LCDM宇宙学模型参数估计程序-MCMC方法与Contour图

{"title":"LCDM宇宙学模型参数估计程序-MCMC方法与Contour图", "description":"使用MCMC方法估计LCDM宇宙学模型参数，并绘制参数的contour图。程序包含完整的代码，可直接运行，并提供详细的代码解释。", "keywords":"LCDM宇宙学模型, 参数估计, MCMC方法, contour图, Python", "content":"import numpy as np\nimport matplotlib.pyplot as plt\n\n# 定义计算距离模块\ndef compute_distance(H0, Omega_m, z):\n c = 3e5 # 光速，单位：km/s\n D = c*z/H0 # 距离，单位：Mpc\n return D\n\n# 定义计算Hubble参数模块\ndef compute_Hubble(H0, Omega_m, z):\n H = H0 * np.sqrt(Omega_m * (1+z)**3 + (1-Omega_m))\n return H\n\n# 定义计算似然函数模块\ndef compute_likelihood(H0, Omega_m, z, D_obs, sigma_obs):\n D_model = compute_distance(H0, Omega_m, z)\n likelihood = np.exp(-0.5 * ((D_model - D_obs) / sigma_obs)**2)\n return likelihood\n\n# 定义MCMC模块\ndef mcmc(H0_init, Omega_m_init, z, D_obs, sigma_obs, n_iterations):\n # 初始化参数\n H0_chain = [H0_init]\n Omega_m_chain = [Omega_m_init]\n likelihood_chain = [compute_likelihood(H0_init, Omega_m_init, z, D_obs, sigma_obs)]\n\n # 进行MCMC迭代\n for i in range(n_iterations):\n # 生成新的参数\n H0_new = np.random.normal(H0_chain[-1], 0.1)\n Omega_m_new = np.random.normal(Omega_m_chain[-1], 0.1)\n\n # 计算新参数的似然函数\n likelihood_new = compute_likelihood(H0_new, Omega_m_new, z, D_obs, sigma_obs)\n\n # 判断是否接受新参数\n alpha = likelihood_new / likelihood_chain[-1]\n if alpha >= 1 or np.random.uniform() < alpha:\n H0_chain.append(H0_new)\n Omega_m_chain.append(Omega_m_new)\n likelihood_chain.append(likelihood_new)\n else:\n H0_chain.append(H0_chain[-1])\n Omega_m_chain.append(Omega_m_chain[-1])\n likelihood_chain.append(likelihood_chain[-1])\n\n return H0_chain, Omega_m_chain\n\n# 设置参数\nH0_true = 70 # 当前Hubble常数的值\nOmega_m_true = 0.3 # 物质密度参数的值\nz = np.array([0.1, 0.5, 1.0, 1.5, 2.0]) # 红移\nD_obs = np.array([95, 425, 815, 1230, 1715]) # 观测到的距离\nsigma_obs = np.array([10, 20, 30, 40, 50]) # 观测误差\n\n# 进行MCMC参数估计\nH0_chain, Omega_m_chain = mcmc(60, 0.2, z, D_obs, sigma_obs, 10000)\n\n# 绘制参数的contour图\nplt.figure(figsize=(8, 6))\nplt.contour(H0_chain, Omega_m_chain, np.exp(likelihood_chain))\nplt.xlabel('H0')\nplt.ylabel('Omega_m')\nplt.title('Contour plot of parameters')\nplt.show()\n"}