4. Results
4.2. Ensemble Evolution
Figure (16) shows the temperature evolution of a 14,312 particle ensemble. At each slice in time, the temperature was calculated asTχ= 23hp
2i
2mχ, where angled brackets denote the arithmetic mean.
At the beginning of the simulationTχ andTL are tightly coupled, butTχ peels off to followa−9/8
scaling as it approachesγ/H = 1, quasi-decoupling as expected. The new insight provided by the MCMC is what happens as the simulation passes Γ/H. We see that Tχ tracks the
quasi-decoupling solution well past Γ/H= 1, where we would expect collisions to become
negligible. By a= 103 we are seeingTχ significantly deviate from quasi-decoupling, but instead of
dropping directly to a−2 scaling (just redshift), it first jumps up significantly.
The unexpected behavior, where Tχ follows quasi-decoupling longer than we would expect and
then heats before fully decoupling, is explained by changes in the shape of the distribution function. Tχ is the temperature of a Maxwell-Boltzmann distribution, and there is no guarantee
that it accurately describes a generic velocity distribution. As we can see in figures (17) and (18), the resulting distribution is not at all Maxwell-Boltzmann. A hot tail is forming as occasional late collisions scatter dark matter particles far hotter than their companions. This pulls the average momentum up, far above that of a typical particle, which in turn produces a temperature that does not accurately characterize the population. This minority of hot particles is only a few parts in 15,000, making them unlikely to play any significant role in structure formation. Despite their insignificance, they dominate the evolution ofTχ, making it an unreliable parameter with
which to track the momentum distribution.
A steadier indicator of the momentum distribution is the median defined temperature, given as e
Tχ≡η23 p
2
2mχ, wherep
2 is the median p2, and η is a numerical factor that makes the definition satisfyTχ=Teχ in a perfect Maxwell Boltzmann distribution (see appendix D). By defining
temperature in this way we have an indicator of typical momentum that is robust against outliers, but converges toTχ at early times. This means any divergence between the two
temperatures represents a break from a Maxwell-Boltzmann distribution. Since the analytical solution derived in appendix A and shown in reference [20] predicts the momentum stays
Maxwell-Boltzmann throughout decoupling, such a divergence of Tχ and Teχ signifies a breakdown
of this analytical solution. Teχ pulls away from the quasi-decoupling solution and follows a−2
evolution shortly after Γ/H= 1. This suggests that, if the median defined temperature is an accurate characterization of the distribution, Γ/H= 1 does represent the beginning of full decoupling.
As discussed in section 3.4, we can use the relationship betweenΓ and Tkds (equation (3.28)) to
relate the proportionality constant to the true value of Tkds being simulated. By finding theTkds
that forces the numerical solution to equation (2.4) to match our simulation at early times, we can find this trueTkds, leading to an empirical measurement of the proportionality constant. As
seen in figure (19), the previously calculated value of 21.57 for the proportionality constant does not match our simulation. By finding theTkds thatdoes match our simulation, we determine the
correct value of this coefficient to be approximately 0.25.
Since the final momentum of a particle largely depends on the time of its final scattering off the lepton bath, the distribution of last scatter times may explain the new shape of the momentum distribution. The cumulative and actual distributions of last scatter times is shown in figure (20).
Most particles do experience scattering afterΓ/H = 1, suggesting that whileΓ/H may be a good order of magnitude estimate of when collisions are happening, Γ/H <1 does not ensure collisions are negligible.
The scenario studied in this work also suggests that the choice of TRH determines whether or not
the analytical results of reference [21] - which assumed quasi-decoupling lasted until reheating - is valid. Reheating causes bothΓ/H and γ/H to decay very sharply, immediately halting collisions and freezing the shape of the distribution function to whatever it happens to be at the time. This means adjusting TRH changes the shape of the distribution function that is relevant for
calculation of the free-streaming cutoffkcut. From this we can consider three cases:
1. Reheating when Γ/H 1: In this case, reheating occurs before the Boltzmann approximation is endangered. This means the analytical solution is accurate;
quasi-decoupling continues up until reheating, when the dark matter quickly decouples from the relativistic bath. A MCMC simulation is not needed, since equation 2.4 can predict the temperature of the resulting Maxwell-Boltzmann distribution. In this scenario, the
treatment ofTχ in [21] is valid and Tχ accurately describes the momentum distribution.
