FERMILAB-PUB-20-242-A, KCL-PH-TH/2020-33

KEK-Cosmo-257, KEK-TH-2231

IPMU20-0070, PI/UAN-2020-674FT

RUP-20-22

# The first three seconds:
A Review of Possible Expansion Histories of the early Universe

Rouzbeh Allahverdi1 Mustafa A. Amin2 Asher Berlin3 Nicolás Bernal4 Christian
T. Byrnes5 M. Sten Delos6 Adrienne L. Erickcek6 Miguel Escudero7 Daniel G.
Figueroa8 Katherine Freese9,10 Tomohiro Harada11 Dan Hooper12,13,14 David I.
Kaiser15 Tanvi Karwal16 Kazunori Kohri17,18 Gordan Krnjaic12 Marek Lewicki7,19
Kaloian D. Lozanov20 Vivian Poulin21 Kuver Sinha22 Tristan L. Smith23 Tomo
Takahashi24 Tommi Tenkanen25,111Corresponding author James Unwin26 Ville
Vaskonen${}^{7,27,\,{\rm a}}$ Scott Watson28 1Department of Physics and
Astronomy, University of New Mexico, Albuquerque, NM 87131, USA  2Department
of Physics & Astronomy, Rice University, Houston, Texas 77005, USA  3Center
for Cosmology and Particle Physics, Department of Physics, New York
University, New York, NY 10003, USA  4Centro de Investigaciones, Universidad
Antonio Nariño, Carrera 3 Este # 47A-15, Bogotá, Colombia  5Department of
Physics and Astronomy, Pevensey II Building, University of Sussex, BN1 9RH, UK
6Department of Physics and Astronomy, University of North Carolina at Chapel
Hill,
Phillips Hall CB3255, Chapel Hill, North Carolina 27599, USA  7Physics
Department, King’s College London, London WC2R 2LS, UK  8Instituto de Física
Corpuscular (IFIC), University of Valencia-CSIC, E-46980, Valencia, Spain
9Physics Department, University of Texas, 2515 Speedway, STOP C1600, Austin,
TX 78712, USA  10Oskar Klein Centre for Cosmoparticle Physics, Stockholm
University, 10691 Stockholm, Sweden  11Department of Physics, Rikkyo
University, Toshima, Tokyo 171-8501, Japan  12Fermi National Accelerator
Laboratory, Theoretical Astrophysics Group, Batavia, IL 60510, USA
13University of Chicago, Kavli Institute for Cosmological Physics, Chicago, IL
60637, USA  14University of Chicago, Department of Astronomy and Astrophysics,
Chicago, IL 60637, USA  15Department of Physics, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA  16Center for Particle Cosmology,
Department of Physics & Astronomy, University of Pennsylvania, Philadelphia,
PA 19104, USA  17Theory Center, IPNS, KEK, and Sokendai, Tsukuba, Ibaraki
305-0801, Japan  18Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa,
Chiba 277-8583, Japan  19Faculty of Physics, University of Warsaw ul. Pasteura
5, 02-093 Warsaw, Poland  20Max Planck Institute for Astrophysics, Karl-
Schwarzschild Str. 1, Garching, 85748, Germany  21Laboratoire Univers &
Particules de Montpellier (LUPM), CNRS & Université de Montpellier (UMR-5299),
Place Eugène Bataillon, F-34095 Montpellier Cedex 05, France  22Department of
Physics and Astronomy, University of Oklahoma, Norman, OK, 73019, USA
23Department of Physics and Astronomy, Swarthmore College, 500 College Ave.,
Swarthmore, PA 19081, USA  24Department of Physics, Saga University, Saga
840-8502, Japan  25Department of Physics and Astronomy, Johns Hopkins
University, 3400 N. Charles Street, Baltimore, MD 21218, USA
<EMAIL_ADDRESS>26Department of Physics, University of Illinois at
Chicago, 845 W Taylor Street, Chicago, IL 60607, USA  27NICPB, Rävala 10,
10143 Tallinn, Estonia<EMAIL_ADDRESS>28Department of Physics,
Syracuse University, Syracuse, NY 13214, USA

