# Dark Matter Is The New BBN

Dan Hooper1,2,3 Huangyu Xiao1 1Fermilab, Theoretical Astrophysics Department,
Batavia, IL 60510, USA  2University of Chicago, Kavli Institute for
Cosmological Physics, Chicago IL, USA  3University of Chicago, Department of
Astronomy & Astrophysics, Chicago IL, USA

###### Abstract

Measurements of the primordial element abundances provide us with an important
probe of our universe’s early thermal history, allowing us to constrain the
expansion rate and composition of our universe as early as $\sim 1\,{\rm s}$
after the Big Bang. Prior to this time, we have essentially no empirical
information on which to base any such claims. In this paper, we imagine a
future time in which we have not only detected the particles that make up the
dark matter, but have measured their mass and annihilation cross section with
reasonable precision. In analogy to the light element abundances, the dark
matter abundance in this scenario could be used to study and constrain the
expansion rate and composition of our universe at the time of dark matter
freeze out, which for a standard thermal relic occurs at $T_{f}\sim
20/m_{\chi}$, corresponding to $t\sim 4\times 10^{-10}\,{\rm s}\times({\rm
TeV}/m_{\chi})^{2}$, many orders of magnitude prior to the onset of Big Bang
nucleosynthesis. As examples, we consider how such measurements could be used
to constrain scenarios which feature exotic forms of radiation or matter, a
ultralight scalar, or modifications to gravity, each of which have the
potential to be much more powerfully probed with dark matter than with the
light element abundances.

††preprint: FERMILAB-PUB-23-291-T

##  I Introduction

Observations of the cosmic microwave background, large scale structure, and
the rate of Hubble expansion have provided us with a detailed description of
our universe’s history, spanning from the time of recombination to the
present. To study and constrain our universe prior to recombination, we rely
almost entirely on measurements of the primordial element abundances [1, 2, 3,
4, 5]. These measurements, which are in good agreement with the predictions of
Big Bang nucleosynthesis (BBN), provide us with our earliest glimpse into our
universe’s thermal history (for a summary, see the Particle Data Group’s
review of BBN [6]). In particular, this data confirms that our universe was
radiation dominated and generally well described by the standard $\Lambda$CDM
cosmological model during and after the era of primordial nucleosynthesis [7,
8, 9, 10, 11, 12]. Furthermore, these measurements allow us to significantly
constrain the composition of our universe and the laws that governed its
evolution as early as the time of proton-neutron freeze out, $\sim 1\,{\rm s}$
after the Big Bang, when the temperature was $\sim 1\,{\rm MeV}$ [13, 14, 15,
16, 17, 18, 19, 20, 21, 22, 23, 24, 25].

Despite everything we have learned from the primordial element abundances, we
have essentially no empirical information to constrain the thermal history of
our universe prior to BBN. Although we can use the Large Hadron Collider (LHC)
and other accelerators to study the interactions of Standard Model particles
at energies which correspond to temperatures as high as $\sim 1\,{\rm TeV}$
($\sim 10^{-12}\,{\rm s}$), new physics that is not significantly coupled to
Standard Model fields remains largely unconstrained by laboratory experiments.
It thus remains possible that new physics may have played an important role in
the evolution of our early universe prior to the onset of BBN. As a few
examples, our early universe may have included an era that was dominated by
exotic matter or radiation [26, 27, 28, 29, 30, 31, 32, 33, 34], or in which
the masses and/or couplings of particles were influenced by an evolving scalar
field [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46]. Unknown phase
transitions may have occurred at very early times [47, 48, 49, 50, 51, 52],
and modifications to general relativity [53, 54, 55, 56, 57, 58, 59, 60, 61]
could have lead to a wide range of exotic dynamics.

In this paper, we imagine a point in the future when the particle species that
makes up the dark matter has been identified and its characteristics have been
broadly measured. If the dark matter is a thermal relic of the early universe,
measurements of its annihilation cross section could be used to estimate the
relic abundance that is predicted for this state (for reviews, see Refs. [62,
63, 64]). This is in close analogy to how our knowledge of various nuclear
reaction rates has been used to predict the abundances of the primordial
elements. Carrying this analogy further, just as the measurements of the light
element abundances have been used to constrain the composition and evolution
of our universe since the time of proton-neutron freeze out, $t\sim 1\,{\rm
s}$, measurements of the dark matter abundance and the characteristics of this
particle species could be used to constrain how our universe expanded and
evolved since the time of dark matter’s thermal freeze out. Dark matter could
thus serve as a probe of our universe’s very early thermal history, allowing
us to constrain times as early as $t\sim 4\times 10^{-10}\,{\rm s}\times({\rm
TeV}/m_{\chi})^{2}$ after the Big Bang, many orders of magnitude prior to the
onset of BBN.

##  II BBN As A Probe of New Physics: A Mini-Review

Historically speaking, the light element abundances provided early and
critical support for the Big Bang theory over steady-state cosmological models
[65, 66, 67, 68, 69]. In more recent decades, such measurements have served as
an important probe of our universe’s early thermal history. In this section,
we briefly review the physics of BBN, and how measurements of the primordial
element abundances have been used to constrain new physics.

Prior to BBN, neutrons and protons were maintained in chemical equilibrium
through the processes $n+\nu_{e}\leftrightarrow p+e^{-}$ and
$n+e^{+}\leftrightarrow p+\bar{\nu}_{e}$. In this equilibrium state, the
neutron-to-baryon ratio, $X_{n}$, was given by

$X_{n}\equiv\frac{n_{n}}{n_{b}}=\frac{1}{1+e^{(m_{n}-m_{p})/T}},$ (1)

where $m_{n}-m_{p}\approx 1.29$ MeV is the neutron-proton mass splitting. The
evolution of the neutron abundance is governed by the following Boltzmann
equation:

$\frac{dX_{n}}{dt}=-\lambda_{np}(1+e^{-(m_{n}-m_{p})/T})(X_{n}-X_{n}^{\rm
eq}),$ (2)

where $\lambda_{np}$ is the neutron-proton conversion rate. In the absence of
new physics, the neutron-to-baryon ratio freezes out and becomes nearly
constant (up to the effects of neutron decay) at a temperature of $T\sim
1\,{\rm MeV}$, after which $X_{n}\approx 0.157$. The production of deuterium
and other light elements starts around $T\sim 0.07$ MeV, by which time neutron
decay has reduced this ratio to $X_{n,\rm BBN}=0.157\,\,e^{-t_{\rm
BBN}/\tau_{n}}\approx 0.122$, where $\tau_{n}\approx 880.2\,{\rm s}$. As the
deuteron abundance increased, other nuclear reactions became possible,
ultimately causing most of the neutrons to become bound within
$\rm\prescript{4}{}{H}e$ nuclei, which has the highest binding energy per
nucleon among all isotopes lighter than carbon. As a consequence, the
resulting helium mass fraction, $Y_{p}$, is approximately equal to half the
neutron abundance at the onset of primordial nucleosynthesis:

$Y_{p}\equiv\frac{4n_{\rm He}}{n_{b}}\approx 2X_{n}\approx 0.245.$ (3)

###  II.1 Exotic Matter or Radiation

The measured abundances of primordial helium and deuterium are in good
agreement with the predictions of standard BBN. Since these predicted
abundances are quite sensitive to the rate of Hubble expansion (and thus to
the total energy density) at the time of proton-neutron freeze out, these
measurements allow us to place significant constraints on the densities of
matter and other energy that were present $\sim 1\,{\rm s}$ after the Big
Bang.

According to the first Friedmann equation, the rate of Hubble expansion is
proportional to the square root of the total energy density (in the absence of
significant curvature). As a result, the presence of any exotic matter or
radiation will increase the expansion rate, causing the neutron-to-baryon
ratio to freeze out of equilibrium earlier, when the equilibrium value of this
ratio was higher. Although increasing the rate of expansion also suppresses
the effectiveness of subsequent fusion processes, this turns out to be a
secondary effect.

In the case of exotic radiation, it is conventional to express the
contribution to the energy density in terms of the effective number of
neutrino species, $\Delta N_{\rm eff}=N_{\rm eff}-3.046$, and the energy
density in photons:

$\displaystyle\rho_{\rm exotic}=\Delta N_{\rm
eff}\times\frac{7}{8}\bigg{(}\frac{4}{11}\bigg{)}^{4/3}\rho_{\gamma}.$ (4)

The ultimate impact of modifying the energy density in exotic radiation is to
change the helium mass fraction by $\Delta Y_{p}\approx 0.013\times\Delta
N_{\rm eff}$, and the relative deuterium abundance by $\Delta(^{2}{\rm H}/{\rm
H})_{p}\approx(0.325\times 10^{-5})\times\Delta N_{\rm eff}$. After performing
a fit to the measured helium and deuterium abundances, allowing both $N_{\rm
eff}$ and the baryon abundance to float, one obtains a constraint on
$2.3<N_{\rm eff}<3.4$ (at the 95% confidence level), consistent with the
Standard Model prediction of 3.046 [10]. In other words, the density of exotic
radiation at the time of proton-neutron freeze out was no more than $\sim
10\%$ of that in photons. Similar constraints have been derived on the density
of exotic matter at the time of proton-neutron freeze out (for an example, see
Ref. [13]).

