Dark Matter from Exponential Growth

Abstract

We propose a novel mechanism for the production of dark matter (DM) from a thermal bath based on the idea that DM particles χ can transform heat bath particles ψ: χψ→χχ. For a small initial abundance of χ, this leads to an exponential growth of the DM number density in close analogy to other familiar exponential growth processes in nature. We demonstrate that this mechanism complements freeze-in and freeze-out production in a generic way, opening new parameter space to explain the observed DM abundance, and we discuss observational prospects for such scenarios.

Full text

Dark Matter from Exponential Growth
Torsten Bringmann ,1,* Paul Frederik Depta ,2,†Marco Hufnagel,3,‡
Joshua T. Ruderman,4,2,5,6,§ and Kai Schmidt-Hoberg 2,∥
1Department of Physics, University of Oslo, Box 1048, N-0316 Oslo, Norway
2Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, D-22607 Hamburg, Germany
3Service de Physique Th´eorique, Universit´e Libre de Bruxelles, Boulevard du Triomphe, CP225, B-1050 Brussels, Belgium
4Center for Cosmology and Particle Physics, Department of Physics, New York University, New York, New York 10003, USA
5Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA
6School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv 69978, Israel
(Received 25 June 2021; revised 20 August 2021; accepted 14 September 2021; published 3 November 2021)
We propose a novel mechanism for the production of dark matter (DM) from a thermal bath based on the
idea that DM particles χcan transform heat bath particles ψ:χψ →χχ. For a small initial abundance of χ,
this leads to an exponential growth of the DM number density in close analogy to other familiar exponential
growth processes in nature. We demonstrate that this mechanism complements freeze-in and freeze-out
production in a generic way, opening new parameter space to explain the observed DM abundance, and we
discuss observational prospects for such scenarios.
DOI: 10.1103/PhysRevLett.127.191802
Introduction.—While the identity and underlying proper-
ties of the dark matter (DM) in our Universe remain
mysterious, its energy density has been precisely inferred
by a series of satellite missions studying the cosmic
microwave background (CMB). Any theoretical descrip-
tion of DM must therefore include a DM production
mechanism that leads to the observed DM relic abundance
of ΩDMh2≃0.12 [1].
A particularly appealing framework for the genesis of
DM, minimizing the dependence on initial conditions, is its
creation out of a thermal bath. The most commonly adopted
paradigm falling into this category is thermal freeze-out
from the primordial plasma of standard model (SM)
particles in the early Universe [2]. However, given the
increasingly strong constraints on this setup, a plethora of
alternate production scenarios with DM initially in thermal
equilibrium have recently been proposed, including “hid-
den sector freeze-out”[3–9],“Forbidden DM”[10,11],
“Cannibal DM”[12,13],“Coscattering DM”[14–16],
“Zombie DM”[17],“Elder DM”[18],“Kinder DM”
[19], and “SIMP DM”[20–22]. Another possibility is that
DM never entered thermal equilibrium at all, in which case
it can be produced via a “leakage”out of a thermal bath,
often referred to as “freeze-in”[23,24]. While a large
number of variants of the freeze-out paradigm have been
suggested, less model building has been performed around
the freeze-in idea (see, however, Refs. [25–32]).
In this Letter, we propose a novel and generic DM
production scenario between these two polarities based on
the idea that a DM particle χcan “transform”a heat bath
particle ψinto another χ; cf. Fig. 1. For a small initial
abundance nχ, as shown below, this results in an expo-
nential growth of the DM abundance. To be in accord with
observations, the exponential growth is required to shut off
before the DM particle χis fully thermalized, so one can
also think of this mechanism as a “failed thermalization.”
Interestingly, the exponential growth of nχcomes to an
end naturally in our framework, so that the observed DM
abundance is readily obtained.
A novel DM production mechanism.—Quantitatively, the
evolution of the DM number density nχis governed by the
Boltzmann equation
FIG. 1. The transformation process leading to exponential
production of DM (χ) from the heat bath (ψ).