###### Abstract

It is commonly assumed that the energy density of the Universe was dominated
by radiation between reheating after inflation and the onset of matter
domination 54,000 years later. While the abundance of light elements indicates
that the Universe was radiation dominated during Big Bang Nucleosynthesis
(BBN), there is scant evidence that the Universe was radiation dominated prior
to BBN. It is therefore possible that the cosmological history was more
complicated, with deviations from the standard radiation domination during the
earliest epochs. Indeed, several interesting proposals regarding various
topics such as the generation of dark matter, matter-antimatter asymmetry,
gravitational waves, primordial black holes, or microhalos during a
nonstandard expansion phase have been recently made. In this paper, we review
various possible causes and consequences of deviations from radiation
domination in the early Universe – taking place either before or after BBN –
and the constraints on them, as they have been discussed in the literature
during the recent years.

###### Contents

  1. 1 Introduction
  2. 2 Causes of nonstandard expansion phases
    1. 2.1 Post-inflation reheating (Authors: M. A. Amin, D. I. Kaiser & K. D. Lozanov)
      1. 2.1.1 Single-field phenomena
      2. 2.1.2 Multifield phenomena
      3. 2.1.3 Future directions
    2. 2.2 Extra stages of inflation (Authors: T. Tenkanen & V. Vaskonen)
      1. 2.2.1 Multistep inflation
      2. 2.2.2 Thermal inflation
    3. 2.3 Heavy particles and dark sectors (Authors: N. Bernal & J. Unwin)
      1. 2.3.1 Nonstandard cosmology from heavy particles
      2. 2.3.2 The transition to radiation domination
      3. 2.3.3 Nonstandard cosmology beyond heavy particles
    4. 2.4 Moduli fields (Authors: K. Sinha & S. Watson)
    5. 2.5 Post-BBN modifications (Authors: T. Karwal & T. L. Smith)
  3. 3 Consequences of nonstandard expansion phases
    1. 3.1 Consequences for dark matter (Authors: A. Berlin, D. Hooper & G. Krnjaic)
    2. 3.2 Phase transitions and baryon asymmetry (Authors: R. Allahverdi & M. Lewicki)
      1. 3.2.1 Phase transition dynamics
      2. 3.2.2 Electroweak baryogenesis
      3. 3.2.3 Constraints on baryogenesis from dilution by reheating
    3. 3.3 Consequences for models of inflation (Authors: K. Freese & T. Tenkanen)
    4. 3.4 Curvaton scenario (Authors: C. Byrnes & T. Takahashi)
      1. 3.4.1 Curvaton perturbations and local non-Gaussianity
      2. 3.4.2 Isocurvature perturbations and the evolution of non-Gaussianity after inflation
    5. 3.5 Formation of microhalos (Author: A. Erickcek)
      1. 3.5.1 Growth of density perturbations
      2. 3.5.2 Cutting off microhalo formation
      3. 3.5.3 The microhalo population
    6. 3.6 Formation of primordial black holes (Authors: T. Harada & K. Kohri)
      1. 3.6.1 Radiation-dominated universe
      2. 3.6.2 Matter-dominated universe
  4. 4 Constraints on nonstandard expansion phases
    1. 4.1 BBN and CMB constraints (Authors: M. Escudero & V. Poulin)
      1. 4.1.1 A brief review of standard BBN
      2. 4.1.2 BBN constraints on nonstandard cosmologies
      3. 4.1.3 Effects of nonstandard expansion rate on the CMB
      4. 4.1.4 Combining CMB with BAO and SN1a
      5. 4.1.5 Probing perturbation dynamics with CMB
    2. 4.2 Observational probes of microhalos (Authors: M. S. Delos & A. L. Erickcek)
      1. 4.2.1 Gravitational signatures
      2. 4.2.2 Dark matter annihilation within microhalos
      3. 4.2.3 The microhalo population within galaxies
    3. 4.3 Observational probes on primordial black holes (Authors: T. Harada & K. Kohri)
      1. 4.3.1 PBH evaporation and its consequences
      2. 4.3.2 Observational constraints on PBHs
      3. 4.3.3 Constraints on the power spectrum of curvature perturbation
      4. 4.3.4 Constraints on curvature perturbations from PBH formation during matter dominance
    4. 4.4 Gravitational wave backgrounds (Authors: D. G. Figueroa & M. Lewicki)
      1. 4.4.1 Inflation
      2. 4.4.2 Cosmic strings
      3. 4.4.3 First order phase transitions
  5. 5 Summary

