Unavoidable CMB spectral features and blackbody photosphere of our Universe
aa r X i v : . [ a s t r o - ph . C O ] J un September 26, 2018 14:7 WSPC/INSTRUCTION FILE sunyaev˙plenary
International Journal of Modern Physics Dc (cid:13)
World Scientific Publishing Company
UNAVOIDABLE CMB SPECTRAL FEATURES AND BLACKBODYPHOTOSPHERE OF OUR UNIVERSE ∗ RASHID A. SUNYAEV
Max Planck Institut f¨ur Astrophysik, Karl-Schwarzschild-Str. 185741, Garching, GermanySpace Research Institute, Russian Academy of Sciences, Profsoyuznaya 84/32117997 Moscow, RussiaInstitute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540, [email protected]
RISHI KHATRI
Max Planck Institut f¨ur Astrophysik, Karl-Schwarzschild-Str. 185741, Garching, [email protected]
Received February 25, 2013Revised March ??, 2013Spectral features in the CMB energy spectrum contain a wealth of information aboutthe physical processes in the early Universe, z . × . The CMB spectral distortionsare complementary to all other probes of cosmology. In fact, most of the informationcontained in the CMB spectrum is inaccessible by any other means. This review outlinesthe main physics behind the spectral features in the CMB throughout the history of theUniverse, concentrating on the distortions which are inevitable and must be present ata level observable by the next generation of proposed CMB experiments. The spectraldistortions considered here include spectral features from cosmological recombination,resonant scattering of CMB by metals during reionization which allows us to measuretheir abundances, y -type distortions during and after reionization and µ -type and i -type(intermediate between µ and y ) distortions created at redshifts z & . × . Keywords : CMB,Cosmology,Origin and formation of the Universe,Background radia-tions,Observational cosmology,Intergalactic matterPACS numbers:98.80.-k,98.80.Bp,98.70.Vc,98.80.Es,98.62.Ra
1. Spectral distortions of CMB
The remarkable measurement of the cosmic microwave background spectrum (CMB)by COBE/FIRAS showed that the CMB is almost a perfect blackbody with tem-perature T CMB = 2 . ± .
001 K and was not able to detect any distortions from ∗ Based on a talk presented at the Thirteenth Marcel Grossmann Meeting on General Relativity,Stockholm, July 2012. 1 eptember 26, 2018 14:7 WSPC/INSTRUCTION FILE sunyaev˙plenary Sunyaev and Khatri a blackbody. However, the standard model of cosmology predicts distortions inthe spectrum from processes which heat, cool, scatter and create CMB photons,throughout most of the history of the Universe. Many of these processes are con-nected with absolutely new physics. The goal of the present paper is not to listall possible sources of distortions from physics beyond the standard model but topoint out the unavoidable distortions predicted in the standard model. These dis-tortions are small but fortunately significant progress in technology in the last twodecades permits an improvement of 2–3 orders of magnitude over COBE/FIRAS. New proposals like Pixie, CoRE and LiteBIRD promise to detect the majorityof the unavoidable spectral distortions we discuss in this paper. Pixie will be ableto make absolute measurements as well as measure anisotropies with an angularresolution of 2 . ◦ . CoRE and LiteBIRD will be able to measure only the frequencydependent anisotropies with high sensitivity and have proposed angular resolutionsof ∼ ′ and ∼ ′ respectively. The detection of these spectral distortions wouldprovide new information about the important properties of the Universe such asreionization and formation of the first stars and galaxies at redshifts 6 . z . z ∼ − . × . z . × due to the dissipation of sound waves inthe primordial plasma. We will also discuss the effect of rapidly cooling baryons andelectrons leading, under some additional conditions, to Bose-Einstein condensationof CMB photons.
2. Line features from the epoch of hydrogen and heliumrecombination
One of the most important phase transitions in the history of the Universe is recom-bination of electrons with protons and helium nuclei to form neutral atoms.
12, 13
Thestanding sound waves, excited by primordial initial perturbations, in the previouslytightly coupled electron-baryon-photon plasma are frozen into the free streamingphotons
6, 14 and are observed today by CMB experiments such as COBE/DMR, BOOMERANG, ACBAR, WMAP, SPT, ACT and many others. a The re-combination of HeIII to HeII at redshift z ≈ z ≈ z ≈ α and 2 s − s and Peebles while Dubrovich pointed outthe importance of ( n, n −
1) transitions in the hydrogen atoms for the observationsof the recombination spectrum. The full recombination spectrum was calculated inRefs 10, 22, 11. Fig. 1 shows the line profiles for the recombination lines. It is in-teresting to note that for hydrogen most of the recombination photons are emitted a A more complete list of CMB experiments is available on NASA LAMBDA website, http://lambda.gsfc.nasa.gov/product/expt eptember 26, 2018 14:7 WSPC/INSTRUCTION FILE sunyaev˙plenary
Unavoidable CMB spectral features and blackbody photosphere of our Universe
Figure is takenfrom Ref. 9. significantly earlier than the last scattering surface (where the Thomson visibilityfunction is peaked). Fig. 2 shows the hydrogen recombination spectrum and Fig. 3demonstrates the contributions from helium to the total recombination spectrum.The x-axes shows the observed frequency today which is redshifted by a factor of ∼ ∼ would have the sensitivity to detect the CMB recombinationspectrum at 5 − σ . eptember 26, 2018 14:7 WSPC/INSTRUCTION FILE sunyaev˙plenary Sunyaev and Khatri ν [GHz] −29 −28 −27 ∆ I ν [ J m − s − H z − s r − ] free−bound emissionbound−bound transitions + 2s spectrumsum of all B a l m e r c on t . P a s c h e n c on t . n max = n split = 100 p o ss i b l e l e v e l o f t o t a l e m i ss i o n e x p e c t e d l e v e l o f e m i s s i o n H Ly2s α α P α Br α P f und c on t . B r ac k e tt c on t . Fig. 2. Spectral features resulting from cosmological recombination of hydrogen at redshift z ∼ α line is redshifted to sub-mm spectral band. Figure is taken fromRef. 10.
