# PDF Exploring Topology of the Universe in the Cosmic Microwave Background

The cosmic textures problem does not arise if the relevant fields do not exists. The real importance of inflation is that it provides a causal model for how seeds for structure formation in the expanding universe came into being. The real universe is only approximately a Robertson-Walker spacetime. The first term does not grow fast enough to generate the structures we see today from statistical fluctuations. This creates an almost scale-free power law spectrum spectrum of scalar and tensor perturbations at the end of the inflationary era Mukhanov, These fluctuations are then processed by pressure gradients during the hot big bang era, generating baryon acoustic oscillations, and giving a spectrum of primordial perturbations on the Last Scattering Surface with acoustic peaks, that then forms the basis for structure formation through gravitational attraction after decoupling.

As stated above, the real importance of inflation is that it gives a mechanism for the generation of an almost scale-free spectrum of primordial perturbations that lead to structure formation in the universe after decoupling. It is the first theory to give such a causal mechanism for the origin of structure. Thus it has to be determined by observations. Furthermore we do not have a good theory as to how the quantum fluctuations generated during the inflationary epoch become classical by the end of inflation decoherence does not do the job, as some have suggested, becasue while it gets rid of entanglement it does not get rid of superpositions.

This is a major lacuna in the theory. A key issue as regards the dynamics of the universe is whether it had a start. This has different answers if we consider General Relativity with classical matter, General Relativity with quantum fields, and quantum gravity. Note that the present cosmological constant will not be able to prevent this, as it is constant, but the matter would have had a much higher energy density in the early universe. It represents an edge to spacetime, where space, time, and matter come into existence all timelike geodesics are incomplete towards the past.

Physics as we know it comes to an end then because spacetime did not exist. Classical General Relativity, Singularity theorems A key issue is whether this prediction is a result of the high symmetry of these spacetimes, and so they might disappear in more realistic models with inhomogeneity and anisotropy.

Many attempts to prove theorems in this regard by direct analysis of the field equations and examination of exact solutions failed. The situation was totally transformed by a highly innovative paper by Roger Penrose in that used global methods and causal analysis to prove that singularities will occur in gravitational collapse situations where closed trapped surfaces occur, a causality condition is satisfied, and suitable energy conditions are satisfied by the matter and fields present.

Stephen Hawking then showed that time-reversed versions of this theorem would be valid in an expanding universe, because time reversed closed trapped surfaces occur in realistic cosmological models; indeed their existence can be shown to be a consequence of the existence of the cosmic microwave blackbody radiation Hawking and Ellis, Thus the prediction of a start to the universe is not dependent on the high symmetries of the Robertson-Walker geometries.

According to classical general relativity, they can be expected to occur in realistic classical cosmological models. John Wheeler emphasized that existence of spacetime singularities - an edge to spacetime, where not just space, time, and matter cease to exist, but even the laws of physics themselves no longer apply - is a major crisis for physics:.

How could physics lead to a violation of itself -- to no physics? Quantum Fields As is shown by inflationary models, it is no longer necessarily true that the energy condition 54 holds once quantum fields are taken into account see This means that despite some contrary claims, it is after all possible there are singularity-free models when this is taken into account, because such fields are expected to be important in the early universe. Quantum gravity This is of course still a prediction of the classical theory of gravitation, even if the matter source is a quantum field.

The real issue is whether there will be a singularity at the start of the universe when we take full quantum gravity into account. It is still not known if quantum gravity solves this issue or not, primarily because we do not know what the true nature of quantum gravity is. There are hints from loop quantum cosmology that quantum gravity might remove the initial singularity, but the issue is not yet resolved. Thus in the end the outcome of the classical singularity theorems is a prediction that there most likely was a quantum gravity era in the very early universe, but not a statement as to whether a physical singularity existed or not.