##  1 Introduction

More than forty years ago, Steven Weinberg famously summarized the state of
early-universe cosmology in his book, The First Three Minutes [1]. Weinberg
began his account by describing an early time when the Universe was filled
with radiation. Since Weinberg’s book first appeared, cosmologists have made
enormous progress in understanding the physics of the very early Universe. Yet
there remains a gap in our understanding of cosmic history – a gap that spans
the first few seconds. How, during that brief time interval, did the energy
density of the universe come to be dominated by relativistic particles? In
other words, what set the conditions at the start of Weinberg’s analysis?

The earliest epoch we can confidently gain information about is inflation, an
early period of accelerated expansion in the very early Universe [2, 3, 4, 5,
6, 7], which makes predictions consistent with high-precision measurements of
the temperature anisotropies in the Cosmic Microwave Background (CMB)222For
alternatives to cosmic inflation, see e.g. Refs. [8, 9]. [10, 11, 12, 13, 14]
. These observations limit the energy scale during inflation to be smaller
than $\mathcal{O}(10^{16})\,{\rm GeV}$, well below the Planck scale $M_{\rm
P}=1.22\times 10^{19}\,{\rm GeV}$. Meanwhile, constraints on the effects of
neutrinos on the CMB, as well as Big Bang Nucleosynthesis (BBN) constraints,
reveal that the Universe had to be radiation dominated (RD) by the time
neutrinos decoupled from the thermal plasma, at the energy scale
$\mathcal{O}(1)\,{\rm MeV}$ [15, 16, 17, 18, 19]. Between these two epochs
stretches a period of cosmic history that is not well constrained by
observations (see Fig. 1). Although the time interval between the end of
inflation and neutrino decoupling is small ($\sim 1\,{\rm sec}$), the energy
scale can drop during this period by almost 20 orders of magnitude and the
Universe may have expanded by up to 60 e-folds [20, 21, 22].

The rate at which the Universe expands depends on the (spatially averaged)
total energy density $\langle\rho\rangle$ and pressure $\langle p\rangle$ of
the combination of fluids that fill the Universe. In particular, in a
spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) Universe filled with
components that can be modelled as perfect fluids, the Einstein field
equations imply that the scale factor $a(t)$ evolves as

$a(t)\propto t^{\frac{2}{3(1+w)}},$ (1)

where $t$ is cosmic time and the equation of state parameter, $w$, is given by

$w=\frac{\langle p\rangle}{\langle\rho\rangle}\,.$ (2)

During inflation the equation of state must be $w\simeq-1$, whereas RD
expansion corresponds to $w\simeq 1/3$. Somehow the Universe must have
transitioned from $w\simeq-1$ to $w\simeq 1/3$ between the end of inflation
and neutrino decoupling.

According to standard cosmology, after an initial inflationary phase the
Universe quickly entered a period of radiation domination, followed by an era
of matter domination, and finally transitioned to the dark energy dominated
period which we reside in today. However, this simple four stage evolution of
the Universe need not necessarily be the case. Indeed, at the moment we do not
have much observational information about the state of the Universe prior to
BBN that occurred during radiation dominance at temperature $T\gtrsim 5\,{\rm
MeV}$ [23, 24, 25, 26, 27, 28, 29]. It is therefore possible that the
cosmological history featured, for example, an additional early matter-
dominated (EMD) epoch due to e.g. slow post-inflationary reheating or massive
metastable particles that dominated the total energy density. Potentially more
exotic scenarios that change the expansion history of the Universe compared to
the standard RD case can also be realized, and they can have important
consequences for the physics of the early Universe.

