# Detectable, defect-free dark photon dark matter

David Cyncynates<EMAIL_ADDRESS>Department of Physics, University of
Washington, Seattle, WA 98195, U.S.A.  Zachary J. Weiner<EMAIL_ADDRESS>Department of Physics, University of Washington, Seattle, WA 98195, U.S.A.

###### Abstract

Ultralight dark photons are compelling dark matter candidates, but their
allowed kinetic mixing with the Standard Model photon is severely constrained
by requiring that the dark photons do not collapse into a cosmic string
network in the early Universe. Direct detection in minimal production
scenarios for dark photon dark matter is strongly limited, if not entirely
excluded; discovery of sub-meV dark photon dark matter would therefore point
to a nonminimal dark sector. We describe a model that evades such constraints,
capable of producing cold dark photons in any parameter space accessible to
future direct detection experiments. The associated production dynamics yield
additional signatures in cosmology and small-scale structure, allowing for
possible positive identification of this particular class of production
mechanisms.

Evidence for cold dark matter abounds in astrophysical and cosmological
observations [1, 2, 3], but not for its fundamental nature—the mass and spin
of its constituents or its interactions with the visible sector. Dark photons
are among the best-motivated candidates for new light degrees of freedom and
are common features of grand unified theories and string theory [4, 5, 6, 7,
8, 9, 10, 11]. They may constitute all the dark matter in scenarios ranging
from minimal gravitational production during inflation [12, 13, 14, 15, 16] to
nonthermal mechanisms involving additional new degrees of freedom [17, 18, 19,
20, 21]. At low energies, a dark photon can interact with the Standard Model
(SM) through kinetic mixing with the ordinary photon, yielding signatures in
cosmology [22, 23, 24, 25, 26, 27], astrophysics [28, 29, 30, 31, 32, 33, 34,
35, 36, 37, 27, 38, 39], and the laboratory [28, 40, 41, 42, 43, 44, 45, 46,
47, 48, 49, 50]; numerous dark matter haloscopes are poised to probe a vast
space of unexplored dark photon masses and kinetic mixing [51, 52, 53, 54, 55,
56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74,
75, 76, 77, 78, 79, 75, 80]. Such theoretical and observational promise
demands understanding whether the parameter space within experimental reach
also allows for consistent dark photon dark matter production.

Recent work demonstrated a stringent upper bound on the kinetic mixing that
allows for viable dark photon dark matter [27]: the dark photon backreacts on
the Higgs responsible for its mass and, with large enough couplings at large
enough energy density, can restore the dark $\mathrm{U}(1)_{D}$ gauge
symmetry. The associated Goldstone boson winds about sites of symmetry
restoration, seeding string vortices that deplete the energy in the cold,
coherent dark electromagnetic fields. Such a defect network dilutes like
radiation and cannot be the dark matter.

The dark gauge coupling $g_{D}$, which controls the strength of backreaction
of dark gauge bosons onto the Higgs, is a free parameter in all production
mechanisms and can simply be tuned small enough to avoid defect formation. But
the dark photon’s kinetic mixing with the Standard Model photon $\varepsilon$
is generated by heavy fermions charged under both $\mathrm{U}(1)_{Y}$ and
$\mathrm{U}(1)_{D}$ and is therefore also proportional to the dark gauge
coupling $g_{D}$ [81]. Fig. 1 shows that, for known production mechanisms, the
prospects for probing the kinetic mixing of dark photon dark matter are
severely limited, if not entirely absent.

Figure 1:  Dark photon dark matter parameter space, including exclusions from
astrophysical [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 27, 38], cosmological
[22, 23, 24, 25, 26, 27], and haloscope observations [51, 52, 53, 54, 55, 56,
57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74] in
gray, and experimental prospects [75, 76, 77, 78, 79, 75, 80] in yellow. For
inflationary production [12], defect formation with a fiducial kinetic mixing
$\varepsilon\sim eg_{D}/16\pi^{2}$ [81, 82] excludes the bulk of parameter
space [27] (red shaded region). Most other models effectively produce dark
photons no later than when the Hubble rate $H_{\star}\approx
m_{\gamma^{\prime}}$, bounding the kinetic mixing to be below the dashed red
line.

