FindMinimum works only if you know the answer

Question

I am processing SNe Ia data and using FindMinimum to try and extract two values: t1 and a0. The FindMinimum value works only if I supply values that are close to the answer. If I provide no starting values or arbitrary starting values (e.g. t1=1, a0=1), then the function doesn't converge. If I give it values close to the answer (e.g. t1=1*^18, a0=1*^-14), then I get an answer. Here's the code:

c = 2.99792*^5;
megaParsec = 3.08567758*^19;
alpha = 0.147;
beta = 3.13;
mag = -19.34;

chiSquared[t1_, a0_] := 
 Sum[residual[i, t1, a0]^2/error[i]^2, {i, 1, Length[data]}]

error[i_] := (distance = observedDistance[i]; 
  distanceModuli = 
   Log10[distance/megaParsec]*5 + 25; (data[[i, 6]]/distanceModuli)*
   distance)

luminousDistance[z_, t1_, a0_] := -((a0*t1^2*z + 
      2*c*((-t1)*(1 + z) + Sqrt[t1^2*(1 + z)]))/(2 + z))

residual[i_, t1_, a0_] := (z = data[[i, 2]]; 
  observedDistance[i] - luminousDistance[z, t1, a0])

observedDistance[i_] := (magnitude = 
   data[[i, 3]] + alpha*(data[[i, 4]] - 1) - beta*data[[i, 5]] - mag; 
  10^((magnitude - 25)/5)*megaParsec)

And here is the data:

data = {{"sn2004s", 0.01, 14.183, 0.973, 0.035, 0.213}, {"sn1999ac", 
    0.01, 14.13, 0.987, 0.056, 0.177}, {"sn1997do", 0.011, 14.317, 
    0.983, 0.056, 0.204}, {"sn2006bh", 0.011, 14.347, 0.814, -0.045, 
    0.168}, {"sn2002dp", 0.011, 14.597, 0.973, 0.113, 
    0.203}, {"sn2005al", 0.012, 14.843, 0.871, -0.073, 
    0.179}, {"sn2001ep", 0.013, 14.904, 0.903, 0.088, 
    0.189}, {"sn1997e", 0.014, 15.118, 0.819, 0.036, 
    0.2}, {"sn2001fe", 0.015, 14.685, 1.077, -0.002, 
    0.194}, {"sn2005bo", 0.014, 15.646, 0.867, 0.236, 
    0.186}, {"sn2002ha", 0.014, 14.703, 0.867, -0.056, 
    0.202}, {"sn2006n", 0.015, 15.09, 0.787, -0.023, 
    0.196}, {"sn1999dq", 0.014, 14.409, 1.103, 0.075, 
    0.183}, {"sn1999aa", 0.016, 14.728, 1.12, -0.052, 
    0.167}, {"sn1992al", 0.014, 14.499, 0.959, -0.087, 
    0.187}, {"sn2001bt", 0.014, 15.317, 0.899, 0.18, 
    0.181}, {"sn2005el", 0.015, 14.842, 0.838, -0.08, 
    0.19}, {"sn1999dk", 0.014, 14.881, 0.991, 0.086, 
    0.198}, {"sn2001v", 0.016, 14.596, 1.111, 0.025, 
    0.176}, {"sn2005kc", 0.015, 15.502, 0.933, 0.176, 
    0.192}, {"sn1994s", 0.015, 14.801, 1.031, -0.037, 
    0.208}, {"sn2001cz", 0.016, 15.083, 1.007, 0.071, 
    0.19}, {"sn2001cn", 0.015, 15.306, 0.933, 0.145, 
    0.191}, {"sn2001bf", 0.015, 14.719, 1.1, 0, 0.215}, {"sn2004eo", 
    0.015, 15.104, 0.88, 0.058, 0.18}, {"sn2004ey", 0.016, 14.676, 
    1.001, -0.107, 0.201}, {"sn2001en", 0.015, 15.095, 0.877, 0.038, 
    0.309}, {"sn2006td", 0.016, 15.735, 0.841, 0.123, 
    0.208}, {"sn1996bv", 0.017, 15.353, 1.064, 0.162, 
    0.234}, {"sn2006ax", 0.017, 14.984, 1.001, -0.091, 
    0.174}, {"sn2001da", 0.017, 15.464, 0.778, 0.078, 
    0.412}, {"sn2000dk", 0.018, 15.361, 0.768, -0.001, 
    0.184}, {"sn1998v", 0.017, 15.105, 0.983, 0.004, 
    0.243}, {"sn1998ef", 0.018, 14.832, 0.892, -0.068, 
    0.203}, {"sn2007ci", 0.019, 15.909, 0.729, 0.066, 
    0.188}, {"sn1992bo", 0.019, 15.79, 0.771, -0.03, 
    0.182}, {"sn2002kf", 0.02, 15.664, 0.862, -0.032, 
    0.222}, {"sn2005ki", 0.02, 15.536, 0.844, -0.067, 
    0.171}, {"sn2003w", 0.021, 15.89, 0.993, 0.134, 
    0.178}, {"sn1992bc", 0.021, 15.145, 1.081, -0.086, 
    0.173}, {"sn2006ej", 0.02, 15.779, 0.853, 0.026, 
    0.204}, {"sn2007bc", 0.022, 15.912, 0.852, 0.011, 
    0.191}, {"sn2002jy", 0.022, 15.758, 1.109, -0.008, 
    0.2}, {"sn2008bf", 0.022, 15.739, 1.034, 0.013, 
    0.178}, {"sn2006bq", 0.022, 16.191, 0.848, 0.079, 
    0.191}, {"sn2006et", 0.022, 16.003, 1.11, 0.166, 
    0.215}, {"sn2006cp", 0.023, 16.015, 1.052, 0.099, 
    0.188}, {"sn2006ar", 0.023, 16.486, 0.903, 0.128, 
    0.198}, {"sn1995ak", 0.022, 15.982, 0.85, 0.011, 
    0.278}, {"sn2006mp", 0.023, 16.009, 1.092, 0.036, 
    0.188}, {"sn2005bg", 0.025, 15.833, 1.044, -0.003, 
    0.185}, {"sn2006ac", 0.023, 16.193, 0.895, 0.08, 
    0.174}, {"sn1994m", 0.025, 16.278, 0.83, 0.043, 
    0.208}, {"sn2000cn", 0.024, 16.554, 0.755, 0.115, 
    0.189}, {"sn2007f", 0.024, 15.914, 1.059, -0.02, 
    0.187}, {"sn2000ca", 0.023, 15.606, 1.062, -0.073, 
    0.194}, {"sn2007qe", 0.024, 16.074, 1.059, 0.067, 
    0.171}, {"sn2006sr", 0.024, 16.157, 0.852, 0.011, 
    0.195}, {"sn1993h", 0.024, 16.766, 0.726, 0.179, 
    0.193}, {"sn2002bf", 0.025, 16.358, 0.936, 0.169, 
    0.232}, {"sn2002he", 0.025, 16.271, 0.82, -0.012, 
    0.214}, {"sn1992ag", 0.025, 16.355, 0.95, 0.18, 
    0.231}, {"sn2005ms", 0.027, 16.18, 1.045, -0.01, 
    0.181}, {"sn1992p", 0.028, 16.097, 1.079, -0.063, 
    0.271}, {"sn2007cq", 0.025, 15.85, 0.938, 0.003, 
    0.198}, {"sn2005na", 0.027, 15.934, 0.95, -0.078, 
    0.183}, {"sn2004gs", 0.027, 17.146, 0.768, 0.167, 
    0.172}, {"sn1999gp", 0.027, 16.044, 1.182, 0.029, 
    0.186}, {"sn2007co", 0.027, 16.491, 0.964, 0.098, 
    0.186}, {"sn1998ab", 0.028, 16.089, 0.982, 0.066, 
    0.194}, {"sn2002de", 0.028, 16.699, 1.062, 0.139, 
    0.223}, {"sn2003u", 0.029, 16.521, 0.791, 0.003, 
    0.227}, {"sn2005eq", 0.029, 16.322, 1.159, 0.026, 
    0.184}, {"sn2001ba", 0.03, 16.244, 1.008, -0.095, 
    0.196}, {"sn1996c", 0.031, 16.654, 1.073, 0.087, 
    0.208}, {"sn2006qo", 0.03, 16.865, 1.048, 0.192, 
    0.18}, {"sn2003ch", 0.03, 16.725, 0.842, -0.001, 0.2}, {"sn1990o",
     0.03, 16.267, 1.047, -0.018, 0.229}, {"sn1997dg", 0.031, 16.84, 
    0.941, -0.024, 0.215}, {"sn2006az", 0.031, 16.517, 0.858, -0.064, 
    0.168}, {"sn2004as", 0.033, 17.011, 1.049, 0.077, 
    0.2}, {"sn2007bd", 0.032, 16.614, 0.844, -0.021, 
    0.18}, {"sn1999cc", 0.032, 16.783, 0.812, 0.015, 
    0.174}, {"sn2006s", 0.033, 16.898, 1.112, 0.074, 
    0.172}, {"sn2006bt", 0.031, 16.971, 1.011, 0.13, 
    0.181}, {"sn2004l", 0.033, 17.385, 0.925, 0.192, 
    0.234}, {"sn2005iq", 0.034, 16.76, 0.878, -0.085, 
    0.17}, {"sn2003iv", 0.035, 17.03, 0.741, -0.04, 
    0.25}, {"sn2006gr", 0.034, 17.009, 1.103, 0.1, 
    0.184}, {"sn2005eu", 0.035, 16.521, 1.101, -0.027, 
    0.213}, {"sn2002hd", 0.036, 16.867, 0.858, 0.081, 
    0.377}, {"sn1992bg", 0.035, 16.749, 0.956, -0.037, 
    0.281}, {"sn1996bl", 0.035, 16.677, 0.979, 0.006, 