Figure 1: The evolution of the comoving Hubble scale $1/(aH)$ as a function of
the scale factor $a$. The expansion history between inflation and the
subsequent, post-BBN era remains unknown. Here $a_{\rm end}$ marks the end of
inflation, $a_{\rm BBN}$ denotes the commencement of BBN, and $a_{\rm eq}$ is
the time of matter-radiation equality at redshift $z\simeq 3400$.

BBN is a cornerstone of the standard cosmological model. At present, the
primordial abundances of helium and deuterium are measured with 1% precision.
This makes BBN a powerful arena to test nonstandard cosmological expansion
histories. A common lore is that the Universe must be radiation dominated
prior to BBN. This is due to the fact that the abundances of light elements
are extremely sensitive to the expansion rate, thermalization of neutrino
background, and the time-evolution of the neutron-to-proton ratio and their
measured values agree very well with the predictions obtained assuming
standard RD expansion. Likewise, measurements of the CMB can be used to
achieve information about the expansion history of the early Universe.
However, the standard flat $\Lambda$CDM model, while successful at explaining
numerous cosmological data sets, currently faces several potential challenges.
There are theoretical motivations to consider physics beyond $\Lambda$CDM,
such as the unknown nature of inflation, dark matter (DM), dark energy, and
mechanisms to explain neutrino masses. As observations have become
increasingly precise, several anomalies have appeared, including the Hubble
tension [30], the $\sigma_{8}$ tension [31], the $A_{\rm lens}$ anomaly [32,
33], the coincidence problem [34, 35], the small-scale structure problem [36],
and the anomalous measurement by EDGES [37]. Theorists are hence motivated in
multiple ways to postulate new physics beyond $\Lambda$CDM.

Quantity / abbreviation | Explanation
---|---
$a$ | Scale factor
BAO | Baryonic acoustic oscillations
BBN | Big Bang Nucleosynthesis
CL | Confidence level
CMB | Cosmic Microwave Background
(C)DM | (Cold) Dark matter
EMD | Early matter-dominated
EoS | Equation of state
GW | Gravitational wave
$H\equiv\dot{a}/a$ | Hubble parameter
$H_{0}$ | Hubble parameter at present
$g_{*}$ | Number of energy degrees of freedom
$g_{*,s}$ | Number of entropy degrees of freedom
LSS | Large scale structure
MD | Matter-dominated
$M_{\rm P}\equiv 1/\sqrt{G}=1.22\times 10^{19}\,{\rm GeV}$ | The Planck mass
$N\equiv\ln(a_{2}/a_{1})$ | Number of e-folds between the time $a_{1}$ and $a_{2}$
$N_{\rm eff}$ | Effective number of neutrinos
NFW | Navarro-Frenk-White profile
$\Omega_{\rm i}\equiv\rho_{\rm i}/\rho_{\rm crit}$ | Energy density fraction of component $i$
PBH | Primordial black hole
PT | Phase transition
RD | Radiation-dominated
$\rho_{\rm i}$ | Energy density of component $i$
$\rho_{\rm crit}\equiv 3H^{2}M_{\rm P}^{2}/(8\pi)$ | Critical energy density
SGWB | Stochastic gravitational wave background
SM | Standard Model
$T_{\rm reh}$ | Reheating temperature
UCMH | Ultracompact minihalo
vev | Vacuum expectation value
$w\equiv\langle p\rangle/\langle\rho\rangle$ | Equation of state parameter
$z$ | Redshift
Table 1: A list of the most common quantities and abbreviations used in the
paper.

Notably, early periods of nonstandard cosmology can alter the cosmological
abundances of particle species. In particular, they can have a significant
impact on our expectations for the DM relic abundance, since nonstandard eras
can change the expansion rate of the Universe [38, 39], lead to entropy
injections that may dilute the DM relic abundance [40], or provide a
nonthermal production mechanism for DM [41, 42]. Furthermore, a period of
nonstandard cosmology may also impact small scale structure formation,
potentially enhancing or impeding the formation of dense ultracompact DM
microhalos [43, 44, 45, 46, 47], axion miniclusters [48, 49], and primordial
black holes (PBH)s [50, 51, 52, 53, 54, 55, 56]. The survival of
nongravitational relics (including, for example and in addition to DM, matter-
antimatter asymmetry [57, 58, 59, 60, 61], primordial magnetic fields [62, 63,
64, 65, 66], or effective number of relativistic species [67, 68]) will depend
on the details of the expansion history and post-inflationary thermalization
of radiation. Along with large-scale gravitational effects associated with the
expansion history, smaller scale gravitational effects – including horizon-
scaled variations of expansion history and associated non-Gaussianities [69,
70, 71, 72, 73, 74, 75, 76, 77, 78], imprints from small-scale clustering of
matter [43, 79, 80], PBHs [81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92,
93], and gravitational waves resulting from violent nonlinear dynamics [94,
95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105] – provide hope for future
observational probes of the time before BBN.