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PHYSICAL REVIEW LETTERS 127, 191802 (2021)
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_
nχþ3Hnχ¼hσvitr nχneq
ψ−hσviann2
χ:ð1Þ
Here, neq
ψis the number density of ψin equilibrium, His the
Hubble rate, and hσvitr (hσvian) is the cross section for the
process χψ →χχ (χχ →χψ), averaged over the phase
space of the initial state. We assume that ψis in equilibrium
with the SM heat bath; this equilibrium can be maintained,
for example, by rapid annihilations of ψψ to SM states. We
note that the zombie collisions of Refs. [17,33] involve a
similar process to Fig. 1, but the roles of the DM and bath
particles are reversed.
As long as neq
χ≫nχ, we can neglect the second term on
the rhs of the above equation. Introducing xψ≡mψ=T and
Yχ≡nχ=s, with sbeing the entropy density of the heat
bath, the solution of the Boltzmann equation is given by
YχðxψÞ≃Y0
χexp Zxψ
x0
ψ
dx
xPðxÞ;ð2Þ
where
PðxÞ¼ ˜
H−1neq
ψhσvitr:ð3Þ
Here, Y0
χdenotes the DM abundance at some initial “time”
x0
ψ, and we have defined
˜
H≡H=½1þð1=3Þdðlog gs
eff Þ=
dðlog TÞ, where gs
eff encodes the entropy degrees of
freedom.
Equation (2) describes exponential growth of the DM
abundance, with growth rate P, as long as P0ðxÞ>0.
For highly relativistic heat bath particles (with neq
ψ∝x−3
ψ),
this is automatically achieved for hσvitr ¼ðσvÞ0
trxk
ψ, with
ðσvÞ0
tr ≃const:and k>1, i.e., infrared (IR) dominated
transformation processes since H∝x−2
ψ. Later, once the
heat bath particles become nonrelativistic, exponential
growth will inevitably come to an end for any value of
kdue to the Boltzmann suppression of neq
ψ, leading to an
asymptotically flat YχðxψÞ. Parametrically, we thus find
Yfinal
χ∼Y0
χexp λ
k−1xk−1
ψ;NRð4Þ
for the final DM abundance, where xψ;NR ∼3and
λ∼6×10−2gψg−1
2
eff mψmPlðσvÞ0
tr, with geff the energy
degrees of freedom and mPl the Planck mass.
We confirm this expectation in Fig. 2, where we show
(with solid lines) the full solution of Eq. (1), adopting for
illustration a constant amplitude jMtrj2¼λ2
tr, which is
realized if χand ψare real scalars with interaction
L⊃ðλtr=3!Þψχ3. We calculate, for simplicity, hσvitr for
a Maxwell-Boltzmann distribution [34], leaving a deter-
mination of its precise phase space distribution for future
work; this leads to hσvitr ∝T−2for T≫mχ,mψ. Starting
from an initial value of the DM abundance (indicated by the
dashed lines), the onset of exponential growth is clearly
visible as Pbecomes larger than ∼0.2for xψ≳0.01 until it
stalls because Pis heavily suppressed again for xψ≳5. The
figure also illustrates an attractive feature of exponential
growth from a phenomenological point of view: the
coupling strength required to match the observed DM relic
abundance is only logarithmically sensitive to the initial
abundance. In the examples shown here, e.g., decreasing
the initial abundance by 4 orders of magnitude (from the
green to the red line) is compensated by a mere increase of
about 22% in λtr.
Exponential growth in nature.—It is intriguing how
closely the evolution of the DM abundance in Fig. 2
mimics other well-known examples of exponential growth
in nature—like, for example, the progression of an illness
after an initial outbreak. In fact, we can formalize this
analogy by referring to the SIR (susceptible, infected, and
recovered) epidemiological model [35], where the number
of infected individuals, I, evolves according to
_
I¼βSI −γI; ð5Þ
with Sthe number of susceptible individuals and βand γ
the infection and recovery rates, respectively. We recognize
that this is simply Eq. (1), in the limit nχ≪neq
χ, after
identifying I↔nχ,S↔neq
ψ,β↔hσvitr, and γ↔3H.