    0.206}, {"sn2000cf", 0.037, 17.05, 0.916, -0.023, 
    0.193}, {"sn2006mo", 0.037, 17.486, 0.759, 0.065, 
    0.207}, {"sn2001eh", 0.037, 16.667, 1.185, -0.004, 
    0.198}, {"sn1999aw", 0.039, 16.791, 1.234, -0.032, 
    0.182}, {"sn2002hu", 0.038, 16.69, 1.045, -0.058, 
    0.183}, {"sn2003fa", 0.039, 16.763, 1.152, -0.01, 
    0.182}, {"sn2001az", 0.04, 16.986, 1.108, -0.049, 
    0.259}, {"sn2005lz", 0.041, 17.674, 0.844, 0.093, 
    0.201}, {"sn1992bl", 0.042, 17.345, 0.815, -0.035, 
    0.24}, {"sn1992bh", 0.042, 17.649, 0.99, 0.065, 
    0.215}, {"sn2004gu", 0.047, 17.439, 1.141, 0.101, 
    0.178}, {"sn2005hc", 0.045, 17.302, 1.078, -0.006, 
    0.18}, {"sn1993ag", 0.049, 17.865, 0.884, 0.085, 
    0.241}, {"sn1995ac", 0.049, 17.091, 1.085, -0.012, 
    0.184}, {"sn1990af", 0.05, 17.796, 0.741, -0.006, 
    0.198}, {"sn1993o", 0.053, 17.656, 0.906, -0.073, 
    0.196}, {"sn1999ao", 0.055, 17.906, 0.95, -0.017, 
    0.212}, {"sn1998dx", 0.054, 17.546, 0.844, -0.088, 
    0.24}, {"sn2006ob", 0.059, 18.302, 0.741, 0.022, 
    0.187}, {"sn2006oa", 0.058, 17.955, 1.131, 0.023, 
    0.196}, {"SDSS3901", 0.063, 18.015, 1.117, 0.051, 
    0.18}, {"sn1992bs", 0.063, 18.317, 0.966, -0.018, 
    0.237}, {"sn2006an", 0.065, 18.195, 1.061, 0.016, 
    0.222}, {"sn2007ae", 0.063, 17.832, 1.198, 0.002, 
    0.234}, {"SDSS10028", 0.065, 18.373, 0.891, 0.054, 
    0.2}, {"SDSS6057", 0.067, 18.641, 0.944, 0.129, 
    0.206}, {"sn2006al", 0.069, 18.485, 0.809, -0.063, 
    0.248}, {"sn1993b", 0.07, 18.497, 0.914, 0.057, 
    0.273}, {"sn2006on", 0.068, 18.494, 1.038, 0.104, 
    0.31}, {"sn1992ae", 0.075, 18.448, 0.944, -0.027, 
    0.266}, {"sn2005ir", 0.075, 18.4, 1.043, 0.014, 
    0.17}, {"sn1999bp", 0.078, 18.422, 1.065, -0.037, 
    0.198}, {"sn1992bp", 0.079, 18.335, 0.877, -0.075, 
    0.212}, {"sn2005ag", 0.08, 18.44, 1.029, -0.014, 
    0.179}, {"SDSS1241", 0.087, 19.103, 0.929, 0.072, 
    0.189}, {"SDSS3592", 0.087, 18.751, 0.975, -0.04, 
    0.178}, {"SDSS6773", 0.09, 18.663, 0.979, -0.011, 
    0.207}, {"SDSS2102", 0.095, 18.634, 1.133, -0.094, 
    0.217}, {"SDSS10434", 0.104, 19.185, 1.01, -0.053, 
    0.224}, {"SDSS3256", 0.108, 19.495, 0.942, -0.029, 
    0.222}, {"SDSS7147", 0.11, 19.516, 0.796, -0.034, 
    0.195}, {"SDSS8719", 0.116, 19.392, 0.992, -0.059, 
    0.199}, {"SDSS5395", 0.117, 19.459, 1.11, 0.002, 
    0.188}, {"SDSS2561", 0.118, 19.813, 0.993, 0.086, 
    0.193}, {"SDSS1371", 0.119, 19.073, 1.072, -0.076, 
    0.187}, {"SDSS5549", 0.121, 19.654, 1.02, 0.033, 
    0.182}, {"SDSS2916", 0.124, 19.937, 0.875, 0.066, 
    0.248}, {"06D2fb", 0.124, 19.772, 0.964, -0.004, 
    0.181}, {"SDSS6406", 0.125, 19.616, 1, 0.026, 0.188}, {"SDSS2992",
     0.127, 20.034, 0.889, 0.127, 0.21}, {"SDSS744", 0.128, 19.793, 
    1.149, 0.08, 0.25}, {"SDSS5751", 0.13, 20.136, 1.068, 0.191, 
    0.179}, {"SDSS1032", 0.13, 20.326, 0.717, 0.088, 
    0.219}, {"SDSS2635", 0.143, 19.83, 1.092, -0.015, 
    0.202}, {"SDSS1794", 0.143, 20.058, 1.136, 0.018, 
    0.244}, {"SDSS8921", 0.145, 19.961, 1.104, 0.007, 
    0.231}, {"SDSS5103", 0.146, 20.377, 0.963, 0.055, 
    0.191}, {"SDSS11300", 0.147, 20.309, 0.862, 0.11, 
    0.243}, {"SDSS10106", 0.147, 20.948, 0.99, 0.2, 
    0.233}, {"SDSS2308", 0.148, 19.587, 1.069, -0.164, 
    0.189}, {"SDSS2031", 0.153, 19.703, 1.049, -0.091, 
    0.195}, {"SDSS5550", 0.156, 19.844, 1.202, -0.055, 
    0.19}, {"SDSS2689", 0.162, 20.254, 1.165, 0.095, 
    0.202}, {"SDSS3087", 0.165, 20.266, 1.056, 0.025, 
    0.196}, {"05D3ne", 0.169, 20.251, 0.809, -0.147, 
    0.218}, {"SDSS5916", 0.172, 20.439, 0.922, 0.014, 
    0.206}, {"SDSS3080", 0.174, 20.236, 0.999, -0.038, 
    0.195}, {"SDSS5350", 0.175, 20.323, 0.913, -0.057, 
    0.248}, {"SDSS5635", 0.179, 20.923, 1.011, 0.002, 
    0.233}, {"SDSS2372", 0.181, 20.58, 1.032, 0.045, 
    0.21}, {"SDSS6936", 0.181, 20.575, 1.003, -0.007, 
    0.208}, {"SDSS1580", 0.183, 20.291, 1.099, -0.014, 
    0.215}, {"05D2ah", 0.184, 20.765, 0.991, 0.019, 
    0.184}, {"SDSS6422", 0.184, 20.274, 1.08, -0.097, 
    0.193}, {"SDSS8213", 0.185, 21.133, 0.923, 0.179, 
    0.226}, {"SDSS5994", 0.187, 20.476, 1.074, -0.041, 
    0.218}, {"SDSS6304", 0.19, 20.952, 0.927, 0.095, 
    0.207}, {"SDSS762", 0.191, 20.657, 1.102, 0.009, 
    0.211}, {"SDSS2440", 0.193, 20.653, 1.051, -0.062, 
    0.212}, {"SDSS7335", 0.198, 21.265, 0.781, 0.067, 
    0.243}, {"SDSS6780", 0.202, 20.947, 0.789, -0.004, 
    0.246}, {"SDSS7243", 0.204, 20.789, 1.079, 0.002, 
    0.237}, {"SDSS3331", 0.206, 21.089, 0.95, 0.076, 
    0.225}, {"04D1dc", 0.211, 21.084, 0.856, 0.023, 
    0.191}, {"SDSS7847", 0.212, 21.225, 1.017, 0.155, 
    0.224}, {"SDSS6933", 0.213, 20.832, 0.995, 0.002, 
    0.202}, {"SDSS8495", 0.214, 20.811, 1.098, -0.001, 
    0.246}, {"SDSS1316", 0.217, 20.907, 1.058, -0.073, 
    0.374}, {"SDSS9467", 0.218, 21.057, 0.83, -0.118, 
    0.269}, {"05D3kx", 0.219, 20.867, 1.069, -0.016, 
    0.179}, {"SDSS7512", 0.219, 21.104, 1.055, 0.027, 
    0.233}, {"SDSS5533", 0.22, 21.173, 0.976, 0.046, 