###  II.2 Varying Masses and Couplings

If any of the masses or couplings that are relevant to the processes of BBN
change with time or environment, this could impact the resulting primordial
element abundances. In particular, the temperature of proton-neutron freeze
out is highly sensitive to the neutron-proton mass splitting, $m_{n}-m_{p}$.
Furthermore, the rates for the various relevant fusion processes can be
affected by the binding energies of deuterium and the other light elements.
These considerations allow us to place a tight constraint of
$|\Delta(m_{n}-m_{p})|/(m_{n}-m_{p})\lesssim 0.02$, as evaluated at the time
of proton-neutron freeze out. Similarly, this information can be used to
significantly constrain any change in the deuteron’s binding energy, $|\Delta
E_{d}|/E_{d}\lesssim 0.1$ [70, 71, 72, 73].

###  II.3 Energy Injection

In many new physics scenarios, energetic particles can be injected into the
universe during or after the era of BBN, potentially impacting the primordial
element abundances. For example, exotic particles could have been decaying
during our universe’s first minutes, creating high-energy photons and other
particles that could be capable of disassociating nuclei.

An energetic photon can disassociate a nucleus in the early universe only if
its energy exceeds the binding energy of its target and it is able to reach
its target before being absorbed through the process of pair production.
Consider, for example, a photon with just enough energy to disassociate a
helium-4 nucleus, $E_{\gamma}=28.3\,{\rm MeV}$. At temperatures greater than a
fraction of a keV, such photons will be efficiently absorbed through pair
production, strongly limiting their ability to disassociate nuclei [74]. At
higher temperatures, nuclei can still be broken up by energetic pions, but not
efficiently by photons.

Once the temperature has dropped enough to make photodisassociation
potentially effective, any energetic photons that are injected into the bath
can break up helium-4 nuclei. This process both reduces the abundance of
helium, and also generates additional deuterium. Roughly speaking, such
scenarios can be ruled out if the number of helium-disassociating photons
exceeds the uncertainty on the measured number of primordial deuterons.

As an example, consider an exotic particle species with a lifetime of $\sim
10^{8}\,{\rm s}$ which produces one 30 MeV photon in each of its decays. If we
assume that each of these photons produces one deuteron through the
disassociation of a helium nucleus, then this would lead to a change in the
deuteron abundance that is given by $\Delta(^{2}{\rm H}/{\rm
H})\sim(\rho_{X}/\rho_{b})(m_{p}/m_{X})$, where $\rho_{X}$ and $m_{X}$ are the
density and mass of the decaying particle. If
$\rho_{X}/\rho_{b}\mathrel{\raise 1.29167pt\hbox{$>$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}3\times 10^{-6}\times(m_{X}/100\,{\rm MeV})$, this
would produce enough additional deuterium to increase its primordial abundance
beyond the measured range, allowing us to place strong constraints on particle
species that were decaying or otherwise injecting energy into the universe
during or shortly after the era of BBN [13, 15, 16, 17, 18, 20, 24].

###  II.4 Constraints on Dark Matter

If the dark matter was in equilibrium with the Standard Model bath in the
early universe, the abundance of that species will freeze out when the rate of
Hubble expansion exceeds that of dark matter annihilation. For a dark matter
candidate that freezes out with an abundance that is equal to the measured
density of dark matter, this freeze out occurs at a temperature approximately
given by $T_{f}\sim m_{\chi}/20$. If the dark matter freezes out of
equilibrium after proton-neutron converting processes do, the energy density
of the dark matter and its annihilation products will significantly contribute
to the total energy density, increasing the rate of Hubble expansion and the
resulting abundance of neutrons, ultimately leading to larger abundances of
primordial helium and deuterium. These considerations allow us to rule out
thermal relics as dark matter candidates if they are lighter than
$m_{\chi}\sim 3-10\,{\rm MeV}$, depending on the number of their internal
degrees-of-freedom, and on whether they annihilate into electrons/photons or
into neutrinos and other invisible radiation [14, 23, 75, 76].

##  III The Abundance of a Thermal Relic

Before turning our attention to examples of new physics scenarios that we hope
to constrain using measurements of the dark matter and its abundance, we will
briefly review the process of dark matter freeze out and the calculation of
dark matter’s thermal relic abundance.

For concreteness, we will consider here a stable dark matter candidate,
$\chi$, of mass $m_{\chi}$, which we take to be a Majorana fermion. The
evolution of the number density of this species in the early universe is
governed by the following Boltzmann equation:

$\frac{dn_{\chi}}{dt}+3Hn_{\chi}=\langle\sigma v\rangle(n_{\chi,{\rm
eq}}^{2}-n_{\chi}^{2}),$ (5)

where $H$ is the rate of Hubble expansion and $\langle\sigma v\rangle$ is the
dark matter’s thermally averaged annihilation cross section. When the $\chi$
population is non-relativistic, the equilibrium abundance of these particles
is given by $n_{\chi,{\rm eq}}=g_{\chi}(m_{\chi}T/2\pi)^{3/2}{\rm
exp}(-m_{\chi}/T)$, where $g_{\chi}=2$ for a Majorana fermion.

The numerical solution to this equation yields a current density that is given
by [77]

$\displaystyle\rho_{\chi}\approx\rho_{\rm DM}\times\bigg{(}\frac{2.2\times
10^{-26}\,{\rm cm}^{3}/{\rm s}}{\langle\sigma
v\rangle}\bigg{)}\,\bigg{(}\frac{80}{g_{\star}(T_{f})}\bigg{)}^{1/2}\,\bigg{(}\frac{x_{f}}{26}\bigg{)},$
(6)

where $\rho_{\rm DM}$ is the measured dark matter density, $g_{\rm
star}(T_{f})$ is the number of relativistic degrees-of-freedom at the
temperature of freeze out, and $x_{f}\equiv m_{\chi}/T_{f}$. This latter
quantity is given by

$\displaystyle x_{f}\approx 26+\ln\bigg{[}$
$\displaystyle\bigg{(}\frac{2.2\times 10^{-26}\,{\rm cm}^{3}/{\rm
s}}{\langle\sigma
v\rangle}\bigg{)}\,\bigg{(}\frac{80}{g_{\star}(T_{f})}\bigg{)}^{1/2}\,\bigg{(}\frac{g_{\chi}}{2}\bigg{)}$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\bigg{(}\frac{x_{f}}{26}\bigg{)}^{3/2}\,\bigg{(}\frac{T_{f}}{10\,{\rm
GeV}}\bigg{)}\bigg{]}.$ (7)

In the case of non-relativistic dark matter particles, the thermally averaged
annihilation cross section that appears in Eqs. 6 and 7 can be expressed as
follows:

$\displaystyle\langle\sigma v\rangle$ $\displaystyle\approx\frac{\int\sigma
v\,v^{2}\,e^{-xv^{2}/4}\,dv}{\int v^{2}\,e^{-xv^{2}/4}\,dv}$ (8)
$\displaystyle=\frac{x^{3/2}}{2\pi^{1/2}}\int\sigma
v\,v^{2}\,e^{-xv^{2}/4}\,dv,$

where $\sigma v$ is the dark matter’s (non-thermally averaged) annihilation
cross section, and $x\equiv T/m_{\chi}$.

##  IV Exotic Matter and Radiation

Figure 1: The fraction of the total energy density that consists of Standard
Model radiation, dark matter, and an exotic species of unstable matter. In
this example, we have taken the exotic matter to be a scalar with 1 degree of
freedom, a mass of $m_{\Phi}=865\,{\rm TeV}$, and a lifetime of
$\Gamma_{\Phi}^{-1}=2.2\times 10^{-6}\,{\rm s}$. We further assume that it
decoupled from the Standard Model bath at a high temperature. Once this
species becomes non-relativistic, it quickly comes to dominate the energy
density of the universe. It then decays, after the dark matter has frozen out
of equilibrium, reheating the universe to a temperature of $T=0.3\,{\rm GeV}$.
The dark matter, in this example, is taken to be a Majorana fermion with a
mass of $m_{\chi}=30\,{\rm GeV}$ and an annihilation cross section of
$\langle\sigma v\rangle=2.2\times 10^{-26}\,{\rm cm}^{3}/{\rm s}$.

As a first example, we will consider those scenarios in which exotic forms of
matter or radiation were present during or after the era of dark matter freeze
out, impacting the expansion history and the resulting thermal relic abundance
of dark matter.

In the case of exotic radiation, the energy density can be expressed in terms
of its contribution to the effective number of relativistic degrees-of-
freedom, $g_{\star}(T)$. The total energy density is related to $g_{\star}(T)$
as follows:

$\displaystyle\rho=\frac{\pi^{2}g_{\star}(T)T^{4}}{30},$ (9)

where $T$ is the temperature of the photons in the bath. In kinetic and
chemical equilibrium, a relativistic ($m\ll T$) fermionic (bosonic) particle
species contributes 7/8 (1) to $g_{\star}$ for each of its internal degrees-
of-freedom.

