An Expanding Universe

In the 1920s E. Hubble and his coworkers recognized that the spectral lines of galaxies were shifted systematically to the red with respect to the position of laboratory wavelengths. The redshift z of a galaxy can be written as z, Vr is the radial velocity  

Xobs denotes the observed wavelength, Xiab the rest wavelength, the index lab means 

The systematic redshift of distant galaxies can be interpreted as an expansion of 

Hanslmeier, Water in the Universe, Astrophysics and Space Science Library 368, DOI 10.1007/978-90-481-9984-6 8, Springer Science+Business Media B.V. 2011 

Cepheids, Novae, Supernovae and other bright objects. For these objects their true luminosities are known, therefore, the distance can be calculated by comparing their measured brightness with the true luminosity. There exist many other methods for the determination of distances of galaxies and again the reader is referred to the textbooks. 

The astronomical evidence that prompted abandonment of the concept of an equilibrium universe, in favour of an expanding one, is largely due to the work of Edwin Hubble, but without the explosive development of physics in the first quarter of the 20th century, his work would not have been possible. As with Newton and Darwin, new ideas were in the air, and Hubble was the one to complete the synthesis of ideas that created a new world view. Some of the other important players include James Clerk Maxwell (1831-79), Johann Balmer (1825-1898), Max Planck (1858-1947), Albert Einstein (1879-1955) and Niels Bohr (1885-1962). Without the insight of these scientists Hubble s observations make no sense. 

The most important development was the ability to make reliable estimates of astronomical distances. To establish the distance to a star, astronomers rely on the phenomenon of steUar parallax. The first successful measurement of this type was performed at the Cape Observatory in the early 19th century. 

Parallax is the apparent shift in the position of an object against a distant background because of a change in the observer s point of view. 

Stars exhibit the same phenomenon. As the earth orbits the sun nearby stars appear to move back and forth against the background of more distant stars. The parallax of the star is the half angle p which defines the shift of the star s apparent position viewed from opposite sides of the earth s orbit around the sun. To derive the distance d from the angular measurement it is necessary to know the distance between planet Earth and the sun, which defines the astronomical unit (All) of distance. If the angle p is measured in seconds of arc, the distance d, defined as 

The distance between Earth and Sun is obtained by measuring the parallax of another planet and use of Kepler s third law for the orbital periods of the planets. The closer an object is to the observer, the larger its parallax. On occasion the minor planet Eros approaches the earth more closely than any of the major planets and measurement of its parallax displacement during diurnal rotation of the earth at such time provides one of the best estimates of the AU. 

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There is no evidence that Minkowski space is flat on the large scale. The assumption of euclidean Minkowski space could therefore be, and probably is an illusion, like the flat earth. In fact, there is compelling evidence from observed spectroscopic red shifts that space is curved over galactic distances. These red shifts are proportional to distances from the source, precisely as required by a curved space-time[52j. An alternative explanation, in terms of an expanding-universe model that ascribes the red shifts to a Doppler... 

Although in Minkowski space, this corresponds to energy and momentum conservation, this is no longer the case in an expanding universe. 

So far, we have been able to build a scenario which solves the horizon, flatness and monopole problems. As we already said, this scenario also explains the existence of an almost scale invariant spectrum in the cosmological perturbations. However, the derivation of this crucial result is significantly more involved than the previous one. First because we need to do a careful study of the cosmological perturbation in the context of general relativity and in an expanding universe. Second because we then need to solve these equations in the specific case of inflation. Third, because perturbation theory only tell of the evolution of cosmological perturbations, so that we need to specify the... 

In 10.4,1 will discuss the evolution of density perturbations in an expanding Universe and in 10.5 the plasma oscillations thereby induced. In 10.6 I will introduce that statistical tools to describe the distribution of CMB tem-pertatures on the sky, and in 10.7 how the cosmological parameters influence the distribution of temperatures. Finally, in 10.8 I will briefly review how we actually analyze CMB data and conclude in 10.8. 

