Measuring cosmological parameters using Type Ia Supernovae

IDC 452: Seminar Delivery Course

Prajakta Mane

MS19054

A supernova is a transient astronomical event that occurs during the last evolutionary stages of a massive star or when a white dwarf is triggered into runaway nuclear fusion.
Is classified according to the light curves and the absorption lines of different chemical elements that appear in the spectra.
Type Ia supernovae occur in binary systems in which one of the stars is a white dwarf.
Have the light curves with a sharp maximum and gradual decline, producing a fairly consistent peak luminosity because of the fixed critical mass at which a white dwarf will explode.

Cosmological parameters governing the evolution of the universe:

H^2 = \left(\frac{\dot{a}}{a}\right)^2 = \frac{8 \pi G}{3} \rho - \frac{\kappa}{a^2}

H^2 = \left(\frac{\dot{a}}{a}\right)^2 = \frac{8 \pi G}{3} \rho - \frac{\kappa}{a^2}

∵ First Friedmann Equation

Taylor expanding a(t):

\frac{1}{a(t)} \approx 1 - H_0 (t-t_0) + \left(\frac{1+q_0}{2}\right) H_0^2 (t-t_0)^2

\frac{1}{a(t)} \approx 1 - H_0 (t-t_0) + \left(\frac{1+q_0}{2}\right) H_0^2 (t-t_0)^2

∵ Independent of cosmological

models

\text{where } q_0 = - \left( \frac{\ddot{a}}{a H^2}\right) = \frac{1}{2} \sum_{i}^{} \Omega_{i, 0} (1+3\omega_i)

\text{where } q_0 = - \left( \frac{\ddot{a}}{a H^2}\right) = \frac{1}{2} \sum_{i}^{} \Omega_{i, 0} (1+3\omega_i)

∵ The Deceleration parameter

Parameters to be found observationally:

H(t), q_0, \rho_i, \omega_i

H(t), q_0, \rho_i, \omega_i

P = \omega \rho

P = \omega \rho

\dot{\rho} + 3 \frac{\dot{a}}{a} (\rho + P)

\dot{\rho} + 3 \frac{\dot{a}}{a} (\rho + P)

∵ Fluid Equation

Standard Candles: Type Ia supernovae are the most useful, precise, and mature tools for determining astronomical distances and can provide constraints on value of the Hubble's constant.

D_L = \left(\frac{L}{4 \pi f}\right)^{\frac{1}{2}} \approx \frac{c}{H_0} z \left(1 + \frac{1-q_0}{2}z\right)

D_L = \left(\frac{L}{4 \pi f}\right)^{\frac{1}{2}} \approx \frac{c}{H_0} z \left(1 + \frac{1-q_0}{2}z\right)

The luminosity distance DL is defined as,

L: Standardized luminosity
f: Flux of the standard candle measured on earth

z: Redshift of the standard candle, found from (1+z) = 1/a(t) relation

Collaborations like High-z supernova Search Team, PI: Perlmutter;

Supernova Cosmology Project, PI: Schmidt and Riess

H0 measurement from low-z supernovae,

q0 measurement from high-z supernovae

Riess et al, 1999

Permutter et al, 1999

Independent Method to Constrain the value of the Hubble's Constant: The time-delay measurements of the lensed, multiply imaged supernovae.

\Delta t = \frac{D_d D_s}{cD_{ds}}(1+z)\Delta\tau

\Delta t = \frac{D_d D_s}{cD_{ds}}(1+z)\Delta\tau

\frac{D_d D_s}{D_{ds}} \alpha \frac{1}{H_0}

\frac{D_d D_s}{D_{ds}} \alpha \frac{1}{H_0}

Collaborations like H0LiCoW, COSMOGRAIL, TDCOSMO

1. Perlmutter, S. and Schmidt B., “Measuring Cosmology with Supernovae.” Lecture Notes in Physics 598 (2003): 195-217.

2. Barbara Ryden, "Introduction to Cosmology" (2003): Chapter 7.

3. Riess, A., et al, "Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant" (1998), Astrophys. J.

4. Perlmutter, S., et al., "Measurements of Ω and Λ from 42 High-Redshift Supernovae", (1999), Astrophys. J.

5. Refsdal S., "On the Possibility of Determining Hubble's Parameter and the Masses of Galaxies from the Gravitational Lens Effect" (1964), MNRAS.