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PROBLEM 7: THE FLATNESS PROBLEMWhy is the universe flat? The universe is flat. Not flat like a disc because that is a twodimensional surface but the shape of spacetime itself for the whole universe is flat. This may not seem surprising but it is. We know that spacetime can be curved. From the theory of relativity, Einstein told us that gravity comes from the curvature of space, so curved space exists all over the universe wherever there is an)' object, big or small. Figure 5.4 shows the Sun curving space around it, causing the Earth to move within the curved space and orbit the Sun. This is gravity. The smaller curvature of space due to the Earth can also be seen in the figure. So if space is curving all over the universe what is the total amount of curvature in the universe? This turns out to be zero. This is what we mean by flat. Why is there no curvature of the universe as a whole? The universe could have any curvature and being flat is just one, very specific, value. So why is the universe flat? This is the Flatness Problem. Figure 5.4: Curved spacetime. The Sun and Earth warp spacetime, represented by the grid, and then move within the warped spacetime. Gravity is the curvature of spacetime. Credit: T. Pyle/Caltech/MIT/LIGO Lab. As cosmologists, it is particularly important that we know the shape of the universe because it affects our calculations of the cosmological parameters and it will determine what will happen to the universe in the future, that is something we would like to know. There are three types of curvature that the universe can have: positive, negative or zero. Figure 5.5 shows what this means in two dimensions. Zero curvature is a flat plane, like a piece of paper. A positive curvature is a shape like a sphere. A negative curvature is a shape like a saddle. The geometry of threedimensional volumes behaves in a similar way to twodimensional surfaces so we will stick with the twodimensional analogy because it is much easier to understand (and to draw). In a flat space parallel lines stay parallel and a triangle will have angles that add up to 180 degrees. This is the geometry that we get taught at school. When we go to a shape with positive curvature then parallel lines do not stay parallel but will meet (like the longitudinal lines of Earth meeting at the North pole) and a triangle’s angles will add up to more than 180 degrees. For the universe, a positive curvature would mean that when we look at further and further distances we would eventually see the same galaxies again and again as we loop round the curved space (just like going round the World and ending up back at home). Space in a positively curved universe would be closed and the universe could end by collapsing back in on itself. In negative curved space the opposite happens and parallel lines get further apart, a triangle’s angles add up to less than 180 degrees and the universe is an open shape and will expand forever. Using the CMB, and these properties of triangles, we can measure the curvature of the universe. The temperature map of the CMB (see Figure 4.2) Figure 5.5: Three possible curvatures of spacetime in two dimensions: a sphere (positive curvature), a saddle (negative curvature) and flat (zero curvature). The shape of a triangle is shown on each surface. Credit: NASA/WMAP Science Team. shows the tiny temperature fluctuations (the anisotropies) that we measure. In section 4.2.2 we saw that these fluctuations were formed from the oscillations of matter in the early universe and are determined by the distance of the sound horizon. We know what this distance should be and we also know when the CMB formed. So we can make a triangle with one side being either end of a ‘blob’ on the CMB anisotropy map (formed from the oscillations) and the other two sides the distance they were from us when they formed. If the universe is flat then the smallest angle in the triangle should be 1 degree. If it is positively curved it will be larger and if it is negative it will be smaller. What we measure is that it is exactly 1 degree to an accuracy of 0.2%. This may not seem very flat but the real problem comes when we extrapolate back to the early universe. The expansion of the universe over time will have exaggerated any curvature. So to get a flat universe to a 0.2% accuracy today then at the beginning of the universe it had to be flat to one part in 10^{GO}. This is like changing the Sun’s mass by adding a couple of particles. To get space to be flat to this accuracy cannot be a coincidence but the ACDM model does not have an explanation for it. This is why the Flatness Problem is a problem. Once again inflation theory comes to the rescue. It solves the Flatness Problem by having a period of massive expansion in the very early universe. Any curvature the universe had before inflation will be enormously expanded. Our observable universe today is just a small part of the total universe so that we are seeing just a small patch of curvature in a bigger curved space. This patch will look flat because it has been stretched so much. This is similar to the way we see the Earth as flat because we are seeing a small patch of a very large sphere. It may be that the universe is not as flat as we think. Recent work by Eleanor di Valentino and team [100] in 2020 reanalysed the CMB data. They looked at all the ‘blobs’ on the CMB map, not just the ones at the sound horizon, and they looked at the gravitational lensing of the CMB light. What they found was that there was more matter in the universe than previously measured. The extra matter is enough to give the universe a positive curvature. It is not very curved, so probably not enough to close the universe (expansion and acceleration means it won’t collapse), and beyond a certain distance expansion will be greater than the speed of light so there will be no lapping of the universe and we won’t be seeing the same galaxy multiple times. As our measurements of matter density, expansion, and dark energy' improve then there is always the possibility that the Flatness Problem may go away but with the evidence we have today the universe looks very flat; flat enough to cause a problem for the ACDM model. Of course the Flatness Problem could be solved by using a different model of the universe than ACDM but we don’t have one that solves it at the moment. Our best solution today is to add inflation to ACDM and for this to happen we need new Cosmological Clues for inflation. PROBLEM 8: THE ANTIMATTER PROBLEMWhere did the antimatter go? Also called The Baryogenesis Problem The universe is made of matter. This seems an obvious statement but from our known laws of physics there should be no matter. In the early universe matter was created. When matter is created, it is always created in pairs of matter and antimatter and very quickly it is destroyed in pairs of matter and antimatter. So if the same amount of matter and antimatter was produced and then destroyed why is there any' matter in the universe? This is the Antimatter Problem. For any' matter to exist in the universe there must have been fractionally more matter than antimatter. So where did that little bit of anti?matter go? Or put another way, why was there more matter than