Update README.md with installation instructions and add GoldbergerWis…

…eRadionSpectra.jl package usage details

README.md

@@ -1,8 +1,10 @@
 # GoldbergerWiseRadionSpectra.jl 
-[![Stable](https://img.shields.io/badge/docs-stable-blue.svg)](https://yardw.github.io/GoldbergerWiseRadionSpectra.jl/stable/) [![Dev](https://img.shields.io/badge/docs-dev-blue.svg)](https://yardw.github.io/GoldbergerWiseRadionSpectra.jl/dev/) [![Build Status](https://github.com/yardw/GoldbergerWiseRadionSpectra.jl/actions/workflows/CI.yml/badge.svg?branch=main)](https://github.com/yardw/GoldbergerWiseRadionSpectra.jl/actions/workflows/CI.yml?query=branch%3Amain) [![Coverage](https://codecov.io/gh/yardw/GoldbergerWiseRadionSpectra.jl/branch/main/graph/badge.svg)](https://codecov.io/gh/yardw/GoldbergerWiseRadionSpectra.jl)
+<!-- [![Stable](https://img.shields.io/badge/docs-stable-blue.svg)](https://yardw.github.io/GoldbergerWiseRadionSpectra.jl/stable/)  -->
+<!-- [![Dev](https://img.shields.io/badge/docs-dev-blue.svg)](https://yardw.github.io/GoldbergerWiseRadionSpectra.jl/dev/)  -->
+[![Build Status](https://github.com/yardw/GoldbergerWiseRadionSpectra.jl/actions/workflows/CI.yml/badge.svg?branch=main)](https://github.com/yardw/GoldbergerWiseRadionSpectra.jl/actions/workflows/CI.yml?query=branch%3Amain) 
+<!-- [![Coverage](https://codecov.io/gh/yardw/GoldbergerWiseRadionSpectra.jl/branch/main/graph/badge.svg)](https://codecov.io/gh/yardw/GoldbergerWiseRadionSpectra.jl) -->
 
-# Usage
-## Installation
+# Installation
 To install the package, you can clone the repository manually or by running the following command in the terminal.
 ```bash
 git clone https://github.com/yardw/GoldbergerWiseRadionSpectra.jl.git
@@ -20,4 +22,22 @@ Then `activate` the current package environment and `instantiate` the package.
 activate .
 instantiate
 ```
-The above commands will automatically install all necessary dependencies and now the package is ready to be used.
+The above commands will automatically install all necessary dependencies and now the package is ready to be used.
+
+# Usage
+After instatiation, you can use the package in the Julia REPL or in a Julia script by running the following command.
+```julia
+using GoldbergerWiseRadionSpectra
+```
+The package provides following functions:
+- `search` : search for the radion mass spectrum in the Goldberger-Wise model, with respect to given parameters
+    - inputs:
+        - `l2`: the backreaction parameter
+        - `g2`: the quadratic brane potential parameter, in unit of the critical value g^2_0 = 4k+2u
+    - outputs:
+        - `m20`: the squared mass of the radion
+        - `m2err`: the error bound of the squared mass
+        - `Fp`: the first-order derivative of the radion profile at the initial position. It can be used to rebuild the radion profile.
+- `m2s_analytical` : calculate the perturbative mass spectrum of the radion in the Goldberger-Wise model, with respect to `l2` and `g2`
+- `extract_sol` : extract the radion profile from the output of `search`
+You can also modify other parameters for Goldberger-Wise model, which includes `yₘ, u, ϕP, ϕT, k, M_IR, γ²₀`, in the `eom.jl` file.