3. Frequency dependent blurring of the CMB anisotropies fromresonant scattering by metals and a way to measure theabundance of metals during the epoch of reionization
Planck mission will provide us with sensitive independent measurements of thecosmic microwave background (CMB) angular fluctuations in different spectralbands. Using just the existing Planck channels, we can find upper limits (and pos-sibly a measurement) of the abundance of several important ions, such as OIII, OI,CII, NII etc, at different redshifts. The sensitivity to the presence of metals betweenthe last scattering surface (LSS) and us comes from the resonant scattering of theprimordial angular fluctuations of CMB by atoms and ions in the intergalactic spacebetween halos and regions inside halos with density smaller than the critical density,when collisions are unimportant. This resonant line scattering by atoms and ions,just like Thomson scattering from electrons, blurs the CMB anisotropies on scalessmaller than horizon, ∆ TT (cid:12)(cid:12)(cid:12)(cid:12) z =0 ( ν, ˆn ) ≈ e − τ LSS ( ν ) ∆ TT (cid:12)(cid:12)(cid:12)(cid:12) LSS ( ˆn ) , (1)where ∆ TT (cid:12)(cid:12) z =0 ( ν, ˆn ) is the observed temperature anisotropy in CMB today in thedirection ˆn , ∆ TT (cid:12)(cid:12) LSS ( ˆn ) is the anisotropy we would see if the optical depth to thelast scattering surface τ LSS ( ν ) was zero, i.e. there was no reionization. Formally,eptember 26, 2018 14:7 WSPC/INSTRUCTION FILE sunyaev˙plenary Unavoidable CMB spectral features and blackbody photosphere of our Universe this solution arises as the boundary term in the line of sight integral solution ofthe first order Boltzmann equation of photons.
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The total optical depth hasa frequency independent part arising from Thomson scattering, τ T ≈ .
087 anda frequency dependent part arising from the scattering with metals, τ X ( ν ). τ X ( ν )was calculated for almost all the important metal species in Ref. 25. Figure 4 fromRef 25 shows the total optical depth as well as the contributions from individualmetal species for two different models of ionization and metal production. Theeffect of the non-zero optical depth to the LSS is to blur the anisotropies, i.e. thehot spots become colder and the cold spots become warmer. The angular powerspectrum is suppressed by a factor of e − τ . The best signal-to-noise in the powerspectrum is, of course, obtainable for the anisotropies around the first acoustic peakof CMB, around angular wavenumber ℓ ≈ ℓ & −
30 andup to the angular resolution limit of Planck ℓ . (cid:10) ∆ T /T hot / cold ( ν ) (cid:11) . This analysis would be similar to the one done by the WMAPteam to detect polarization generated at the last scattering surface. They found12 ,
387 hot spots and 12 ,
628 cold spots with rms temperature fluctuation of 83 . Sunyaev and Khatri
NII OI TotalHistory BTotalHistory ACII OIII
Case I op ti ca l d e p t h i n li n e s observed frequency (GHz) l og o f m e t a l a bund a n ce ( s o l a r fr ac ti on ) intensegalaxy formation −5−4−3−2 BA Redshift
WMAPreionization
Fig. 4. Optical depth to the last scattering surface from resonant scattering on metals for the finestructure transitions of NII (205 . µ m), OI (63 . µ m), OIII (88 . µ m) and CII (157 . µ m) lines,for two different histories of ionization and metal production, are shown. Individual contributionsare also shown for history A. Figure is taken from Ref 25. µ K. In our case, the resonant scattering of the radiation will decrease the spotbrightness of hot spots at the frequency of observation, ν obs , if there is significantamount of the corresponding ion at redshift (1 + z ) = ν res /ν obs , where ν res is therest frame frequency of the resonant line. Similarly, resonant scattering will decreasethe amplitude ∆ T of the cold spots, increasing their brightness. Fig. 4 shows thatresonant scattering is unable to blur the fluctuations at low frequencies, becausethey correspond to high redshifts ( z >
30) where there are negligible amount ofmetals, for the lines of most abundant ions. By comparing the brightness of hotand cold spots at different frequencies, we have an opportunity to measure theeptember 26, 2018 14:7 WSPC/INSTRUCTION FILE sunyaev˙plenary
Unavoidable CMB spectral features and blackbody photosphere of our Universe abundance of ions during the epoch of reionization. The analysis is limited not bysensitivity but by the calibration error. A detection of an optical depth of 10 − using the hottest and coldest spots with temperature fluctuation ∼ µ K wouldrequire a calibration accuracy of ∼ . µ K. Any difference between two frequencychannels above the calibration error would be due to resonant scattering and willconstrain the presence of metals between us and the last scattering surface. Possiblelimits are shown in Fig. 5 for the most important metal species. X i on / X s o l a r IonPossible limits on abundance of Ions from Planck z=2.4z=1.3z=0.4 z=12.3z=7.9z=4.4 z=9.2z=5.8z=3.1 z=35.6z=23.4z=14 z=32.2z=21.2z=12.5 z=22.8z=14.8z=8.6
353 GHz217 GHz143 GHz
Fig. 5. Possible limits on metals from Planck if the difference in brightness between 100GHZchannel and 353 ,
217 and 143GHz channels of ∼ . µ K for hot/cold spots with ∆ T = 100 µ K(0 .
