Measuring the Cosmological Parameters

We have known since the late 1920s that the universe is expanding.Quantifying the expansion is done conventionally in terms of twonumbers. H0, the Hubble constant, measures the current expansion rate of theuniverse, and q0 is the rate at which the expansion is slowing, or decelerating,because of the self-gravitational pull of all the matter in the universe.The standard cosmological solutions of Einstein's equations of generalrelativity are specified by H0 and q0. H0−1, the inverse of the Hubble constant, is a measure of the current ageof the universe, while q0 is a measure of how long the universe will continue to expand.

Two additional quantities that affect the expansion are the cosmologicalconstant, Λ, the vacuum energy density of the universe, and Ω, the ratio of the total mass/energy density in the universeto the critical density, which is required to just bring the expansionto a halt in the infinite future. Consider first the case where Λ = 0. If Ω < 1, the self-gravity of the universe is insufficient ever to stopits expansion (an “open” universe). If Ω > 1, the expansion will eventually stop and theuniverse will collapse (a “closed” universe).

The Λ term represents a strange phenomenon. As noted, it measures theenergy density of a vacuum, which remains constant as the universeexpands, unlike ordinary matter and radiation whose densities decreasewith expansion. A non-zero vacuum energy density would mean thatenergy is present in an empty universe even in the absence of particlesor radiation. Though it seems odd, such a possibility is consistentwith Einstein's theory of gravitation. The key point about Λ is that such energy generates gravity even withoutnormal matter or radiation—hence, gravity from a vacuum. Becauseof this phenomenon, and because Λ remains constant as the universeexpands (a vacuum cannot be diluted), the existence of non-zero Λ radically changes the dynamics of the universe. This is the keyconcept that underlies inflation, which is discussed in section V. If there is currently no vacuum energy density in the universe,then Λ = 0 and q0 = Ω/2; most cosmologists believe that Λ is 0, but understanding why it is so small is a profound questionof fundamental physics.

The Hubble constant, H0

The Hubble constant measures how fast the universe is expanding today.In addition, the age of the universe can be expressed as approximately(2/3)H0−1 (the precise value depends on Ω and Λ). The accurate determination of H0 has occupied astronomers for several decades, and the scientificmotivation for finding an accurate value of this critical constanthas become ever stronger. Another key use of H0 is to estimate the physical distance and size of objects that havemeasurable redshifts. For example, the size of the largest structuresin the universe is related to the distance that light could havetraveled in the time up to the epoch when matter began to dominateover radiation. The corresponding size scale today is an importantrelic of the Big Bang, but its value is proportional to H0−2 and therefore suffers from the current uncertainty. An accurate measurementof H0 is crucial for assessing whether the detailed models of theevolution of structure in the universe can be reconciled with a widerange of observations.

Current estimates of H0 range between 45 km/s per megaparsec (a megaparsec is about 3 millionlight-years) and 90 km/s per megaparsec. A value near 45 to 50 isconsidered low and 80 to 90 is considered high. If Λ = 0 and Ω = 1 (the theoreticallypreferred values for these parameters), then H0 = 50 km/s per megaparsec means an age of 13.3 billion years, andH0 = 90 km/s per megaparsec means 7.4 billion years. A fundamentalreality check comes from requiring the oldest stars in our galaxyto be younger than the age of the universe. This requirement, a logicalnecessity, sets an upper limit to H0. Astronomers' best estimates of the age for such globular cluster stars are near15 billion years, in conflict with

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