All of the above aspects are topics of this paper. In the following we review
various possible causes and consequences of deviations from radiation
domination in the early Universe – taking place either before or after BBN –
and the constraints on them, as they have been discussed in the literature
during the recent years. We emphasize that this paper is not an introduction
to the topic and the underlying physics but a review of recent advances
related to nonstandard cosmological epochs. Several interesting proposals,
regarding for example the generation of DM or matter-antimatter asymmetry
during a nonstandard expansion phase, warrant further research. We hope that
this review helps in guiding this endeavour.

The paper is organized into three sections: Sec. 2 concentrates on the causes,
Sec. 3 on the consequences, and Sec. 4 on the constraints on nonstandard
expansion eras. Each section is further divided into subsections, which
discuss the general topic of the section in more detail from both the model-
building as well as observational point of view. Finally, in Sec. 5, we
summarize our discussion. The paper contains various quantities and
abbreviations that are used throughout the paper. They are laid out in Table
1.

##  2 Causes of nonstandard expansion phases

In this section, we will discuss several causes of nonstandard expansion eras,
taking place either before or after BBN. Such causes of nonstandard expansion
can broadly be grouped as (i) modifications to the properties of the standard,
$\Lambda$CDM, components, and (ii) the introduction of new components
altogether. These two categories can be further refined into modifications to
the background and/or perturbative evolution. These categories are not
exclusive, and some models may fall within multiple descriptions, as we will
discuss.

###  2.1 Post-inflation reheating (Authors: M. A. Amin, D. I. Kaiser & K. D.
Lozanov)

In this section, we focus on the period right after inflation. For reviews,
see Refs. [106, 107, 108, 109, 110]. In some of the simplest models of
inflation consistent with observations [12, 13, 14], inflation is driven by a
slowly rolling scalar field with a potential $V(\phi)\propto\phi^{\alpha}$.
The shape of the potential affects the characteristics of the primordial
perturbations which ultimately seed the CMB anisotropies; for values of the
inflaton field relevant to scales probed by the CMB, observations constrain
power potential to be more shallow than quadratic ($\alpha<2$). The dynamics
at the end of inflation, on the other hand, are determined by the shape of the
potential near its minimum, as well as by the inflaton’s couplings to other
fields, including SM fields and/or intermediaries (see Fig. 2).333We consider
a power-law behavior of the potential during inflation for highlighting the
relative flatness (compared to quadratic) of the potential for large field
values. In some models, such as those with “Hill-top” potentials [111] or
“Natural Inflation” models [112] this might not be the best parametrization.

####  2.1.1 Single-field phenomena

We first consider typical effects in single-field models that have negligible
couplings to other fields. As shown in Fig. 2, at large field values the
potential takes the form $V(\phi)\propto\phi^{\alpha}$ with $\alpha<2$ (to be
consistent with observations), near the minimum we assume a power-law of the
form $V(\phi)\propto|\phi|^{2n}$, and we assume that the potential
interpolates smoothly between these two forms at an intermediate scale
$|\phi|\sim M$.

Figure 2: The energy stored in the almost homogeneous inflaton is tranferred
to its own perturbations, as well as other fields, and eventually to the SM
fields and DM.