01% measurement) will be detected.
The limits on the metal abundance obtained in this way from the blurring ef-fect are density averaged limits. Sensitivity of Planck in the HFI channels ν =100 , ,
217 GHz is ∼ . , . , . µK ) respectively at ℓ = 220 using just 50% ofsky for 14 months of operation and much better when using all the data and aver-aging over all ℓ modes. Planck thus has, in principle, opportunity to get independentupper limits to the difference in optical depth to LSS seen by different frequencychannels, if the frequency channels can be calibrated relatively to each other pre-cisely. The most promising method of calibration is by using the orbital dipole ofthe Planck spacecraft. The motion of the earth and Planck satellite in the solarsystem can be measured with exquisite precision. In fact, for the WMAP satellite,the orbital dipole is known at a precision of ∼ . and it is thus, in principle,eptember 26, 2018 14:7 WSPC/INSTRUCTION FILE sunyaev˙plenary Sunyaev and Khatri possible to achieve a calibration precision at the level of 10 − for the hottest/coldestspots with temperature fluctuation of 100 µ K, improving the numbers presented inFig. 5 by a factor of 100! An additional precise source of calibration with well de-fined spectrum is also provided by the y -type distortion quadrupole induced by ourmotion with respect to the CMB. y -type spectral distortions The primordial plasma of our Universe is of very low density. Average numberdensity of photons exceeds that of the electrons by a factor of 10 . Under thesecircumstances, Compton scattering of radiation on free thermal electrons (takinginto account the Doppler shift of photon energy, the recoil effect and double Comp-ton scattering) is the most important physical process responsible for the inter-action of matter and radiation in the Universe. Our Universe is optically thin tobremsstrahlung absorption and emission of CMB photons almost up to the timeof positron-electron annihilation which occurred at redshifts z ∼ .The interaction of CMB blackbody photons (temperature T ) with the plasma(temperature T e ) is described by the Kompaneets equation. The Kompaneetsequation is the Fokker-Planck approximation of Boltzmann equation with Comptonscattering, ∂n∂y γ = 1 x ∂∂x x (cid:18) n + n + T e T ∂n∂x (cid:19) , (2)where we have defined y γ ( z, z max ) = − Z zz max dz k B σ T m e c n e TH (1 + z ) , (3) x = hν/ ( k B T ) is the dimensionless frequency, h is the Planck’s constant, ν is fre-quency, k B is the Boltzmann’s constant. Also, H ( z ) is the Hubble parameter, n e isthe electron number density and z max is the maximum redshift at which we startthe calculation. The three terms in the right-hand side brackets in Eq. (2) describethe change in photon frequency due to recoil ( δν/ν ∼ − hν/ ( m e c )(1 − cos θ )),induced recoil and Doppler effect (( δν/ν ) rms ∼ k B T e / ( m e c )) respectively. The(induced) recoil and Doppler effects cancel each other if the photon spectrum isa Planck/Bose-Einstein spectrum with temperature T and T e = T , making theright-hand side vanish.The y -type distortion is the solution of the Kompaneets equation in the minimalcomptonization limit, y γ ≪
1. This solution, in the small distortion limit, is easilyobtained by approximating the occupation number on the right hand side of Eq. (2)by Planck spectrum ( n pl = 1 / ( e x − n pl + n pl2 = − ∂n pl ∂x . The resultingdifferential equation is now trivial to integrate, giving ( for y ≪ n y ( x ) = y xe x ( e x − (cid:20) x (cid:18) e x + 1 e x − (cid:19) − (cid:21) , (4)eptember 26, 2018 14:7 WSPC/INSTRUCTION FILE sunyaev˙plenary Unavoidable CMB spectral features and blackbody photosphere of our Universe x n ( x ) xBlack Bodyy-distortion Fig. 6. y -type distortion. Compton scattering of CMB blackbody photons with hot thermalelectrons up-scatters low frequency photons to higher frequencies. Occupation number n ( x )multiplied by x , making it proportional to intensity, as a function of dimensionless frequency x = hν/k B T is shown, where T is the temperature of the blackbody spectrum. This effect hasbeen observed by several instruments including Planck, ACT and SPT in the direction of manyclusters of galaxies. where the amplitude of the distortion is proportional to the electron pressure (inthe limit T e ≫ T ), integrated along the line of sight, y = − Z z min z max k B σ T m e c n e ( T e − T ) H (1 + z ) d z (5) ≈ T e ≫ T − Z z min z max k B σ T m e c n e T e H (1 + z ) d z ≡ y e , (6)For constant temperature T e , y is proportional to the integrated Thomson opticaldepth, y ≈ k B T e m e c τ = 1 . × − T e τ . If the total energy injected into the CMB is∆ E/E γ , where E γ is the energy density of the CMB, then the y -type distortion isgiven by y = (1 / E/E γ . Eq. (5) also shows that we have a positive distortionfor T e > T and a negative distortion for T e < T . y -type distortion is shown in Figs.6 and 7eptember 26, 2018 14:7 WSPC/INSTRUCTION FILE sunyaev˙plenary Sunyaev and Khatri -1.5-1-0.5 0 0.5 1 1.5 2 2.19 3.83 1 10
