In the first part of this thesis we investigate the cosmological evolution of ultralight axion-like (ULA) scalar fields with potentials of the form V(φ) = m2f2[1−cos (φ/f)]2, with particular emphasis on the deviation in their behavior of the corresponding small-φ power law approximations for these potentials: V(φ) ∝φ2n. We show that in the slow-roll regime, when ˙φ2/2V(φ), the full ULA potentials yield a more interesting range of quintessence possibilities than the corresponding power-law approximations. In the second part we study the decay of particles in the future of the accelerating universe.
Professor Sheldon as my advisor, and Professor Kephart and Professor Taylor as sources of information in the two areas of physics that border my interest in cosmology: particle theory and general relativity. Because observations suggest that nearly 70% of the universe's energy density is in the form of this exotic negative pressure component, understanding its nature is a necessary step in understanding the nature of our universe. In this chapter we provide background information on the theoretical cosmology necessary to perform the calculations we plan to perform in the following chapters.
In the third chapter, we study the decay of particles in the future of an accelerated universe.
Friedman-Robertson-Walker Cosmology
Dark energy is among the greatest mysteries of our time; although the accelerated expansion of the universe was confirmed through observation, the theoretical cause of this acceleration, called "dark energy", eludes us. Additional motivation for studying dark energy comes from the Hubble tension, the discrepancy between direct local measurements of the Hubble parameter and the value inferred within the ΛCDM model from measurements of the cosmic microwave background (CMB). Then, we can introduce the Hubble constant, which characterizes the expansion rate of the universe, the observable size of the universe, and the age of the universe's expansion.
We often model the matter and energy in the universe after a perfect fluid, and therefore we must obtain their evolution equations using Einstein's equation. We use Einstein's equations (1.7) and (1.8) to derive the evolution of the fluids in our universe. Then we can calculate the density by inserting our expressions fora(t) and ˙a(t) into equation (1.7).
We can also express the flatness of the universe in terms of the sum of the densities of each of the fluids, matter, radiation, and dark energy (which is also called vacuum energy).
Observational Evidence
The Hubble tension
Apart from the current accelerated expansion of the universe, another phenomenon that potentially requires a physical explanation is the 'Hubble tension', the discrepancy between direct local measurements of the Hubble parameter and the value inferred from measurements of. The possibility that a scalar field may transiently contribute to the energy density has been proposed as a possible solution to the Hubble stress. In these scalar field solutions to the Hubble voltage, the universe is never dominated by the energy density of the scalar field; instead, the scalar field density reaches about 10%.
Different Models of Dark Energy
- The Cosmological Constant
- Quintessence
- Slow rolling ULA fields
- Oscillating ULA fields
In these scalar field solutions for the Hubble stress, the universe is never dominated by the energy density of the scalar field; instead, the scalar field density reaches about 10%. of the total density in the universe and then decays away. Motivated both by the need for a theoretical understanding of dark energy, as well as for the Hubble stress, we study scalar fields with potentials of the form. 17], using this approach can cause the evolution of the scalar field to deviate significantly from the evolution for the full potential of Eq.
We highlight the differences between the development of the ULA models and the corresponding power-law approximations and discuss how these differences affect both the quintessence and Hubble. Finally, after the field reaches the bottom of the potential, it will undergo rapid oscillations with frequency ν H. Recall that these potentials are the limiting case of ULA potentials when φf.
Although this is a simple calculation, the results are of little use in describing the resulting density evolution. If this occurs during a radiation-dominated period, then this maximum density relative to the background is reached when w = 1/3, well beyond the validity of the slow-rolling approach. However, the subsequent fluctuation and decay of the energy density of the scalar field can be usefully described analytically, as shown in the next subsection.
Rapidly fluctuating scalar fields, in which the period of the scalar field oscillation is much shorter than the Hubble time, were first systematically investigated by Turner [28], followed by many others. For n = 1, this result for 1 + w can be expressed in terms of elliptic integrals, but this gives little insight into the behavior of the equation of state parameter. 4-6 provide a comparison of the oscillation frequency for the ULA potential with the oscillation frequency for the corresponding power potential of small Θm.
In turn, these rapidly fluctuating ULA scalar fields form an important component of early dark energy models that could resolve the Hubble tension. Our results provide insight into the evolutionary behavior of the models examined in Ref. In particular, the fact that it decreases with increasing Θm for ULA potentials results in a slower decrease of the mean density of oscillations at low redshift for large values of Θm.