Direct detection in most parameter space would therefore point to a nonminimal
dark sector.

In this letter we describe an extension of the Abelian-Higgs model that
realizes cold dark photon dark matter with kinetic mixing detectable by any
planned or proposed laboratory experiment. We discuss additional signatures in
cosmology and small-scale structure that could corroborate the nonminimality
of the model. A companion article [83] explores further generalizations
thereof, discusses the implications of defect formation on existing dark
photon production mechanisms in detail, and enumerates more ad hoc means to
generate a hierarchy between the kinetic mixing and the dark gauge coupling.

The Abelian-Higgs theory is described by the Lagrangian111 We use natural
units in which $\hbar=c=1$ and the reduced Planck mass
$M_{\mathrm{pl}}=1/\sqrt{8\pi G}$, fix a cosmic-time Friedmann-Lemaître-
Robertson-Walker (FLRW) metric
$\mathrm{d}s^{2}=\mathrm{d}t^{2}-a(t)^{2}\delta_{ij}\mathrm{d}x^{i}\mathrm{d}x^{j}$
with $a(t)$ the scale factor, and employ the Einstein summation convention for
spacetime indices. Dots denote derivatives with respect to cosmic time $t$,
and the Hubble rate is $H\equiv\dot{a}/a$.

$\displaystyle\mathcal{L}_{\mathrm{AH}}$
$\displaystyle=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}D_{\mu}\Phi\left(D^{\mu}\Phi\right)^{\ast}-V_{\Phi}(\Phi),$
(1)

where $A$ is the dark photon, $\Phi$ is the dark Higgs field,
$D_{\mu}=\partial_{\mu}-\mathrm{i}g_{D}A_{\mu}$ is the gauge covariant
derivative, and the Higgs potential $V_{\Phi}$ is the usual symmetry breaking
potential,

$\displaystyle V_{\Phi}(\Phi)$
$\displaystyle=\frac{\lambda}{4}\left({\left|\Phi\right|}^{2}-v^{2}\right)^{2}.$
(2)

In the broken phase, the dark photon acquires a mass
$m_{\gamma^{\prime}}=g_{D}v$ and contributes
$g_{D}^{2}|\Phi|^{2}A_{\mu}A^{\mu}/2$ to the Higgs’s effective potential. If
this contribution (which coincides with the dark photon’s energy density
$\rho_{A}$ when it is nonrelativistic) exceeds $\lambda v^{4}$, then the dark
photon backreacts strongly onto the Higgs and seed topological defects. If
dark photon dark matter is produced at a Hubble rate $H_{\star}$, evading
defect formation requires222 This threshold assumes the dark photon is
composed of nonrelativistic modes; we discuss its generalization in Ref. [83].

$\displaystyle g_{D}$ $\displaystyle\lesssim
10^{-14}\lambda^{1/4}\left(\frac{m_{\gamma^{\prime}}}{{\mathrm{\mu
eV}}}\right)\left(\frac{H_{\star}}{{\mathrm{\mu eV}}}\right)^{-3/8}.$ (3)

In the minimal setup, kinetic mixing is generated by loops of a few heavy
fermions with $\mathcal{O}(1)$ charge [81] with $\varepsilon\sim
g_{D}e/16\pi^{2}$, so this constraint strongly limits direct detection
prospects, as illustrated in Fig. 1. The general considerations leading to Eq.
3 motivate two possible solutions: either modulate the parameters of the
Abelian-Higgs theory to raise the threshold for defect formation or delay
production as late as possible, such that the dark photon never has enough
energy density to exceed the defect formation threshold.