20 124 217 500 100 ( - / ε ) ∆ I ν ( - W m - s t e r - H z - ) x=h ν /kTFrequency(GHz) µ =5.6x10 -5 Y=10 -5 Fig. 7. y -type and µ -type distortions. Difference in intensity from the blackbody spectrum isshown. The distortions shown correspond approximately to the COBE/FIRAS limits. Both typesof distortions shown have the same energy density and number density of photons. Figure fromRef. 35. y -type distortions from reionization and WHIM Present observations indicate that the Universe was reionized between redshifts of6 . z . when the first stars and galaxies flooded the Universe with ultra-violet radiation. The ionizing radiation also heated the gas to temperatures wellabove the CMB temperature, with the electron temperature in the ionizing regions T e ∼ K. Late time structure formation shock heated the gas to even highertemperatures, . T e . K, creating the warm-hot intergalactic medium(WHIM). The optical depth, τ , to the last scattering surface is well constrainedby CMB observations to be τ ≈ . ± . T e ≈ K, we expect y ∼ − . However if a significant fraction of baryons endup in the WHIM at z .
3, as expected from recent simulations,
39, 40 we expect the y -distortions from the WHIM to dominate over those from reionization. In anycase, these distortions would be easily detected by PIXIE and the next genera-tion CMB experiments like CoRE, ACTPol and SPTPol should also be able todetect the fluctuations in the y -type distortions from the the WHIM. The rate of y -type distortion injection with redshift is shown in Fig. 8 for a simple model wherereionization happens between 8 < z <
15 and the density averaged temperature offree electrons is assumed to be T e = 10 K for z > T e = 10 / (1 + z ) . K ateptember 26, 2018 14:7 WSPC/INSTRUCTION FILE sunyaev˙plenary
Unavoidable CMB spectral features and blackbody photosphere of our Universe -7 -7 -7 -7 -( + z ) d Y / d z redshift (z) ReionizationWHIM+Galaxy groups Fig. 8. Sky averaged thermal y -type distortions created after recombination. Initially the gas iscolder than the CMB because non-relativistic baryons cool faster as T e ∝ (1 + z ) compared tophotons T ∝ (1 + z ).
12, 13, 37
Once first stars form and reionization starts, gas is heated above theCMB temperature and much larger positive y -type distortions are created. d y/ dln(1 + z ) is plottedwhich is approximately equal to the y -type distortions created in redshift interval δz ∼ z . It islikely that the y -type distortions from WHIM would dominate over the reionization contribution.This figure is taken from Ref 35. z < The contribution from the WHIM to the y -type distortions dominates overthose from reionization in this model. y -type distortion from averaging of blackbodies in our CMBsky The temperature of the CMB in the sky is not constant but is a function of direc-tion. The most significant anisotropy comes from our peculiar motion with respectto the CMB rest frame (i.e the frame in which the CMB dipole is zero). Our pe-culiar velocity, or equivalently the CMB dipole seen in our rest frame, was mea-sured very precisely by COBE and WMAP experiments and has an amplitude∆ T dipole . ± .
008 mK , v/c ≈ . × − . In addition, there are small scaleanisotropies of amplitude 10 µ K in the microwave sky due to the presence of pri-mordial density perturbations. If we average the CMB intensity over all angles orpart of the sky, either explicitly to improve the sensitivity or because of the finitebeam of the telescope, we will inevitably mix the blackbodies of different tempera-tures. It was shown in Ref. 46 that mixing of blackbodies gives rise, at the lowesteptember 26, 2018 14:7 WSPC/INSTRUCTION FILE sunyaev˙plenary Sunyaev and Khatri x n ( x ) xT+ ∆ T T- ∆ TT[1+( ∆ T/T) ] TAverage(Y) µ Fig. 9. Spectrum resulting from averaging two blackbodies with temperatures T ± ∆ T . We haveused large value of ∆ T to make changes visible, but have used formula with only lowest ordernon-vanishing correction to the blackbody spectrum valid for small distortions. We have plottedintensity (up-to a numerical constant), n ( x ) x ∝ I ν . Figure taken from Ref.44. non vanishing order, to a y -type distortion and that the thermal y -type distor-tion from comptonization is just superposition of blackbodies. This result is quitegeneral and is also applicable if the source of electron motion is not thermal butkinetic, i.e. there is a y -type distortion arising from peculiar motion of baryons. Asa simple example, superposition of two blackbodies with temperatures T ± ∆ T isshown in Fig. 9 (intensity) and Fig. 10 (effective temperature). The resulting spec-trum after averaging can be recognized as a blackbody with higher temperature T (cid:16) (cid:0) ∆ TT (cid:1) (cid:17) with a y -type distortion of amplitude (cid:0) ∆ TT (cid:1) /
44, 47
For the dipoleanisotropy in our sky, we get y ≈ . × − and averaging of the rest of the smallerscale anisotropies (without dipole) contribute y ≈ × − . The y -type spectrumprovides, in principle, a high precision source to calibrate CMB experiments. µ -type distortions The thermal capacity of electrons and baryons is negligible in comparison with thatof photons and Compton interaction rapidly establishes a Maxwellian distributionof electrons with temperature T e defined by the radiation (photon) field.