The problem we are running into here is what we will call the physics horizon. Many phenomena such as baryogenesis and reheating at the end of inflation still need to be understood in better detail, but we can't test the relevant physics in the laboratory or in colliders at present, or perhaps ever. And no matter how we improve colliders, we have no hope of reaching the Planck energy to explore experimentally the nature of quantum gravity. Thus there are energies beyond which we will never be able to test physics for both experimental and economic reasons.

The physics the other side of this horizon will always be speculative. We will be able to make well-informed guessses as to what its nature is likely to be, but we will never be able to directly test such hypotheses. We may be able to test some of their implications for cosmology, such as whether they imply an inflationary era or not; but it is extremely unlikely we will ever be able to prove that what we hypothesize about the relevant physics at such very early eras is the only possibility.

This applies of course specifically to any theories we may have about creation of the universe. The core feature of any scientific enterprise is the way we can experimentally or observationally test our proposed theories. Cosmology now has an abundance of extraordinarily sensitive observations with which to test its models; nevertheless, there are strict limits to what such tests can achieve. Key observable variables resulting are. Galaxies, radios sources, and qso's do not do the job because of their intrinsic variability. However Supernovae of type SN1A have been found to be reliable standard candles, because the rate of decay after the Supernova maximum is related to its intrinsic luminosity.

Observations of gravitational waves from black hole binary merges have the potential to provide high quality standard candles in the future. A power series derivation of observational relations in generic cosmological models is given by Kristian and Sachs Kristian and Sachs, A series of further phenomena arise in these cases resulting both directly from the model being inhomogeneous, and also from the natures of the structures that arise and reflect the nature of the inhomogeneous cosmological context that lead to their existence.

## [astro-ph/] Exploring Topology of the Universe in the Cosmic Microwave Background

The standard model of cosmology is based on nullcone data electromagnetic radiation of all wavelengths coming to us up the past light cone, with an associated lookback time, together with geological data, deriving from massive particles that originated near our past light cone a very long time ago. Supernovae Supernovae provide good standard candles, determining the deceleration parameter and hence showing the universe is accelerating at recent time Figure 3 , and so dark energy must be present, in agreement with the conclusion from structure formation studies.

Their angular size on the LSS constrains the background cosmological model. The spectrum of this radiation is a perfect black body spectrum, confirming the origin of this radiation from a plasma with matter-radiation equilibrium in the early universe. The Cosmic Blackbody Radiation angular power spectrum has a series of acoustic peaks corresponding to the matter acoustic peaks, extremely well modeled by structure formation theories based on an initial almost scale-free primordial spectrum Durrer, , Peter and Uzan H-P, , see Figure 5 a.

This radiation will be polarised because of interactions with matter, and the Cosmic Blackbody Radiation E-mode polarisation spectrum also has acoustic peaks Figure 5 b. These spectra, and particularly the positions and relative heights of the peaks, importantly constrains inflationary universe models Ade et al. There will also be B-mode polarisation if gravitational waves are present; presence or absence of such modes is a key test of inflationary models.

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The present best values provide evidence against a quadratic inflaton potential, as is required for the simplest version of chaotic inflation. The concordance model is characterised by a set of parameters that are determined by taking all these observations into account. Not all groups use precisely the same set, but a very useful comprehensive survey is given by the Particle Data Group Lahav and Liddle, with measured values updated annually.

The main ones given by the Planck group Ade et al. We know the neutrino background must be present, even though we cannot measure it. This is an example of how good limits on background model parameters follow from the observations related to the perturbed model. These values directly place limits on the slow roll parameters 43 via 52 , and hence constrain inflationary models Ade et al. In particular they give an upper limit on the energy scale of inflation. An important future effort will be getting more precise limits on this ratio by observations of B-mode Cosmic Microwave Backround polarisation.

Neutrino parameters Cosmological models with and without neutrino mass have different primary Cold Dark Matter power spectra, so one obtains limits on the number of neutrinos and on neutrino masses. The Planck team state Ade et al. This is a striking demonstration of how cosmological observations can help determine parameters of the standard model of particle physics.

