Simultaneous Falsification of LCDM and Quintessence with Massive, Distant Clusters

TL;DR

This work assesses whether LCDM with Gaussian initial conditions and quintessence can be falsified by the existence of massive, distant galaxy clusters. By combining expansion-history constraints from SN, CMB, BAO, and $H_0$ with a growth–cluster framework, it propagates into robust, model-wide limits on cluster abundances using the growth function $G(z)$ and a halo mass function, while accounting for parameter and sample variance and Eddington bias. The main finding is that, under conservative criteria (e.g., a cluster mass ~3 times the typical exclusion threshold in surveys of ~300 deg$^2$), no presently known cluster falsifies LCDM or quintessence; systematic mass measurements, rather than cosmological priors, dominate the exclusion power. The results provide practical fitting formulas to evaluate exclusion risks for any given $(M,z)$ and survey area, and highlight that mass calibration is the critical bottleneck for falsification claims, with implications for interpreting future cluster surveys and reconciling previous claims of tension.

Abstract

Observation of even a single massive cluster, especially at high redshift, can falsify the standard cosmological framework consisting of a cosmological constant and cold dark matter (LCDM) with Gaussian initial conditions by exposing an inconsistency between the well-measured expansion history and the growth of structure it predicts. Through a likelihood analysis of current cosmological data that constrain the expansion history, we show that the LCDM upper limits on the expected number of massive, distant clusters are nearly identical to limits predicted by all quintessence models where dark energy is a minimally coupled scalar field with a canonical kinetic term. We provide convenient fitting formulas for the confidence level at which the observation of a cluster of mass M at redshift z can falsify LCDM and quintessence given cosmological parameter uncertainties and sample variance, as well as for the expected number of such clusters in the light cone and the Eddington bias factor that must be applied to observed masses. By our conservative confidence criteria, which equivalently require masses 3 times larger than typically expected in surveys of a few hundred square degrees, none of the presently known clusters falsify these models. Various systematic errors, including uncertainties in the form of the mass function and differences between supernova light curve fitters, typically shift the exclusion curves by less than 10% in mass, making current statistical and systematic uncertainties in cluster mass determination the most critical factor in assessing falsification of LCDM and quintessence.

Figures (11)

• Figure 1: Predicted mean, full-sky abundance $\bar{N}(M,z)$ of clusters above mass and redshift thresholds $M$ and $z$, respectively, for flat $\Lambda$CDM models that fit current CMB+SN+BAO+$H_0$ data. Vertical dotted lines are plotted at $\bar{N}_{S.95}(f_{\rm sky}=1)$, the 95% CL sample variance limit for a full-sky survey. For $M=2\times 10^{15}\,h^{-1}\,M_{\odot}$, we shade the lower 95% of each distribution; exclusion at the 95% joint CL for a cluster of this mass in the full sky occurs at the redshift for which all of the shaded area lies to the left of the vertical $\bar{N}_{S.95}(f_{\rm sky}=1)$ line (in this case, $z\approx 0.9$). Probability distributions here and in later figures are normalized so that ${\rm max}[P(\log\bar{N})] = 1$.
• Figure 2: Dependence of the flat $\Lambda$CDM mass threshold at $z=1$ on sample variance $s$ and parameter variance $p$ confidence levels and on the sky fraction $f_{\rm sky}$. Individual variations are computed using the fitting functions of Appendix \ref{['sec:fit']} with one parameter varied at a time and the remaining parameters fixed to the fiducial values of $s=0.95$, $p=0.95$, and $f_{\rm sky}=1$.
• Figure 3: Predicted mean, full-sky abundance of clusters with $M>10^{15}\,h^{-1}\,M_{\odot}$ and $z>1.48$, for flat and nonflat $\Lambda$CDM, flat quintessence without early dark energy, and nonflat quintessence with early dark energy. The vertical dotted line marks $\bar{N}_{S.95}(f_{\rm sky}=1)$, i.e. a $10^{15}\,h^{-1}\,M_{\odot}$ cluster at $z=1.48$ observed anywhere in the sky would exclude all models in $P(\log\bar{N})$ to the left of the dotted line with a significance of at least 95% sample CL. The lowest-$\bar{N}$ 95% of each distribution is shaded.
• Figure 4: $M(z)$ exclusion curves. Even a single cluster with ($M, z$) lying above the relevant curve would rule out both $\Lambda$CDM and quintessence. Upper panel: flat $\Lambda$CDM 95% joint CL for both sample variance and parameter variance for various choices of sky fraction $f_{\rm sky}$ from the MCMC analysis (thin solid curves) and using the fitting formula from Appendix \ref{['sec:fit']} (thick dashed curves; accurate to $\lesssim 5\%$ in mass). Lower panel: Two of the most anomalous clusters detected to date, compared with the 95% joint CL exclusion curve for 300 deg$^2$ which approximates the total survey area for each cluster. We show the X-ray determined masses with and without Eddington bias correction (black solid points with thick error bars and red open points with thin error bars, respectively, offset in redshift by $\pm 0.01$ for clarity).
• Figure 5: Impact of systematic errors in cluster mass determination and mass function amplitude on the mean number of clusters in the full sky with $M>10^{15}\,h^{-1}\,M_{\odot}$ and $z>1.5$ for flat $\Lambda$CDM. A fractional change in mass determination can change the number of clusters by orders of magnitude. Conversely, a factor of two change in the mass function amplitude near this mass and redshift changes the mass limits by only a few percent.