48, 49
If theeptember 26, 2018 14:7 WSPC/INSTRUCTION FILE sunyaev˙plenary
Unavoidable CMB spectral features and blackbody photosphere of our Universe T e ff ( K ) xObserved Frequency (GHz)T+ ∆ TT- ∆ T T[1+( ∆ T/T) ] TAverage(Y) µ Fig. 10. Same as Fig. 9, but effective blackbody temperature defined by n ( ν, T eff ) ≡ e hν/ ( k B T eff) − is plotted. Figure taken from Ref.44. At high redshifts y γ becomes significant (seeFig. 11, Eq. 3) and comptonization converts y -type distortions created by the mixing of blackbodiesto intermediate-type (or i -type) and µ -type distortions. spectrum of photons is Bose-Einstein spectrum then the equilibrium electron tem-perature is exactly equal to the radiation temperature. The other way around, thetimescale for the establishment of equilibrium Bose-Einstein spectrum for photonsthrough Compton scattering is comparatively much longer compared to the age ofthe Universe at redshifts z . × , see Fig. 11 and the discussion below.Bose-Einstein spectrum with chemical potential parameter µ b and temperature T e , n BE = 1 / ( e hν/ ( k B T e )+ µ −
1) = 1 / ( e xT/T e + µ −
1) is the equilibrium solution of theKompaneets equation, as can be readily verified by substituting it in Eq. (2). Thereference temperature T now corresponds to a blackbody spectrum with the samenumber density of photons as the Bose-Einstein spectrum, and the factor T /T e isthere because we have defined the dimensionless frequency x with reference to T .For small distortions, µ ≪
1, we can expand the Bose-Einstein spectrum around thereference blackbody giving, for the µ -type distortion (using T e /T − µ/ . n µ = µe x ( e x − (cid:16) x . − (cid:17) . (7) b Our definition of µ is equivalent to the negative of the usual statistical physics chemical potentialdivided by temperature, making it dimensionless and we will call it chemical potential parameter. eptember 26, 2018 14:7 WSPC/INSTRUCTION FILE sunyaev˙plenary Sunyaev and Khatri -7 -6 -5 -4 -3 -2 -1 y γ ( z i n j ) z inj µ -typey-type intermediate-type B l ac kbody P ho t o s ph e r e Fig. 11. The Compton parameter y γ , Eq. 3, is plotted. At z & y γ > y which is always smaller than unity (as verified byCOBE/FIRAS ). This figure is taken from Ref 35. The intensities of y and µ -type distortions are plotted in Fig. 7. The µ parameteris related to the fractional energy injected into the CMB by
50, 51 µ = 1 . E/E γ .The µ -type distortion is thus the solution of the Kompaneets equation in thesaturated comptonization limit, for y γ ≫
1. The function y γ (0 , z inj ) is plotted inFig. 11. If y γ &
1, comptonization is very efficient in establishing a Bose-Einsteinspectrum if energy is injected at redshift z inj , while for y γ ≪ y -typedistortion can be created from the heating or cooling of the CMB. We should notethat, on the contrary, the y -parameter in Eq. (5) describes the amplitude of injectedenergy/ y -type distortion and is always much smaller than unity for the physicalprocesses considered in this paper. Negative y and µ distortions from Bose-Einsteincondensation of CMB It was recognized long ago that in the early Universe, when baryons and photonsare tightly coupled, the photons must transfer energy to the baryons to keep themin equilibrium (until z ∼ T e ∝ (1 + z ) )than radiation ( T ∝ (1 + z )) with the expansion of the Universe. This is simplybecause the baryons are non-relativistic with an adiabatic index of 5 / / Unavoidable CMB spectral features and blackbody photosphere of our Universe -2x10 -26 -1x10 -26 -26 -26 -26 -26 ∆ I ν ( W m - s t e r - H z - ) x=h ν /kT ν (GHz)-y+y CMBx10 -8 |y|=10 -9 Fig. 12. The negative y -type distortions resulting from cooling of CMB. The expected magnitudesof y and µ parameters for CMB is shown in red in Tables 1 and 2. Figure from Ref. 44. of the CMB gives rise to spectral distortions ( y, µ and intermediate-type) whichare exactly the negative of the distortions caused by the heating of the CMB.Figure 12 shows the y -type distortion resulting from the cooling of the CMB. The µ -type distortion resulting from the cooling of the CMB is shown in Fig. 13 alongwith a positive distortion resulting from an equivalent amount of heating. There is,however, one very important physical difference between the heating and the coolingof the CMB. The cooling of the CMB results in an excess in the number densityof photons compared to the blackbody radiation with the same energy density,and this can be recognized as the condition for the Bose-Einstein condensation ofphotons to happen.
44, 50
In the case of the CMB, of course, the condensing photons,moving to lower frequencies due to the stimulated scattering and the recoil effect,are efficiently destroyed by bremsstrahlung and double Compton absorption,
37, 44 and no actual photon condensate is formed.
