and $k_c=\infty$). Constraints are shown for all data combined (forefront, yellow/orange contours), excluding the lensing correlation (red, dot-dashed) and excluding both lensing and galaxy-ISW correlations (blue, dashed). The effect of a constant modification on correlations around the first acoustic CMB peak leads to tight constraints, limited to deviations $\lesssim 3\%$ from GR at the 95% confidence level for all data combined. Driven by their opposing effects on the well-measured first peak, the principle degeneracy direction is well described by constant ($Q-R$).

Figure 6

As in Fig. 5 but giving the 68% and 95% confidence limits for allowing scale-independent, but time-evolving modifications in Model I, that become important at late times ($0\le s\le 3$, $k_c=\infty$). The principle degeneracy direction is well described by constant $Q_0(1+R_0)/2$. Constraints on the time-evolving modification are primarily driven by their effect on the cosmic-variance limited, late-time ISW effect in the CMB. As such, they are improved by the inclusion of the galaxy-ISW cross-correlation data, but remain far weaker than for the time-independent modification shown in Fig. 5.

Figure 7

Impact of allowing different modifications to the growth history on large and small scales for Model I. The figure shows 68% and 95% confidence limits on large (left panel) and small scales (right panel) using all data combined. In each figure, constraints are shown for the scale-independent scenario (forefront, red contours), and the scale-dependent scenario in which distinct deviations from GR are allowed on large and small scales ($Q_0$, $R_0$, $Q_{\infty }$, and $R_{\infty }$ are all varied with crossover scale $k_c={10}^{-2}{\mathrm{Mpc}}^{-1}$ and $0\le s\le 3$) [blue, dashed].

Figure 8

One-dimensional marginalized 68% and 95% confidence limits on the total matter power spectrum today for $\Lambda \mathrm{CDM}$ (black contour lines) and the two scenarios using Model I from Fig. 7 (filled red contours). The upper panel shows the constraints for the scale-independent modification, while the lower panel shows constraints if both large and small scale growth histories are modified independently. The data allows a broader range of growth histories, tending to suppress power on large scales relative to the $\Lambda \mathrm{CDM}$ scenario, constrained by the ISW. By comparison, the matter power spectrum on smaller scales is only marginally effected by the modified growth history. This is even the case when the subhorizon scales are modified independently of the superhorizon scales.

Figure 9

One-dimensional marginalized 68% and 95% confidence limits for Model I for the growth parameters $f$, $g_0$ and $\gamma$ at two scales $k=5\times {10}^{-3}{\mathrm{Mpc}}^{-1}$ and $k=0.05{\mathrm{Mpc}}^{-1}$ (left and right panels in each figure, respectively). $\Lambda \mathrm{CDM}$ (red contours) is compared to Model I with $0\le s\le 3$ (blue contours). [Left figure] Constraints for the scale-independent modification, with $k_c=\infty$. (Right) Constraints for the scale-dependent modification varying $\{Q_0,R_0,Q_{\infty },R_{\infty }\}$ with a transition scale $k_c={10}^{-2}{\mathrm{Mpc}}^{-2}$. The modified gravity models can have a markedly broader range of growth rates than $\Lambda \mathrm{CDM}$, especially on large scales.

Figure 10

One-dimensional marginalized 68% and 95% confidence limits for Model II with $k_x={10}^{-2}{\mathrm{Mpc}}^{-1}$. (Left) The matter power spectrum for Model II (red contours) compared to $\Lambda \mathrm{CDM}$ (black full and dashed lines). [Right] The growth parameters $f$, $g_0$ and $\gamma$, and $Q$ defined in (12) at $k=5\times {10}^{-3}{\mathrm{Mpc}}^{-1}$ for $\Lambda \mathrm{CDM}$ (red contours) and Model II (blue contours). On large scales the suppression of growth can lead to decay of ${\Delta }_c$, with $f<0$, for which $\gamma$ is undefined. For redshifts at which the MCMC chains include viable models with $f<0$, we do not give constraints on $\gamma$.