    0.21}, {"SDSS3452", 0.23, 20.799, 1.092, -0.068, 
    0.211}, {"SDSS10449", 0.244, 20.995, 1.109, 0.039, 
    0.285}, {"SDSS3377", 0.245, 20.791, 1.12, -0.06, 
    0.219}, {"05D3mq", 0.246, 21.521, 0.912, 0.034, 
    0.204}, {"SDSS3451", 0.25, 20.958, 1.055, -0.038, 
    0.22}, {"06D3gn", 0.25, 21.892, 0.949, 0.16, 0.186}, {"SDSS3199", 
    0.251, 21.539, 1.164, 0.03, 0.224}, {"SDSS5717", 0.252, 21.355, 
    1.175, -0.012, 0.211}, {"SDSS5736", 0.253, 21.421, 0.949, 0.011, 
    0.211}, {"SDSS9032", 0.254, 21.345, 1.075, 0.041, 
    0.271}, {"SDSS9457", 0.257, 21.453, 0.978, 0.022, 
    0.302}, {"SDSS1112", 0.258, 21.563, 0.918, 0.022, 
    0.302}, {"SDSS8046", 0.259, 21.633, 1.049, 0.077, 
    0.249}, {"SDSS6108", 0.259, 21.537, 1.065, 0.063, 
    0.246}, {"SDSS3241", 0.259, 21.01, 1.057, -0.168, 
    0.251}, {"SDSS1253", 0.262, 21.23, 0.871, -0.09, 
    0.253}, {"SDSS2017", 0.262, 21.286, 1.148, -0.085, 
    0.26}, {"04D3ez", 0.263, 21.697, 0.891, 0.089, 0.184}, {"05D1hk", 
    0.263, 21.184, 1.168, -0.006, 0.192}, {"SDSS2422", 0.265, 21.144, 
    1.112, -0.156, 0.215}, {"SDSS2943", 0.265, 21.372, 1.055, 0.007, 
    0.24}, {"SDSS6315", 0.267, 20.919, 0.971, -0.166, 
    0.233}, {"06D3fp", 0.268, 21.748, 0.999, 0.104, 0.18}, {"03D4cj", 
    0.27, 21.052, 1.124, -0.063, 0.184}, {"SDSS6192", 0.272, 21.698, 
    0.826, -0.018, 0.268}, {"SDSS5957", 0.28, 21.453, 0.983, -0.089, 
    0.245}, {"06D3dt", 0.282, 22.168, 0.986, 0.117, 0.192}, {"03D4ag",
     0.285, 21.277, 1.111, -0.043, 0.19}, {"SDSS2165", 0.288, 21.604, 
    1.063, -0.096, 0.245}, {"SDSS2789", 0.29, 21.576, 0.901, -0.077, 
    0.27}, {"03D3ba", 0.291, 21.984, 1.084, 0.146, 
    0.238}, {"SDSS6249", 0.294, 21.821, 1.086, 0.064, 
    0.256}, {"SDSS10550", 0.3, 22.02, 1.162, 0.105, 0.372}, {"06D4dh",
     0.303, 21.449, 1.052, -0.126, 0.185}, {"SDSS11864", 0.303, 
    22.299, 1.015, 0.07, 0.432}, {"SDSS5966", 0.31, 21.798, 1.02, 
    0.002, 0.315}, {"SDSS6699", 0.311, 21.796, 0.872, -0.126, 
    0.277}, {"SDSS5844", 0.311, 21.571, 1.015, -0.099, 
    0.254}, {"SDSS6649", 0.314, 21.598, 1.09, -0.057, 
    0.251}, {"SDSS7475", 0.322, 21.535, 1.025, -0.123, 
    0.251}, {"05D2ab", 0.323, 22.001, 0.987, -0.013, 
    0.191}, {"SDSS6924", 0.328, 21.633, 1.076, -0.041, 
    0.265}, {"03D1fc", 0.332, 21.866, 1.048, 0.016, 0.194}, {"04D3kr",
     0.337, 21.957, 1.127, -0.004, 0.184}, {"SDSS2533", 0.34, 21.79, 
    1.191, -0.04, 0.278}, {"04D3nh", 0.34, 22.142, 1.059, 0.009, 
    0.184}, {"03D1bp", 0.347, 22.421, 0.88, 0.002, 0.192}, {"04D2mc", 
    0.348, 22.58, 0.845, 0.142, 0.205}, {"05D2ie", 0.348, 22.249, 
    0.988, -0.046, 0.198}, {"SDSS9207", 0.35, 22.062, 1.126, 0.022, 
    0.322}, {"05D2hc", 0.35, 22.693, 0.931, 0.057, 0.194}, {"05D2mp", 
    0.354, 22.417, 1.138, 0.058, 0.208}, {"03D3bl", 0.355, 22.951, 
    1.002, 0.241, 0.211}, {"04D2fs", 0.357, 22.437, 1.01, 0.081, 
    0.191}, {"04D3fk", 0.358, 22.537, 0.96, 0.11, 0.184}, {"04D1hd", 
    0.369, 22.166, 1.071, -0.06, 0.179}, {"04D2cf", 0.369, 22.491, 
    0.882, 0.015, 0.265}, {"05D3jr", 0.37, 22.663, 0.902, 0.096, 
    0.189}, {"03D3ay", 0.371, 22.293, 1.054, -0.018, 
    0.234}, {"05D4bm", 0.372, 22.22, 1.02, -0.041, 0.183}, {"05D4fo", 
    0.373, 22.463, 0.924, -0.022, 0.183}, {"05D4cw", 0.375, 22.145, 
    0.911, -0.12, 0.187}, {"SDSS7779", 0.381, 21.943, 1.124, -0.049, 
    0.25}, {"SDSS5737", 0.393, 22.439, 1.27, 0.144, 
    0.339}, {"SDSS8707", 0.395, 22.272, 1.11, -0.088, 
    0.26}, {"05D4ff", 0.402, 22.615, 0.932, 0.028, 0.192}, {"06D3ed", 
    0.404, 22.615, 0.963, -0.036, 0.182}, {"05D4dt", 0.407, 22.808, 
    0.891, -0.023, 0.186}, {"06D4cq", 0.411, 22.562, 1.04, -0.005, 
    0.184}, {"04D2fp", 0.415, 22.559, 1.034, -0.012, 
    0.197}, {"05D2dw", 0.417, 22.488, 1.125, 0.021, 0.198}, {"05D3cf",
     0.419, 22.965, 0.97, 0.045, 0.203}, {"04D4gg", 0.424, 22.753, 
    1.131, 0.124, 0.207}, {"05D2cb", 0.427, 23.407, 1.1, 0.193, 
    0.206}, {"04D1rh", 0.435, 22.582, 1.085, -0.015, 
    0.203}, {"06D4co", 0.437, 22.521, 0.96, -0.027, 0.183}, {"06D2gb",
     0.442, 23.008, 0.829, 0.03, 0.23}, {"06D3df", 0.442, 22.685, 
    1.122, 0.021, 0.195}, {"03D3aw", 0.449, 22.654, 1.066, -0.053, 
    0.241}, {"04D2gb", 0.45, 22.916, 0.831, 0.042, 0.21}, {"04D3gt", 
    0.451, 23.259, 0.976, 0.223, 0.192}, {"03D3cd", 0.461, 22.593, 
    1.208, 0.012, 0.293}, {"05D3lc", 0.461, 22.982, 0.913, -0.021, 
    0.187}, {"03D4au", 0.468, 23.817, 1.048, 0.158, 0.239}, {"05D3mx",
     0.47, 23.043, 0.832, -0.057, 0.202}, {"04D4jr", 0.47, 22.642, 
    1.16, -0.026, 0.195}, {"04D3df", 0.47, 23.521, 0.787, 0.108, 
    0.199}, {"04D4ju", 0.472, 23.771, 1.045, 0.184, 0.214}, {"05D2bv",
     0.474, 22.719, 0.989, -0.096, 0.197}, {"05D2ac", 0.479, 22.677, 
    1.133, -0.012, 0.191}, {"05D3dd", 0.48, 22.941, 0.985, -0.015, 
    0.203}, {"05D1ix", 0.49, 22.879, 1.054, -0.034, 0.198}, {"03D1ax",
     0.496, 22.992, 0.925, -0.062, 0.196}, {"05D4af", 0.499, 23.108, 
    1.01, -0.013, 0.222}, {"06D2bk", 0.499, 23.273, 1.054, 0.036, 
    0.234}, {"03D1au", 0.504, 23.012, 1.137, 0.017, 0.203}, {"05D4av",