6. Beyond µ and y : Intermediate-type distortions The y -type distortions are created at redshifts z . . × , y γ . .
01, when thecomptonization is minimal. At z & × , the Compton parameter y γ & µ -type distortion, which is the solution of the Kompaneets equation, Eq. (2) inthe saturated comptonization limit. The energy injected between the redshifts 1 . × . z . × is only partially comptonized as the Compton parameter is 0 . . eptember 26, 2018 14:7 WSPC/INSTRUCTION FILE sunyaev˙plenary Sunyaev and Khatri -2e-26-1e-26 0 1e-26 2e-26 3e-26 4e-26 0.01 0.1 1 10 1 10 100 1000 ∆ I ν ( W m - s t e r - H z - ) x=h ν /kT ν GHz - µ + µ CMBx10 -8 | µ |=5.6x10 -9 Fig. 13. The negative µ -type distortions resulting from cooling of CMB. The expected magnitudesof y and µ parameters for CMB is shown in red in Tables 1 and 2. Figure from Ref. 44. y γ .
2, see Fig 11. The spectrum created from the energy injection in this redshiftrange is therefore in between y -type distortion (minimal comptonization) and a µ -type distortion (saturated comptonization). Figure 14 shows the spectrum thatwould be obtained if the energy is injected at redshifts corresponding to different y γ parameters. In practice, the energy is more likely to be injected continuously overa redshift range rather than instantaneously at a single redshift and we expect theobserved spectrum to be a linear superposition (with appropriate weights) of thedistortions for different values of y γ . Since the intermediate-type spectrum dependson the redshift of energy injection, there is additional information here compared tothe y and µ -type distortions which only remember the total energy injected and notthe exact time of energy injection. The rather simple behavior of the intermediate-type distortions with respect to the energy injection redshift opens up the possibilityto measure the redshift dependence of the energy injection rate. For example, incase of Silk damping or dark matter annihilation we the energy injection rate is apower law in redshift, d Q/ d z ∝ (1 + z ) α and with the intermediate-type distortionwe can measure the parameter α and thus the spectral index of the primordial powerspectrum. Combining the intermediate-type distortions and the µ -type distortionsalso allows to distinguish between a power law energy injection and the exponentialdependence of energy injection rate on the redshift expected from particle decay.Intermediate-type distortions are explored in detail in Ref. 35.eptember 26, 2018 14:7 WSPC/INSTRUCTION FILE sunyaev˙plenary Unavoidable CMB spectral features and blackbody photosphere of our Universe -1.5-1-0.5 0 0.5 1 1.5 2 1 1020 100 1000 ∆ I ν ( - / y ) ( - W m - s t e r - H z - ) xObserved Frequency (GHz)x x min x max y-typey γ =0.01y γ =0.05y γ =0.1y γ =0.2y γ =0.5y γ =1y γ =2 µ -type Fig. 14. Intermediate-type distortions are shown for different values of Compton parameter( y γ , z inj ) = (0 . , . × ) , (0 . , . × ) , (0 . , . × ) , (0 . , . × ) , (0 . , . × ) , (1 , . × ) , (2 , . × ). Figure is taken from Ref. 35.
7. Blackbody photosphere of the Universe
The blackbody spectrum of CMB is created dynamically in the early Universe,initially because of the complete thermal equilibrium between the photons andthe electrons/positrons (and other particles at even higher redshifts) through paircreation and annihilation and bremsstrahlung. Subsequent adiabatic expansion ofthe Universe preserves the blackbody spectrum, except small BEC effects describedin section 5.1 above. Once e ± pair production becomes inefficient at z ∼ − ,double Compton scattering ( γ + e − ↔ γ + γ + e − ) and bremsstrahlung ( Z + e − ↔ γ + Z + e − ) become the dominant mechanism of photon absorption and emission atlow frequencies while Compton scattering ( γ + e − ↔ γ + e − ) efficiently redistributesthe photons in energy (comptonization). The redistribution of energy among theavailable photons by comptonization establishes a Bose-Einstein spectrum with achemical potential parameter µ at z & while the emission/absorption of photonsby double Compton and bremsstrahlung drives the chemical potential parameter tozero at z & creating a Planck spectrum.Understanding the creation of the blackbody spectrum therefore requires solv-ing the Kompaneets equation, describing comptonization, with the source termsarising from bremsstrahlung and double Compton scattering.
59, 60
A quite accu-rate solution to this partial differential equation for comptonization with a sourceeptember 26, 2018 14:7 WSPC/INSTRUCTION FILE sunyaev˙plenary Sunyaev and Khatri -( + z ) d Q / d z redshift (z)10 -10 -8 -6 -4 -2 B l a c kb o d y Su r f a ce L a s t S c a tt er i n g Su r f a ce Blackbody Photosphere
Blackbody visibility e + e - -> γγ BBN µ -typei-typey-type Silk Damping, n s =0.96Cooling/BEC Fig. 15. Comparison of Silk damping (heating) and Bose-Einstein condensation (cooling) ratesin standard cosmology. Also shown is the blackbody visibility function defining the blackbodyphotosphere, where the distortions from the blackbody are exponentially suppressed (see section 7for details). Since heating of photons from electron-positron annihilation happens deep inside theblackbody photosphere, the photon spectrum hardly deviates from a blackbody even though thecomoving energy density of photons is more than doubled. This situation is to be contrasted withneutrinos, heating of which due to electron positron annihilation is just 1% but it results is signif-icant deviations from a Fermi-Dirac distribution which are important for big bang nucleosynthesis(BBN).