Figure 2: The evolution of the dark matter number density to entropy ratio as
a function of temperature for a dark matter candidate that is a Majorana
fermion with a mass of $m_{\chi}=30\,{\rm GeV}$ and an annihilation cross
section of $\langle\sigma v\rangle=2.2\times 10^{-26}\,{\rm cm}^{3}/{\rm s}$.
The blue curve represents the case of standard thermal freeze out, while the
orange and red curves illustrate the results that are obtained in scenarios in
which there was an era in the early universe that was dominated by exotic
matter, resulting in the reheating of the universe to a temperature of either
0.5 or 0.3 GeV.

From Eq. 6, it can be shown that (up to logarithmic corrections to $x_{f}$;
see Eq. 7) the dark matter abundance scales with $g_{\star}^{-1/2}$, as
evaluated at the temperature of freeze out. Thus, if we were to measure the
dark matter’s annihilation cross section to a precision of
$p\equiv\Delta\langle\sigma v\rangle/\langle\sigma v\rangle$, we could
constrain the value of $g_{\star}(T_{f})$ to lie within a range of $\Delta
g_{\star}/g_{\star}\sim 2p$. For an example in which $T_{f}=1\,{\rm GeV}$
(corresponding to $g_{\star}\approx 72$), a cross section measurement with a
precision of $p=0.2$ would be sensitive to the presence of 30 (33) bosonic
(fermionic) exotic relativistic degrees-of-freedom in equilibrium with the
thermal bath.

Next, we will consider the case of an exotic species of matter, $\Phi$, that
was present in the early universe. If such particles are long-lived and were
not in equilibrium in the early universe, their abundance will evolve as
$\rho_{\Phi}\propto a^{-3}$, while the energy density in radiation drops as
$\rho_{\rm rad}\propto a^{-4}\,g_{\star}/g^{4/3}_{\star,S}$, where
$g_{\star,S}$ is the number of relativistic degrees-of-freedom in entropy. As
a result of this evolution, the decoupled matter will make up an increasingly
large fraction of the total energy density as the universe expands,
potentially leading to an era of early matter domination. When the exotic
species eventually decays, it will heat the Standard Model bath. If the
temperature after those decays is greater than a few MeV, BBN will proceed as
usual, making any such scenario consistent with the measurements of the light
element abundances.

To calculate the dark matter’s thermal relic abundance in this class of
scenarios, we will consider the following system of Boltzmann equations:

$\begin{split}\frac{d\rho_{\Phi}}{dt}&+3H\rho_{\Phi}=-\Gamma\rho_{\Phi},\\\
\frac{d\rho_{\rm rad}}{dt}&+4H\rho_{\rm rad}=\Gamma\rho_{\Phi},\\\
\frac{dn_{\chi}}{dt}&+3Hn_{\chi}=\langle\sigma v\rangle(n_{\chi,{\rm
eq}}^{2}-n_{\chi}^{2}),\end{split}$ (10)

where $\Gamma$ is the decay width of $\Phi$. Note that in calculating the
Hubble rate, $H$, we must take into account the total energy density,
including that of the $\Phi$ population.

Figure 3: The potential of a future measurement of the dark matter’s
annihilation cross section to constrain early matter dominated scenarios. The
shaded regions illustrate the regions of the $m_{\Phi}-T_{\rm rh}$ plane that
could be excluded by a measurement of $\langle\sigma v\rangle$ to a precision
of either 50% or 20%. Such a measurement could be used to rule out scenarios
which feature a long matter dominated era which ends after dark matter freeze
out. In the left (right) frame, we take the mass of the dark matter particle
to be $m_{\chi}=$30 GeV (300 GeV). Along the right axis of each frame, we show
the fraction of the energy density in exotic matter, as evaluated at
$T=m_{\chi}$.

In Fig. 1, we show an example of an early matter dominated scenario, including
a heavy scalar with $g_{\Phi}=1$ degrees of freedom, a mass of
$m_{\Phi}=865\,{\rm TeV}$, a lifetime of $\Gamma_{\Phi}^{-1}=4.5\times
10^{-6}\,{\rm s}$, and that decoupled from the Standard Model bath at a high
temperature. Once this species becomes non-relativistic, it quickly comes to
dominate the energy density of the universe. In decaying after dark matter
freeze out, the exotic particles reheat the universe to a temperature of
$T=0.3\,{\rm GeV}$.

Note that so long as the exotic matter dominated the energy density prior to
its decay, the reheating temperature is directly related to the decaying
particle’s width, $\Gamma$. This can estimated by treating the decays as
instantaneous (occuring at $t=\Gamma^{-1}$), and equating the energy density
of the universe before the decays,

$\displaystyle\rho_{\Phi}=\frac{M^{2}_{\rm Pl}}{6\pi t^{2}}=\frac{M^{2}_{\rm
Pl}\Gamma^{2}}{6\pi},$ (11)

to the energy density of the radiation bath afterwards,

$\displaystyle\rho_{\rm rad}=\frac{\pi^{2}g_{\star}(T_{\rm rh})T_{\rm
rh}^{4}}{30},$ (12)

yielding the following temperature of reheating:

$\displaystyle T_{\rm rh}=\bigg{(}\frac{5\Gamma^{2}M^{2}_{\rm
Pl}}{\pi^{3}g_{\star}(T_{\rm rh})}\bigg{)}^{1/4}.$ (13)

In Fig. 2, we show an example of how dark matter freeze out could be impacted
by the presence of long-lived exotic matter that was out-of-equilibrium in the
early universe. In this case, the dark matter is taken to be a 30 GeV Majorana
fermion with an annihilation cross section of $\langle\sigma
v\rangle=2.2\times 10^{-26}\,{\rm cm}^{3}{\rm s}^{-1}$. In the cases shown,
the exotic matter impacts the resulting dark matter abundance in two distinct
ways. First, the density of the exotic matter increases the total energy
density at the time of freeze out, causing the dark matter to decouple earlier
and increasing the resulting relic abundance. Second, the decays of the exotic
matter increase the energy density of Standard Model radiation, diluting the
abundance of dark matter. This will be the case so long as the thermal relic
in question freezes out of equilibrium before the end of the early matter
dominated era. For lower values of $T_{\rm rh}$, the matter dominated era
lasts longer, allowing the exotic matter to make up a larger fraction of the
total energy density before decaying, and thus leading to a greater degree of
dilution and to a lower abundance of dark matter.

In Fig. 3, we illustrate how one could constrain scenarios featuring an early
matter dominated era by measuring the dark matter’s annihilation cross section
to a precision of either 50% or 20%. In each frame, we show the regions of the
$m_{\phi}-T_{\rm rh}$ plane that could be excluded by such a measurement.

##  V Ultralight scalars coupled to dark matter

Ultralight scalars have been motivated by a wide range of theoretical
considerations, and include examples such as the QCD axion [78] and string
theory moduli [79, 80]. Such particles can have large occupation numbers in
phase space, allowing them to be described as classical waves. With small
couplings to Standard Model fields, it can be challenging to detect these
particles in laboratory experiments. Although ultralight scalars with
universal couplings are constrained by fifth-force experiments [81], supernova
cooling [82], and black hole superradiance [83, 84], these constraints are
typically weaker than those derived from measurements of the primordial
element abundances [85].

Figure 4: The evolution of the dark matter number density to entropy ratio as
a function of temperature for a dark matter candidate that is a Majorana
fermion with a mass of $m_{\chi}=30\,{\rm GeV}$ and an annihilation cross
section of $\langle\sigma v\rangle=2.2\times 10^{-26}\,{\rm cm}^{3}/{\rm s}$.
The blue curves represents the case of standard thermal freeze out, while the
red curve illustrates the results that are obtained in a scenario in which the
interactions of an ultralight scalar modifies the expansion rate during dark
matter freeze out. In this plot, we adopt $m_{\phi}=10^{-10}$ eV and
$\Lambda=10^{13.8}\,{\rm GeV}$, corresponding to
$\langle\phi\rangle^{2}/\Lambda^{2}\approx 0.1$. For these parameters, the
interactions of the ultralight scalar increase the resulting dark matter’s
relic abundance by a factor of 2.7.

In this section, we will focus on how an ultralight scalar coupled to dark
matter could impact the expansion rate during the time of dark matter freeze
out. In this context, we will consider a model that is described by the
following Lagrangian:

$\mathcal{L}\supset\frac{1}{2}(\partial_{\mu}\phi)^{2}-\frac{1}{2}m_{\phi}^{2}\phi^{2}-\frac{\phi}{\Lambda}m_{\chi}\bar{\chi}\chi,$
(14)

where $\chi$ is the fermionic dark matter candidate, $\phi$ is the ultralight
scalar, and $\Lambda$ is a free parameter. The ultralight scalar, $\phi$, will
have a vacuum expectation value (VEV) in the early universe,
$\langle\phi\rangle$, and constraints from the light element abundances
require that this VEV should not be too large during the BBN era,
$\langle\phi\rangle^{2}/\Lambda^{2}\lesssim 0.1$ [85, 86]. At earlier times,
however, the value of $\langle\phi\rangle$ may have been much larger. The era
of dark matter freeze out thus provides us with an powerful opportunity to
probe the existence of ultralight scalars.