During inflation, the inflaton field $\phi(t,{\bf
x})=\varphi(t)+\delta\phi(t,{\bf x})$ will be dominated by the spatially
homogeneous component, with $|\delta\phi/\varphi|\ll 1$. After slow-roll
evolution has ended, the homogeneous condensate $\varphi(t)$ will oscillate
around the mininum of its potential. The (time-averaged) equation of state
will be determined by the power $n$ [113],

$w=\frac{n-1}{n+1}\,,$ (3)

which for $n=2$ gives $w=1/3$, and for $n=1$, yields $w=0$.

(Almost) spatially homogeneous, oscillating scalar fields are unstable. They
become spatially inhomogeneous either due to self-interactions, or
gravitational clustering. In particular, the effective mass of the inflaton
fluctuations, $m_{\rm eff}^{2}=V_{,\phi\phi}(t)$, depends on $\varphi(t)$.
Here $V_{,\phi\phi}\equiv{\rm d}^{2}V(\phi)/{\rm d}\phi^{2}|_{\phi=\varphi}$.
The quasi-periodic oscillations of $\varphi(t)$ can drive a resonant transfer
of energy from the inflaton condensate into shorter-wavelength fluctuations
[114, 115, 116]. For reviews, see Refs. [106, 107, 108, 109, 110]. For a
single inflaton field minimally coupled to gravity (and neglecting coupled
metric perturbations), the equation of motion for Fourier modes of the
inflaton fluctuations takes the form

$\delta\ddot{\phi}_{k}+3H\delta\dot{\phi}_{k}+\left[\frac{k^{2}}{a^{2}}+V_{,\phi\phi}(t)\right]\delta\phi_{k}=0,$
(4)

where overdots denote derivates with respect to cosmic time $t$,
$H\equiv\dot{a}/a$, and $k$ is comoving wavenumber. As $\varphi(t)$ oscillates
quasi-periodically at the end of inflation, the modes $\delta\phi_{k}(t)$
behave like oscillators with time-varying frequencies. In the limit
$|V_{,\phi\phi}|\gg H^{2}$, solutions to Eq. (4) generically take the form

$\delta\phi_{k}(t)={\cal P}_{k}^{+}(t)\,e^{\mu_{k}t}+{\cal
P}_{k}^{-}(t)\,e^{-\mu_{k}t},$ (5)

where ${\cal P}_{k}^{\pm}(t)={\cal P}_{k}^{\pm}(t+\tau)$, with $\tau$ the
period of oscillation of $V_{,\phi\phi}(t)$. The quantity $\mu_{k}$ is known
as the Floquet exponent. It is generally a complex function of $k$, $\tau$,
and the amplitude of $V_{,\phi\phi}(t)$. Modes $\delta\phi_{k}(t)$ whose
wavenumbers $k$ lie within so-called “resonance bands” $\Delta k$, for which
${\rm Re}(\mu_{k})\neq 0$, will become exponentially amplified due to
parametric resonance. Such exponential growth corresponds to a rapid,
nonadiabatic transfer of energy from the homogeneous condensate into higher-
momentum inflaton particles. Moreover, the resonance process can involve the
collective decay of multiple inflatons from the condensate into higher-
momentum particles, making the process intrinsically nonperturbative.

Figure 3: For single-field inflaton potentials with quadratic minima which
flatten as we move away from the minimum ($n=1$, $M\ll M_{\rm P}$ in Fig. 2),
the almost homogeneous inflation field fragments rapidly into dense solitons
(called oscillons) after inflation due to an attractive self-interaction. The
oscillons then cluster gravitationally. The equation of state $w$ throughout
this phase is approximately zero. Each oscillon has fixed physical size and
amplitude, neither of which redshift with expansion. The above boxes are
comoving and roughly horizon-sized initially. Lighter colors represent higher
densities. Figure based on Ref. [79].

