# Why isn't there a high spike visible in the CMBR, due to a massive recombining of electrons and protons

This answer made me wonder about something. The CMBR has its origin in the beginning of the universe. The gas of electrons, protons, etc. was in thermal equilibrium. When the electrons and protons recombined, the photons within the gas were set free, the CMBR "came into being", and this photon gas carried a black body signature along.
But weren't there also photons created due to the combining of the electrons and protons? Why isn't a sharp peak observed in the CMBR, due to this massive recombining?

• Re a possible lack of any beginning, the motive for the syllable "re"" in "recombination" isn't specified in the 1st several of the sources that searches by the entire word bring up, but may have some root that led to Penrose's specification of "anomalous spots of significantly raised temperature" as "Hawking points" (spots in the CMBR which he considers to have resulted from the release of Hawking radiation by black holes at the end of each of an endless sequence of aeons into which time can be divided.) They were a factor in his 2020 Nobel Prize. See arxiv.org/pdf/1808.01740.pdfe . Apr 3, 2021 at 17:20

Cosmic Microwave Background (CMB) photons were produced by the combining of protons and electrons. The main reason there is no obvious spike at some particular frequency/wavelength, is because the electrons and protons had a continuum of kinetic energies prior to (re)combination and the majority of photons that escape to form the CMB arise from scattering processes that have a continuum of energies and no spectral lines.

There should however be an effect whereby recombination took place to excited levels in the hydrogen atoms that subsequently decay. Recombination (of hydrogen; the recombination of helium took place at higher redshifts and temperatures) took place at $$\sim 3000$$ K and the first excited levels is at 10.2 eV; thus $$\exp(-E/k_BT)$$ is $$\sim 10^{-17}$$. What this means is that hydrogen atoms in excited states will immediately start to decay towards the ground state.

These emitted photons are resonantly and strongly absorbed by other (nearby) hydrogen atoms - so the universe is still optically thick to these photons long after the hydrogen atoms were formed. It turns out (see Dubrovich & Shakhvorostova) that Lyman alpha emission does not result in a net reduction to the ground state because of this. Instead, it is slower 2-photon processes, which result in a continuum, that are the dominant excited state depopulation mechanism.

In practice then, what we expect to see is weak emission lines that are mostly from lower energy transitions. But these are smeared out because the recombination process takes place over a range of redshifts and look-back times (e.g. between redshifts of 1200-1400 for most of the hydrogen transitions).

A radiative transfer calculation is provided by Sathyanarayana et al. (2015) and the recombination part of the spectrum is shown below. This is an additive contribution to the main part of the CMB spectrum caused by scattering processes. For comparison, the CMB spectrum peaks at about $$330\times 10^6$$ Jy/sr at frequencies of about 140 Ghz. So these lines are fluctuations of order 1 part in $$10^9$$. Current instrumentation/data has precisions of more like 1 part in $$10^5$$ in terms of spectral intensity, so these "lines" are orders of magnitude below current thresholds for detection.

Recombination spectrum predicted from the CMB. This is to be added to the main CMB continuum, which is about 9 orders of magnitude brighter [adapted from Sathyanarayana et al. (2015)].

The recombination does not happen instantaneously everywhere, but is a somewhat gradual process. That means that there is still significant amounts of charged particles to scatter photons through Compton scattering, broadening peaks. Also, it turns out that there are a lot of energy levels to choose from (especially in helium), making the process smear out wavelengths. See slide 54 in this deck for a simulated spectrum.

Note that cosmologists are actually looking for these peaks on top of the CMB blackbody spectrum since they do contain a lot of information, it is just that they are weak (<1%) and the signal is noisy.

In the context of the previous question you linked to, there are several reasons that there isn't a significant contribution to the CMB spectrum at large due to recombination, some of which related to peak broadening are discussed well in other answers. But it is indeed true that recombination releases radiation through its primary reaction $$e^-+p^+ \leftrightarrow H+\gamma$$, and this radiation is part of the CMB: so how could this release not have a significant impact on the spectrum when some $$75 \%$$ of the protons and electrons in the universe combined in quick succession?