Also shown is the nuclear binding energy released during helium production in BBNcalculated using Kawano’s modification of Wagner’s code . term responsible for the emission and absorption of photons was found analyti-cally in Ref. 51. Although Ref. 51 only considered bremsstrahlung as the emis-sion/absorption mechanism, the double Compton scattering cross section is similarenough to bremsstrahlung that their method of solution could be applied immedi-ately to the double Compton scattering also. This was done in Ref. 62 for a lowbaryon density Universe such as ours, where double Compton scattering dominatesover bremsstrahlung. Recently, corrections to this solution were computed in Ref. 63and a solution with an accuracy of ∼
1% including both bremsstrahlung and dou-ble Compton processes was presented. The solution describes the evolution of thechemical potential parameter of the Bose-Einstein spectrum created at redshift z and is given by µ ( z = 0) = µ ( z ) e −T T ( z ) ≈ "(cid:18) z z dC (cid:19) / (8)where, z dC ≈ . × defines the blackbody surface , behind which µ is exponen-tially suppressed. The Bose-Einstein spectrum with the effect of this exponentialsuppression of chemical potential at low frequencies is shown in Fig. 16. The preciseanalytic solution for the blackbody visibility function derived in Ref. 63 G = e −T ,eptember 26, 2018 14:7 WSPC/INSTRUCTION FILE sunyaev˙plenary Unavoidable CMB spectral features and blackbody photosphere of our Universe µ -type distortions including thelow frequency part where bremsstrahlung and double Compton effects create Rayleigh-Jeans spec-trum with temperature equal to the electron temperature defined by the interaction with µ -typeradiation field . including the effects of bremsstrahlung and double Compton, is plotted in Fig. 15,curve ’d’. Thus any perturbations away from the Planck spectrum are suppressedexponentially at z & × . The electron-positron annihilation at z ∼ − more than doubles the entropy and energy in photons, but the deviations resultingfrom the Planck spectrum never rise above a tiny value of ∼ − . This demon-strates how difficult, almost impossible, it is to create deviation from the Planckspectrum at z & × , in the blackbody photosphere.
8. Energy release in the early Universe8.1.
Spectral distortions from Silk damping: A view of inflationspanning 17 e-folds!
Primordial adiabatic perturbations excite standing sound waves on entering thehorizon in the early Universe in the tightly coupled electron-baryon-photoneptember 26, 2018 14:7 WSPC/INSTRUCTION FILE sunyaev˙plenary Sunyaev and Khatri fluid.
6, 14, 64
On small scales, there is shear viscosity and thermal conductivity in -9 -8 -7 -6 -5 -4 -3 -4 -3 -2 -1 k P ( k ) k Mpc -1 CMB best fit + Ly- α W e a k er c o n s t r a i n t s B l a c kb o d y P h o t o s ph ere y-typelow redshift confusion limited Intermediate-type µ -typeCOBE y limit COBE µ limitPixiePixiePixie6 e-folds 7 e-folds17 e-folds Fig. 17. Spectral distortions can deliver 7 additional e-folds extending our view of inflation from6 (at present) to 17 e-folds. Constraints on the primordial power spectrum of initial curvatureperturbation P ( k ) from CMB and Ly- α observations of SDSS
65, 66 are shown along withcurrent constraints from COBE spectral distortion measurements and future constraints fromthe proposed experiment Pixie . the fluid which dissipates the energy in the sound waves, suppressing the fluctua-tions.
14, 67, 68
Microscopically, photon diffusion through the plasma creates a localquadrupole, which is dissipated by shear viscosity, and relative motion between thephotons and the baryons creates a local dipole, which is dissipated by thermal con-duction or Compton drag. This Silk damping of primordial fluctuations is alreadyobserved by the current CMB experiments SPT and ACT with high precision.The power which disappears from the fluctuations appears in the average CMB spec-trum or monopole as y , µ and intermediate-type spectral distortion.