All of these effects are included in codes like CMBFAST (Seljak and Zal-darriaga 1996 Zaldarriaga and Seljak, 2000, http //www.cmbfast.org/) and CAMB (Lewis et al., 2000, http //camb.info/) which solve the combined Boltzmann and linearized Einstein equations in an expanding Universe. These codes allow one to calculate the CMB temperature power spectrum for a given model. A sample of spectra for various input cosmological parameters is shown in Figure 10.2. 

Man is. . . related inextricably to all reality, known and unknowable. .. plankton, a shimmering phosphorescence on the sea and the spinning planets and an expanding universe, all bound together by the elastic string of time. It is advisable to look from the tidal pool to the stars and then back to the tide pool again. 

The positive value for the change in entropy means that there will be more disorder. An expanding universe and a food fight are sure signs of more disorder. 

The Friedmann equation (eq. 2.9) relates the time-dependence of the scale factor to that of the density. The Einstein equations yield a second relation among these which may be thought of as the surrogate for energy conservation in an expanding universe. 

Wu, C. H., Apweiler, R., Bairoch, A., et al. (2006) The Universal Protein Resource (UniProt) an expanding universe of protein information. Nucleic Acids Res. 34, D187-D191. 

The assumptions on which the model rests are too crude to be realistic. In particular, the assumption of a universal time coordinate directly contradicts the basis of general relativity. To avoid the problem de Sitter repeated the calculation based on relativistically curved space-time, with the surprising result of an empty universe with variable radius. The traditional interpretation of this result as an expanding universe is not unique. It could just as well imply space-time with continuous curvature, characteristic of projective space. In the event, both solutions were soon superseded by an expanding-universe cosmology based on a Doppler interpretation of galactic redshifts. 

Theories like those of Lemaitre or Friedmann, which predict an expanding universe, are all based on forcing an affine metric, such as the Robertson-Walker metric, on the projective geometry of space-time. This has the effect of splitting local Minkowski space into separate space and time coordinates, without the natural complex relationship that ties space and time together. 

These new methods of measuring astronomical distances inaugurated modern astronomy and astrophysics and the interpretation of such measurements in terms of an expanding universe led on to the modern standard cosmology. 

One of the misleading statements, used to introduce most accounts of big-bang theory, is that the theory is based on general relativity. It is, more precisely, based on the assumption that the universe expands. The only connection between expanding universe and general relativity occurs in de Sitter s 1917 solution (Section 6.3) in terms of a variable hypersphere, which is characteristic of curved four-dimensional space-time. To define this variability as an expanding universe it is necessary to separate time from space coordinates, an operation which is not allowed in relativity theory. The expansion of such a Newtonian universe is deduced from galactic redshifts, but as stated by Parish (1981) ... 

In an expanding universe 2ct defines the particle horizon that limits the independent region over which a homogeneous equilibrium state can develop at time t. The radius of the initial homogeneous region, at a temperature of... 

There is no more important concept in standard cosmology than Hubble s constant, which relates spectroscopically measured redshifts in galactic light to the estimated distance of the source and its inferred rate of recession. This definition specifies a proportionality constant only for Doppler shifts. It also occurs for redshifts with a linear distance dependence due to other causes, in which case it has no connotation to an expanding universe. It is undefined for redshifts, independent of distance. 

After two thousand years we are back in a similar situation. While the evidence for an expanding universe was accumulating and the quantum theory was being formulated, theoretical advances with far-reaching consequences for cosmology were made elsewhere. Advances which led to the formulation of the theory of relativity, the only theory that enquires into the nature of matter and its distribution through space. 

These derivatives provide additional sources and destinations for inflation-linked flows, lower the barrier to entry, and bring in an expanded universe of market participants. This has, over time, enhanced and made more efficient the underlying bond market. 

The unexpected appearance of complex operators is also associated with nonzero commutators and reflects the essential two-dimensional representation in MP Minkowski space-time. In four-dimensional space-time, all commutators are non-zero, as appropriate for wave motion of both quantum and relativity theories. An important consequence is that local observation has no validity on global extrapolation, as evidenced by the appearance of cosmical red shifts in the curved manifold and the illusion of an expanding universe. 

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