The evolution of $\phi$ is governed by the following differential equation:

$\ddot{\phi}+3H\dot{\phi}+m_{\phi,\rm eff}^{2}\,\phi=0.$ (15)

The effective scalar mass, $m_{\phi,\rm eff}$, receives contributions from any
non-relativistic particle species that it interacts with. At early times,
however, Standard Model particles disappear from the thermal bath as they
become non-relativistic, and thus do not significantly contribute to the value
of $m_{\phi,\rm eff}$. The dark matter, in contrast, is stable, and thus can
impact $m_{\phi,\rm eff}$ more substantially:

$m_{\phi,\rm eff}^{2}=m_{\phi}^{2}+\frac{2\Theta_{\chi}}{\Lambda^{2}},$ (16)

where

$\displaystyle\Theta_{\chi}$ $\displaystyle=\rho_{\chi}-3p_{\chi}$ (17)
$\displaystyle=2\frac{m_{\chi}^{2}T^{2}}{2\pi^{2}}\int_{m_{\chi}/T}^{\infty}dx\frac{\sqrt{x^{2}-m_{\chi}^{2}/T^{2}}}{e^{x}+1}.$

Figure 5: Regions of the $m_{\phi}$, $\Lambda^{-1}$ plane in which
$\langle\phi\rangle^{2}/\Lambda^{2}>0.1$ at the time of proton-neutron freeze
out (labeled “BBN”) [85], or at the time of dark matter freeze out (labeled
“DM”). The region above the blue curve is excluded by BBN. We have taken the
dark matter candidate to be a Majorana fermion with a mass of
$m_{\chi}=30\,{\rm GeV}$ and an annihilation cross section of $\langle\sigma
v\rangle=2.2\times 10^{-26}\,{\rm cm}^{3}/{\rm s}$. From this comparison, it
is clear that dark matter could potentially be used to probe large regions of
parameter space that is inaccessible to constraints based on the primordial
element abundances.

From Eq. 15, it follows that the field will start to oscillate when
$m_{\phi,\rm eff}\sim H$, after which it behaves as a population of non-
relativistic particles, $\rho_{\phi}\propto a^{-3}$. Prior to this transition,
the value of $\phi$ remained approximately fixed at its initial value,
constituting a large contribution to the overall energy density. Writing the
number density and energy density of the scalars as
$n_{\phi}=m_{\phi}\phi^{2}/2$ and $\rho_{\phi}=n_{\phi}m_{\phi,{\rm
eff}}=m_{\phi}m_{\phi,{\rm eff}}\phi^{2}/2$, we can express the value of
$\langle\phi\rangle^{2}/\Lambda^{2}$ at the time of dark matter freeze out as
follows:

$\frac{\langle\phi\rangle^{2}}{\Lambda^{2}}\approx\frac{2\rho_{\phi,0}}{\Lambda^{2}m_{\phi}m_{\phi,\rm
eff}}{\rm min}\left(a_{\rm fo}^{-3},a_{\rm crit}^{-3}\right),$ (18)

where $\rho_{\phi,0}$ is the energy density of ultralight scalars in the
current universe. For concreteness, we will take $\rho_{\phi,0}$ to be 1% of
the total dark matter density. The quantities $a_{\rm fo}$ and $a_{\rm crit}$
are the scale factors evaluated at the time of dark matter freeze out and when
$m_{\phi,\rm eff}=H$, respectively.

Although the value of $\langle\phi\rangle/\Lambda$ can impact a variety of
effective couplings, we will focus here on its impact on the Hubble rate
through the value of $M_{\rm Pl}$. The effective Planck mass is modified
through the interactions of the ultralight scalar as $M^{2}_{\rm
Pl,\,eff}\approx M^{2}_{\rm Pl}\,(1-2\langle\phi\rangle^{2}/\Lambda^{2})$,
where $M_{\rm Pl}\approx 1.22\times 10^{19}\,{\rm GeV}$ is the standard value
of the Planck mass.

By decreasing the value of the effective Plank mass, the interactions of the
ultralight scalar increase the Hubble rate, and thus cause dark matter to
freeze out earlier than it would have otherwise. An example of this is shown
in Fig. 4, for the case of $m_{\phi}=10^{-10}$ eV and $\Lambda=10^{13.8}\,{\rm
GeV}$, corresponding to $\langle\phi\rangle^{2}/\Lambda^{2}\approx 0.1$ at the
time of dark matter freeze out. As in the previous section, we have considered
here a dark matter candidate that is a Majorana fermion with a mass of
$m_{\chi}=30\,{\rm GeV}$ and an annihilation cross section of $\langle\sigma
v\rangle=2.2\times 10^{-26}\,{\rm cm}^{3}/{\rm s}$. For this choice of
parameters, the interactions of the ultralight scalar increase the dark
matter’s relic abundance by a factor of 2.7.

In Fig. 5, we show the regions of the $m_{\phi}$, $\Lambda^{-1}$ plane in
which the value of $\langle\phi\rangle^{2}/\Lambda^{2}$ is greater than 0.1 at
the time of proton-neutron freeze out [85], and at the time of dark matter
freeze out. From this, it is clear that dark matter could potentially be used
to probe large regions of this parameter space that is inaccessible to
constraints based on BBN.

##  VI Modified Gravity Scenarios

Modifications to general relativity have been motivated by many
considerations, including those of dark energy and dark matter, as well as by
challenges encountered in efforts to renormalize gravity [60]. In some classes
of modified gravity theories, the effects are most significant in the early
universe, leading to strong constraints from BBN [87, 88, 89], and perhaps one
day from the freeze out of dark matter. In this section, we briefly consider a
few examples of such scenarios.

###  VI.1 $f(G)$ gravity

As a first example, we will consider a class of modified gravity models in
which the Einstein action is modified by the quantity, $f(G)$, which is a
function of the Gauss-Bonet invariant, $G$ [90]. The modified Einstein action
in such models is given by

$S=\int d^{4}x\,\sqrt{-g}\left[\frac{M^{2}_{\rm
Pl}}{128\pi^{2}}R+f(G)\right],$ (19)

where $g$ is the determinant of the metric tensor, and $R$ is the Ricci
scalar. The quantity, $G$, is given by [90]

$G=R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}.$ (20)

The new $f(G)$ term leads to a modification of gravity which takes the form of
a new contribution to the energy density:

$\begin{split}\rho_{f(G)}=&\frac{1}{2}\Bigl{[}-f(G)+24H^{2}(H^{2}+\dot{H})f^{\prime}(G)\\\
&-24^{2}H^{4}(2\dot{H}^{2}+H\ddot{H}+4H^{2}\dot{H})f^{\prime\prime}(G)\Bigl{]}.\end{split}$
(21)

Adopting the following form $f(G)=\alpha G^{n}$, where $n\neq 1$, one finds
that this contribution to the energy density scales as follows:

$\frac{\rho_{f(G)}}{\rho_{\rm SM}}\propto\alpha\,T^{8n-4}.$ (22)

Thus, if $n>1/2$, the exotic component of the energy density will be most
significant at the highest temperatures and earliest times, making dark matter
freeze out a powerful probe of this class of scenarios.

###  VI.2 Running Vacuum Model

Models in which gravitational anomalies are coupled to axion fields [91] can
lead to time-dependent contributions to the energy density. The exotic energy
density in such “running-vacuum” models can be expressed as [92, 93]

$\rho_{\rm RVM}=a+bH^{2}+cH^{4}+...$ (23)

where the $a$ term corresponds to the cosmological constant, and the $b$ and
$c$ terms represent deviations from the $\Lambda{\rm CDM}$ paradigm. In the
early universe, the $cH^{4}$ term will generally be most significant, such
that

$\frac{\rho_{\rm RVM}}{\rho_{\rm SM}}\propto cH^{2}\propto cT^{4}.$ (24)

One again, this implies that probes of our universe’s very early thermal
history could be used to place strong constraints on such scenarios.

In addition to those we have considered here, there are many other well-
motivated theories of modified gravity which share these general features,
including $f(P)$ gravity [94], and Gauss-Bonnet-Dilaton Gravity [95]. In
scenarios in which the effects of new physics are most pronounced at high
temperatures, we could expect dark matter freeze out to potentially provide
very strong constraints.

##  VII Discussion and Summary

In this paper, we have imagined a time in the future when we have not only
detected the particle that makes up the dark matter of our universe, but have
measured its mass and annihilation cross section with a reasonably high degree
of precision. Combining this information with the measured dark matter
abundance, one could probe the expansion rate of our universe at the time of
dark matter’s decoupling. This is in close analogy to how the light element
abundances have been used to probe the era of of Big Bang nucleosynthsis, as
early as $t\sim 1\,{\rm s}$ after the Big Bang. For dark matter in the form of
a generic thermal relic, freeze out occurs at a temperature of $T_{f}\sim
m_{\chi}/20$, corresponding to $t\sim 4\times 10^{-10}\,{\rm s}\times({\rm
TeV}/m_{\chi})^{2}$. This process could thus potentially be used to probe the
thermal history of our universe at much earlier times than BBN.

