Redshift Study Guide

📖 Core Concepts Redshift (z) – fractional increase in wavelength: \(z = \dfrac{\lambda{\text{obs}}-\lambda{\text{emit}}}{\lambda{\text{emit}}}\). Positive \(z\) = redshift, negative \(z\) = blueshift. Wavelength ratio – \(1+z = \dfrac{\lambda{\text{obs}}}{\lambda{\text{emit}}}\). Three physical origins Doppler redshift – motion of source relative to observer. Gravitational redshift – photon climbing out of a potential well. Cosmological redshift – stretching of space (expansion of the Universe). Scale factor (a) – size of the Universe relative to today; cosmological redshift satisfies \(1+z = \dfrac{1}{a}\). Hubble’s law (low‑z) – recession velocity \(v \approx H0 D\); connects distance \(D\) to redshift via \(v = cz\) for small \(z\). --- 📌 Must Remember Small‑velocity Doppler: \(z \approx \dfrac{v{\parallel}}{c}\). Transverse Doppler (small \(v\perp\)): \(z \approx \tfrac{1}{2}\left(\dfrac{v\perp}{c}\right)^2\). Relativistic Doppler (line‑of‑sight): \(1+z = \gamma(1+\beta)\) with \(\beta = v/c\), \(\gamma = 1/\sqrt{1-\beta^2}\). Gravitational redshift (Schwarzschild): \(z = \left(1-\dfrac{2GM}{rc^2}\right)^{-1/2} - 1\). Cosmic microwave background redshift: \(z{\text{CMB}} \approx 1089\). Typical photometric redshift uncertainty: \(\Delta z \sim 0.5\). --- 🔄 Key Processes Compute spectroscopic redshift \[ z = \frac{\lambda{\text{obs}}}{\lambda{\text{emit}}} - 1 \] Convert redshift to recession velocity (low‑z) \[ v = cz \] Apply Hubble’s law for distance \[ D = \frac{v}{H0} = \frac{cz}{H0} \] Relativistic Doppler calculation (line‑of‑sight) \[ 1+z = \sqrt{\frac{1+\beta}{1-\beta}} \quad\Longrightarrow\quad \beta = \frac{(1+z)^2-1}{(1+z)^2+1} \] Gravitational redshift from potential difference (weak field) \[ \frac{\Delta\lambda}{\lambda} \approx \frac{\Delta\Phi}{c^2} \] Photometric redshift estimation – match observed broadband colors to a library of template spectra; pick the template with minimum χ². --- 🔍 Key Comparisons Doppler vs. Cosmological redshift Cause: relative motion vs. expansion of space. Formula: \(z \approx v/c\) (low‑v) vs. \(1+z = 1/a\). Directionality: can be positive or negative (blueshift) for Doppler; always positive for pure cosmological expansion. Gravitational redshift vs. Gravitational blueshift Redshift: photon climbs out → wavelength ↑. Blueshift: photon falls in → wavelength ↓. Dependence: only on potential difference \(\Delta\Phi\); independent of emission angle. Spectroscopic vs. Photometric redshift Spectroscopic: uses discrete line wavelengths → high precision (Δz ≲ 0.001). Photometric: uses broadband colors → lower precision (Δz  0.5), relies on templates. --- ⚠️ Common Misunderstandings “Redshift always means the object is moving away.” Only true for pure Doppler redshift; cosmological redshift also stretches light even if peculiar motion is toward us. “Higher z = greater velocity directly via \(v=cz\).” Valid only for low z (≲0.1). At larger z, relativistic and cosmological effects dominate; use proper relativistic formulas or the Hubble‑law integral. “Gravitational redshift depends on the photon’s path angle.” It depends solely on the potential difference, not on the emission/reception angle. --- 🧠 Mental Models / Intuition “Stretching a rubber band” – imagine space as a rubber band being stretched; every wavelength on the band lengthens proportionally → cosmological redshift. “Climbing a hill with a ball of light” – the photon loses kinetic energy climbing out of a gravitational well, so its “color” shifts red; rolling downhill gives a blueshift. “Speed limit vs. time dilation” – for fast‑moving sources, the extra redshift beyond the classical Doppler shift comes from time dilation (the source’s clock runs slower). --- 🚩 Exceptions & Edge Cases Transverse Doppler effect – a source moving perpendicular to the line of sight still shows a redshift (\(z \approx \tfrac{1}{2}(v\perp/c)^2\)). Peculiar velocities – local motions (e.g., Andromeda’s blueshift) can partially cancel the cosmological redshift; must be subtracted when measuring Hubble flow. Strong gravitational fields – near a black‑hole horizon the Schwarzschild formula predicts \(z \to \infty\); photons are infinitely redshifted. --- 📍 When to Use Which Low redshift (z ≲ 0.1): use \(v \approx cz\) and Hubble’s law \(D = cz/H0\). Moderate‑to‑high redshift (z > 0.1): apply the relativistic Doppler formula or the cosmological relation \(1+z = 1/a\); use Mattig formula for distance if \(\Lambda = 0\). Strong gravitational fields: use the full Schwarzschild redshift expression; weak‑field approximations (\(\Delta\lambda/\lambda \approx \Delta\Phi/c^2\)) are insufficient. Only photometry available: adopt photometric redshift methods, aware of large uncertainties. --- 👀 Patterns to Recognize “Line‑of‑sight velocity → linear shift of all spectral lines.” If every line moves by the same factor, the shift is Doppler (or cosmological). “Systematic wavelength‑dependent distortion (e.g., “fingers of God”). Indicates redshift‑space distortions from galaxy cluster velocity dispersion. “Same line appears at two different redshifts in the same spectrum.” Possible gravitational lensing or multiple‑source superposition. --- 🗂️ Exam Traps Choosing \(v = cz\) for \(z=0.5\). This ignores relativistic and cosmological corrections; answer will be off by > 30 %. Assuming any blueshift must be gravitational. Most observed blueshifts (e.g., Andromeda) are Doppler from peculiar motion. Mixing up the scale factor and redshift sign: Remember \(a = 1/(1+z)\); a larger \(z\) means a smaller \(a\). Treating photometric redshift as precise. Test‑makers may give a photometric \(z\) and expect you to note the large uncertainty. Neglecting transverse Doppler contribution when a source moves perpendicular to the observer; a small positive redshift may still appear. ---