Khatri and Wandelt Reply: Our calculations and resultsin [1] are correct. That we have to measure the 21-cmbrightness temperature Tb with cosmic microwave back-ground (CMB) temperature TCMB as a standard is implicitin the definition of the 21-cm brightness temperature. The21-cm brightness temperature is defined as the differenceof the observed temperature (which is a function of fre-quency) and the CMB temperature (which is independentof frequency). Thus we agree that the quantity to bemeasured is Tb=TCMB, but this has no effect on any ofour calculations or conclusions in [1].

Before proceeding let us remark that in any theory withdimensionful quantities we have to fix a few conversionfactors between different dimensions, for example, thespeed of light c, Boltzmann’s constant kB, Planck’s con-stant h, etc. We will fix these conversion factors andmeasure all quantities in the same physical units.

Cosmological measurements are different from lab orquasar measurements because in cosmology we can findstandards which remain invariant with respect to the varia-tion of microphysical parameters like the fine-structureconstant on time scales of the order of the age of theUniverse. In the present case of 21-cm cosmology therelevant standard is provided by the CMB temperature.The CMB temperature and the baryon-to-photon ratio � ¼nB=n�, where nB is the baryon number density and n� isthe photon number density is fixed (apart from negligiblechanges due to scattering at later times) when the tempera-ture of the Universe is around 0.5 MeV and the electronsand positrons have annihilated. In addition, the relativenumber density of protons or hydrogen nH is fixed oncethe big bang nucleosynthesis is completed around T ¼0:01 MeV. We reiterate that �, TCMB, and nH are strictconstants after recombination, redshift z & 1100, in stan-dard cosmology apart from trivial evolution due to theexpansion of the Universe. In particular, these are notaffected by the variation in the fine-structure constant �or other microphysical constants. Although nH is changedby processing by stars at late times, this does not affect ourcalculations which depend only on nH during the dark agesand which can be constrained using the 21-cm signal.

Any data analysis constraining � will have to margin-alize over the cosmological parameters including the pri-mordial nH and �. These parameters do not depend onalpha since we do not derive them from any fundamentalphysics but use them as free parameters to be determinedby a joint fit of cosmological observations including 21-cmobservations of the dark ages, CMB, and big bang nucleo-synthesis. Once fixed during very early Universe theseparameters are independent of � although we do notknow their exact values. The unique frequency dependenceof the signal due to variation in �means that we should notexpect significant degeneracies with other cosmologicalparameters. We agree (and mention in [1]) that extending

our work to variations of other fundamental constantswould be of interest.Thus we can think of measuring the 21-cm transition

frequency �21 (and also the electron mass me) in the unitsof CMB temperature which gives �21 / �4 used in [1].This is, in fact, what physically also happens. The absorp-tion of the CMB photons by neutral hydrogen amounts to ameasurement of �21 with respect to the CMB blackbodyspectrum. Note that the same arguments cannot be used forintensity of 21-cm radiation I�21

since the intensity of theCMB radiation is also a function of frequency and not astrict constant as the CMB temperature, i.e., the CMBintensity at �21 depends on �21.Similar arguments apply to the comments by Flambaum

and Porsev [2] following their Eq. (2). In their dimension-less parameter XH, � is a fixed constant after recombina-tion. Although we do not know the origin or the exact valueof this constant, it is nevertheless independent of � or anyother microphysics at z & 1100. We can form another di-mensionless quantity �21=TCMB and we have XH /TCMB=�21 / ��4, which is consistent with the calculationin [1]. We have already taken the effect of � on recombi-nation and hence on the ionization fraction xe into accountin [1] following previous CMB calculations. The change inxe of course affects the spin temperature TS and hence the21-cm signal. The effect of recombination on � is negli-gible for our purpose as has been shown in the recentprecision calculations of recombination (with changes inTCMB � nK compared to �mK 21-cm signal). We remarkagain that all quantities in XH except �21 are fixed bycosmology and are independent of microphysics, in par-ticular �, during and after recombination at z & 1100. Thearguments of Flambaum and Porsev [2] about ignoring XH

are thus incorrect.

Rishi Khatri*Department of AstronomyUniversity of Illinois at Urbana-Champaign1002 W. Green Street, Urbana, Illinois 61801 USA

Benjamin D. Wandelt†

Institut d’Astrophysique de ParisUniversite Pierre et Marie Curie (Paris 6)98 bis, boulevard Arago, 75014 Paris, FranceDepartment of AstronomyUniversity of Illinois at Urbana-Champaign1002 W. Green Street, Urbana, Illinois 61801 USA

Received 10 May 2010; published 12 July 2010DOI: 10.1103/PhysRevLett.105.039002PACS numbers: 98.70.Vc, 06.20.Jr, 98.80.Es

*rkhatri2@illinois.edu†bwandelt@uiuc.edu

[1] R. Khatri and B.D. Wandelt, Phys. Rev. Lett. 98, 111301(2007).

[2] V. V. Flambaum and S.G. Porsev, preceding Comment,Phys. Rev. Lett. 105, 039001 (2010).

PRL 105, 039002 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending16 JULY 2010

0031-9007=10=105(3)=039002(1) 039002-1 � 2010 The American Physical Society