The Cosmic Triangle: Revealing the State of the Universe

Abstract

The "cosmic triangle" is introduced as a way of representing the past, present, and future status of the universe. Our current location within the cosmic triangle is determined by the answers to three questions: How much matter is in the universe? Is the expansion rate slowing down or speeding up? And, is the universe flat? A review of recent observations suggests a universe that is lightweight (matter density about one-third the critical value), is accelerating, and is flat. The acceleration implies the existence of cosmic dark energy that overcomes the gravitational self-attraction of matter and causes the expansion to speed up.

Figures (8)

• Figure 1: The Cosmic Triangle represents the three key cosmological parameters -- $\Omega_m$, $\Omega_{\Lambda}$, and $\Omega_k$ -- where each point in the triangle satisfies the sum rule $\Omega_m + \Omega_{\Lambda} + \Omega_k = 1$. The blue horizontal line (marked Flat) corresponds to a flat universe ($\Omega_m + \Omega_{\Lambda} = 1$), separating an open universe from a closed one. The red line, nearly along the $\Lambda=0$ line, separates a universe that will expand forever (approximately $\Omega_{\Lambda}>0$) from one that will eventually recollapse (approximately $\Omega_{\Lambda}<0$). And the yellow, nearly vertical line separates a universe with an expansion rate that is currently decelerating from one that is accelerating. The location of three key models are highlighted: standard cold-dark-matter ( Scdm) is dominated by matter ($\Omega_m=1$) and no curvature or cosmological constant; flat ($\Lambda$ cdm), with $\Omega_m=1/3$, $\Omega_{\Lambda}=2/3$, and $\Omega_k=0$; and Open CDM ( Ocdm), with $\Omega_m=1/3$, $\Omega_{\Lambda}=0$ and curvature $\Omega_k=2/3$. (The variant, tilted Tcdm model is identical in its position to Scdm).
• Figure 2: The Cosmic Triangle Observed represents current observational constraints. The tightest constraints from measurements at low red shift (clusters, including the mass-to-light method, baryon fraction, and cluster abundance evolution), intermediate red shift (supernovae), and high red shift (CMB) are shown by the three color bands (each representing 1-$\sigma$ uncertainties). Other tests discussed in the paper are consistent with but less constraining than the constraints illustrated here. The cluster constraints indicate a low-density universe; the supernovae constraints indicate an accelerating universe; and the CMB measurements indicate a flat universe. The three independent bands intersect at a flat model with $\Omega_m\sim 1/3$ and $\Omega_{\Lambda}=2/3$; the model contains a cosmological constant or other dark energy.
• Figure 3: The evolution of cluster abundance as a function of red shift is compared with observations Bahcall98 for massive clusters (above $10^{15}$ solar masses within a 2 Mpc radius, assuming $H_0=65$ km s$^{-1}$ Mpc$^{-1}$). Only the $\Lambda$ cdm and Ocdm fit well the observed cluster abundance at $z\sim 0$, although the Tcdm fits much better than the Scdm model. See also Fig. 4. All four models are normalized to the cosmic microwave background fluctuations on large scales (see text). The observational data points Bahcall98 (with 1- and 2-$\sigma$ error-bars) show only a slow evolution in the cluster abundance, consistent with low $\Omega_m$ models and inconsistent with $\Omega_m=1$.
• Figure 4: The mass power spectrum represents the degree of inhomogeneity in the mass distribution as a function of wavenumber, $k$. (The wavenumber is inversely proportional to the length scale; small scales are to the right (large k's), and large scales are to the left (small k's)). COBE measurements of the CMB anisotropy (boxes on left) and measurements of cluster abundance at $z \sim 0$ (boxes on right) impose different quantitative constraints for each model; the constraints have been color-coded to indicate the model to which they apply. All curves are normalized to the CMB fluctuations on large scales ( i.e., curves are forced to pass through the COBE error boxes on left). Note that the COBE-normalized Scdm model significantly overshoots the cluster constraint (green box on right). The data points with open circles and 1$\sigma$ error bars represent the APM galaxy red shift survey Peac97; if one assumes bias, then this set of points can be shifted downwards to match the model, but the shape of the spectrum suggested by the data is unchanged.
• Figure 5: Supernovae as standard candles: the relation of observed brightness (in logarithmic units of "magnitude") vs. red shift for Type Ia supernovae observed at low red shift by the Calan-Tololo Supernova Survey and at high red shift by the Supernova Cosmology Project is presented (with 1$\sigma$ error bars) and compared with model expectations. (Brighter is down and dimmer is up.) (All $\Omega_m=1$ flat models yield identical predictions in this method, thus Tcdm is identical to Scdm.) The strong gravitational pull exerted by $\Omega_m=1$ models (such as Tcdm or Scdm), decelerates the expansion rate of the universe and produces an apparent 'brightening' of high red shift SNIa, whereas the effect of a cosmological constant accelerating the expansion rate (as in $\Lambda$ cdm) is seen as a relative 'dimming' of the distant SNIa caused by their larger distances. The lower-right plot shows a close-up view of the expected deviations between the three models as a function of red shift. The background color (and shading of the data points) indicates the region for which the universe's expansion would accelerate (yellow) or decelerate (red) for $\Omega_m \sim 0.2$. (Higher values of $\Omega_m$ would extend the yellow accelerating-universe region farther down on this plot.) Similar results are found by the HZS team Riess98, as discussed in the text. The results provide evidence for an accelerating expansion rate.