Cosmological constant

As shown before (Vermeil, 1917) equation (6.4), with cosmological constant, is the mathematically correct form of the gravitational field equations. 

Although the cosmological constant does not feature in standard cosmology any more it has a prominent meaning in the supportive field theories of physics, associated with virtual particles in the vacuum. In inflation theory it is more precisely identified as a form of antigravity - the same role initially envisaged for it by Einstein. 

Another attractive feature of the cosmological constant is an ability to eliminate the gravitational singularity at zero separation, more popularly ascribed to quantum gravity. Maybe it is antigravity and maybe it is not constant, like a short-range exclusion principle, inversely proportional to distance. 

The cosmological constant poses a major dilemma for standard cosmology and particle physics. In a popular article of twenty years ago (Abbott, 1988), with the byhne  

In cosmological models A is routinely ignored because of its vanishingly small value, but in particle physics it has a non-zero value, arising from the presence of infinitely many virtual particles that constantly fluctuate in and out of the vacuum. Experimental estimates of spatial curvature predict 

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The mass of the universe is in general evenly distributed it acts as Einstein s cosmological constant and accelerates expansion. Poor Agrees well with most of the recent measurements, but the evidence is still thin, and theoretical problems are still unsolved. 

In a cosmology with significant cosmological constant, one can similarly define Qa = A/3/P. 

From Eq. (4.9), show that the contribution of the cosmological constant (aka... 

If, on average, a total column density of 1020 H I atoms is found in the interval of redshift 2.0 to 2.5 along one line of sight by counting damped Lyman-a systems, make an estimate of 2m for the two Universe models without cosmological constant discussed in the text. 

Show that, in a flat Universe model with a cosmological constant, Eq. (12.17) reduces to... 

The critical density is traditionally dehned as that density which separates the closed (finite) universe from the open (infinite) universe in the simplest model available, i.e. in a universe without cosmological constant or quintessence. It corresponds to a universe with zero total energy, where the kinetic energy due to expansion is exactly balanced by gravitational potential energy. The value of the critical density is 10 gcm, which amounts to very httle when compared to a chunk of iron ... 

We note several very general formulations of the problem. A striking example of this is Ya.B. s 1967 paper [14 ], in which he considers the possibility of a theory in which the bare photon field is absent, while the observed electromagnetic field is created entirely by quantum fluctuations of a vacuum. This bold idea, which extends to electrodynamics an earlier idea about gravitational interaction (in part, under the influence of Ya.B. s papers on the cosmological constant), has not yet been either proved or disproved. However, both ideas have elicited lively discussion in the scientific literature. 

Recently, Ya.B. has been working on a complete cosmological theory which would incorporate the creation of the Universe (1982) [45 ]. Let us mention finally that it was Ya.B. who recently gave a profound formulation of the question of the cosmological constant, i.e., the energy density in Minkovsky space (see Section 9). More precisely, the question is formulated thus is the Minkovsky space a self-consistent solution of the equations for all possible fields and the equations of general relativity ... 

J. V. Narlikar, J. C. Pecker, and J. P. Vigier, Some consequences of a spatially varying cosmological constant in a spherically symmetric distribution of matter, J. Astrophys. Astron. 12(1), 7-16 (1991). 

Such a term is called the cosmological constant, and has been historically introduced by Einstein, as a modification of his original theory. We have then... 

When the cosmological constant is zero, there are three types of solutions ... 

The Pressure-less Friedmann Models of the Universe with a cosmological constant... 

Let us suppose that the field behaves like a cosmological constant for some amount of time. In this case, Eq.(7.24) tells us that the expansion is almost exponential (H Const, a oc eHt). Moreover, the relation between a and r/ is... 

In order inflation to proceed for a sufficiently large amount of time, we want the scalar field to behave like a cosmological constant. We therefore first impose that the is much closer w = — 1 than w = — and that it is necessarily... 

Before continuing the study of the dynamics of the inflationary phase, let us focus on one specific example of inflationary scenario chaotic inflation. Historically, this was not the first model that was proposed but we think it was the first to provide a satisfactory scenario. The main difficulty with inflation is to have the slow roll conditions to be satisfied at some epoch. Indeed, as we saw, one need to put the field away from the minimum of its potential for the inflaton to behave like a cosmological constant. The first inflationary model ( Guth 1981) supposed a potential like that of Eq. (7.28) where the field slowly moved away from its minimum because of a phase transition. However, this led to a number of difficulties, see for example Ref. (Liddle Lyth). Fortunately, it was soon realized that it was not necessary to have a time dependent potential for inflation to proceed. Linde (Linde 1985) noticed that inflation could start as soon as the Universe would exit the Planck era. The idea was that it is reasonable to suppose that at the end of the Planck era (when p > ), no large-scale correlation could be expected in the scalar field, so that one could expect very irregular (hence, chaotic) initial condition with... 

There are two problems which appear when one considers the possibility of a small but non zero cosmological constant. The first one has to do with the... 

The biggest mystery related to the cosmological constant lies there indeed, from very general quantum held theory considerations, any particle with momentum k should have a contribution E Tiuo to the total energy even in its lowest energy state, so that one expects that any species should contribute to the vacuum energy density according to... 

As we have seen during the course on inflation, a scalar field can behave as a cosmological constant when its kinetic term becomes negligible in front of its potential term. However, the features of the scalar field we are interested in differ significantly from an inflationary scalar field in the former case, we want a field that is negligible at early times and which dominates afterwards, whereas in the latter case, it is the contrary. Historically, the first scalar field dark energy model was aimed to address the possibility to have some components with a constant equation of state parameter w other than 0 (matter), 1/3 (radiation), —1/3 (curvature) and —1 (cosmological constant) (Ratra Peebles 1988). 

A second relation involving a can in principle be derived when we know the current value of the dark energy equation of state parameter wbe- Indeed, since the potential does not possess a local minimum, the field never stops, so that it never behaves exactly as a cosmological constant. Moreover, even if its equation of state parameter w decays (without ever reaching) toward —1, the rate at which this transition occurs depends on the steepness of the potential the steeper the potential, the slowest w goes toward —1. In particular, one finds, at the epoch Qq 0.7,... 

As compared to a cosmological constant, quintessence modifies the late time evolution of the expansion rate of the universe. It can therefore affect the luminosity distance of supernovae as well as the angular distance of CMB patterns. However the most dramatic difference between a quintessence models and the now standard ACDM scenario comes when one considers structure formation. 

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