These possibilities may be realized by extending the Abelian-Higgs theory Eq.
1 with couplings to a singlet scalar $\phi$ as

$\displaystyle\begin{split}\mathcal{L}&=-\frac{W(\phi)}{4}F_{\mu\nu}F^{\mu\nu}+\frac{X(\phi)}{2}D_{\mu}\Phi\left(D^{\mu}\Phi\right)^{\ast}\\\
&\hphantom{{}={}}+Y(\phi)V_{\Phi}(\Phi)+\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi).\end{split}$
(4)

We discuss concrete choices of the coupling functions $W$, $X$, and $Y$ below.
The dark Higgs and photon are made canonical via the rescalings
$\Psi=\sqrt{X(\phi)}\Phi$ and $\mathcal{A}_{\mu}=\sqrt{W(\phi)}A_{\mu}$.
Written in terms of the canonical fields, the Higgs’s potential is

$\displaystyle Y(\phi)V_{\Phi}(\Phi)$ $\displaystyle=\frac{\lambda
Y(\phi)}{4X(\phi)^{2}}\left({\left|\Psi\right|}^{2}-X(\phi)v^{2}\right)^{2},$
(5)

and its covariant derivative is

$\displaystyle\sqrt{X(\phi)}D_{\mu}\Phi$
$\displaystyle=\partial_{\mu}\Psi-\frac{\mathrm{i}g_{D}\mathcal{A}_{\mu}\Psi}{\sqrt{W(\phi)}}-\frac{\partial_{\mu}\sqrt{X(\phi)}}{\sqrt{X(\phi)}}\Psi.$
(6)

The form of Eqs. 5 and 6 motivates absorbing the $\phi$ dependence of the
theory into its fundamental parameters as

$\displaystyle g_{D}(\phi)$ $\displaystyle=g_{D}/\sqrt{W(\phi)},$ (7a)
$\displaystyle\lambda(\phi)$ $\displaystyle\equiv\lambda Y(\phi)/X(\phi)^{2},$
(7b) $\displaystyle v(\phi)$ $\displaystyle\equiv v\sqrt{X(\phi)}.$ (7c)

If the scalar is homogeneous, i.e., $\phi(t,\mathbf{x})=\bar{\phi}(t)$, then
its cosmological evolution permits the theory’s parameters to vary over
cosmological history.

The threshold for defect formation then depends on the scalar as
$\lambda(\phi)v(\phi)^{4}=\lambda v^{4}Y(\phi)$. One might be tempted to
arrange for $Y(\phi)\gg 1$ to simply raise the threshold for defect formation
as high as needed. This solution is no different than simply taking large
$\lambda$ in the bare Abelian-Higgs Lagrangian, and it requires the Higgs to
be a composite degree of freedom because fundamental Higgs scattering violates
perturbative unitarity for $\lambda\gtrsim 4\pi$. Such issues aside, a still
more appealing solution would utilize the scalar field’s dynamics to not only
prevent defect formation but also generate the dark photon relic abundance.

We turn to dynamical mechanisms that evade defect formation by delaying
production, illustrated by Fig. 2.

Figure 2:  Illustration of dark photon production that avoids defect formation
by delaying the time of production. After it is produced and becomes
nonrelativistic, the dark matter has a known energy density at any scale
factor $a$ (indicated by the green line) extrapolated from its present-day
value. For any choice of model parameters, its energy density would exceed the
threshold for defect formation (in the red shaded region) at some early time.
By sufficiently delaying the production of dark photons, i.e., until some
critical Hubble rate $H_{\star}$ (indicated by the blue line), they never
trigger defect formation (per the solid black curve). Dark photons produced
too early (as in the dashed black curve) collapse into a string network that
is not viable cold dark matter.

In general, the mass sets a kinematic barrier for particle production;
resonant production with rolling scalars effectively requires the scalar’s
mass $m_{\phi}\gtrsim m_{\gamma^{\prime}}$ for efficient dark photon
production. As the scalar starts rolling when $H\sim m_{\phi}\gtrsim
m_{\gamma^{\prime}}$, in these scenarios production typically occur no later
than $H\sim m_{\gamma^{\prime}}$. On the other hand, scalar couplings offer a
means to suppress the dark photon’s mass in the early Universe: via Eq. 7,

$\displaystyle m_{\gamma^{\prime}}(\phi)$ $\displaystyle\equiv
m_{\gamma^{\prime}}\sqrt{X(\phi)/W(\phi)}.$ (8)

In such a scenario, $X(\phi)$ and/or $W(\phi)$ must evolve so that the
theory’s parameters take on their bare values at the present day. However, the
scalar couplings generate derivative interactions that cannot necessarily be
neglected as $\bar{\phi}$ evolves. This feature is precisely what enables
production of a relic abundance of dark photons via tachyonic resonance,
familiar from other dark photon models [17, 18, 19, 20].