This mathematically exact correspondence motivates us to
further introduce
FIG. 2. Top: Number density of χrelative to the entropy density
of the heat bath (solid lines) for mχ¼1GeV, mψ¼2GeV and
different values of the transformation coupling λtr. For each value
of λtr, we fix the initial abundance of χ(dashed lines) such that
the final abundance corresponds to the observed DM density.
Bottom: Rvalue corresponding to the abundance evolution in the
top panel.
PHYSICAL REVIEW LETTERS 127, 191802 (2021)
191802-2

R≡βS
γ¼nψhσvitr
3H¼
˜
H
3HP; ð6Þ
where the final equality follows from Eq. (3).Rmeasures
the number of transformation processes that each DM
particle undergoes per Hubble time and is, through Eqs. (2)
and (6), directly related to the final DM abundance.
Initial abundance.—In the above discussion, we have
deliberately remained agnostic about the origin of Y0
χand
simply treated this quantity as a free input parameter.
We now outline various physical mechanisms that could
generate such an initial DM abundance.
The first class of initial DM production mechanisms
takes place much earlier than the typically rather short
period where ψψ →χψ processes dominate. This includes
well-studied examples such as UV-dominated freeze-in
[36,37] or direct production from the decay of the inflaton
or other heavy particles [38] but could also be related to
more exotic examples like false vacua after a phase
transition in the dark sector [39,40] or by the evaporation
of black holes. Common to all these scenarios is that the
final DM abundance is independent of how exactly the
initial abundance is set: the only phenomenologically
relevant input is the DM abundance at the onset of the
era of exponential growth, thus providing a direct map to
the generic situation depicted in Fig. 2.
In the second class of relevant scenarios, the initial and
exponential phases of DM production are intertwined. This
is particularly relevant for IR dominated freeze-in rates [23]
that are too small to explain the observed DM abundance
without a subsequent phase of exponential growth. A nice
feature of the mechanism proposed here is in fact that a
nonvanishing freeze-in contribution due to the transforma-
tion coupling λtr is automatically built-in, as discussed
below. In general, the Boltzmann equation including 2→2
freeze-in processes becomes
_
nχþ3Hnχ≃hσvitrnχneq
ψþhσvifi ðneq
ψÞ2;ð7Þ
where hσvifi is the total cross section for ψψ →χχ
and ψψ →χψ. Since neq
ψ≫nχ, a necessary condition
for transformation processes to be non-negligible compared
to traditional freeze-in is thus hσvifi ≪hσvitr. Once the two
terms on the rhs of the above equation are of a similar size,
on the other hand, ψψ →χψ will very quickly take over
due to the exponential growth of nχ.
We show the evolution of the DM abundance for this
scenario in Fig. 3, assuming for simplicity that all ampli-
tudes are constant. We also choose the same masses and
transformation couplings as in Fig. 2to facilitate compari-
son. Instead of fixing the initial abundance, however, we
now fix the freeze-in coupling to result in the correct relic
abundance (thus taking a vanishing DM abundance as the
initial condition). The above-discussed three phases—
freeze-in, transformation, and the final flattening of the
abundance evolution curve—are clearly visible in the figure.
We stress that this brings a new perspective to the widely
studied freeze-in mechanism, which can trigger a subsequent
phase of exponential growth in a rather natural way. It
therefore becomes possible to satisfy the relic density
constraint with significantly smaller couplings λfi than
generally assumed, opening up new parameter space where
freeze-in is relevant for setting the DM energy density.
