Chapter 6 Supernova cosmology

Application of the Luminosity Distance

Two main applications

• The luminosity distance $d_L$ is a function of redshift $z$, and there are usually two applications

  • Given observed flux $F_{obs}$, if we know the redshift of a certain galaxy and universe parameters well, we can calculate the luminosity distance, and then the absolute luminosity - luminosity function

  • Given observed flux $F{obs}$, for some certain objects (Cepheids, SNe) we know the absolute luminosity and thus the luminosity distance - estimation of $\Omega_m,\Omega\Lambda$

Luminosity function

• Describes the number of galaxies per unit volume with luminosity in the range $[L, L+d L]$

• Schechter function

  $$\Phi(L) d L=\phi^{*}\left(\frac{L}{L^{*}}\right)^{\alpha} e^{-L / L^{*}} \frac{d L}{L^{*}}$$

  • $\alpha<0$ - Faint end slope (diverges at the faint end - there must be a turn-over)

  • $L^*$ - Characteristic (fiducial) luminosity

  • $\phi^*$ - Overall normalisation

• Luminosity-weighted luminosity function - $L\Phi(L)$

  • Converge if $\alpha>-2$

• Well fitted by a Schechter function in the local universe

Distance modulus

$$M-m=2.5\lg\left(\frac{d_{L,0}}{d_L}\right)^2=5\lg\frac{d_{L,0}}{d_L}$$

For cosmology cases, 10 pc is too small and thus not practical, while 1 Mpc is usually used as $d_{L,0}$

$$m=M+5\lg d_L(\text{Mpc})+25$$

For small $z$,

$$d_L=(1+z)r_1\approx\frac{c}{H_{0}}\left[z+\frac{1}{2}\left(1-q_{0}\right) z^{2}+\cdots\right]$$

$$m=M-5 \log H_{0}+5 \log c z+\cdots+25$$

Type Ia Supernovae

• Nuclear explosions of carbon/oxygen white dwarfs in binary systems

• The peak luminosity appears to be well-correlated with decay time

  The larger $L_{peak}$, the slower the decay

  $$M_{B} \approx 0.8\left(\Delta m_{15}-1.1\right)-19.5$$

  • $M_B$ - peak absolute luminosity in B-band

  • $\Delta m_{15}$ - the observed change in apparent magnitude 15 days after the peak

• The maximum light of Type Ia Supernovae appears to be dimmer than that in an empty universe ($\Omegak=1$) and that in a matter dominate universe ($\Omega_m=0.25, \Omega\Lambda=0$)

• $\Omegam\sim0.25, \Omega\Lambda\sim0.75$ - accelerating universe

Parameter estimation

• A sample of $n$ SN measurements

  • Magnitude $m_i$

  • Typical magnitude error $\pm \sigma_{m,i}$

  • Redshift $z_i$ (error can be neglected)

  • Try to estimate $\left(\Omega{\mathrm{m}, 0}, \Omega{\Lambda, 0}, M\right)$

    • Fix $M$ (with sufficient precision) and estimate $\Omega$s

    • Fit simultaneously for all three

• MLE

  • Likelihood function (assumption of Gaussian distribution)

    $$L(\theta) \propto \exp \left[-\frac{1}{2} \sum_{i=1}^{n}\left(\frac{m\left(z_{i} ; \theta\right)-m_{i}}{\sigma_{m, i}}\right)^{2}\right]$$

    where

    $$\theta =\left(\Omega_{\mathrm{m}, 0}, \Omega_{\Lambda, 0}, M\right)$$
  • We are not interested in $M$ so we take integration over $M$ to get the 2-D likelihood function

    $$L\left(\Omega_{\mathrm{m}, 0}, \Omega_{\Lambda, 0}\right)=\int d M L\left(\Omega_{\mathrm{m}, 0}, \Omega_{\Lambda, 0}, M\right)$$
  • We can generate $L$ in the phase space of $\left(\Omega{\mathrm{m}, 0}, \Omega{\Lambda, 0}\right)$ to get the estimator, and compare the allowed region (in the phase space) under certain confident levels with other tests (cluster, CMB, BAO, etc.) to narrow it down