     0.509, 23.558, 1.095, 0.185, 0.204}, {"05D2dy", 0.51, 22.913, 
    1.099, -0.1, 0.211}, {"04D2mj", 0.513, 23.783, 1.154, 0.171, 
    0.216}, {"04D1pg", 0.515, 23.57, 1.092, 0.119, 0.218}, {"05D3ci", 
    0.515, 23.564, 1.166, 0.16, 0.278}, {"04D4in", 0.516, 22.902, 
    1.169, -0.034, 0.201}, {"06D3el", 0.519, 22.913, 1.08, -0.082, 
    0.196}, {"04D2gc", 0.522, 23.327, 1.133, 0.037, 0.219}, {"06D2ca",
     0.531, 23.301, 1.099, 0.06, 0.249}, {"06D2cc", 0.532, 23.468, 
    0.944, 0.089, 0.256}, {"05D2eb", 0.534, 23.006, 1.113, -0.036, 
    0.233}, {"05D4ek", 0.536, 23.303, 1.059, 0.081, 0.21}, {"05D4be", 
    0.537, 22.916, 1.1, -0.114, 0.199}, {"04D4bq", 0.55, 23.347, 
    1.095, 0.134, 0.23}, {"04D3hn", 0.552, 23.503, 0.935, 0.096, 
    0.21}, {"06D2ck", 0.552, 23.447, 1.05, -0.002, 0.249}, {"06D4bo", 
    0.552, 23.231, 1.098, -0.028, 0.227}, {"05D1ee", 0.559, 23.556, 
    0.953, 0.023, 0.222}, {"04D1hx", 0.56, 23.715, 1.042, 0.143, 
    0.215}, {"05D1kl", 0.56, 24.154, 1.075, 0.164, 0.246}, {"05D1cc", 
    0.563, 23.496, 0.981, -0.005, 0.211}, {"05D1dn", 0.566, 23.317, 
    1.129, 0.018, 0.222}, {"03D4gl", 0.571, 23.314, 1.238, 0.039, 
    0.831}, {"05D2dt", 0.574, 23.656, 1.024, 0.045, 0.249}, {"06D3et",
     0.576, 23.512, 0.861, -0.039, 0.213}, {"05D3jq", 0.579, 23.322, 
    1.161, 0.035, 0.209}, {"05D3gp", 0.58, 23.521, 0.946, -0.054, 
    0.286}, {"03D4gf", 0.58, 23.336, 1.095, 0.01, 0.236}, {"05D1dx", 
    0.58, 23.304, 1.078, -0.027, 0.205}, {"03D1aw", 0.582, 23.584, 
    1.098, 0.001, 0.235}, {"04D1jg", 0.584, 23.272, 1.028, -0.079, 
    0.219}, {"04D1kj", 0.585, 23.345, 1.034, -0.052, 
    0.211}, {"05D4ej", 0.585, 23.746, 1.034, 0.02, 0.222}, {"04D1sa", 
    0.585, 23.559, 0.94, -0.064, 0.237}, {"05D1hm", 0.587, 24.102, 
    1.126, 0.155, 0.244}, {"05D4bf", 0.589, 23.627, 1.029, 0.018, 
    0.235}, {"04D2mh", 0.59, 23.403, 1.163, 0.041, 0.215}, {"04D1oh", 
    0.59, 23.388, 1.01, -0.048, 0.227}, {"03D4gg", 0.592, 23.413, 
    1.098, 0.046, 0.264}, {"05D3lr", 0.6, 23.854, 1.02, 0.096, 
    0.251}, {"05D4ef", 0.605, 23.832, 0.839, -0.054, 
    0.225}, {"05D2he", 0.608, 23.953, 1.044, 0.073, 0.244}, {"03D4dy",
     0.61, 23.268, 1.127, -0.06, 0.231}, {"04D3do", 0.61, 23.577, 
    0.906, -0.097, 0.213}, {"03D1dt", 0.612, 23.3, 1.048, -0.054, 
    0.273}, {"04D4an", 0.613, 24.046, 0.987, 0.025, 0.27}, {"05D1ck", 
    0.617, 24.074, 1.007, 0.111, 0.231}, {"04D2an", 0.62, 23.597, 
    0.991, -0.019, 0.295}, {"04D3co", 0.62, 23.757, 0.936, 0.019, 
    0.245}, {"03D4dh", 0.627, 23.39, 1.13, -0.045, 0.225}, {"04D4fx", 
    0.629, 23.501, 1.115, 0.01, 0.23}, {"05D2ci", 0.63, 23.612, 0.901,
     0.045, 0.263}, {"05D1cb", 0.632, 23.715, 0.967, -0.001, 
    0.222}, {"03D4at", 0.634, 23.733, 1.019, -0.008, 
    0.261}, {"04D1pu", 0.639, 24.024, 0.843, 0.094, 0.291}, {"05D2ec",
     0.64, 23.672, 0.994, -0.063, 0.248}, {"05D4ag", 0.64, 23.895, 
    1.055, 0.068, 0.287}, {"05D3ax", 0.643, 23.62, 1.071, -0.034, 
    0.28}, {"04D3cy", 0.643, 23.8, 0.978, -0.011, 0.257}, {"05D3lb", 
    0.647, 23.896, 1.04, 0.035, 0.224}, {"05D3kt", 0.648, 23.965, 
    0.979, 0.085, 0.23}, {"04D1sk", 0.663, 24.058, 1.028, 0.1, 
    0.251}, {"05D3hs", 0.664, 23.501, 1.045, -0.141, 
    0.234}, {"05D3mh", 0.67, 24.106, 1.084, 0.053, 0.272}, {"03D1co", 
    0.679, 24.088, 1.084, -0.019, 0.266}, {"05D2bt", 0.68, 23.521, 
    1.067, -0.113, 0.239}, {"06D3cc", 0.683, 24.067, 1.082, 0.006, 
    0.312}, {"04D4ic", 0.687, 24.121, 0.9, -0.01, 0.276}, {"06D3em", 
    0.69, 24.377, 1.006, 0.149, 0.259}, {"05D1ke", 0.69, 23.611, 
    1.035, -0.077, 0.23}, {"03D4cz", 0.695, 24.045, 0.818, -0.071, 
    0.292}, {"05D2ck", 0.698, 24.474, 0.733, -0.015, 
    0.277}, {"04D4ib", 0.699, 23.595, 1.101, -0.093, 
    0.233}, {"04D2iu", 0.7, 24.246, 0.806, 0.021, 0.288}, {"06D4ba", 
    0.7, 23.761, 1.065, -0.081, 0.28}, {"05D2le", 0.7, 23.961, 
    1.054, -0.012, 0.247}, {"05D4cq", 0.701, 23.73, 1.077, -0.058, 
    0.24}, {"05D4bj", 0.701, 24.103, 1.012, 0.059, 0.249}, {"04D1si", 
    0.702, 23.867, 0.994, -0.003, 0.244}, {"04D4hu", 0.703, 23.92, 
    1.005, -0.078, 0.241}, {"05D3gv", 0.715, 24.001, 0.911, -0.043, 
    0.254}, {"05D3jh", 0.718, 23.753, 0.927, -0.096, 0.23}, {"06D3gh",
     0.72, 23.926, 1.034, -0.021, 0.27}, {"04D1aj", 0.721, 23.904, 
    1.03, -0.004, 0.266}, {"05D4ev", 0.722, 24.259, 0.896, -0.014, 
    0.259}, {"06D3do", 0.725, 23.898, 1.096, -0.044, 
    0.281}, {"06D3bz", 0.727, 23.959, 0.923, -0.057, 
    0.268}, {"04D2gp", 0.732, 24.219, 0.849, -0.091, 0.29}, {"06D4bw",
     0.732, 23.904, 1.066, 0.005, 0.25}, {"05D2fq", 0.733, 23.997, 
    1.084, -0.033, 0.249}, {"05D2ct", 0.734, 24.385, 1.042, 0.093, 
    0.293}, {"04D1pp", 0.735, 23.998, 0.865, -0.064, 
    0.236}, {"05D3jk", 0.736, 23.717, 1.082, -0.079, 
    0.227}, {"05D1eo", 0.737, 24.316, 0.854, -0.038, 
    0.255}, {"04D2ja", 0.74, 24.129, 1.04, -0.131, 0.281}, {"04D3fq", 
    0.742, 24.116, 0.956, -0.03, 0.264}, {"04D2kr", 0.744, 23.865, 
    1.057, -0.02, 0.247}, {"05D3jb", 0.745, 23.939, 1.102, -0.025, 