In the presence of a homogeneous scalar, the linearized equation of motion for
the transverse polarizations of the dark photon $\mathcal{A}_{\pm}$ is

$\displaystyle 0$
$\displaystyle=\ddot{\mathcal{A}}_{\pm}+H\dot{\mathcal{A}}_{\pm}+\omega_{\pm}^{2}\mathcal{A}_{\pm},$
(9)

with an effective squared frequency

$\displaystyle\omega_{\pm}^{2}$
$\displaystyle=\frac{k^{2}}{a^{2}}+m_{\gamma^{\prime}}^{2}\frac{\bar{X}}{\bar{W}}-\frac{H}{2}\frac{\dot{\bar{W}}}{\bar{W}}-\frac{\partial_{t}^{2}\sqrt{\bar{W}}}{\sqrt{\bar{W}}}$
(10)

[with the shorthand $\bar{W}=W(\bar{\phi})$ and $\bar{X}=X(\bar{\phi})$].
Whenever $\omega_{\pm}^{2}$ is negative (i.e., due to the coupling terms
outweighing the mass and momentum contributions), the transverse dark photon
modes grow exponentially. Time derivatives of $\bar{W}$ are necessarily
negative in Eq. 10 as $\bar{W}$ decreases from the large values that suppress
$m_{\gamma^{\prime}}$ at early times. Since the transverse modes are
derivatively coupled only to the kinetic function $W(\phi)$, we set
$X(\phi)=1$ and $Y(\phi)=1$ from here on out.333 The longitudinal mode has
derivative couplings via $X(\phi)$, though efficient production requires that
$X$ decreases with time; we discuss this and other possibilities in Ref. [83].

To illustrate the production mechanism, we present a concrete example and
compute the dark photon relic abundance, present the expanded parameter space
accessible to direct detection experiments, and discuss other constraints on
the model. We consider a so-called runaway potential [84, 85, 86] for the
scalar where

$\displaystyle V(\phi)$ $\displaystyle=M^{2}f^{2}e^{-\phi/f}\equiv
m_{\phi}^{2}f^{2}e^{-\left(\phi-\phi_{0}\right)/f}.$ (11)

The latter equality defines $m_{\phi}$ as the scalar’s effective mass at its
homogeneous initial condition $\bar{\phi}=\phi_{0}$; the scalar remains frozen
until $H\approx m_{\phi}$. An approximate solution to the scalar’s homogeneous
equation of motion,
$0=\ddot{\bar{\phi}}+3H\dot{\bar{\phi}}+V^{\prime}(\bar{\phi})$, is

$\displaystyle\bar{\phi}(t)$
$\displaystyle=\phi_{0}+f\ln\left[1+(m_{\phi}t)^{2}\right].$ (12)

Full solutions exhibit moderate oscillations in $\ln t$ about Eq. 12. Without
loss of generality, we take $\phi_{0}<0$ and assume a coupling of the form

$\displaystyle W(\phi)$ $\displaystyle=1+e^{-\beta\phi/f},$ (13)

constructed such that $W\approx 1$ after $\phi$ crosses zero, which occurs at
$m_{\phi}t_{\mathrm{d}}\approx e^{-\phi_{0}/2f}$ when the Hubble rate is
$H_{\mathrm{d}}=1/2t_{\mathrm{d}}$.

To compute the abundance of dark photons sourced by the rolling scalar, we may
first approximate the solution to Eq. 9 to $\mathcal{O}([k/m_{\phi}]^{2})$ as

$\displaystyle\frac{\mathcal{A}_{\pm}(t,k)}{\mathcal{A}(0,k)}\approx\left[1+(m_{\phi}t)^{2}\right]^{-\beta/2}\left[1+(k/m_{\phi})^{2}\delta\mathcal{A}(t)\right].$
(14)

Equation 14 solves the $k\to 0$ limit of Eq. 9 when $\delta A=0$; plugging Eq.
14 back into Eq. 9 yields a leading-$k$ correction of