To further illustrate the last point, we show in Fig. 4a
full “phase diagram”of the combination of couplings λtr
and λfi that allow the production of DM from the heat
bath for a fixed mass ratio of mψ=mχ¼1.2. At each point
in this plane, we thus adjust the mass mχsuch that
ΩDMh2¼0.12 (with dashed lines indicating isocontours
of mχ). Depending on the couplings, the relic abundance
can be set via different mechanisms. In the green (red)
region, the relic is mainly set via freeze-in (freeze-out) of
the process ψψ ↔χχ. In the blue (yellow) region, the relic
density is instead mainly set via freeze-in (freeze-out) of the
process ψχ ↔χχ. In regions where two colors overlap,
more than one production mechanism can lead to the
correct relic abundance, albeit for different masses. In the
gray regions, either of the production mechanisms would
require fully thermalized scalars with mψ<0.5MeV—for
the mass ratio mψ=mχ¼1.2adopted for the purpose of this
figure—which is in conflict with constraints from big bang
nucleosynthesis (BBN) [41]. The blue region is bounded
toward large values of λtr since such couplings would—for
exponential production—require masses above the unitar-
ity limit of a thermal particle mψ>140 TeV [42].
FIG. 3. As in Fig. 2but now with a vanishing initial DM
abundance and, on top of the transformation interaction, freeze-in
production based on a constant matrix element. The coupling λfi
for the latter is chosen such that the final abundance of χ
corresponds to the observed DM density. Dashed lines show
the would-be abundance from freeze-in alone (when setting
λtr ¼0, for which λfi ≈5.81 ×10−11 would give ΩDM h2¼0.12).
PHYSICAL REVIEW LETTERS 127, 191802 (2021)
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There is an irreducible 2→4freeze-in contribution
ψψ →4χ, with cross-section scaling as λ4
tr, which we
estimate to dominate over 2→2freeze-in within the light
blue region. We neglect 2→4processes in Fig. 4, which
only have a logarithmic effect on the value of λtr that results
in the observed relic density. Finally, we note that λtr
generates λfi radiatively and, in the absence of fine-tuning,
we expect λfi ≳λ2
tr=ð4πÞ2. This bound is satisfied except in
the light blue region where 2→4processes are relevant.
Discussion.—We stress that exponential growth due to
processes as depicted in Fig. 1is by no means restricted to
specific model realizations but is a general mechanism of
DM production that essentially interpolates between the
traditionally considered freeze-in and freeze-out regimes. At
first glance it may seem worrisome that the dark matter
density depends exponentially on the transformation cross
section, implying that the cross section must be carefully
chosen to match observation. But this can be turned around:
in fact it implies that this mechanism is highly predictive, as
manifested by the logarithmic sensitivity of the necessary
cross section to the initial conditions. We also note that
exponential sensitivity is quite common in nature and, for
example, can result from renormalization group flows where
IR parameters can be exponentially sensitive to UV param-
eters: the proton mass, e.g., depends exponentially on the
size of the strong coupling at high energies [43].
It is worth pointing out that the mechanism proposed
here works for a large range of different particle masses, not
just the specific choice displayed in the examples above.
Limits from BBN or CMB on new light degrees of freedom
[41,44], as encountered in Fig. 4, could be significantly
lowered by considering the possibility of ψbeing a SM
particle. Larger values of mψwould not affect the DM
velocity dispersion at T∼mχ(but may sharpen model-
dependent constraints from the decay of ψ, see below).
Even a reverted mass hierarchy is possible, mχ>m
ψ,in
which case it would be mχrather than mψthat determines
when the evolution of nχ=s starts to flatten. For mχ>m
ψ,
DM in general becomes unstable—but not necessarily on
cosmological timescales. Such highly suppressed decays
may potentially be visible in late-time observables related
to cosmological structure formation or cosmic rays, making
corresponding scenarios even more attractive.
Let us finish by considering the phenomenological
consequences for a simple, concrete model realization,
where ψis a real scalar and couples to the SM via a quartic
Higgs portal coupling λhψjHj2ψ2=2. The relic density of ψ
is then determined by standard freeze-out, exactly as for
scalar singlet DM [45], and we fix λhψby requiring
Ωψ¼0.1ΩDM. This implies a unitarity limit of around
mψ≈44 TeV [42] and a direct detection limit from
Xenon1T [46] of mψ>1TeV (obtained by rescaling
the results from Ref. [45]); next-generation direct detection
experiments will fully explore masses up to the unitarity
limit [47]. In Fig. 5, we show these constraints in the plane
mχ−λtr. Here, we fix λfi such that the dominant DM
FIG. 4. Phase diagram of transformation (λtr ) and freeze-in (λfi)
couplings that can result in the correct DM abundance for a fixed
mass ratio mψ=mχ¼1.2. Colored regions indicate the respective
mechanism that is responsible for thermal production, while
dashed lines show the required value of mχ. Gray regions would
require a new heat bath particle ψtoo light to be compatible with
constraints from BBN [41]. In the light blue region an additional
2→4freeze-in contribution is expected.