    0.235}, {"04D3ks", 0.75, 23.855, 1.075, -0.03, 0.257}, {"04D4im", 
    0.751, 23.852, 1.148, 0.043, 0.245}, {"04D3oe", 0.756, 24.06, 
    0.908, -0.161, 0.258}, {"05D2nt", 0.757, 24.099, 1.156, -0.006, 
    0.247}, {"05D3mn", 0.76, 24.043, 0.987, -0.021, 0.244}, {"06D3gx",
     0.76, 23.885, 1.043, -0.089, 0.296}, {"05D4cn", 0.763, 24.102, 
    1.111, 0.015, 0.243}, {"05D1if", 0.763, 24.059, 1.019, -0.035, 
    0.243}, {"05D3hh", 0.766, 24.319, 1.07, 0.018, 0.279}, {"04D1qd", 
    0.767, 24.228, 1.007, 0.012, 0.245}, {"04D1de", 0.768, 24.144, 
    1.101, -0.079, 0.241}, {"04D4id", 0.769, 24.212, 1.074, -0.1, 
    0.276}, {"04D1pc", 0.77, 24.553, 0.97, 0.067, 0.256}, {"05D4bi", 
    0.775, 24.051, 1.098, -0.094, 0.268}, {"04D1jd", 0.778, 24.425, 
    1.01, 0.051, 0.258}, {"05D4cs", 0.79, 23.967, 1.135, -0.061, 
    0.236}, {"03D4fd", 0.791, 24.232, 1.072, -0.024, 
    0.288}, {"04D1ks", 0.798, 24.145, 1.079, 0.076, 0.251}, {"05D3dh",
     0.8, 24.203, 1.009, 0.078, 0.26}, {"03D1fq", 0.8, 24.512, 
    0.87, -0.045, 0.272}, {"05D3cx", 0.805, 23.913, 1.051, -0.11, 
    0.266}, {"05D3ha", 0.805, 24.388, 0.972, 0.09, 0.287}, {"05D4gw", 
    0.808, 24.481, 1.025, 0.047, 0.288}, {"05D4dy", 0.81, 24.617, 
    0.956, -0.083, 0.275}, {"04D3ny", 0.81, 24.262, 1.024, 0.011, 
    0.31}, {"04D4dm", 0.811, 24.402, 0.981, 0.021, 0.267}, {"04D3mk", 
    0.813, 24.294, 0.954, -0.104, 0.254}, {"04D3nc", 0.817, 24.293, 
    1.139, -0.022, 0.28}, {"06D2ce", 0.82, 24.215, 1.219, 0.018, 
    0.333}, {"04D3lu", 0.822, 24.377, 0.934, -0.084, 0.25}, {"05D1cl",
     0.83, 24.353, 1.242, -0.018, 0.269}, {"04D3cp", 0.83, 24.111, 
    1.049, -0.18, 0.256}, {"04D2al", 0.836, 24.319, 0.933, -0.04, 
    0.372}, {"Elvis", 0.839, 24.397, 0.985, 0, 0.263}, {"05D4fg", 
    0.839, 24.195, 1.024, -0.102, 0.254}, {"06D2ga", 0.84, 24.29, 
    1.176, -0.001, 0.356}, {"05D1az", 0.842, 24.254, 1.184, 0.015, 
    0.269}, {"05D4hn", 0.842, 24.196, 1.122, 0.048, 0.319}, {"04D1hy",
     0.85, 24.307, 1.11, -0.026, 0.254}, {"06D4ce", 0.85, 24.205, 
    1.168, -0.059, 0.303}, {"05D3kp", 0.85, 24.139, 1.131, -0.082, 
    0.243}, {"05D4dw", 0.855, 24.438, 1.046, 0.008, 0.278}, {"04D1ff",
     0.86, 24.243, 1.08, 0.046, 0.257}, {"05D1iz", 0.86, 24.392, 
    1.095, -0.082, 0.336}, {"05D1er", 0.86, 24.618, 1.042, 0.062, 
    0.29}, {"03D1bk", 0.865, 24.345, 1.018, -0.167, 0.263}, {"05D1em",
     0.866, 24.283, 1.017, -0.1, 0.26}, {"04D4ii", 0.866, 24.399, 
    1.165, 0.039, 0.284}, {"03D1ew", 0.868, 24.359, 1.036, -0.036, 
    0.284}, {"05D2nn", 0.87, 24.395, 0.879, -0.147, 0.321}, {"04D4bk",
     0.88, 24.327, 1.181, -0.061, 0.28}, {"05D3cq", 0.89, 24.22, 
    0.96, -0.151, 0.278}, {"05D2by", 0.891, 24.568, 1.132, -0.008, 
    0.29}, {"03D4di", 0.899, 24.314, 1.146, -0.043, 0.29}, {"05D3ht", 
    0.901, 24.417, 1.118, -0.095, 0.282}, {"04D3gx", 0.91, 24.666, 
    0.94, -0.11, 0.293}, {"04D1ow", 0.915, 24.366, 1.004, -0.124, 
    0.268}, {"05D2bw", 0.92, 24.4, 0.972, -0.111, 0.319}, {"05D2ay", 
    0.92, 24.672, 0.983, 0.014, 0.349}, {"05D2ob", 0.924, 24.822, 
    1.103, 0.033, 0.318}, {"03D4cy", 0.927, 24.705, 1.103, -0.032, 
    0.346}, {"06D2cd", 0.93, 24.876, 1.166, 0.051, 0.49}, {"04D4jy", 
    0.93, 24.765, 1.022, -0.076, 0.345}, {"04D4ih", 0.934, 24.43, 
    1.03, -0.166, 0.276}, {"Vilas", 0.935, 24.473, 1.036, -0.028, 
    0.272}, {"04D4hf", 0.936, 24.811, 1.081, 0.028, 0.35}, {"05D3la", 
    0.936, 24.494, 0.967, -0.083, 0.267}, {"03D4cx", 0.949, 24.464, 
    0.947, 0.019, 0.331}, {"04D1pd", 0.95, 24.734, 1.039, 0.022, 
    0.298}, {"04D3ml", 0.95, 24.56, 1.117, -0.077, 0.29}, {"04D3nr", 
    0.96, 24.587, 0.99, 0.005, 0.299}, {"05D3km", 0.96, 24.773, 
    1.022, -0.123, 0.28}, {"04D4jw", 0.961, 24.848, 0.892, -0.173, 
    0.365}, {"Patuxent", 0.97, 25.026, 0.962, -0.129, 0.356}, {"Ombo",
     0.975, 24.891, 1.208, 0.018, 0.271}, {"05D2my", 0.981, 24.688, 
    1.13, -0.026, 0.305}, {"04D3lp", 0.983, 25.023, 0.816, -0.044, 
    0.378}, {"04D1rx", 0.985, 24.77, 1.081, -0.062, 0.307}, {"04D1iv",
     0.998, 24.624, 1.152, -0.074, 0.285}, {"06D4cl", 1, 24.578, 
    1.13, -0.065, 0.296}, {"04D3dd", 1.002, 25.234, 1.112, -0.016, 
    0.412}, {"Strolger", 1.01, 24.99, 1.195, -0.077, 0.433}, {"Eagle",
     1.02, 24.968, 1.017, -0.061, 0.289}, {"Ferguson", 1.02, 24.867, 
    1.023, 0.007, 0.329}, {"04D4dw", 1.031, 24.546, 1.127, -0.08, 
    0.326}, {"06D3en", 1.06, 24.756, 0.858, -0.139, 0.358}, {"Gabi", 
    1.12, 25.121, 1.048, -0.037, 0.278}, {"Greenberg", 1.14, 24.727, 
    1.09, -0.055, 0.325}, {"Lancaster", 1.23, 26.054, 0.969, 0.073, 
    0.335}, {"Torngasek", 1.265, 25.757, 1.04, 0.028, 
    0.354}, {"Aphrodite", 1.3, 25.691, 1.058, 0.013, 0.284}, {"Borg", 
    1.34, 25.87, 1.192, 0.09, 0.389}, {"Sasquatch", 1.39, 25.956, 
    1.193, 0.112, 0.535}, {"Primo", 1.54992, 25.7576, 
    0.168369, -0.197134, 0.6329682}, {" GND13Sto ", 1.8, 26.1369, 
    0.527158, -0.0156538, 0.9455797}, {"SN UDS10Wil", 1.914, 
    26.2, -0.5, -0.071, 0.85}, {"GND12Col", 2.25, 26.791, 1.1517, 
    0.0421647, 1.381078}};