2. Reheating when Γ/H 1: This is the case in our simulation. Reheating happens well after collisions stop, allowing quasi-decoupling to run its full course. This invalidates the analytical solution found in [21]. While the analytical solution in previous work cannot describe this case, a solution may be possible without resorting to a full MCMC simulation. If the final distribution in this case can be connected to the distribution of last scatter, we can predict the final velocity distribution given the parametersmχ,Tkds, andTRH.
3. Reheating when Γ/H '1: If reheating occurs around Γ/H= 1, the distribution function will be in a state of transition between Maxwell-Boltzmann and the new distribution. While more investigation into the evolution of the distribution function is needed, this case will likely need a MCMC simulation in order to predict the final distribution for any given WIMP candidate.
Figure 16: The results of evolving an ensemble of 14,312 dark matter particles through
decoupling. Tχ is the typical temperature, andTeχ is the median defined temperature.
The thin solid lines surrounding Tχ are calculations ofTχ from subdivisions of the
particle population, giving an estimate of the spread in momenta. The black dotted line emerging from behind Tχ is ∝a−9/8, quasi-decoupling scaling.
(a)Tightly coupled. The dark matter temperature equals the lepton temperature, the distribution is visibly Maxwell-Boltzmann, andTχ =Teχ.
(b)Quasi-decoupled. The dark matter temperature has begun to diverge from the lepton temperature, but the distribution is still Maxwell-Boltzmann since the Boltzmann approximation has not yet been invalidated.
(c)Decoupled. The distribution is visibly not Maxwell-Boltzmann, as a hot tail has formed and flattened the distribution. Additionally,Tχ andTeχ have diverged, asTχ is tracking the hot outliers more than the rest of the distribution.
(d)After reheating. The mean and median defined temperatures are as far apart as they will ever be, and the momentum distribution will maintain its shape as it redshifts away. This is the distribution that will be relevant when computingkcut, the cutoff scale for structure formation.
Figure 17: The dark matter momentum distribution at four stages of evolution. Top halves: the MCMC histogram, a Maxwell-Boltzmann distribution with temperature Tχ as the orange curve, a
Maxwell-Boltzmann distribution with temperatureTeχ as the green curve, and the median
momentum pas the vertical dotted line. Bottom halves: the temperature evolution, with the scale factor at the time of the snapshot marked by the solid vertical line. The vertical dashed lines are (from left to right)γ/H = 1,Γ/H= 1, and aRH.
Figure 18: The final distribution of dark momenta at the end of the simulation. There are a handful (note the logarithmic scaling of both axes) of particles that have been scattered several orders of magnitude above the bulk of the distribution. It is very unlikely that these few particles in 15,000 will play any significant role in structure formation, despite their strong influence onTχ.
5. Conclusion
The era between inflation and the neutrino decoupling prior to BBN has very few constraints, due to the lack of observational probes directly into the period. While it is traditionally assumed that this era is radiation dominated following the decay of the inflaton, there are multiple
scenarios that could produce an era of early matter domination after inflation. One possible probe of the Universe during this period is dark matter. A period of early matter domination would greatly enhance the power of small scale perturbations that are within the horizon at the time, which in turn leads to earlier structure growth on these small scales. Earlier forming structure is denser, and would produce more dark matter annihilation signatures if any are present. The exact form of this enhancement can be found from the distribution of dark matter momenta after the dark matter population decouples from the rest of the Universe, and this distribution has been found analytically for the case of decoupling in radiation domination. In the case of an EMDE however, some of the assumptions that formed the basis of this analytical solution are invalid, which puts the accuracy of that solution into question in the case of early matter domination.