$\displaystyle\delta\mathcal{A}(t)$
$\displaystyle=\int_{0}^{m_{\phi}t}\mathop{}\\!\mathrm{d}x\,(1+x^{2})^{\beta}\,_{2}F_{1}(1/4,\beta,5/4,-x^{2}).$
(15)

We set $a=1$ when $H=m_{\phi}$. Taking $x\gg 1$, the growth rate of the
small-$k$ modes scales as
$\mathcal{A}_{\pm}\propto(k/m_{\phi})^{2}(m_{\phi}t)^{\beta+1/2}$.
Amplification continues roughly until any other term in Eq. 9 becomes
important, i.e., at

$\displaystyle t_{\star}(k)$
$\displaystyle\approx\min\left\\{\frac{1}{2H_{\mathrm{d}}}\left(\sqrt{\beta^{2}+\frac{\beta}{2}}\frac{2H_{\mathrm{d}}}{m_{\gamma^{\prime}}}\right)^{\frac{1}{\beta+1}},\frac{2\beta^{2}+\beta}{2k^{2}/m_{\phi}}\right\\}.$
(16)

Modes with larger wave number stop growing earlier, and the power spectrum of
$\mathcal{A}_{\pm}$ peaks near

$\displaystyle\frac{k_{\star}}{m_{\phi}}\approx
2^{\frac{\beta}{2(\beta+1)}}\frac{m_{\gamma^{\prime}}}{\sqrt{H_{\mathrm{d}}m_{\phi}}}\left(\frac{H_{\mathrm{d}}}{m_{\gamma^{\prime}}}\sqrt{\beta\left(\beta+\frac{1}{2}\right)}\right)^{\frac{2\beta+1}{2\beta+2}}.$
(17)

As $m_{\gamma^{\prime}}\gg H_{\mathrm{d}}$ in general, all amplified modes are
highly nonrelativistic once $W(\phi)\to 1$. Provided $\beta$ is greater than a
few, the time of production Eq. 16 reduces to $t_{\star}\approx
1/2H_{\mathrm{d}}$. After this time, the mode amplitude of the nonrelativistic
modes decreases as the mass continues to increase $\mathcal{A}\propto W^{1/4}$
per the WKB approximation. Integrating the spectrum of $\mathcal{A}$ over wave
number up to $k_{\star}$ to estimate $\rho_{A}(t_{\star})\approx
m_{\gamma^{\prime}}^{2}\left\langle\mathcal{A}^{2}\right\rangle$ at
production,

$\displaystyle\frac{\rho_{A}(t_{\star})}{H_{\mathrm{d}}^{4}}$
$\displaystyle\equiv\mathcal{N}_{\beta}\left(\frac{m_{\phi}}{H_{\mathrm{d}}}\right)^{2\beta-1}\left(\frac{H_{\mathrm{d}}}{m_{\gamma^{\prime}}}\right)^{\frac{\beta-5}{\beta+1}},$
(18)

where the $\beta$-dependent coefficient is

$\displaystyle\mathcal{N}_{\beta}$
$\displaystyle=\frac{(2\beta^{2}+\beta)^{\frac{3(3\beta+1)}{2(\beta+1)}}\Gamma(5/4)^{2}\Gamma(\beta-1/4)^{2}}{3\pi^{2}(1+4\beta)^{2}2^{\frac{2\beta^{2}-\beta}{\beta+1}+\frac{7}{2}}\Gamma(\beta)^{2}}.$
(19)

This result reproduces full numerical solutions within one to two orders of
magnitude for $-\phi_{0}/f\gtrsim 5$ [below which the errors are dominated by
the approximation made in Eq. 12]. More detailed exposition is provided in
Ref. [83]. Note that the success of the mechanism is not unique to these
particular choices of $W$ and $V$—the salient features are a substantial mass
suppression at early times [so that $m_{\gamma^{\prime}}(\phi_{0})\ll
m_{\phi}$] and a coupling function that evolves faster with $\phi$ than the
potential (so that tachyonic resonance is efficient). We discuss other
possibilities in Ref. [83].