And here is the function that selects the minimum value:

FindMinimum[{chiSquared[t1, a0]}, {{t1, 1*^16}, {a0, 1}}]

This will return:

{111076., {t1 -> 6.65636*10^18, a0 -> 3.87928*10^-14}}

Which is pretty close, but if you remove the initializers, or set t1 to 1, then it's unable to find a minimum.

FindMinimum[{chiSquared[t1, a0]}, {t1, a0}]

{1.49713*10^7, {t1 -> 72.0749, a0 -> -2.21105*10^7}}

How do I get the FindMinimum function to work if I don't know the answer?

Answer 1

This is more of an extended comment than an answer. First, building off @JackLaVigne's work, I think it's better to work in log of your parameters, so I defined a new objective function

chiSquared2[t1pow_, a0pow_] := Sum[residual[i, 10^t1pow, 10^a0pow]^2/error[i]^2, {i, 1, Length[data]}]

This still doesn't work well inside FindMinimum without a starting guess:

FindMinimum[{chiSquared2[t1pow, a0pow]}, {t1pow, a0pow}]
(* {1.49713*10^7, {t1pow -> 8.10751, a0pow -> -22095.5}} *)

With a good starting guess, it arrives at the same answer you found:

FindMinimum[{chiSquared2[t1pow, a0pow]}, {{t1pow, 18}, {a0pow, -14}}]
(* {111076., {t1pow -> 18.8232, a0pow -> -13.4112}} *)

Let's have a look at the objective function:

ContourPlot[Log10[chiSquared2[t1pow, a0pow]], {t1pow, 0, 20}, {a0pow, -16, 2},
Epilog -> {Red, Point[{18.823, -13.4112}]}]

Looks like a large, flat plain in dark blue, bordered by a ramp in the northeast corner. If you start at (1,1) it'd be hard to know which direction takes you down to the minimum at the red dot. Our success with FindMinimum tells us the minimum is somewhere around that little horn in the landscape. Let's zoom in there:

ContourPlot[Log10[chiSquared2[t1pow, a0pow]], {t1pow, 16, 20}, {a0pow, -20, -12},
Epilog -> {Red, Point[{18.823, -13.4112}]}]

Enhance!

ContourPlot[Log10[chiSquared2[t1pow, a0pow]], {t1pow, 18.0, 19.2}, {a0pow, -14, -13},
Epilog -> {Red, Point[{18.823, -13.4112}]}]

So you can see why a good starting guess is required.

Answer 2

Let's apply Jim Baldwin's idea of scaling the parameters so that they are in the same ball park.

I will use but not re-copy data from the question.

c = 2.99792*^5;
megaParsec = 3.08567758*^19;
alpha = 0.147;
beta = 3.13;
mag = -19.34;

I have some small modifications to your functions.

observedDistance[i_] := Module[
  {
   magnitude
   },
  magnitude = 
   data[[i, 3]] + alpha*(data[[i, 4]] - 1) - beta*data[[i, 5]] - mag;
  10^((magnitude - 25)/5)*megaParsec
  ]

luminousDistance[z_, t1_, a0_] := -(a0*t1^2*z + 
    2*c*(-t1*(1 + z) + Sqrt[t1^2*(1 + z)]))/(2 + z)

residual[i_, t1_, a0_] := Module[
  {
   z = data[[i, 2]]
   },
  observedDistance[i] - luminousDistance[z, t1, a0]
  ]

error[i_] := Module[
  {
   distance = observedDistance[i],
   distanceModuli
   },
  distanceModuli = Log10[distance/megaParsec]*5 + 25; 
  distance*data[[i, 6]]/distanceModuli
  ]

Update (get ball park values)

In order to get values that are in the ballpark Manipulate can sometimes be used (this won't work for all functions because they may have a restricted range). Your question indicates that you would like to allow your parameters to have a very wide range.