antimatter in the early universe? We have the Big Bang Nucleosynthesis theory that predicts the hydrogen and helium abundance in the universe. We would like a similar theory that explains how much atomic matter formed and why it is 5% of the total energy budget. Such a theory is called ‘baryogenesis’ but we do not have one. It is the hope that if we can find out why matter and antimatter were different in the early universe then maybe that will provide a prediction for the exact amount of matter in the universe today and the baryogenesis problem will be solved. How much difference does there need to be between matter and antimatter? A rough estimate for this is based on how much light and how much matter we see in the universe. For each matter particle in the universe today there are one billion light particles (light comes in packets which we call photons and can be considered as particles). When matter annihilates with antimatter it produces light. We estimate that for each one billion matter particles that annihilated then only one matter particle survived. This gives us a difference of one billion plus one particles of matter to each one billion antimatter particles. A very small, but very important difference. It is this difference that allows there to be any matter in the universe today. The idea of antimatter has been around since the 1880s but the modern concept of antimatter was predicted by Paul Dirac in 1928 [101]. Dirac was working on combining relativity with quantum theory for the electron and he found an elegant mathematical solution but there were extra terms that looked like the opposite particle of the electron. These turned out to be the antimatter of electrons  the positron. Positrons were discovered four years later by Carl D. Anderson [102] when looking at cosmic rays and received the Nobel prize in 1936 for this discovery. Every known particle has an antimatter particle; they are identical in their mass and in every way except they have the opposite charge and spin. Antimatter annihilates rapidly with matter so they are only seen from the interactions with other particles such as in high energy particle accelerators or from cosmic rays. The starting assumption in physics is that the laws of nature are the same for antimatter as for matter. This seems a good place to start but if it was true then there would be no matter. So there must be at least some small difference in the way matter and antimatter behave. We call this an asymmetry; the behaviour between matter and antimatter is not symmetrical. There are three types of symmetry that could be broken by antimatter:
We can combine these actions; a CP symmetry changes the particle’s charge and reflects it in a mirror. If CPT is applied to a matter particle it becomes it’s own antimatter particle. In 1967, Andrei Sakharov [103] formulated three conditions that are necessary for matter to exist today:
The third condition is met in the early universe as it is cooling down and rapidly changing temperature. The second condition requires C symmetry violation and, although theoretically it could be possible, is has not been observed. There have been observations of CP violations. The discovery that the neutral kaon particle shows CP violation [104] won James Cronin and Val Fitch the Nobel prize in 1980. More recently in 2019 [105], evidence was found at CERN for CP violation in the charm quark particle. The first condition requires that the total baryon number must be the same before and after any interaction. If the baryon number of the universe at the Big Bang is assumed to be zero then conservation means that it has to stay zero and there will be as much antimatter as matter, so for there to be a difference then conservation of baryon number must be violated. There is a mechanism for this in particle physics discovered by Gerard ‘t Hooft in 1976 [106]. By adding small corrections to quantum theory, an effect called the Spharelon can destroy the conservation of baryon number; antimatter baryons can turn into leptons (electron type particles) and baryons can turn into antimatter leptons. Spharelons are hypothetical and there is no evidence that they exist and there is no evidence for condition 1. In addition, the calculations show that they would not be strong enough to create the numbers of observed baryons. So it is theoretically possible to meet the three Sakharov Conditions. The problem is that the effects are too small to be able to explain the amount of baryons that there are in the universe today. So maybe a different condition is needed. A possible extra condition is to say that CPT symmetry needs to be violated. This may produce a strong enough effect to create the quantity of baryons observed. The ALPHA experiment at CERN is testing CPT symmetry by creating antimatter hydrogen and comparing it’s properties with normal matter hydrogen (NASA estimated that a gram of antihydrogen would cost $62 trillion to make). The evidence rules out the known interactions of the Standard Model of particle physics to solve the Antimatter Problem so another approach is needed. Perhaps it could be solved by a new hypothetical particle such as the inflaton that comes out of inflation theory or the axion that is proposed as a dark matter particle. We will have to wait for any evidence of these to be able to know whether they do. Alternatively, a new theory for the unification of the forces (GUTs) or quantum gravity may provide an answer but a suitable model does not exist yet. All the above discussion is based on the amount of matter and antimatter being created equally just after the Big Bang and then particle interactions changing the proportions. Of course an alternative theory is that they were created unequally, but we do not have any ideas as to why that would happen within the laws of physics. We know the conditions that are needed to solve the Antimatter Problem, and there are known particle interactions that do meet these conditions, but they cannot explain the quantity of baryons that exist today. There are other ideas for new particles that could explain the asymmetry between matter and antimatter but there is no evidence for these. So the Antimatter Problem remains an unsolved problem. We have seen that the ACDM model does not explain all the evidence we have: the flatness, the uniformity and the lack of antimatter. We have also seen that it introduces new concepts that we cannot explain; dark matter, dark energy, the Big Bang, and primordial fluctuations. This is why ACDM is a model and not a theory, there are still anomalies that need to be solved. Inflation theory can explain three of the problems and if inflation is verified then the Horizon Problem, the Flatness Problem, and the Cosmic Web Problem could be solved. The Big Bang Problem may never be solved but we may be able to develop hypotheses consistent with the laws of physics. We continue to hunt for dark matter and explore dark energy. Maybe if we find an axion it could solve the Dark Matter Problem and the Antimatter Problem. Solving these Cosmological Problems may require new technology, or new physics or adding something else to the ACDM model. Solving these problems is what makes science exciting. 
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