Current constraints on the cosmic growth history

TL;DR

The paper investigates whether cosmic acceleration signals can be explained by deviations from General Relativity using a phenomenological growth framework with scale- and time-dependent factors $Q(k,a)$ and $R(k,a)$ that modify the Poisson equation and the relation between Newtonian potentials. By combining expansion-history data (SN1a, BAO, CMB) with growth probes (galaxy clustering, weak lensing, ISW cross-correlations), it shows that early-time modifications affecting CMB acoustic peaks are tightly constrained (effective Newton's constant within about 3% of GR), whereas late-time modifications remain weakly constrained yet consistent with $\Lambda$CDM at 95% confidence. The analysis reveals that growth on large and small scales can diverge under modified histories, offering a distinctive signature for future surveys with wide angular coverage to probe. A remaining degeneracy between $Q$ and $R$ means breaking the constraint requires additional observables such as velocity and cross-correlation measurements, and upcoming surveys are well positioned to significantly tighten these tests of cosmic growth history.

Abstract

We present constraints on the cosmic growth history with recent cosmological data, allowing for deviations from Lambda CDM as might arise if cosmic acceleration is due to modifications to GR or inhomogeneous dark energy. We combine measures of the cosmic expansion history, from Type 1a supernovae, baryon acoustic oscillations and the CMB, with constraints on the growth of structure from recent galaxy, CMB and weak lensing surveys along with ISW-galaxy cross-correlations. Deviations from Lambda 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 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.

Figures (10)

• Figure 1: The effect of changing $Q$ [left panels] and $R$ [right panels] in Model I, defined in (\ref{['QRdef']}). 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 $\Lambda$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 $\Delta_{xH}=0 (GR), \pm 5\times 10^{-5}$ in Model II defined in (\ref{['delxdef']}) 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 cut off, $k_x=0.01 Mpc^{-1}$, 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 $\Lambda$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 figure \ref{['fig1']} 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 anti-correlation 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, $\phi$ 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 $\dot\phi$ and $\dot\psi$ and boosts the temperature dipole anisotropy, raising the acoustic peaks. In contrast to the time varying evolution in figure \ref{['fig3']}, 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% confirndence 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 $(Q-R)$.