First I will take data and generate the magnitude from it and call it dataM:

dataM = Table[observedDistance[i], {i, Length@data}];

We will look at the logarithm of this distance or magnitude.

ListPlot[Transpose@Join[{data[[All, 2]], Log10[dataM]}]]

We create a Manipulate to scan the parameters from 10^-40 to 10^40. We are going to use the exponent from -40 to 40 as the input to the Manipulate.

Manipulate[
 DynamicModule[
  {
   minLogChi = 
    Min[21.1479, Log10[luminousDistance[0.01, 10.^t1Exp, 10.^a0Exp]]],
   maxLogChi = 
    Max[23.6936, Log10[luminousDistance[2.25, 10.^t1Exp, 10.^a0Exp]]]
   },
  Show[
   Plot[
    Log10[luminousDistance[z, 10.^t1Exp, 10.^a0Exp]],
    {z, 0.01, 2.25},
    PlotStyle -> Red
    ],
   ListPlot[
    Transpose@Join[{data[[All, 2]], Log10[dataM]}],
    PlotStyle -> Black
    ],
   PlotRange -> {{0, 2.25}, {minLogChi, maxLogChi}}
   ]
  ],
 {{t1Exp, 18.0}, -40, 40, Appearance -> "Open"},
 {{a0Exp, -14.0}, -40, 40, Appearance -> "Open"}
 ]

If you scroll the the exponents you will find that you can't get much bigger than 19 for t1Exp (or you get an error). Also you can't get much bigger than -13 for a0Exp.

You can type in values fort1Exp and/or a0Exp or make a second Manipulate with a tighter range.

Either way you will come to the conclusion that t1Exp = 18 is best. Also a0Exp must be less than or equal to -13.

Now proceed to the next phase.

Scaled Parameters

The key modification is to scale the parameters that are being optimized so they are within approximately one to ten.

chiSquared[t1Scaled_, a0Scaled_] := Module[
  {
   t1 = t1Scaled*10^18,
   a0 = a0Scaled*10^-14
   },
  Sum[residual[i, t1, a0]^2/error[i]^2, {i, 1, Length[data]}]
  ]

Now FindMinimum works fine without providing starting values

sol = FindMinimum[{chiSquared[t1Scaled, a0Scaled]}, {t1Scaled, a0Scaled}]

(* {111076., {t1Scaled -> 6.65636, a0Scaled -> 3.87928}} *)

You need to apply the scaling factor in order to get the unscaled parameters.

{t1, a0} = {t1Scaled*10^18, a0Scaled*10^-14} /. sol[[2]]

(* {6.65636*10^18, 3.87928*10^-14} *)

Answer 3

The root of the problem is the magnitude calculation which involves a Log10 operation, not the disparity between the power of the variables. The Log10 operation creates discontinuities in the solution space which the Mathematica solvers were not able to navigate.

The solution: instead of framing the problem in terms of the magnitude, I converted all the calculations and input values to kilometers. This made the solution space continuous and FindMinimum was able to find a solution that was reasonably close using very rough guesses for the initial conditions (t0 = 1*^14, a0 = 1). Then I used NMinimize and the original Log10 based calculations with a set of tight ranges around the result of the FindMinimum solution to find the absolute minimum in terms of magnitude and the magnitude errors.

Many thanks to Jack LaVigne for the effort he put into the scaled parameters solution but I think the real issue is that Mathematica's FindMinimum doesn't handle imaginary numbers well. The package from Original Labs, as it turns out, has a similar problem when the formula involves a Log10 operation: it can't navigate around the discontinuities.

Update:

Here is the code for the version that calculates the minimums using a continuous solution space by working in km instead of distance moduli (a notation similar to magnitude):

c = 299792.; 
megaParsec = 3.08567758*^19; 
alpha = 0.147; 
beta = 3.13; 
mag = -19.34; 
chiSquared[t1_, a0_] := Sum[residual[i, t1, a0]^2/error[i]^2, {i, 1, Length[data]}]
error[i_] := observedDistance[i, data[[i,6]]]
luminousDistance[z_, t1_, a0_] := -((a0*t1^2*z + 2*c*((-t1)*(1 + z) + Sqrt[t1^2*(1 + z)]))/(2 + z))
residual[i_, t1_, a0_] := observedDistance[i, 0.] - luminousDistance[data[[i,2]], t1, a0]
observedDistance[i_, error_] := (magnitude = data[[i,3]] + alpha*(data[[i,4]] - 1) - beta*data[[i,5]] - mag + error; 10^((magnitude - 25)/5)*megaParsec)
FindMinimum[{chiSquared[t1, a0]}, {{t1, 1*^16}, {a0, 1}}]

This will put you in the neighborhood using very rough guesses. The output is:

{3.00375, {t1 -> 6.76415*10^18, a0 -> 3.82816*10^-14}}

Which is close (like all the solutions that attempt to change the scale), but not the minimum. The chi-square on this solution is 0.74. Now that we're in the neighborhood, I can switch back to the Log10 version (the errors from Astronomy papers are always quoted in distance moduli, so it's the only valid test for Chi-Square). Here's the version using distance moduli:

c = 299792.; 
megaParsec = 3.08567758*^19; 
alpha = 0.147; 
beta = 3.13; 
mag = -19.34; 
chiSquared[t1_, a0_] := Sum[residual[i, t1, a0]^2/data[[i,6]]^2, {i, 1, Length[data]}]
luminousDistance[z_, t1_, a0_] := (distance = -((a0*t1^2*z + 2*c*((-t1)*(1 + z) + Sqrt[t1^2*(1 + z)]))/(2 + z)); 5*Log10[distance/megaParsec] + 25)
residual[i_, t1_, a0_] := observedDistance[i] - luminousDistance[data[[i,2]], t1, a0]
observedDistance[i_] := data[[i,3]] + alpha*(data[[i,4]] - 1) - beta*data[[i,5]] - mag
NMinimize[{chiSquared[x, y], 
  1*^17 < x < 1*^19 && 1*^-15 < y < 1*^-13}, {x, y}]

This version truly finds the minimum. The output is:

{342.207, {x -> 6.64112*10^18, y -> 3.87624*10^-14}}

And produces a chi-square of 0.72. So the solution is to first convert the formula to give a continuous solution space. Then the FindMinimum algorithms can navigate into the neighborhood of the minimum. Once you have the neighborhood, you can use NMinimize to find the absolute minimum.