For single-field models with $n=1$, equation of state remains $w=0$ with or
without fragmentation of the homogeneous condensate. Even in this simple case,
the post-inflation epoch can host interesting nonlinear physics. In
particular, for $M\ll M_{\rm P}$, as the field explores the flatter-than-
quadratic region of the potential ($\phi\gtrsim M$ with $\alpha<2$), broad
self-resonance drives copious production of oscillons [117, 118, 119, 120,
121, 122, 123]. Oscillons are highly overdense, spatially localized,
exceptionally long-lived field configurations [124, 125, 126, 127, 128, 129,
130] where the field amplitude in their cores does not redshift as the
universe expands. The rapid formation of oscillons from an almost homogeneous
condensate at the end of inflation, and their slower gravitational clustering
[79] is shown in Fig. 3.444Note that oscillons are long-lived (with lifetimes
$\sim 10^{8}$ oscillations of the field [130] in some models) but not
perfectly stable and will evolve by slowly emitting scalar radiation. In
regions of parameter space that do not yield such resonant phenomena
($M\gtrsim M_{\rm P}$), the inflaton condensate will fragment and cluster due
to gravitational interactions on a typical time-scale $\Delta N_{\rm
frag}\simeq 5$ [131, 132, 80], although the equation of state remains $w\simeq
0$. Either of these single-field scenarios with $n=1$ would yield a long
duration during which the Universe remained MD rather than transitioning to RD
expansion, and hence could lead to conflicts with observational constraints
similar to moduli fields discussed in Sec. 2.4.

For $n\neq 1$, fragmentation of the condensate is also inevitable (independent
of the value of $M$). Resonant amplification of small inhomogeneities will
again fragment the homogeneous condensate. Unlike the $n=1$ case, here the
equation of state will approach that of radiation, $w=1/3$ after sufficient
time, even in the absence of couplings to other fields [133]. This counter-
intuitive result arises because for the $n>1$ case, the resonant band
structure is such that the transfer of energy from the homogeneous condensate
to relativistic waves of the inflaton field continues to be efficient as the
homogeneous, oscillating field approaches the minimum of the potential due to
expansion. This leads to a universe filled with relativistic waves of the
inflaton field, with an equation of state $1/3$ [133, 121]. Note that the
produced waves are relativistic, and remain so, because the effective mass of
the field at the minimum of the potential is zero for $n>1$. The number of
e-folds between the end of inflation and the beginning of RD epoch in this
scenario is given by [133, 121]

$\Delta N_{\rm{rad}}\gtrsim\frac{n+1}{3}\ln\left[\frac{k}{\Delta
k}\frac{\sqrt{8\pi}\,M}{M_{\rm P}}\frac{|4-2n|}{n+1}\right]\qquad n\neq
0,\,2\,,$ (6)

where $\Delta k/k$ is the relative width of the narrow resonance band. Eq. (6)
holds for $n\neq 0,\,2$ and $M\lesssim M_{\rm P}$, and yields
$\mathcal{O}(1)\lesssim\Delta N_{\rm rad}\lesssim{\cal O}(10)$. For $n=2$, Eq.
(3) indicates that the homogeneous evolution already has $w=1/3$. In that
case, if $M\ll M_{\rm P}$, then $\Delta N_{\rm rad}\lesssim 1$ due to broad
self-resonance. If the inflaton field couples to other light fields, the
duration to RD epoch will be shortened. In this sense, Eq. (6) provides an
upper bound on the duration between the end of inflation and onset of RD
epoch. This upper bound can significantly reduce the uncertainty in
inflationary observables such as the tensor-to-scalar ratio $r$ and spectral
tilt $n_{\rm s}$ [133].

Our discussion of single field models assumes that the coupling to other
fields is sufficiently weak to not play a major role in the single field
dynamics for a significant number of e-folds after inflation. Of course, the
inflaton must couple to other fields (even if very weakly) to eventually yield
a radiation-dominated, thermal universe filled with Standard Model particles
before BBN. For example, to prevent an indefinite MD period for the $n=1$
case, couplings to other relativistic daughter fields must be invoked to drain
energy away from the inflaton. This is true whether the universe is filled
with oscillons, or includes gravitationally bound inflaton structures, or even
if the inflaton remains homogeneous. For the $n>1$ case, we can get to
radiation domination without couplings to other fields, however, couplings to
other fields must be again invoked to eventually lead to the requisite
particle content, and thermalization before BBN. The detailed nature of the
decay and thermalization will depend on the nature of the coupling to daughter
fields (which may be Standard Model fields or intermediaries), the
homogeneous/inhomogeneous state of the field, and could be perturbative or
non-perturbative.555For an example of a non-perturbative, explosive gauge
field production from mergers of oscillons see Ref. [134].

