Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 73 / 1

A kinematic redshift is the relativistic Doppler effect arising from relative motion of emitter and receiver. In a local inertial frame, recession decreases the received frequency and approach increases it. A cosmological redshift instead compares photons emitted and received by comoving observers at different cosmic times: expansion of the scale factor stretches their wavelengths. It exists even when both observers have zero peculiar velocity. Both are frequency ratios measured by observers, but a large cosmological redshift should not be inserted into a flat-spacetime radial-speed formula to define the galaxy's unique recession speed.

Take $v>0$ for recession. By the definition of kinematic redshift, $1+z_{\mathrm{kin}}={\nu }_e/{\nu }_0$. Inverting the supplied relativistic Doppler effect and expanding with $\beta =v/c$ givesThusA negative radial velocity gives a blueshift at this order.

For a radial null geodesic in FLRW spacetime, the FRW metric gives $cdt/a(t)=\pm dr/\sqrt{1-k{r}^2}$. The emitter and receiver have fixed comoving coordinates, so two successive wave crests satisfy the same comoving radial distance relation:Subtracting the two expressions in the short-wave-period limit yields $\delta t_e/a(t_e)=\delta t_0/a(t_0)$. Comoving observers measure these intervals as proper time; since $\nu =1/\delta t$,For a fixed reception epoch, the cosmological redshift factor is inversely proportional to the emission scale factor. This also shows the associated cosmological time dilation.

In the intended comparison of emission and reception epochs, with negligible peculiar velocities, the Earth-observed cosmological redshifts correspond to $a_A=a_0/2$ and $a_B=a_0/10$. A photon emitted at the B epoch and received at the A epoch consequently hasThese factors divide, whereas the redshifts themselves do not subtract. There is a geometrical qualification to this usual answer: the two Earth-observed redshifts alone do not specify an arbitrary actual B-to-A observation. The assumed events must be connected by a null geodesic, as when A is on the same radial photon path between B and Earth. If the galaxies have an unspecified angular separation, or A observes at a different cosmic time, the B emission event reaching A is different and its redshift is not determined by these two numbers alone.

The combined kinematic and cosmological redshift is most cleanly found by passing through the emitter's local comoving frame. The distance estimate then requires an explicit approximation to the Hubble law.

Let ${\nu }_c$ be the photon frequency in the local comoving frame just after emission. The kinematic redshift gives ${\nu }_e/{\nu }_c=1+z_{\mathrm{kin}}$, and propagation in the FRW metric gives ${\nu }_c/{\nu }_0=1+z_{\mathrm{cosm}}$. Multiplication givesEquivalently, $z_{\mathrm{tot}}=z_{\mathrm{cosm}}+z_{\mathrm{kin}}+z_{\mathrm{cosm}}{z}_{\mathrm{kin}}$. Both redshifts being small is what permits their approximate addition. Here the observer's own peculiar velocity is assumed already corrected to the Hubble flow frame.

In the local, nonrelativistic Hubble law approximation, $c{z}_{\mathrm{tot}}\simeq H_{0}r+v_p$. Ignoring the peculiar velocity gives an inferred distance $r_{\mathrm{est}}=r+v_p/H_0$, soThe requested accuracy therefore requiresAt exactly $120\mathrm{Mpc}$ the worst-case error is $5\%$, so “better than” requires a strictly larger distance. This is the conventional leading-order estimate. If one retains the exact Doppler factor and the product term while treating $r=c{z}_{\mathrm{cosm}}/H_0$ as exact, the positive-velocity case gives $|\Delta r|/r=(1+z_{\mathrm{cosm}})q/z_{\mathrm{cosm}}$, with $q=\sqrt{(1+600/c)/(1-600/c)}-1$. It requires $r>(c/H_0)q/(0.05-q)\simeq 125\mathrm{Mpc}$. That small refinement is conditional on the distance-redshift prescription; the simple $120\mathrm{Mpc}$ answer uses the local approximation consistently.

For Planck law expressed per unit frequency, put $x=h\nu /(k_{B}T)$. At fixed temperature, the frequency dependence is $B_{\nu }\propto T^3{x}^3/(e^x-1)$. Differentiation givesThe positive interior maximum therefore satisfies $3(1-e^{-x_{\ast }})=x_{\ast }$, with $x_{\ast }=2.821439\dots$ independent of temperature. The zero solution of the rearranged equation is a boundary, not the interior maximum. Hence the frequency form of Wien's displacement law isThis frequency peak is not $c$ divided by the peak wavelength of $B_{\lambda }$: the two spectral densities differ by the frequency-to-wavelength Jacobian.

Motion toward incoming cosmic microwave background radiation produces a blueshift. The photon occupation function of a blackbody depends on $h\nu /(k_{B}T)$, so the same Doppler factor multiplies both frequency and effective temperature. Looking forward along the Sun's peculiar velocity, one obtainsTaking the mean cosmic microwave background temperature as about $2.73\mathrm{K}$ givesThe numerical total inherits the rounded mean temperature; the robust estimate is the positive millikelvin dipole increment.

A K correction accounts for observing a redshifted spectrum through a fixed photometric passband, rather than the corresponding emitted band. In the same R-band astronomical magnitude convention,The luminosity distance already incorporates bolometric dimming; the additional correction depends on spectral shape and band response. At $z=3$, observed R-band light at $660\mathrm{nm}$ was emitted at $165\mathrm{nm}$, in the far ultraviolet. It is not a direct measurement of rest-frame R-band luminosity.

To fix the sign, let $L_{\nu }$ be emitted spectral luminosity and approximate R as narrow. Counting photon energy, arrival time and the transformed frequency interval gives $F_{{\nu }_o}=(1+z)L_{(1+z){\nu }_o}/(4\pi d_L^2)$. ThusFor an ordinary old elliptical galaxy, the far-ultraviolet spectrum is weak relative to the optical, so typically $L_{\nu }(165\mathrm{nm})/L_{\nu }(660\mathrm{nm})<1/4$. Then $K_R>0$: neglecting it overestimates the numerical absolute magnitude and underestimates the R-band luminosity. For a young, unobscured star-forming galaxy, hot stars make the far-ultraviolet strong, with an approximately flat $L_{\nu }$ spectrum often serving as a useful model. It gives $K_R=-2.5{\log }_{10}4\simeq -1.51$: neglecting it underestimates the numerical absolute magnitude and overestimates the R-band luminosity. Strong dust extinction or an unusual spectrum can reverse this latter sign; a galaxy label alone does not prove an exact correction. The band-integrated convention is given in Hogg and collaborators' definition of the K-correction.