FIG. 5. Phenomenological consequences of one possible reali-
zation of exponential production, with ψcoupling to the SM via
the Higgs portal. For each value of mχand λtr , the couplings λfi
and λhψare fixed such that ψ(χ) contributes 10% (90%) to the
observed DM abundance. Green and blue areas correspond, as in
Fig. 4, to different thermal productions regimes of χ. The purple
area is excluded by direct detection [46], while in the yellow area
unitary constraints would imply an overproduction of ψ. Red
lines (and hatched area) indicate where to naturally expect CMB
signatures. Light blue shading as in Fig. 4.
PHYSICAL REVIEW LETTERS 127, 191802 (2021)
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component satisfies Ωχ¼0.9ΩDM and is mostly produced
by transformation (blue) or freeze-in (green) processes (in
the white area, Ωχ¼0.9ΩDM may also be possible, but
only via freeze-out or semiannihilations). Finally, the decay
of ψto SM particles can impact the observed CMB power
spectrum; for a 10% DM subcomponent, this leads to a
constraint of τψ≳1023 s[48], projected to tighten by a
factor of ∼3with CMB-S4 [49]. Let us for simplicity adopt
a fixed mass splitting of mψ−mχ¼100 GeV, such that
only decays ψ→χb¯
bare relevant. These are dominated by
either 2-loop or 3-loop diagrams, depending on the relative
strength of transformation and freeze-in couplings. The
naive size of these loops results in decay widths Γ2∼
Pλ2
trλ2
fiλ2
hψy2
b=ð4πÞ8=mψand Γ3∼Pλ6
trλ2
hψy2
b=ð4πÞ12=mψ,
respectively, where ybis the bottom Yukawa coupling
and P≃½m4
ψ−m4
χþ4m2
ψm2
χlogðmχ=mψÞ=½512π3m2
ψis
the phase space in the limit mψ−mχ≫2mb. We use
these expressions in Fig. 5to indicate, with red lines, where
to naturally expect CMB signatures in this model.
Conclusions.—We have introduced a novel type of DM
production mechanism, where an initially tiny DM abun-
dance is enhanced due to a process where DM particles
convert bath particles into more DM particles. The DM
abundance grows exponentially with time in stark contrast
to models of freeze-in where the abundance grows only as a
power law. Our mechanism complements both freeze-in
and freeze-out thermal production scenarios in a generic
way. Concrete model realizations lead, already in their
simplest forms, to interesting phenomenological conse-
quences. Further, and detailed, exploration of this new way
of producing DM from the thermal bath thus appears highly
warranted.
We thank Michael Geller for helpful conversations,
including at a primordial stage. This work is supported
by the Deutsche Forschungsgemeinschaft under Germany’s
Excellence Strategy—EXC 2121 “Quantum Universe”—
390833306, the F.R.S.—FNRS under the Excellence of
Science (EoS) Project No. 30820817—be.h “The H boson
gateway to physics beyond the Standard Model,”and the
National Science Foundation under Grant No. NSF PHY-
1748958. J. T. R. is further supported by the NSF CAREER
Grant No. PHY-1554858, NSF Grant No. PHY-1915409,
an award from the Alexander von Humboldt Foundation,
and the European Research Council (ERC) under the EU
Horizon 2020 Programme (ERC-CoG-2015—Proposal
n. 682676 LDMThExp).
*Corresponding author.
torsten.bringmann@fys.uio.no
†Corresponding author.
frederik.depta@desy.de
‡Corresponding author.
marco.hufnagel@ulb.ac.be
§Corresponding author.
ruderman@nyu.edu
∥Corresponding author.
kai.schmidt-hoberg@desy.de
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