37, 44, 47, 69–73
InFig. 15, The heating rate of CMB from sound wave dissipation is compared withthe cooling due to the energy transfer to baryons for several different power spectraallowed by the current CMB data. It is interesting to note that the cases wherecooling dominates over heating for µ -type distortions are still allowed if there isnon-zero running of the spectral index, although for most of the region of the param-eter space heating dominates over cooling. These distortions can be added togetherlinearly since they are very small. The µ -type and the intermediate-type spectraldistortions are created by the dissipation of modes with comoving wavenumbers8 . k . Mpc − . Most of this range of scales is inaccessible to any other cos-mological probe. When combined with the information from the CMB anisotropieseptember 26, 2018 14:7 WSPC/INSTRUCTION FILE sunyaev˙plenary Unavoidable CMB spectral features and blackbody photosphere of our Universe n S -0.04-0.03-0.02-0.0100.010.02 n r un WMAP7ACTSPT n S = 1 σ σ σ σ WMAP7 WMAP7 with running possible constraints from µ b a l a n ce d i n j ec t i o n s ce n a r i o s σ lower limit Fig. 18. Possible constrains with PIXIE on spectral index and its running. 1 − σ limits are shownalong with the 1 − σ limits if the sensitivity of PIXIE is improved by a factor of 2 −
10. Best fitparameters from WMAP, ACT and SPT are also shown. Figure is taken from Ref. 47. and the large scale structure, this gives us a view of inflation spanning 17 e-foldscompared with 6 − × − Mpc − . k . . − ) without thespectral distortions, as depicted in Fig. 17. The CMB spectral distortions are thusmeasuring the primordial power spectrum on extremely small scales and are veryimportant to our understanding of the physics of the initial conditions. Possibleconstraints from PIXIE are shown in Fig. 18. Recently, precise calculations of thesedistortions were done in Ref. 47 including previously ignored effects such as secondorder Doppler effect and fitting formulae for y and µ -type parameters were alsoprovided. Distortions for several other non-standard and general initial conditionswere calculated in Refs. 74, 75 and the effect of non-Gaussian initial conditions on µ − T correlations was pointed out in Refs. 76, 77. A census of unavoidable µ and y -type distortions in standardcosmology A census of unavoidable µ and y -type distortions in standard cosmology is givenin Tables 1 and 2. Public codes KYPRIX and CosmoTherm are now availableto calculate the evolution of CMB spectral distortion by numerically solving theKompaneets equation with bremsstrahlung and double Compton terms, starting inthe blackbody photosphere at z & × . Many of the processes shown in Tables1 and 2 were first calculated in Ref. 37. The CosmoTherm code also includes higheptember 26, 2018 14:7 WSPC/INSTRUCTION FILE sunyaev˙plenary Sunyaev and Khatri precision calculations of Silk damping and other important cases of energy injection.A fast and precise Mathematica code to calculate the spectral distortions for user-defined energy injection rates, taking advantage of the analytic solution describedabove and pre-computed numerical templates described in the next section, is alsopublicly available c .We stress here again the importance of blackbody photosphere, see Fig. 15. TheCMB spectrum does not carry any information about even strong energy injectiondeep inside the blackbody photosphere. Any injected energy behind the blackbodysurface, z & × , is almost completely thermalized without any observabletraces in the CMB spectrum. The best example of the strength of equilibriumrestoring processes (Compton and double Compton scattering, bremsstrahlung) athigh redshifts is the huge energy release, ∆ E/E ∼ O (1), from electron positronannihilation. The resulting deviations from the complete equilibrium (blackbody, T e = T γ = T ions ) even in this extreme case are just 10 − !. Table 1. Census of µ distortions in standard cosmology.The adiabatic cooling of matter results in negative distor-tions shown in red. Table is taken from Ref. 63.Process µ electron-positron annihilation 10 − BBN tritium decay 2 × − BBN Be decay 10 − WIMP dark matter annihilation 3 × − f γ m WIMP
Silk damping 10 − − − Adiabatic cooling of matter andBose-Einstein condensation − . × − Table 2. Census of y -type distortions in standard cosmology. y -typedistortion from the mixing of blackbodies in our CMB sky are alsoshown. Adiabatic cooling of matter creates negative distortions shownin red. Reionization/WHIM contributions after recombination dominate.Table is taken from Ref. 63.Process y WIMP dark matter annihilation 6 × − f γ m WIMP
Silk damping 10 − − − Adiabatic cooling of matter andBose-Einstein condensation − × − Reionization/WHIM 10 − − − Mixing of blackbodies: CMB ℓ ≥ × − eptember 26, 2018 14:7 WSPC/INSTRUCTION FILE sunyaev˙plenary Unavoidable CMB spectral features and blackbody photosphere of our Universe Constraining new fundamental physics with spectraldistortions
Most of the talk has been concentrated on the spectral distortions expected in stan-dard cosmology. The physics beyond the standard model provides numerous newpossible sources of heating for CMB. In particular, it is clear from Tables 1 and 2that in standard cosmology we expect µ and intermediate-type distortions with anamplitude of 10 − − − . A distortion of larger magnitude, if detected, would thusundoubtedly be a signal of new physics. In addition, the most important standardsource of energy injection, Silk damping, has a power law dependence on redshift,resulting in almost an equal amount of energy in µ -type and intermediate-type dis-tortions. Thus if one of the µ -type or the intermediate type distortion is strongerthan the other, it would again signify new physics. Some of the possible sources of en-ergy injection from new physics are dark matter decay and annihilation, decayof cosmic strings and other topological defects, cosmic string wakes and oscillatingsuperconducting cosmic strings, photon-axion inter-conversion,
85, 86 violationof reciprocity relation, dissipation of primordial magnetic fields, quantum wavefunction collapse, and evaporating primordial black holes.
9. The future looks good for CMB spectroscopy
Three proposed next generation CMB experiments PIXIE, COrE and LiteBIRD would have the sensitivity to detect most of the spectral features considered in thisreview. Current specifications of PIXIE, in particular, with 400 frequency channelsand absolute calibration, are optimal to detect the global y , µ and intermediate-type distortions. COrE and LiteBIRD would have the sensitivity and the angularresolution to detect metals during the epoch of reionization and angular fluctuationsin the y -type sky from the warm-hot intergalactic medium. In the near future,therefore, we expect many interesting results from the observations of the spectralfeatures in the CMB. References
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