One could reasonably ask how realistic it is that we could measure the dark
matter’s annihilation cross section with enough precision to meaningfully
probe the thermal history of the early universe. As an illustrative example,
consider a scenario motivated by the observed Galactic Center Gamma-Ray Excess
[96, 97, 98, 99]. This signal is consistent with arising from annihilating
dark matter particles with a mass of $m_{\chi}\sim 40-50\,{\rm GeV}$ and an
annihilation cross section consistent with that expected of a thermal relic,
$\langle\sigma v\rangle\sim(1-2)\times 10^{-26}\,{\rm cm}^{3}/{\rm s}$ [100].
In the spirit of this study, we consider a time when a large space-based
gamma-ray telescope (such as the proposed Advanced Particle-astrophysics
Telescope [101]) has confirmed a dark matter interpretation of of the Galactic
Center Gamma-Ray Excess by measuring the corresponding signal from several
dwarf spherioidal galaxies. Such a measurement would ultimately be limited to
our ability to determine the $J$-factors of these dwarf galaxies, the best of
which are currently measured to a precision of around 30% (see, for example,
Ref. [102]). By combining the observations of several such dwarf galaxies, it
seems plausible that we could determine dark matter’s low-velocity
annihilation cross section to a precision on the order of tens of percents.

The finite temperature at the time of dark matter freeze out somewhat
complicates this story, as the annihilation cross section as evaluated at
$T_{f}$ will be different in some models than it is in the modern, low
temperature universe. For example, many dark matter models feature both
$s$-wave and $p$-wave contributions to the annihilation cross section,
allowing us to write $\langle\sigma v\rangle\approx a+bv^{2}.$ If the $p$-wave
term is not negligible at the time of dark matter freeze out, the relic
abundance we infer from the $s$-wave term alone will not match the measured
abundance. Fortunately, the results of direct detection experiments could be
helpful in clarifying such a situation. From the combination of the
information that could be provided by indirect and detection experiments, it
is plausible that the spin and Lorentz structure of the dark matter and its
interactions could be reasonably established. Among dark matter candidates
that are capable of generating the Galactic Center excess, for example, one
could consider a scalar Higgs portal model [103, 104], or dark matter that is
a fermion with mixed vector and axial-vector couplings [105]. While these
models would produce potentially indistinguishable gamma-ray signals, they
would lead to very different rates at future direct detection experiments,
enabling us to identify the correct underlying model and to correctly infer
the $p$-wave contribution to the dark matter’s annihilation cross section at
the time of its freeze out.

Once these characteristics of the particles that make up the dark matter have
been established, a vast range of new physics scenarios could be effectively
constrained and studied. In this paper, we have considered scenarios featuring
exotic forms of radiation or matter, models with a ultralight scalar, and
several theories of modified gravity, each of which have the potential to be
far more powerfully probed with dark matter than by other means, including
BBN. This list is by no means intended to be inclusive, but rather reflects
the tip of the iceberg of what we could learn from such a series of
measurements. For decades, our measurements of the light element abundances
have given us confidence that our universe was radiation dominated and
otherwise adhered to the predictions of the $\Lambda$CDM model since $\sim
1\,{\rm s}$ after the Big Bang. Prior to this time, we have no empirical
information with which to ground any such claims. In a future in which dark
matter’s nature has been adequately measured, we could extend this kind of
understanding back in time by many orders of magnitude, mapping out our
universe’s thermal history as early $\sim 4\times 10^{-10}\,\times\,({\rm
TeV}/m_{\chi})^{2}\,{\rm s}$ after the Big Bang.

## Acknowledgements

We would like to thank Gordan Krnjaic, Lian-Tao Wang, and Xucheng Gan for
helpful discussions. DH and HX are supported by Fermi Research Alliance, LLC
under Contract DE-AC02-07CH11359 with the U.S. Department of Energy.

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#  Quantum Metric in Step Response

Nishchhal Verma  Department of Physics, Columbia University, New York, NY
10027, USA  Raquel Queiroz<EMAIL_ADDRESS>Department of Physics,
Columbia University, New York, NY 10027, USA  Center for Computational Quantum
Physics, Flatiron Institute, New York, New York 10010, USA

###### Abstract

Quantum geometry of Bloch wavefunctions has gained considerable interest with
the discovery of moiré materials that exhibit bands flattened by quantum
interference. The quantum metric, the symmetric part of the quantum geometric
tensor, influences several observables, such as the dielectric constant,
superfluid stiffness and optical spectral weight. However, a direct
measurement of the metric itself has remained elusive so far. In linear
response functions such as the conductivity, the matrix elements of the metric
typically appear convoluted with energy prefactors, preventing finding an
observable that is directly proportional to the total quantum metric. The only
observable that may extract it is the integrated optical spectral weight
weighted by the inverse frequency, a generalized sum rule known as the Souza-
Wilkens-Martin (SWM) sum rule. However, the sum rule comes with experimental
challenges, such as requiring a large spectrum of frequency resolution. In
this work, we propose relaxation from constrained equilibrium as a method to
directly measure the symmetric part of the time-dependent quantum geometric
tensor (tQGT), which at $t=0$ is the quantum metric. Additionally, we comment
on other geometric properties of insulators that are absent in the frequency
expansions of conductivity in insulators but can, in principle, be revealed in
step response.

## Introduction.—

To first approximation, insulators are materials in which electrons are
tightly bound to the ions. These electrons do not form conducting channels and
only exhibit small fluctuations around the atom, leading to trivial temporal
dynamics. Topological insulators challenge this notion [1]. The electrons
exhibit fluctuations at the scale of the unit cell and yet lead to robust edge
phenomena with quantized transport, similar to Landau levels. Topology
obstructs the local picture of bound electrons in a topological insulator [2].

We can understand the zero-point motion of bound electrons as the generator of
quantum geometry in the electron wavefunction [3, 4]. Unlike topology, quantum
geometry is ubiquitous in materials and affects many ground state properties
and excitations. For instance, it affects the mass of bound states [5, 6],
electron-phonon coupling [7], and may even play an important role in the
search for high-temperature superconductors [8]. The discovery of moiré
materials has further brought quantum geometry into the spotlight [9]. The
flat bands emerge from the destructive interference of electronic paths over
the moiré length scale, leading to substantial zero-point motion and, hence,
large quantum geometry. While indirect effects of quantum geometry are
ubiquitous [10, 11], it is worth noting that the metric itself has never been
directly measured.

This work aims to find an experimental setup that can directly measure the
symmetric part of the quantum geometric tensor, that is, the quantum metric. A
hint towards the setup comes from the Souza-Wilkens-Martin sum rule [12] where
one can identify $\sigma(\omega)/\omega$ as a response to an electric field
that follows a $1/\omega$ frequency dependence. Alternatively, in the time
domain, it corresponds to a constant electric field which abruptly goes to
zero at $t=0$, that is $E(t)\propto\Theta(-t)$. In the following, we will show
that relaxation from constrained equilibrium provides a setting to obtain
quantum metric. Interestingly, step response also includes geometric
quantities that are otherwise absent in conductivity, such as the orbital
magnetic moment[13, 14].

We begin by reviewing the quantum geometric tensor (QGT) and quantum metric in
linear response theory. We then introduce the time-dependent QGT (tQGT) as a
unifying principle for the geometric properties of insulators via sum rules.
Next, we discuss linear response theory and the concept of step response. We
derive a generalized fluctuation-dissipation theorem (FDT) for the virtual
contribution to the dipole-dipole correlation. This derivation crucially
depends on the fermionic nature of the distribution function. The generalized
FDT relates symmetric and antisymmetric response functions at high
temperatures, providing a framework for studying both symmetric and
antisymmetric components of the tQGT.

## Quantum metric in Linear Response.—

As we discussed, the zero-point motion of electrons in insulators traces its
origin back to the fundamental relation between position and momentum in
quantum mechanics. It is well known that electric fields couple to the dipole
operator and lead to dipolar fluctuations in isolated atoms [15]. A
generalization to electrons in a lattice involves the virtual dipole-dipole
correlation function

$\mathcal{Q}_{\mu\nu}(t-t^{\prime})=\left\langle\hat{r}_{\mu}(t)\;\hat{Q}\;\hat{r}_{\nu}(t^{\prime})\right\rangle$
(1)

where $\hat{r}$ is the position operator and $\hat{Q}$ is the projector into
unoccupied states, and therefore its inclusion in (1) selects the _inter-band_
dipole transitions. The expectation value is taken in the ground state
$\langle\cdot\rangle={\rm Tr}[\hat{P}\cdot]$ where $\hat{P}=1-\hat{Q}$ is the
ground state projector. For Bloch electrons, the $t=0$ value corresponds to
the quantum geometric tensor with quantum metric $g_{\mu\nu}={\rm
Re}[\mathcal{Q}_{\mu\nu}(0)]$ and Berry curvature $\Omega_{\mu\nu}={\rm
Im}[\mathcal{Q}_{\mu\nu}(0)]/2$. The tensor in eq. (1) is therefore named the
time-dependent quantum geometric tensor (tQGT)[16, 3].