The relic abundance of dark photon dark matter is

$\displaystyle\frac{\Omega_{\gamma^{\prime}}}{\Omega_{\mathrm{DM}}}$
$\displaystyle=\mathcal{N}_{\beta}\left(\frac{m_{\phi}}{H_{\mathrm{d}}}\right)^{2\beta-1}\left(\frac{H_{\mathrm{d}}}{m_{\gamma^{\prime}}}\right)^{\frac{\beta-5}{\beta+1}}\frac{H_{\mathrm{d}}^{2}}{M_{\mathrm{pl}}^{2}}\sqrt{\frac{H_{\mathrm{d}}}{H_{\mathrm{eq}}}}$
(20)

where $H_{\mathrm{eq}}\approx 2.2\times 10^{-28}\,\mathrm{eV}$ is the Hubble
rate at matter-radiation equality. Production may occur as late as desired by
choosing $m_{\phi}$ or $\phi_{0}$ to set $t_{\mathrm{d}}$. To produce dark
photons heavier than the scalar, their initial mass
$m_{\gamma^{\prime}}(\phi_{0})\approx m_{\gamma^{\prime}}e^{\beta\phi_{0}/2}$
must be smaller than $m_{\phi}$, requiring $\beta\phi_{0}$ to be sufficiently
large (and negative). Achieving the correct relic abundance turns out to place
a more stringent requirement of $-\beta\phi_{0}/f$ in the range of $150-250$,
with a corresponding mass suppression more than enough to produce dark photons
with any mass of interest. For cosmology to proceed as observed, the dark
matter must exist by, say, a redshift $z\sim 10^{6}$ or $10^{7}$, making
decoupling at $H_{\mathrm{d}}=10^{-22}\,\mathrm{eV}$ a useful benchmark. As
evinced by Fig. 3, such late production is sufficient for viable dark photon
dark matter in reach of any future experiment.

Figure 3:  Parameter space available to the kinetically coupled scalar.
Constraints from defect formation [Eq. 3, shaded red] assume production no
earlier than $H_{\star}=10^{-22}\,\mathrm{eV}$. Nonthermal production
occurring when the Hubble rate is near the dark photon’s mass, e.g. via axion
or scalar oscillations [17, 18, 19, 20, 21], is viable below the dashed red
line. The green and blue regions indicate where enhanced DM substructure at
$k_{\star}$ [Eq. 17] could be probed by future astrometric [87] and
photometric [88, 89] surveys, respectively. Assuming weak gravity conjectures
are true and apply at the displaced field values required by the kinetic
coupling, the dashed black line indicates the maximum achievable gauge
coupling for which local quantum field theory holds during, e.g., high-scale
inflation.

Other resonant production mechanisms (via oscillating pseudoscalars [17, 18,
19, 20] or scalars [21]) require the system to reach a regime where the dark
photon backreacts on the scalar sourcing it—otherwise, the dark matter would
mostly comprise scalars rather than vectors. These nonlinear dynamics can only
be understood with 3D simulations, and the energy exchange that occurs at
backreaction often results in a rough equipartition between the dark photon
and scalar. At attractive feature of scenarios involving runaway scalars is
that they become energetically subdominant without relying on nonlinear
dynamics. The runaway scalar solution uniquely tracks the background such that
its relative abundance is

$\displaystyle\Omega_{\phi}(t)$
$\displaystyle\equiv\frac{\rho_{\phi}(t)}{3H(t)^{2}M_{\mathrm{pl}}^{2}}\approx\left(\frac{2f}{M_{\mathrm{pl}}}\right)^{2}$
(21)

in both the radiation- and matter-dominated epochs [90, 91, 92]. Comparing to
the abundance of the dark photons
$\Omega_{\gamma^{\prime}}=\rho_{\gamma^{\prime}}/3H^{2}M_{\mathrm{pl}}^{2}$
provides a reasonable proxy to assess whether backreaction is important. At
production (when the two decouple), the dark photons have an abundance
$\Omega_{\gamma^{\prime}}(t_{\star})\sim\left[H_{\mathrm{eq}}/H_{\star}\right]^{3/2}\sim
10^{-10}\left[H_{\star}/10^{-21}\,\mathrm{eV}\right]^{-3/2}$. The scalar’s
decay constant need not be anywhere near $M_{\mathrm{pl}}$ to arrange for its
energy density to well above the dark photon’s.