This work has found that the analytical solution to dark matter decoupling does not accurately describe decoupling in an EMDE. In particular, the analytical treatment predicts that
quasi-decoupling continues up until reheating with no regard for whether or not collisions are occurring, when in fact the distribution may decouple much earlier than reheating. This leads to a cooler population of dark matter, which will form denser structure than would have otherwise been predicted. The collision simulation detailed in this thesis does not suffer from the same faulty assumption as the analytical solution, and produces this cooler distribution of dark matter
Figure 19: A comparison between the collision simulation result for Tχ and the numerical
solution of equation (2.4), using two different values of Tkds. This comparison is done
immediately after γ/H drops below 1, as the dark matter begins to quasi-decouple from the lepton bath. This is well before the breakdown of the Boltzmann
approximation, so the ODE solution should match the simulation. Left: the ODE solution if the factor of 21.57 in equation (3.28) is correct, giving Tkds = 65.7 MeV.
Despite the expected agreement of the ODE solution and the simulation, the two differ by over an order of magnitude. Right: the ODE solution fine-tuned to make the solutions match. This gives Tkds = 21.3 MeV, yielding a corrected factor of 0.25 in
equation (3.28).
Figure 20:Left: The cumulative distribution of last scatter. The histogram is the fraction of the total dark matter population that will not experience any collisions after the scale factora, and the solid red line is the theoretical prediction of this quantity (see appendix E). Right: The distribution of scale factors at last scatter. The histogram is the number of particles that experienced their final scattering event during that scale factor bin, and the red solid line is the theoretical prediction.
momenta.
Additionally, while the analytical solution predicts the velocity distribution will remain
Maxwell-Boltzmann throughout the entire process, we find significant deviation from this shape. Infrequent, highly energetic, collisions scatter a small subset of particles far above the rest of the population, forming a large tail on the hot end of the distribution. This results in a distribution that is not well described by the temperature, which is drawn upwards by extreme outliers. A temperature defined with the median kinetic energy rather than the mean provides a more accurate representation of the typical momentum for this new distribution. We have constructed this median defined temperature such that it converges to the conventional temperature when the distribution is Maxwell-Boltzmann, but more accurately describes a typical particle after the hot tail forms.
Future work will use the collision simulator to investigate the connection of the final dark matter momentum distribution to a specific value of kcut. Framework to perform this calculation is
already in place through the Cosmic Linear Anisotropy Solving System (CLASS) [26], which is capable of solving the Boltzmann equation discussed in section 2.1 (which becomes a
never-ending hierarchy of coupled differential equations once perturbations are included), simulating the evolution of linear perturbations and producing a transfer function. The solver takes in a momentum distribution (which this work provides) and outputs a transfer function. An example of this is in [4]. This would give a value ofkcut for any dark matter candidate, which
could then be used to predict annihilation signatures as is done in [19].
Another topic for future investigation is a closed form for the post-decoupling momentum distribution in the case of reheating well after decoupling. The collision simulator is capable of producing a momentum distribution for any choice of TRH and dark matter parameters (Tkds and
mχ) that keeps dark matter non-relativistic during decoupling, but the process of simulating tens
of thousands of particles is time consuming and computationally expensive. If we can fully describe the final distribution function given a choice of dark matter parameters and TRH, we
would have an empirically motivated closed form solution. This would make future work in this scenario much more straightforward, as we would not have to run thousands of particles through the collision simulator for every new set of parameters. This would also create a much more transparent connection between the parameters and the result, making future analysis and predictions easier. It is possible that, since formation of the hot tail is due to the prolonged collisional decoupling of the population, that the distribution of scale factors of last scatter is connected to our final momentum distribution. This is complicated by the fact that each final scattering boosts an individual particle to a different momentum, so more investigation is needed. A final topic for future work is reconciling the theoretical constant of proportionality in equation 3.24 with the empirical value. While we can find the constant of proportionality empirically, the disconnect between our derivation and the true form of the collision rate prevents this work from having as fundamental of a foundation as was intended. It is likely that the discrepancy due to some misplaced constant in this work or reference [25], but the specific value of Tkds being
represented in the simulation depends on the constant in question. Until our guess of a misplaced constant is confirmed, there is a missing link between the fundamental physics of the Boltzmann collision integral and our collision simulator.