Introducing time dependence in Eq. (1) serves to unify several geometric
properties of materials in one formalism. Namely, it was shown that the
orbital magnetic moment [13], dielectric function [16], and optical mass [17,
18, 19] appear as various time derivatives of Eq. (1) and can be effectively
extracted via generalized sum rules of optical conductivity $\sigma(\omega)$
[3]. The reason behind this unification is that the anti-symmetric part of the
tQGT $\mathcal{Q}^{\rm
as}_{\mu\nu}(t)\\!=\\!\mathcal{Q}_{\mu\nu}(t)\\!-\\!\mathcal{Q}_{\mu\nu}(t)^{\dagger}$
can be extracted from linear response and it is related to the optical
conductivity by

$\sigma_{\mu\nu}(t)=\dfrac{\pi
e^{2}}{\hbar}\;\Theta(t)\;\partial_{t}\mathcal{Q}^{\rm as}_{\mu\nu}(t).$ (2)

The formalism outlines a consistent way to get relations between various sum
rules, which have proved useful for a variety of materials [16, 20, 21, 22].
Crucially, all geometric properties of insulators are not independent of each
other and are fixed by the definition of the projectors $\hat{P}$ and
$\hat{Q}$. This realization also permits a consistent interpolation between
_ab-initio_ and tight-binding methods, by a controlled truncation of the
Hilbert space.

Eq. (2) suggests that the Kubo formula for conductivity contains the matrix
elements for quantum geometry. It can be seen explicitly in the Fourier domain
when we expand in powers of frequency $\omega$,

$\sigma_{\mu\nu}(\omega)=\frac{e^{2}}{i\hbar}\sum\limits_{m\neq
n}f_{nm}\omega_{mn}\hat{r}^{nm}_{\mu}\hat{r}^{mn}_{\nu}\sum\limits_{p=0}^{\infty}\left[\dfrac{\omega}{\omega_{mn}}\right]^{p}$
(3)

where $f_{nm}=f_{n}-f_{m}$ and $f_{n}$ are the occupation factors,
$\hbar\omega_{mn}=E_{n}-E_{m}$ are the energy differences between states and
$\hat{r}_{nm}^{\mu}\hat{r}_{mn}^{\nu}$ are the matrix elements of the position
operator. The power series in $\omega/\omega_{mn}$ arises from the Taylor
expansion of $1/(\omega-\omega_{mn})$. The position matrix elements can be
split into symmetric and anti-symmetric combinations
$\hat{r}^{nm}_{\mu}\hat{r}^{mn}_{\nu}=g^{mn}_{\mu\nu}+i\Omega^{mn}_{\mu\nu}/2$
where $g^{mn}_{\mu\nu}$ and $\Omega^{mn}_{\mu\nu}$ are related to the quantum
metric and Berry curvature. It is instructive to notice that the anti-symmetry
of $f_{nm}$ and $\omega_{mn}$ forces the metric $g^{mn}_{\mu\nu}$ contribution
to vanish at zeroth order in frequency. More generally, the metric matrix
elements always appear in the conductivity with odd powers of $\omega_{mn}$
and, therefore, they always appear convoluted by energy prefactors in
nonresonant response. This fact makes it a challenge to find the total quantum
metric $g_{\mu\nu}=\sum_{m,n}f_{n}(1-f_{m})g_{\mu\nu}^{mn}$ directly in a
linear response function. Similarly, in non-linear response, matrix elements
of $g_{mn}^{\mu\nu}$ appear naturally in resonant responses such as in shift
and injection currents [23], but never in an integrated form in a non-resonant
response.

An exception can be made for Landau levels of free electrons. Here, the only
nonvanishing dipole matrix elements are between consecutive levels
$\hat{r}^{mn}\propto\delta_{m,n\pm 1}$ leading to a single transition
frequency $\omega_{c}=eB/m$, and therefore $\omega_{mn}$ in Eq. (3) can be
factored out of the sum. This results in non-resonant optical conductivity
that captures both the integrated quantum metric and Berry curvature to all
orders.

Correlations can also help factor out the energy prefactor. Superfluid
stiffness, which appears as the weight of the delta function in optical
conductivity, is given by the quantum metric times an interaction scale in a
flat band superconductor within mean-field theory [6]. While the result is
sensitive to competition with other correlated states [24] and applies for
exactly flat bands in certain lattices, it raises several important questions
about geometric contributions to the effective mass of the Cooper pairs [25,
26, 27, 17]. Notably, it presents a clear violation of the Ferell-Glover-
Tinkham sum rule [28, 29]. Similar analysis has also been done for excitons,
which are electron-hole bound states [30, 31, 32]. Other than correlations, it
has been suggested that disorder can extract quantum metric in DC response. In
the limit where electron-scattering $\tau^{-1}$ rate is larger than the
bandwidth $w$, the DC conductivity was shown to carry a universal quantum
metric contribution [33]. It was later pointed out that the order of limits
$w\rightarrow 0$ and $\tau\rightarrow 0$ is crucial in obtaining the said
universal contribution [34]. Overall, there has been mounting evidence that
quantum metric is hidden in matrix elements of the optical conductivity, and
it is revealed only when an interaction or disorder scales the energy factors
out of the Kubo formula.

The energy scale can also be introduced in sum rules with frequency pre-
factors. This is exactly what the Souza-Wilkens-Martin sum rule accomplishes:
the integral over all positive frequencies of the optical conductivity
weighted by $1/\omega$ is exactly equal to the quantum metric, independent of
the dispersion. Crucially, the SWM sum rule includes only the positive
frequencies in its integral. If the integral were taken over all frequencies,
the sum rule would be amenable to Kramers-Kronig relations and make an
appearance in the imaginary part of the conductivity. The restriction to
positive frequencies does not allow that to happen. This feature is not
limited to quantum metric but all quantities shown marked in red in Fig. 1b.
They are accessible only via generalized sum rules but not in low-frequency
expansions of linear response [3].

## tQGT in Insulators.—

Gapless systems like metals have a diverging quantum metric owing not to
interband transition but rather to the presence of a Fermi surface [35, 36].
Therefore, this work focuses on insulators with a finite gap. The gap removes
intraband diagonal matrix elements of the position operator (which have a
gauge redundancy), reducing the expression to _inter-band_ terms only. While
the formalism is quite general and applies to any gapped quantum system [3],
we will consider non-interacting electrons for a simpler presentation. It
means that the system admits a Bloch represetation with states $|\psi_{m,{\bf
k}}\rangle=e^{i{\bf k}\cdot\hat{{\bf r}}}|u_{m,{\bf k}}\rangle$ where $m$ is
the band and ${\bf k}$ is the crystal momentum. Expanding Eq. (1) in the cell
periodic part of the Bloch states, we get

$\mathcal{Q}_{\mu\nu}(t)=\int_{\bf k}\sum\limits_{m\neq
n}f_{n}(1-f_{m})\hat{r}_{\mu}^{nm}\hat{r}_{\nu}^{mn}e^{i\hbar\omega_{mn}t}$
(4)

where $\hat{r}_{\mu}^{nm}=\langle u_{n,{\bf k}}|\hat{r}|u_{m,{\bf k}}\rangle$
are the position matrix elements given by Blount [37],
$\hbar\omega_{mn}=E_{m,{\bf k}}-E_{n,{\bf k}}$ are the band energies. We have
suppressed momentum labels for brevity, but it should be noted that all terms
in the equation have the same momentum ${\bf k}$. The occupation function
$f_{n}$ is the usual Fermi factor $f[E]=1/(1+e^{\beta(E-\mu)})$.

Figure 1:  a. Schematic showing the step response setup with a drive $f(t)$
that is slowly turned on from $-\infty$ and is shut down at $t=0$. The system
then relaxes from a constrained equilibrium state to the ground state. b.
Quantum geometric properties obtained from various generalized sum rules of
real and imaginary parts of longitudinal and Hall conductivity with various
$\eta$. The circled quantities appear in frequency expansions as a result of
Kramers-Kronig relations, as opposed to the boxed quantities.