At large enough $f$ the scalar could have an observable impact on cosmology.
In the radiation era, the scalar effectively increases the Hubble rate as
small-scale CMB modes enter the horizon, enhancing diffusion damping of the
photon-baryon plasma [93, 94, 95]. Bounds on extra radiation content from the
CMB largely derive from this effect and currently amount to a bound $\Delta
N_{\mathrm{eff}}/N_{\mathrm{eff}}\lesssim 5\%$ to $10\%$.444 The scalar is not
precisely equivalent to free-streaming neutrinos—genuine constraints would
require a proper treatment of the dynamics of its perturbations. In addition,
runaway scalars redshift like matter after matter-radiation equality,
providing enhanced and distinctive phenomenology compared to pure radiation
(which instead becomes energetically subdominant.)  Current measurements
therefore already limit $f$ to be (roughly) below $M_{\mathrm{pl}}/10$, while
CMB-S4, which projects sensitivity to $\Delta
N_{\mathrm{eff}}/N_{\mathrm{eff}}\sim 1\%$, would probe yet smaller decay
constants $f/M_{\mathrm{pl}}\sim 0.03$.

Resonant production mechanisms in general feature dark matter with a density
power spectrum sharply peaked at order unity on some characteristic scale. In
the case of axion or scalar oscillations, the scalar mass is what sets this
characteristic wave number [17, 18, 19, 20, 21]. On the other hand, the vector
mass sets a kinematic barrier below which resonant enhancement is typically
inefficient. The kinetic coupling suppresses the vector mass during
production, allowing for peak scales of order $m_{\phi}$ which can be far
below the present-day dark photon mass.

Density fluctuations at the power spectrum peak collapse shortly after matter-
radiation equality, forming dense small scale structure at astrophysically
relevant scales [96]. The typical peak wave number is set by the Hubble rate
at production [per Eq. 17], which in the large-$\beta$ limit corresponds to
structures with mass

$\displaystyle M$ $\displaystyle\sim 2\times
10^{9}M_{\odot}\left(\frac{\beta}{10}\right)^{-3}\left(\frac{m_{\phi}}{10^{-22}{\mathrm{eV}}}\right)^{-3/2}.$
(22)

That minihalos can be much more massive than expected from the dark photon’s
mass itself provides a signature that distinguishes this model from other
resonant production mechanisms. While the presence of such massive
substructure does not guarantee that the dark photon has kinetic mixing of any
particular size, if an experiment measures a kinetic mixing larger than
possible for other nonthermal production mechanisms, substructure in the dark
matter halo would necessarily be so massive. Figure 3 illustrates the mass-
coupling parameter space for which upcoming astrometric [87] and photometric
[88, 89] surveys would be able to probe such extremal substructure.

In the class of models we discuss, the kinetic mixing evolves in the early
Universe, but there is no reason a priori that the evolution must have stopped
before the present day. The constant term in Eq. 13, for instance, could
itself be promoted to a slow function of $\phi$, such that early-Universe
probes would measure a kinetic mixing different from laboratory ones. Over
long enough timescales, laboratory experiments could probe drifts in the
kinetic mixing as well.

The success of the scalar production mechanism effectively relies on the
Abelian-Higgs theory being extremely weakly coupled at early times, which
ostensibly would run afoul of so-called weak gravity conjectures (WGCs) [97].
While the application of the WGC in its various forms to effective field
theory and Higgsed gauge symmetries is subtle [98, 99, 100, 101], arguments
that gravity becomes strongly coupled at a scale $\Lambda_{\mathrm{UV}}\sim
g_{D}^{1/3}M_{\mathrm{pl}}$ are considered robust [102, 103]. Requiring that
gravity remains weakly coupled at the highest energy scale probed by any
production scenario therefore places a lower limit on $g_{D}$, which, in
conjunction with upper bounds from defect formation, can be quite
constraining. Requiring the energy scale of inflation to be above
$\Lambda_{\mathrm{UV}}$ limits viable inflationary production to
$m_{\gamma^{\prime}}\gtrsim 40\,{\mathrm{MeV}}$.

