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### Abstract

When I started my research in cosmology, the $\Lambda$CDM model was the standard paradigm of cosmology. Now as I am writing this thesis, despite it still being the only standard one, the $\Lambda$CDM model is challenged by the ever more precise modern observations.One of the most significant challenges is called the Hubble tension, the discrepancy between the locally measured Hubble constant $H_0$ and its value inferred from the early universe Cosmic Microwave Background (CMB) measurements assuming the $\Lambda$CDM model.
During my Ph.D., Hubble tension increased from $\sim 3\sigma$ to $\sim 5\sigma$ thanks to the increasing measurement precision.
The possible new physics behind this tension is attractive, making it one of the focuses of modern cosmology.
In this thesis I provide some thoughts and examples as the possible solutions for the Hubble tension based on my work with collaborators during my Ph.D.Unfortunately, so far we are still not sure if this tension is real or is due to systematic measurement errors, neither do we have the perfect physical solutions.
I hope this work will at least provide some useful intuitions towards the true solutions for the Hubble tension if it is real.
Chapter 1 is provided as an overview of this thesis.
In Chapter 2, I propose the Acoustic Dark Energy (ADE) model as a possible solution for the Hubble tension. As an extension to $\Lambda$CDM, ADE model introduces an additional simple scalar field whose energy density is only important near matter radiation equality. It can simultaneously reduce the Hubble tension and fit the CMB data better than $\Lambda$CDM. Planck 2015 CMB data are used in the analysis.
In Chapter 3, I test the ADE model with the latest CMB data from Planck 2018 and the Atacama Cosmology Telescope. ADE passes the test to retain the ability to reduce the Hubble tension.
In Chapter 4, I consider the interaction between Early Dark Energy and dark matter.It is naively expected that this interaction could suppress the matter density growth and reduce the $S_8$ tension that is generically worsened in Early Dark Energy-like models including ADE.
However, I find the dark energy - dark matter interaction induces a matter-philic fifth-force, hence limiting its ability to reduce the $S_8$ tension.
In Chapter 5, I study the phenomenological imprints of modified gravity before recombination with a parameterized approach. I explore its impacts on various cosmological observables including the CMB, weak lensing and matter power spectrum. As an application, I found this early time modified gravity can also help to reduce the Hubble tension in the perturbation level.
I conclude and talk about possible future directions to solving the Hubble tension in Chapter 6.

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