Many geometric properties of insulators arise from the time derivatives of the
tQGT,

$\mathcal{S}_{\mu\nu}^{\eta}=\int\limits_{0}^{\infty}d\omega\;\dfrac{\sigma^{\rm
abs}_{\mu\nu}(\omega)}{\omega^{1-\eta}}=\dfrac{\pi
e^{2}}{\hbar}\left[(-i\hat{\partial}_{t})^{\eta}\mathcal{Q}_{\mu\nu}(t)\right]_{t=0}$
(5)

with the absorptive part of the conductivity being $\sigma^{\rm
abs}_{\mu\nu}=(\sigma_{\mu\nu}+\sigma_{\nu\mu}^{*})/2$ [3]. Sum rules for
different $\eta$ have longitudinal and Hall parts, all of which are summarized
in Fig. 1b. Most relevant are the $\eta=0$ sum rule, which defines the quantum
metric $g$ [12] and Chern number $\mathcal{C}$; the $\eta=1$ sum rule defines
the plasma frequency or the effective optical mass $n/m_{g}$ (where $n$ is
total density) and the orbital magnetic moment $\mu_{M}$ [14], and lastly
$\eta=-1$ which defines $\chi_{e}$ and $\chi_{m}$. While $\chi_{e}$ is the
electric susceptibility, related to the capacitance of the insulator [16],
$\chi_{m}$ represents a torsion constant related to the chiral-magnetic
effect. As a consequence of Kramers-Kronig, the encircled quantities in Fig.
1b can be found in the frequency expansion of conductivity. If the insulating
gap is $E_{g}$, the sub-gap conductivity admits the expression

$\sigma_{\mu\nu}(\omega\ll
E_{g})=\dfrac{e^{2}}{\hbar}\big{(}\mathcal{C}\epsilon_{\mu\nu}+\delta_{\mu\nu}i\omega\chi_{e}\big{)}$
(6)

featuring both $\mathcal{C}$ and $\chi_{e}$ [16]. The optical mass also makes
an appearance, albeit at high frequencies as an inductive piece $\sim
in/(m_{g}\omega)$. Such high-frequency regimes are typically accessible only
for superconductors, where $n/m_{g}$ is replaced by the superfluid stiffness
[28, 29]. In principle, $n/m_{g}$ appears in a nondissipative response (in
this case at high frequency), in stark contrast to the quantum metric $g$,
orbital magnetic moment $\mu_{M}$ and $\chi_{m}$, which can only be captured
via sum rules over the entire spectrum. As we emphasized earlier, the reason
behind these omissions traces back to Eq. (2), which includes only the anti-
symmetric part of $\mathcal{Q}_{\mu\nu}(t)$.

## Step Response Theory.—

Let us briefly recall the assumptions behind linear response theory. We
consider the system described by $\mathcal{H}=\mathcal{H}_{0}-f(t)\hat{B}$
where $\mathcal{H}_{0}$ is the unperturbed Hamiltonian. The drive function
$f(t)$ introduces a perturbation with operator $\hat{B}$. We assume that the
drive is switched on slowly such that $f(t)\propto e^{-0^{+}|t|}$ for
$t\rightarrow-\infty$. Assuming that the system is in its ground state at
$t=-\infty$, linear response theory gives the expectation value of a different
operator $\hat{A}$ as

$\langle\hat{A}(t)\rangle-\langle\hat{A}\rangle_{0}=\int
dt^{\prime}\chi_{AB}(t-t^{\prime})f(t^{\prime})$ (7)

where the response function is given by the anti-symmetric part of the
correlation function
$\chi_{AB}(t-t^{\prime})=i\Theta(t-t^{\prime})\mathcal{R}^{-}_{AB}(t-t^{\prime})$
where

$\mathcal{R}^{-}_{AB}(t)=\hat{A}(t)\hat{B}(0)-\hat{B}(0)\hat{A}(t)=\left\langle\left[\hat{A}(t),\hat{B}(0)\right]\right\rangle.$
(8)

The $\Theta(t-t^{\prime})$ enforces causality and the factor of $i$ makes
$\chi_{AB}(t)$ hermitian. We can further insert
$f(t)=e^{i(\omega_{0}-i0^{+})t}$ to study the response function at a given
driving frequency $\omega_{0}$.

The calculation of step response begins by considering the drive
$f_{s}(t)=f_{0}\Theta(-t)$ [38]. As depicted in Fig. 1a, the drive takes the
system to a constrained equilibrium state from which it relaxes back to the
unperturbed ground state as the drive is turned off. We thus focus on
$\hat{A}(t)$ for $t>0$, insert drive function $f_{s}(t)$ into Eq. (7) and find

$\langle\hat{A}(t)\rangle-\langle\hat{A}\rangle_{0}=\mathcal{R}_{AB}(t)f_{0}$
(9)

where the relaxation function $\mathcal{R}_{AB}(t)$ is given by

$\mathcal{R}_{AB}(t)=\int d\omega\;e^{-i\omega
t}\;\dfrac{\mathcal{R}^{-}_{AB}(\omega)}{\omega-i0^{+}}.$ (10)

This equation is perhaps not surprising since $\Theta(-t)$ is a highly
singular function $\sim 1/\omega$ in the frequency domain. It thus mixes
various frequency channels as a consequence of the convolution theorem.
Typically, $\mathcal{R}^{-}_{AB}/\omega$ is related to the Fourier transform
of the anti-commutator
$\mathcal{R}^{+}_{AB}(t)=\langle\\{\hat{A}(t),\hat{B}(0)\\}\rangle$ by a
fluctuation-dissipation relation [38]. We derive these relations in the next
section.

## Generalized fluctuation-dissipation relations.—

We begin by recalling that the commutator $\mathcal{R}^{-}_{AB}(t)$ and the
anti-commutator $\mathcal{R}^{+}_{AB}(t)$ can always be defined for a given
correlation function $\mathcal{C}_{AB}(t)=\langle\hat{A}(t)\hat{B}(0)\rangle$.
These quantities are not independent and are related by the fluctuation-
dissipation theorem. A standard derivation follows from the representation of
these quantities on the exact basis

$\mathcal{R}^{\pm}_{AB}(t)=\sum\limits_{m,n}\mathcal{F}_{mn}^{\pm}e^{i\hbar\omega_{mn}t}A_{nm}B_{mn}$
(11)

where $\mathcal{F}_{mn}^{\pm}=f_{m}\pm f_{n}$ and $\\{f_{n}\\}$ are the
occupation factors. Since these occupation factors follow
$f_{n}/f_{m}=e^{-\beta\hbar\omega_{nm}}$, we can switch to the Fourier domain
by replacing $e^{i\hbar\omega_{mn}t}$ with $\delta(\omega-\omega_{nm})$ and
derive the standard fluctuation-dissipation theorem [39]

$\mathcal{R}^{-}_{AB}(\omega)=\tanh\left(\dfrac{\beta\hbar\omega}{2}\right)\mathcal{R}^{+}_{AB}(\omega).$
(12)

The most important consequence of the fluctuation-dissipation relation is that
the relaxation function in Eq. (10) reduces to $\mathcal{R}^{+}_{AB}(t)$ in
the classical limit where
$\tanh(\beta\hbar\omega/2)\approx\beta\hbar\omega/2$, that is
$\mathcal{R}_{AB}(t)=\beta\hbar\mathcal{R}_{AB}^{+}(t)$. In sum, a combination
of the fluctuation-dissipation relation and the classical limit provides a
recipe to measure the anti-commutator $\mathcal{R}^{+}_{AB}(t)$.

We now explore whether the same formalism can be applied to measure the
symmetric part of tQGT $\mathcal{Q}^{s}_{\mu\nu}(t)$. We first notice that
$\mathcal{Q}^{s}_{\mu\nu}(t)$ and $\mathcal{Q}^{as}_{\mu\nu}(t)$ have quite
different pre-factors because of the complimentary projector. For instance,
the symmetric part takes the expression

$\mathcal{Q}^{s}_{\mu\nu}(t)=\int_{\bf k}\sum\limits_{m\neq
n}\mathcal{F}_{nm}^{+}\hat{r}_{\mu}^{nm}\hat{r}_{\nu}^{mn}e^{i\hbar\omega_{mn}t}$
(13)

where $\mathcal{F}_{nm}^{\pm}=f_{n}(1-f_{m})\pm f_{m}(1-f_{n})$. The anti-
symmetric part similarly has

$\mathcal{F}_{nm}^{-}=f_{n}(1-f_{m})-f_{m}(1-f_{n})=f_{n}-f_{m}.$ (14)

These factors are different from those in Eq. (11). However, owing to the
fermionic nature of electrons, we can use the Fermi distribution functions
that crucially satisfy

$\dfrac{f_{m}(1-f_{n})}{f_{n}(1-f_{m})}=e^{\beta\hbar\omega_{nm}}$ (15)

as a consequence of detailed balance condition for optical transitions.
Therefore, despite complicated pre-factors, we get

$\mathcal{Q}^{as}_{\mu\nu}(\omega)=-\tanh\left(\dfrac{\beta\hbar\omega}{2}\right)\mathcal{Q}^{s}_{\mu\nu}(\omega).$
(16)

The only difference between the usual fluctuation-dissipation theorem Eq. (12)
and the generalized one in Eq. (16) is the minus sign.