Density Parameters The density parameters of the concordance model determined by the various observations are shown in Figure 6. The supernova data represent direct measurements of the parameters via the geometry of the background models. The Baryon Acoustic Oscillations and Cosmic Microwave Background data represent measurements of these parameters via the effects of the background model on structure formation, through the coefficients in The data are compatible with each other, but it is the latter that provide the tightest limits. The amount of baryonic matter is determined by nucleosynthesis theory and observations Uzan J-P, , which also place limits on the number of neutrinos.

The key point resulting is that the dominant dark matter is not ordinary baryonic matter: its nature is unknown. Furthermore the nature of dark energy is also unknown. However these two components dominate the dynamics of the universe Figure 6 b. The overall conclusion is that due to a great many very advanced large scale observational projects, cosmology is now a very data rich subject with many observations supporting the concordance model see Figures A variety of different kinds of data all agree on the same basic model, which gives it much more credibility than if there were just a few items supporting the overall model.

There are unresolved issues about the physics involved, but that is because this is a work in progress. A key finding is the existence in cosmology of limits both to causation, represented by particle horizons , and to observations, represented by visual horizons.

They will exist in universe models that accelerate forever, but are irrelevant to observational cosmology. Thus it is an absolute limit to causal effects. Much confusion about their nature was cleared up by Rindler in a classic paper Rindler, , with further clarity coming from use of Penrose causal diagrams for these models Hawking and Ellis, These show that particle horizons occur if and only if the initial singularity is spacelike.

There are many statements in the literature that such horizons represent motion of galaxies away from us at the speed of light, but that is not the case; they occur due to the integrated behaviour of light from the start of the universe to the present day. We cannot see all the way to the particle horizon, because the early universe was opaque. The comoving visual horizon is the most distant matter we can detect by electromagnetic radiation of any kind Ellis, Maartens and MacCallum, It necessarily lies inside the particle horizon. The visual horizon size can be 42 billion light years in an Einstein de Sitter model with a Hubble scale of 14 billion years, because of the changing expansion rate of the universe given by There will be corresponding horizons for neutrino observations, arising from the neutrino decoupling time, and gravitational waves, corresponding to the time of ending of gravitational wave equilibrium with other matter and radiation in the very early universe.

However cosmological observations to those distances by directly detecting neutrinos and primordial gravitational waves would appear very unlikely. For practical purposes the observational limit is given by the visual horizon, the comoving matter comprising that horizon being the matter seen by COBE , WMAP , and Planck satellites Figure 7. The size of the particle horizon at the time of decoupling represents the largest scale at which matter see in Figure 7 can have been causally connected at that time. A partial solution is provided by inflationary universe models, when the expansion at these very early times was almost exponential, as in However one must be cautious here: 62 is valid only in a Robertson-Walker geometry, or approximately in an almost Robertson-Walker geometry, and will be inapplicable if the universe is not spatially homogeneous to begin with.

Furthermore, having causal contact is necessary but not sufficient: one also needs a mechanism to create uniformity. In a genuinely inhomogeneous model, the problem remains open, and inflation does not provide a solution. As mentioned above, complex topologies are possible for all three spatial curvatures, resulting in altered number counts and Cosmic Microwave Background Radiation anisotropy patterns if the smallest identification scale is less than the size of the visual horizon.

Then we live in a "small universe" where we have seen right round the universe since last scattering. This would result in several observational signatures, in particular there would be identical circles of temperature fluctuations in the Cosmic Microwave Background Radiation sky that would depend explicitly in the specific spatial topology. The simplest such models have been ruled out by the Planck observations, but some complex topologies might still be viable. Checking all such possibilities is a massive observational task.

The Robertson-Walker models have an exceptionally simple geometry. Other geometries have been explored, and this is an important exercise, for one cannot put limits on anisotropy and inhomogeneity if one does not have anisotropic and inhomogeneous models where one can compute observational relations which one can compare with the data. One also needs to explore the outcomes of alternatives to standard General Relativity Theory to see if they can do away with the need for dark energy or dark matter.