While it is unclear whether WGCs constrain gauge couplings with displaced
moduli, were it so they would effectively constrain the initial condition of
the scalar field via the combination $\beta\phi_{0}/f$. Achieving the dark
matter relic abundance for a given dark photon mass then requires increasing
$H_{\mathrm{d}}$ [via Eq. 20], diminishing the extent to which the scenario
evades defect formation. Combining the WGC’s lower bound on $H_{\mathrm{d}}$
with the upper bound from defect formation Eq. 3 places an upper bound on the
present-day dark gauge coupling. These bounds are set by conditions in the
early Universe, i.e., requiring the cutoff scale of quantum gravity to exceed
the energy scales of Big Bang Nucleosynthesis, inflation (if measured), or of
the SM plasma during dark photon production. Written in terms of the maximum
Hubble scale $H_{\mathrm{max}}$, which for BBN is $\sim
10^{-15}\,\mathrm{eV}$,

$\displaystyle\frac{g_{D}}{3\times 10^{-18}}$
$\displaystyle\lesssim\left(\frac{m_{\gamma^{\prime}}}{10^{-15}\,\mathrm{eV}}\right)^{25/22}\left(\frac{H_{\mathrm{max}}}{10^{-15}\,\mathrm{eV}}\right)^{-9/22}.$
(23)

Equation 23 takes fiducial values $\beta=25$ and $\lambda=1$ (and is not
particularly sensitive to either). This bound is weaker than the defect
formation bound but would eliminate most (but not all) of the prospective
parameter space if the inflationary Hubble scale is $H_{\mathrm{inf}}\sim
10^{14}\,\mathrm{GeV}$ [see Fig. 3].

Weak gravity conjectures also motivate ultralight dark photons receiving their
mass from the Higgs mechanism rather than the Stückelberg mechanism. In
supersymmetry, Stuückelberg fields are accompanied by radial degrees of
freedom with mass $\sim m_{\gamma^{\prime}}/g_{D}$ just like the Higgs, a fact
conjectured to hold in any theory of quantum gravity [102]. In this case,
inflationary production of dark photons with a Stückelberg mass is just as
constrained as that with a Higgs mass (since the radial modes are produced
during inflation if they are too light). It would be worth understanding
whether the radial mode plays an important role in other production mechanisms
as well.

Though dark matter’s current phenomenological relevance resides only at low
energies and late times, identifying its fundamental nature offers myriad
opportunities to inform high-energy physics. Understanding the mechanisms
underlying its mass and its early-Universe production reshapes the
implications of direct detection of dark photons. We explore these
implications more broadly in companion work [83], with this letter
highlighting a scenario whose nonminimality, aside from enabling direct
detection in the laboratory, offers promising and complementary observational
signatures. These results also motivate searching for signatures of dark
photon dark matter from purely gravitational interactions, especially in the
near-fuzzy regime $10^{-22}\,\mathrm{eV}\lesssim m_{\gamma^{\prime}}\lesssim
10^{-15}\,\mathrm{eV}$ [104, 105, 106, 107, 108, 109] where phenomenology can
depend on the spin of dark matter [110, 111, 112, 113, 114, 115, 116] Further
investigation of consistent dark photon cosmologies will deepen our
understanding of the theoretical motivation for and implications of ultralight
dark matter detection.

We thank Peter Adshead, Benoit Assi, Masha Baryakhtar, Adrienne Erickcek,
Isabel Garcia Garcia, Anson Hook, Junwu Huang, Justin Kaidi, Justin Khoury,
and Mark Trodden for insightful discussions. D.C. and Z.J.W. are supported
through the Department of Physics and College of Arts and Science at the
University of Washington. This material is partially supported by a grant from
the Simons Foundation and the hospitality of the Aspen Center for Physics.
This work was completed in part at the Perimeter Institute. Research at
Perimeter Institute is supported in part by the Government of Canada through
the Department of Innovation, Science and Economic Development Canada and by
the Province of Ontario through the Ministry of Colleges and Universities. The
dark photon parameter space limits and projections quoted above are compiled
in Ref. [117].

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