Well, the largest immediate reason (which dwarfs the peak broadening effects in other answers) is simply that there weren't nearly enough electrons and protons to make a leading-order-discernible "flash". The baryon-to-photon ratio $$\eta := \frac{n_b}{n_\gamma}$$, the ratio of the number density of baryons to the number density of photons, is constrained tightly by both primordial abundance measurements and the CMB anisotropy power spectrum to be around $$\eta \approx 6 \cdot 10^{-10}$$, meaning that there are roughly $$10^9$$ photons for every baryon in the universe. This means that when $$\sim 75 \%$$ of the protons and electrons in the universe combined into hydrogen, they released on the order of $$n_b$$ photons (in density), a billionth of the photons already present.

Of course, the more relevant comparison is the energy emitted: each hydrogen atom formation released about the hydrogen binding energy of $$B_H \approx 13.6$$ eV, so an energy density of $$\sim 0.75 \cdot (13.6 \text{ eV}) n_b$$ was released in total, in comparison to the energy density $$\sim 2.7 n_\gamma k_B T$$ of the priorly existing radiation. Since recombination happened at a temperature of about $$k_B T \approx 0.3$$ eV, the ratio of released radiation energy to present radiation energy was about $$\frac{7.5 \cdot 13.6}{3} \eta \sim 2 \cdot 10^{-8},$$ utterly negligible to the spectrum's broad shape.

While the value of $$\eta$$ seems the most glaring culprit in the physical universe, the mechanisms described in the other answers are actually sufficient to mean the spectrum would be quite close to black body no matter the value of $$\eta$$. This is because increasing $$\eta$$ to allow a significant contribution to the radiation increases the temperature at which recombination occurs while decreasing the temperature at which photons decouple from matter though Compton scattering, yielding a longer time window for thermodynamic equilibrating to a black body spectrum.

• So the number of photons created during positron-electron annihilation (the photon epoch) exceeds the number of photons created by electron-proton recombination (in short)? Mar 30, 2021 at 10:55
• @DescheleSchilder That's essentially the spirit, though photons were still present before the photon epoch and would have been sourced from a myriad of reactions in addition to electron-positron annihilation. Around that same time, for example, radiation was produced during nucleosynthesis (this would have been a small source similar to recombination). Other particle-antiparticle annihilations also occurred each time the plasma temperature passed below the threshold of some particle's mass. I'm not positive of exactly when the bulk of the photons are thought to have been produced. Mar 30, 2021 at 16:57
• Were photons created in the way quarks and leptons were created? Mar 30, 2021 at 21:12
• @DescheleSchilder I believe the current perspective is that virtually all of the standard model particles, along with electromagnetic radiation, were originally created at the end of inflation in the decay of the inflaton field(s), a process known as reheating. My impression is that this is not yet well understood, the details are quite model-dependent, and verification of specific inflation and reheating models is currently well outside the scope of experimental verification. This is a topic of active research. Mar 30, 2021 at 21:33
• More clear it can't get...+100! (If I could) Mar 30, 2021 at 21:39

Your history of the universe is very sketchy, the Big Bang model is much more complicated, having an inflation period in the beginning

and protons form at 1 microsecond, and photon decoupling, i.e. no longer any electromagnetic interactions at 380.000 years. At that time there are not only protons combining to hydrogen, but also other nuclei grabbing electrons and becoming neutral, thus there is not just the release of the photon series of hydrogen creation (many energy levels), but of other atoms too, there are other nuclei, a statistical mixture that does not have distinctive peaks at the level of accuracy of the CMB frequency measurement plot at present.

• So the combined recombination of electrons and positively charged particles does contribute to the CMBR? Mar 29, 2021 at 10:26
• Ofcourse it contributes, statistically, any peaks are smoothed over because of their great number. Mar 29, 2021 at 10:44
• What other nuclei? You mean helium? Helium recombined before hydrogen atoms were formed. Mar 29, 2021 at 14:15
• Anna, what nuclei are made in primordial nucleosynthesis besides helium and deuterium? Are you referring to the 1 part in 10 billion of lithium-7 or something even rarer? All those rarer things have combined before the "epoch of recombination" too. Mar 29, 2021 at 18:08
• "and photon decoupling, i.e. no longer any electromagnetic interactions at 380.000 years. At that time there are not only protons combining to hydrogen, but also other nuclei grabbing electrons". Make up your mind. He nuclei recombined at $z=6000$ (40 kyr) and He ions disappeared at $z=2000$ (190 kyr). Any of the vanishingly rare heavier nuclei recombined before either. H recombination and photon decoupling started at $z=1300$ (380 kyr). Mar 29, 2021 at 20:36