Special cases Locally Rotationally Symmetric models allow higher symmetries. This family of models allows a rich variety of non-linear behaviour, including. Dynamical systems methods can be used to derive phase planes showing the dynamical behaviour of these solutions and the relations of families of such models to each other Wainwright, If a cosmological model is generic, it should include Bianchi anisotropic modes as well as inhomogeneous modes, and may well show mixmaster behaviour.

These models are discussed in the Scholarpedia article " Bianchi universes , by Pontzen. The growth of inhomogeneities may be studied by using exact spherically symmetric solutions, enabling study of non-linear dynamics. The solutions generically have a matter density that is radially dependent, as well as a spatially varying spatial curvature and bang time. These models can be used to study how a spherical mass with low enough kinetic energy breaks free from the overall cosmic expansion and recollapses to form a local bound system.

They can have a complex singularity behaviour, but this is unrealistic as the early universe will not be pressure free. Near the singularity they can however be velocity dominated. However although the supernova and number count observations can be explained in this way, detailed observational studies based in the kinematic Sunyaev-Zeldovich effect show this possibility is unlikely.

## The Universe Is Flat — Now What?

They are based in the realisation that most of the universe is in fact empty space, so instead of using perturbed FLRW models one attempts to patch empty Schwarzschild solutions together in such a way that the average distance between them changes with time. One can then show that this distance obeys a Friedmann like equation and examine observational properties of such models. Other dynamics than general relativity might be applicable on the cosmological scale. This class of models span only a subclass of the more general scalar-tensor theories.

Some cosmologists propose a multiverse exists: it is suggested the observed expanding universe domain bounded by the visual horizon is just one of billions of such domains, in each of which different physics or different cosmological parameters obtain, i. The reasons for proposing this are primarily,. Because of the existence of visual and physical horizons discussed above, the supposed existence of these other universe domains is not observationally testable in the usual sense.

Nevertheless it is claimed by some that the observed small value of the cosmological constant about orders of magnitude less than the value suggested by quantum field theory should be taken as adequate proof that they exist. However this is not the majority view of working cosmologists. Claims of infinities Some multiverse enthusiasts insist that there are not just a finite number of other such domains but an infinite number, each containing an infinite number of galaxies. This is certainly not an observationally testable scientific claim, nor does it inevitably follow from established local physics.

It should be regarded as metaphysical speculation rather than established science. Standard cosmology is a major application of GR showing how matter curves spacetime and spacetime determines the motion of matter and radiation. It is both a major success, showing how the dynamical nature of spacetime underlies the evolution of the universe itself, with this theory tested by a plethora of observations Ade et al.

### Relationship to the Big Bang:

While much of cosmological theory the epoch since decoupling follows from Newtonian gravitational theory, this is not true of the dynamics of the early universe, where pressure plays a key role in gravitational attraction: thus for example Newtonian Theory cannot give the correct results for nucleosynthesis.

The theory provides a coherent view of structure formation with a few puzzles, for example the core-cusp problem , and hence of how galaxies come into existence. A worry about differing values for the Hubble constant obtained by different methods Addison et al. It raises the issue that the universe not only evolves but at least classically had a beginning, whose dynamics lies outside the scope of standard physics because it lies outside of space and time. Limits to physical cosmology The domain of validity of the models has been discussed above.

We can only be sure of their validity as geometric and dynamic models as regards the matter within the domain we can see, curtailed by the visual horizon. The smallest such d is called the diameter of the universe, in which case the universe has a well-defined "volume" or "scale. Assuming a finite universe, the universe can either have an edge or no edge. Many finite mathematical spaces, e. Spaces that have an edge are difficult to treat, both conceptually and mathematically.

Namely, it is very difficult to state what would happen at the edge of such a universe. For this reason, spaces that have an edge are typically excluded from consideration. However, there exist many finite spaces, such as the 3-sphere and 3-torus , which have no edges. Mathematically, these spaces are referred to as being compact without boundary. The term compact basically means that it is finite in extent "bounded" and complete. The term "without boundary" means that the space has no edges. Moreover, so that calculus can be applied, the universe is typically assumed to be a differentiable manifold.