  • $\Omega{\mathrm{m}, 0} \simeq 0.3, \Omega{\Lambda, 0} \simeq 0.7, \Omega_{\mathrm{k}, 0} \simeq 0$

Alternative Explanations

Astrophysical dimming

Evolution

• The properties of SNIa events have evolved with cosmic time (different $z$)

  • SN at high $z$ - younger population

  • Lower metallicity (abundances of carbon/oxygen)

• So far, no evidence

Intersellar dust

• Supernovae in distant galaxies are more (or less) dimmed by dust than local supernovae

• Normal dust not only dim but also redden the light, then by comparing the colors of local/distant SNe, we may assess the importance of the effect

• No difference in reddening

Grey dust

• Strange particles - significantly larger than the wavelength of light, scatter light of all wavelength similarly

Assessments

• Push the measurement of SN Ia light curves to redshifts $z > 1$

  • Back in $z\sim1$, the $\Lambda$ term begins to become comparable to the matter term

  • For $z>1$, the $\Lambda$ term is negligible and SNe should be brighter again

Dark Energy

• The energy density related to the cosmological constant (natural units)

  $$\rho_{\Lambda} \equiv \frac{\Lambda}{8 \pi G}=\mathrm{constant}$$

  The fluid equation

  $$\dot{\rho}_\Lambda=-3(\rho_\Lambda+p_\Lambda) \frac{\dot{a}}{a}=0\Rightarrow p_{\Lambda}=-\rho_{\Lambda}$$
• We introduce constant $\omega_i$

  $$p_i=w_{i} \rho_{i}$$

  Then the fluid equation becomes

  $$\frac{\dot{\rho}_{i}}{\rho_{i}}=-3\left(1+w_{i}\right) \frac{\dot{a}}{a}\Rightarrow \rho_{i} \propto a^{-n_{i}}$$

  where $n{i}=3\left(1+w{i}\right)$, then we have a useful property

  $$\frac{\Omega_{i}}{\Omega_{j}} \propto a^{-\left(n_{i}-n_{j}\right)}$$
• Recall our discussions in Chapter 2

  • Dust (matter)

    $$\rho_{m} \propto a^{-3}\Rightarrow\omega_m=0$$
  • Radiation

    $$\rho_{r} \propto a^{-4}\Rightarrow\omega_r=\frac13$$
  • Cosmological constant

    $$\rho_\Lambda\propto a^0\Rightarrow \omega_\Lambda=-1$$
  • Curvature

    $$\Omega_{k} \equiv-k /(a H)^{2}\Rightarrow\rho_k\propto a^{-2}\Rightarrow \omega_k=-\frac13$$
• Hubble parameter

  $$H(a)=H_{0}\left(\sum_{i} \Omega_{i, 0} a^{-n_{i}}\right)^{1 / 2}$$
• Deceleration parameter

  $$q(t)=-\frac{1}{H^{2}} \frac{\ddot{a}}{a}$$

  • Friedmann equation

    $$\frac{\ddot a}{a}=-\frac{4\pi G}{3}(\rho+3p)$$

    Then

    $$\begin{align*} q(t)&=\frac{4\pi G}{3H^2}(\rho+3p)=\frac12\frac{\rho+3p}{\rho_c}\\ &=\frac12\sum_{i}(1+3\omega_i)\Omega_i\\ &=\frac12\sum_{i}(n_i-2)\Omega_i \end{align*}$$
• Recall that

  • $q<0$ , acceleration of expansion - $n_i<2$, cosmological constant

    • $q>0$, slowing down of expansion - $n_i>2$, matter, radiation

    • $q=0$, linear expansion - $n_i=2$, empty universe

• Most recent results of $\omega$

  $$\omega=-1.007\pm0.081$$
• The evolution of $\omega$

  $$w(a)=w_{0}+w_{a}(1-a)$$

  $$w_{0}=-1.02 \pm0.12,\ w_{{a}}=0.07 \pm 0.6$$