Furthermore, large initial values of Θm for n = 2-3 yield an oscillation frequency that increases as the scalar field energy density (and thus Θm) decreases, which is the opposite of the evolution of ν for very small initial values of Θm.
Discussion
These simulations also suggest that the oscillation frequency is much larger for then= 2 and n= 3 ULA potentials with initial values of Θm near π than for very small initial values of Θm. For rapidly oscillating scalar fields, the oscillation-averaged value of the equation of state parameter w corresponding to the power law approximation deviates from the value ofw in the ULA potential for φ/(f π) > 0.1, where the ULA potential gives a value for w much smaller than that corresponding power law potential, andw→ −1 asφ/f →π. In the same way, the power law approximation for the oscillation frequency ν deviates from the ULA frequency for φ/(f φ) >0.2, where the ULA potential gives a smaller oscillation frequency.
Furthermore, the dependence of the oscillation frequency on the oscillation amplitudeφm is more complex for the ULA potentials; for n = 2 and n = 3, the oscillation frequency increases with φm, reaches a maximum, and then decreases asφm/f →π. We emphasize that the standard action potential (n= 1) was previously investigated, as well as the behavior of the n > 1 potentials in the limit where they are well approximated by a power law potential. What is new here is the treatment of the latter cases with the full ULA potential.
The second part of this thesis will deal with the different but related idea of decaying matter in the universe. 36], Scherrer and Krauss showed the surprising result that radiation can never dominate matter in a universe dominated by cosmological constants. This is surprising, because it is not true if the universe is not dominated by a cosmological constant.
In this part of the thesis, we seek to generalize this result to a class of models more general than a universe dominated by the cosmological constant. We consider a non-relativistic component of the energy density, ρM (matter), which decomposes into a relativistic component, ρR (radiation). If the decay products do not have significant interactions with anything else, the equations for the evolution of matter and radiation are.
We study the evolution of r as a function of time to investigate whether it increases above 1 or not.
Particle Decay in Different Cosmologies
Cosmological Constant
Quintessence
The Big Rip
Then, we can define ˜t to be t/tm, which removes the final instance of tm not appearing as a multiple of 1/τ. Now, we can clearly see that the behavior ofr(t) does not depend on the two parameters separately, but a combination of them. The behavior is shown in two different figures due to the inequality waves inr(t) once tm/τ >1.
Discussion
Assume that we have a scalar field satisfying the slow roll conditions described in Eq. In this case, the equation of motion of the scalar field, 1.30, and the equation for the Hubble parameter 1.7 can be transformed into simpler forms by performing the following change of variables. Now it is useful to rewrite these equations so that they are expressed entirely in terms of observable quantities, Ωφ,γ andλ.
This change of variables is only valid if, for all points, dΩφ/da 6= 0, which we have in the present case. Then we can Eq. A.17) by replacing γ with γ0 and keeping only first-order terms in λ (since others will be almost zero). Then we can transform this equation into a linear differential equation with the change of variables2 =γ.
So then we can write our equation for ds/dΩφ, ds. A.22) The resulting solution, expressed in terms of w, is then:
Equation 2.10
Substituting these values into Eq. where ρT and pT are the total density and total pressure respectively, we arrive at the following equation. Since the matter is pressureless, the total pressure is approx. whereρφ0 is the density of the dark energy. In our case, with a scalar field of the form of Eq. 2.1), this happens when the field is near the top of its potential hill.
We can then approximate the potential by expanding it to the maximum, which we call φ∗. If we assume that w ≈ −1, then the scale factor is well approximated by its value in the ΛCDM model. Type Ia Supernova Discoveries atz¿ 1 from the Hubble Space Telescope: Evidence for Past Deceleration and Constraints on Dark Energy Evolution”.
Early Dark Energy Resolution for the Hubble Tension in the Face of Weak Lensing Surveys and Lensing Aberrations”.
![Figure 1.1: The luminosity distance H 0 d L versus the redshift in a flat cosmology. The Black points are data from [7], while the red are data from the Hubble Space Telescope](https://thumb-ap.123doks.com/thumbv2/123dok/11039390.0/9.918.200.711.137.660/figure-luminosity-distance-versus-redshift-cosmology-hubble-telescope.webp)