A mathematical object that possesses all these properties, compact without boundary and differentiable, is termed a closed manifold. The 3-sphere and 3-torus are both closed manifolds. The curvature of the universe places constraints on the topology. If the spatial geometry is spherical , i.

For a flat zero curvature or a hyperbolic negative curvature spatial geometry, the topology can be either compact or infinite. In general, local to global theorems in Riemannian geometry relate the local geometry to the global geometry. If the local geometry has constant curvature, the global geometry is very constrained, as described in Thurston geometries. In a universe with zero curvature, the local geometry is flat. The most obvious global structure is that of Euclidean space , which is infinite in extent.

Flat universes that are finite in extent include the torus and Klein bottle.

Moreover, in three dimensions, there are 10 finite closed flat 3-manifolds, of which 6 are orientable and 4 are non-orientable. These are the Bieberbach manifolds. The most familiar is the aforementioned 3-torus universe. In the absence of dark energy, a flat universe expands forever but at a continually decelerating rate, with expansion asymptotically approaching zero. With dark energy, the expansion rate of the universe initially slows down, due to the effect of gravity, but eventually increases.

The ultimate fate of the universe is the same as that of an open universe. A flat universe can have zero total energy. This was proposed by Jean-Pierre Luminet and colleagues in [7] [17] and an optimal orientation on the sky for the model was estimated in A hyperbolic universe, one of a negative spatial curvature, is described by hyperbolic geometry , and can be thought of locally as a three-dimensional analog of an infinitely extended saddle shape. There are a great variety of hyperbolic 3-manifolds , and their classification is not completely understood.

Those of finite volume can be understood via the Mostow rigidity theorem. For hyperbolic local geometry, many of the possible three-dimensional spaces are informally called "horn topologies", so called because of the shape of the pseudosphere , a canonical model of hyperbolic geometry. An example is the Picard horn , a negatively curved space, colloquially described as "funnel-shaped". When cosmologists speak of the universe as being "open" or "closed", they most commonly are referring to whether the curvature is negative or positive. These meanings of open and closed are different from the mathematical meaning of open and closed used for sets in topological spaces and for the mathematical meaning of open and closed manifolds, which gives rise to ambiguity and confusion.

In mathematics, there are definitions for a closed manifold i. A "closed universe" is necessarily a closed manifold. An "open universe" can be either a closed or open manifold. If one applies Minkowski space -based special relativity to expansion of the universe, without resorting to the concept of a curved spacetime , then one obtains the Milne model. Any spatial section of the universe of a constant age the proper time elapsed from the Big Bang will have a negative curvature; this is merely a pseudo-Euclidean geometric fact analogous to one that concentric spheres in the flat Euclidean space are nevertheless curved.

Spatial geometry of this model is an unbounded hyperbolic space. The entire universe is contained within a light cone , namely the future cone of the Big Bang. The apparent paradox of an infinite universe contained within a sphere is explained with length contraction : the galaxies farther away, which are travelling away from the observer the fastest, will appear thinner.

It is incompatible with observations that definitely rule out such a large negative spatial curvature. However, as a background in which gravitational fields or gravitons can operate, due to diffeomorphism invariance, the space on the macroscopic scale, is equivalent to any other open solution of Einstein's field equations. From Wikipedia, the free encyclopedia. The local and global geometry of the universe. For the Bee Gees song, see Edge of the Universe song. This article has multiple issues.

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April Early universe. Subject history. Discovery of cosmic microwave background radiation. Religious interpretations of the Big Bang theory. Main article: Observable universe. See also: Distance measures cosmology. Main article: Milne model. Ellis; H. Theoretical and Observational Cosmology. Bibcode : ASIC.. Retrieved 16 March Yoo LXXIV1: Topology of the universe and the cosmic microwave background radiation. The early universe and the cosmic microwave background: theory and observations.

Bibcode : eucm.