## Step Electric Field.—

We consider an external electric field $E_{\nu}$ which couples to the
Hamiltonian via dipolar coupling
$\mathcal{H}=\mathcal{H}_{0}-eE_{\nu}(t)\hat{r}_{\nu}$ where $e$ is the
electron charge and $E_{\nu}(t)$ describes the step drive shown in Fig. 1a. It
is a constant $E_{\nu}$ for $t<0$ and zero for later times. We next compute
the polarization from dipole moment
$\mathcal{D}_{\mu}=-e\langle\hat{r}_{\mu}\rangle$ to linear order in the field
and find

$\mathcal{D}_{\mu}(t)=-\left(e^{2}\int
dt^{\prime}i\Theta(t-t^{\prime})\mathcal{R}^{-}_{\mu\nu}(t-t^{\prime})f_{s}(t^{\prime})\right)E_{\nu}$
(17)

where
$\mathcal{R}^{-}_{\mu\nu}(t-t^{\prime})=\langle[\hat{r}_{\mu}(t),\hat{r}_{\nu}(t^{\prime})]\rangle$
is the usual commutator which can be written out explicitly in terms of Bloch
states

$\mathcal{R}^{-}_{\mu\nu}(t)=\int_{\bf k}\sum\limits_{m\neq
n}\mathcal{F}_{mn}^{-}\hat{r}_{\mu}^{nm}\hat{r}_{\nu}^{mn}e^{i\hbar\omega_{mn}t}$
(18)

where $\mathcal{F}_{mn}^{-}$ are the Fermi factors defined in Eq. (14). We
note that $\mathcal{R}^{-}_{\mu\nu}(t)$ is equal to the anti-symmetric part of
tQGT $\mathcal{Q}^{as}_{\mu\nu}(t)$. As a result, the generalized fluctuation-
dissipation relation in Eq. (16) connects $\mathcal{Q}^{as}_{\mu\nu}(t)$ to
the symmetric part $\mathcal{Q}^{s}_{\mu\nu}(t)$. Putting it all together, we
find that the polarization $\mathcal{D}_{\mu}(t)$ responds to the step
electric field $\mathcal{D}_{\mu}(t)=\mathcal{R}_{\mu\nu}(t)E_{\nu}$ where the
relaxation function is given by

$\mathcal{R}_{\mu\nu}(t)=\int d\omega\;e^{-i\omega
t}\;\dfrac{\tanh(\beta\hbar\omega/2)}{\omega-i0^{+}}\mathcal{Q}^{s}_{\mu\nu}(\omega).$
(19)

This equation is the main result of our work. In the classical limit,
$\beta\hbar\omega\ll 1$, where $\tanh(x)\sim x$ we get

$\displaystyle\mathcal{R}_{\mu\nu}(t)=\beta\hbar\mathcal{Q}^{s}_{\mu\nu}(t)/2.$
(20)

In an experiment, the measurement will likely include the relaxation function
from which the symmetric part of tQGT will need to be extracted. To that end,
we must invert the integral using the convolution theorem. We find that
$\mathcal{Q}^{s}_{\mu\nu}(t)=\int
dt^{\prime}K(t-t^{\prime})\;\mathcal{R}_{\mu\nu}(t^{\prime})$ where the kernel
is given by

$K(t)=\dfrac{\sqrt{2\pi^{3}}}{\beta^{2}\hbar^{2}}{\rm cosech}(\pi
t/\beta\hbar)^{2}.$ (21)

Lastly, we comment on different moments of the relaxation function. Similar to
tQGT, different moments carry different information about different geometric
properties of the system. The longitudinal component of the $t=0$ value is
related to the quantum metric. Similarly, the Hall component of the derivative
at $t=0$ is related to the orbital magnetic moment. In sum, the relaxation
function thus acts as the generating function for the boxed objects in Fig.
1b.

## SSH chain.—

Figure 2: a. 1D SSH chain with two orbitals $A$ and $B$ in the unit cell that
is denoted by the box. The dashed lines indicate staggered hopping with
$t_{\pm}=(1\pm\delta)$ which leads to a gap in the band structure. The system
is an insulator at half-filling. b. The relaxation function for different
temperatures $\beta$ as a function of time. The orange dashed line is the
symmetric part of the tQGT $\mathcal{Q}^{s}(t)$. As expected, the relaxation
function $\mathcal{R}(t)$ coincides with $\beta\mathcal{Q}^{s}(t)$ in the
classical limit $\beta\ll 1$. It also follows the behavior as the system goes
through the topological phase transition with $\delta$. Interestingly,
$\mathcal{R}(t)$ approaches the susceptibility $\chi_{e}$ in the opposite
quantum limit with $\beta\gg 1$.

As a concrete example to illustrate the step response, we turn to the 1D SSH
chain [40]. We consider two orbitals in a unit cell described by the
Hamiltonian

$\mathcal{H}=\sum\limits_{i}(1+\delta)c^{\dagger}_{i,A}c^{\phantom{\dagger}}_{i,B}+(1-\delta)c^{\dagger}_{i,A}c^{\phantom{\dagger}}_{i-1,B}+{\rm
h.c.}$ (22)

where $t$ is the hopping amplitude and $\delta$ denotes the imbalance between
inter- and intra-cell hopping (see Fig. 2a), and $\hbar$ is set to 1. The
Bloch Hamiltonian in momentum space can be written as $H_{k}={\bf
d}(k)\cdot\boldsymbol{\sigma}$ where $\\{\sigma_{i}\\}$ are the Pauli matrices
and

$d_{x}(k)=t_{+}+t_{-}\cos k,\;d_{y}(k)=t_{-}\sin k,\;d_{z}(k)=0$ (23)

where $t_{\pm}=(1\pm\delta)$. The staggering $\delta$ introduces a gap in the
band structure, rendering the system an insulator at half-filling.

The insulator, however, has bounded oscillations, which manifest in a non-
trivial tQGT. The symmetric part $\mathcal{Q}^{s}(t)$ is plotted in Fig. 2b
with a dashed orange line. Importantly, we see that it can be closely
approximated by the relaxation function in the classical limit. The relaxation
function in this model is given by

$\mathcal{R}(t)=\int dk\;\dfrac{\tanh(\beta|{\bf d}(k)|)}{2|{\bf
d}(k)|}\;g_{12}(k)\;e^{-i2|{\bf d}(k)|t}$ (24)

where $g_{12}(k)=|\langle u_{-,k}|\partial_{k}u_{+,k}\rangle|^{2}$ is the
position matrix element between states that are defined by
$H_{k}|u_{\pm,k}\rangle=\pm|{\bf d}(k)||u_{\pm,k}\rangle$. The classical limit
proceeds by replacing $\tanh(\beta|{\bf d}|)\rightarrow\beta|{\bf d}|$ which
gives the intended result $\mathcal{R}=\beta\mathcal{Q}^{s}(t)/2$.

In the opposite limit, where $\beta\gg 1$ so that $\tanh(\beta|{\bf
d}|)\rightarrow 1$, we find that $\mathcal{R}(0)$ approaches $\chi_{e}$

$\mathcal{R}(0)=\chi_{e}=\int dk\;\dfrac{g_{12}(k)}{2|{\bf d}(k)|}$ (25)

which has been recently related to the geometric capacitance in insulators
[16] since it enters in the steady state polarization
$\mathcal{P}\propto\epsilon_{0}\chi_{e}E$ and hence directly affects the
energy stored $\sim\mathcal{P}E=\epsilon_{0}\chi_{e}E^{2}$ inside the material
[41].

## Discussion.—

The zero-point motion of bound electrons in insulators reflects quantum
geometry and necessitates a nonvanishing quantum metric in the electron
wavefunction. However, probing the quantum metric directly in a material has
remained a challenge. One of the reasons, as we outline in this work, is that
linear response to a DC electric field will necessarily pick up the
antisymmetric part of the dipole-dipole correlator for fermions, while the
quantum metric belongs to the symmetric part. We, therefore, propose a
response function in which the electric field itself is antisymmetric in
frequency. In particular, we show that relaxation in the bulk dipole moment
can directly probe the quantum metric. It can be understood as follows. The
fully polarized the medium at $t=0$ exhibits longitudinal dipole-dipole
oscillations as the field is turned off and the system relaxes back to its
unperturbed equilibrium state. The relaxation function thus obtained also
captures the other geometric quantities that are hidden in linear response.
Heuristically, the step electric field $E(t)\propto\Theta(t)$ induces response
at $1/\omega$ which is odd in frequency, and leverages the SWM sum rule to
yield the quantum metric in the relaxation function $\mathcal{R}(t)$.

The relaxation function $\mathcal{R}(t)$ contains a convolution of the
symmetric part of the tQGT $\mathcal{Q}^{s}(t)$ with $\tanh(\beta\omega)$
which reduces to
$\mathcal{R}_{\mu\nu}(t)=\beta\hbar\mathcal{Q}^{s}_{\mu\nu}(t)/2$ in the
classical high-temperature limit. The limit allows for the direct measurement
of the quantum metric $g_{\mu\nu}=\mathcal{Q}_{\mu\nu}^{s}(t=0)$. In the
opposite (low-temperature) limit, we found that the dipole relaxation function
approaches the susceptibility $\chi_{e}$, which appears in nondissipative
linear response. It remains to be seen whether such an experiment can be
performed in a realistic setup. The high-temperature limit would be the most
challenging to implement, as the temperature needs to be the largest energy
scale in the system. There may be simplifications when a subset of bands form
a closed subspace such that the high temperature limit requires temperatures
higher than the bandwidth. Lastly, we note that the kernel in Eq.(21) is
highly non-linear and error propagation can pose significant challenge to a
realistic experiment.

## Acknowledgement.—

Work on quantum geometric properties of quantum materials is supported as part
of Programmable Quantum Materials, an Energy Frontier Research Center funded
by the U.S. Department of Energy (DOE), Office of Science, Basic Energy
Sciences (BES), under award DE-SC0019443. N.V. greatly benefited from
discussions with D. Kaplan during the Aspen Center for Physics workshop on
“Quantum Matter Through the Lens of Moiré Materials”. The Flatiron Institute
is a division of the Simons Foundation.

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