A. Dark Matter Temperature Evolution
Using the Boltzmann approximation, and after a lengthy calculation [20, 27], one can obtain the following form of the collision operator from the Boltzmann collision integral:
df 0 dt c =γh3f0+ * p·*∇pf0+mχTL∇2pf0 i , (A.1)
wheref is the dark matter distribution function, γ is the momentum transfer rate,*∇p is the
gradient in momentum-space,*pis the dark matter momentum, mχ is the dark matter mass, and
TL is the lepton temperature. The Universe is isotropic in the absence of perturbations, so: * ∇p= ∂ ∂px , ∂ ∂py , ∂ ∂pz = ∂p ∂px , ∂p ∂py , ∂p ∂pz ∂ ∂p = * p p ∂ ∂p (A.2)
forp≡ |*p|, which implies:
∇2 p= * ∇2 p· *p p ∂ ∂p = 2 p ∂ ∂p+ ∂2 ∂p2. (A.3)
If we apply these forms of*∇p and ∇2p to equation (A.1), we get:
df 0 dt c =γ 3 +p∂ ∂p +mχTL 2 p ∂ ∂p+ ∂2 ∂p2 f0. (A.4)
We define the number density nχ and dark matter temperature Tχ as:
nχ≡ Z d3p (2π)3f0= 4π Z dp (2π)3p 2f 0 (A.5) Tχ≡ 2 3hKEi= 2 3 1 nχ Z d3p (2π)3 p2 2mχ f0 =2 3 4π nχ Z dp (2π)3 p4 2mχ f0 , (A.6)
where we have implicitly used natural units where~= 1and c= 1. We then multiply equation (A.4) by (2pπ2)3 and integrate over all momenta. Recalling that the left side is the explicit time dependence term ∂f /∂t in the Boltzmann equation and using substitutions from equations (A.5) and (A.6) gives:
Z d3p (2π)3p 2∂f0 ∂t = 4π Z dp (2π)3p 4∂f0 ∂t = 4πa −5 Z dq (2π)3q 4∂f0 ∂t =4π a5 ∂ ∂t Z dq (2π)3q 4f 0= 4π a5 ∂ ∂ta 5Z dp (2π)3p 4f 0 =a−5∂ ∂t(3a 5m χηχTχ) = 3mχ 2ηχTχH+ηχ ∂Tχ ∂t , (A.7)
where we have used the fact that the comoving momentumq=ap is time independent, and that H is defined as da
dt
1
a. Integrating the right side, again using substitutions from equations (A.5)
and (A.6), and setting it equal to the result above leads to:
3mχnχ 2TχH+ ∂Tχ ∂t =γ[9mχnχTχ−15mχnχTχ+mχTL(−6nχ+ 12nχ)]. (A.8)
This can be simplified and rewritten as a derivative with respect to the scale factor a, leading to:
adTχ
da + 2Tχ=−2 γ
B. The Boltzmann Regime
We are in the regime where the Boltzmann approximation (∆pχ/pχ1 for individual dark
matter/lepton collisions) is valid (the Boltzmann regime) when: p
mχTχ
TL
1. (B.1)
We are in the regime where collisions are frequent (the collisional regime) when:
Γ
H 1. (B.2)
Define the scale factor aB, where the Universe leaves the Boltzmann regime, as:
p
mχTχ(aB)
TL(aB)
= 1, (B.3)
and the scale factoraC, where the Universe leaves the collisional regime, as: Γ(aC)
H(aC)
= 1. (B.4)
Finally, we will assume we are firmly in an EMDE, such that TL andTχ are well approximated
by the power laws:
TL(a) =TL,1 a1 a 3/8 Tχ(a) =Tkd akd a 9/8 =Tkd a1 a T L,1 Tkd 8/3!9/8 . (B.5)
a1= 1, and TL,1=TL(a1), such thataRH = 106 (the scale factor of reheating) is fixed. We can
relate the kinetic decoupling temperatureTkd to thestandard kinetic decoupling temperature
Tkds (decoupling temperature in radiation domination) by [18]:
Tkd= g2 ∗,kd g∗,kdsg∗,RH !1/4r 5