# 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 $$ * $M\_B$ - peak absolute luminosity in B-band * $\Delta m\_{15}$ - the observed change in apparent magnitude 15 days after the peak * The maximum light of Type Ia Supernovae appears to be dimmer than that in an empty universe ($\Omega*k=1$) and that in a matter dominate universe ($\Omega\_m=0.25, \Omega*\Lambda=0$) * $\Omega*m\sim0.25, \Omega*\Lambda\sim0.75$ - accelerating universe ### Parameter estimation * A sample of $n$ SN measurements * Magnitude $m\_i$ * Typical magnitude error $\pm \sigma\_{m,i}$ * Redshift $z\_i$ (error can be neglected) * Try to estimate $\left(\Omega*{\mathrm{m}, 0}, \Omega*{\Lambda, 0}, M\right)$ * Fix $M$ (with sufficient precision) and estimate $\Omega$s * **Fit simultaneously for all three** * MLE * Likelihood function (assumption of Gaussian distribution) $$ 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 $$