####  2.1.2 Multifield phenomena

In models in which the inflaton field $\phi$ couples directly to other fields
(which we may collectively denote as $\chi$), the $\chi$ fields typically
acquire $\phi$-dependent effective masses, $m_{\chi}^{2}=m_{\chi}^{2}(\phi)$.
After inflation, these couplings can drive even more efficient transfer of
energy out of the inflaton condensate than in single-field models. Such fast
processes can hasten the transition to $w=1/3$, though in some cases an
incomplete decay of the inflaton can yield a later phase of matter-dominated
expansion, with $w\sim 0$, prior to neutrino decoupling.

For scalar fields $\chi$ that are minimally coupled to gravity, modes
$\chi_{k}$ obey an equation of motion of the same form as Eq. (4), with
$V_{,\phi\phi}\rightarrow m_{\chi}^{2}(\varphi(t))$. Couplings between $\phi$
and $\chi$ are typically less constrained by observations than are self-
couplings of $\phi$, and hence there can be broad resonance bands (with
$\Delta k\gtrsim m_{\chi}$) within which the Floquet exponents for modes
$\chi_{k}$ are large (${\rm Re}(\mu_{k})\sim{\cal O}(m_{\phi})$, where
$m_{\phi}$ is the mass of the oscillating inflaton). Given the nonperturbative
nature of such resonances, these decays can produce daughter particles much
more massive than the inflaton [116, 106, 108, 107, 109, 110]. In some models
(such as those involving spontaneous symmetry breaking), a tachyonic
instability may occur, for which $([k^{2}/a^{2}]+m_{\chi}^{2})<0$, which
likewise yields a rapid, nonperturbative amplification of certain modes
$\chi_{k}$ [135, 136, 137, 138, 139, 140].

Models with multiple scalar fields $\phi^{I}$ that have nonminimal couplings
to gravity can have even richer resonance structure, since (upon performing a
conformal transformation to the Einstein frame) the field-space manifold
acquires nontrivial curvature [141]. In such models, couplings can arise
between the fields from noncanonical kinetic terms as well as from direct
couplings in the interaction potential. For large dimensionless nonminimal
couplings $\xi_{I}\geq{\cal O}(10^{2})$, as one finds in popular models like
Higgs inflation [142], these couplings can yield especially efficient
resonances at the end of inflation [143, 144, 145, 146, 147, 148, 149, 150,
151, 152, 153, 154]. The rapid production of particles with wavenumber $k\geq
H$ can drive $w\rightarrow 1/3$ within $\Delta N_{\rm rad}\sim{\cal O}(1)$
after the end of inflation [155, 156].

The exponential amplification of modes driven by parametric resonance and/or
tachyonic instability does not persist indefinitely. Rather, the phase of
rapid growth ends due to backreaction on the inflaton oscillations from the
produced particles, as well as from rescattering among modes $\chi_{k}$ of
different $k$. Such backreaction effects typically grow quickly, within
$\Delta t_{\rm br}\sim{\cal O}(1)\times\mu_{k}^{-1}\ll H^{-1}$ [106, 108, 107,
109, 110]. In many models, backreaction grows to be significant before
resonance can completely transfer energy from the inflaton condensate into
decay products, which can leave a significant fraction of the total system
energy, $\gtrsim{\cal O}(10\%)$, in the inflaton field.

The subsequent evolution of the equation of state is largely model dependent.
For example, in models with minimally coupled scalar fields and potentials of
the form
$V(\phi,\chi)=m_{\phi}^{2}\phi^{2}/2+\kappa\phi\chi^{2}+g^{2}\phi^{2}\chi^{2}$,
the couplings between $\phi$ and $\chi$ can give rise to a long-lived,
nontrivial period of expansion with $0<w\lesssim 1/3$, depending on the values
of the coupling constants $\kappa$ and $g$ [157, 139, 140] (see Ref. [158] for
recent long-duration simulations). At late times in such models, the expansion
of the Universe will dilute the energy of relativistic decay products so much
that the energy remaining in the inflaton condensate can dominate the total
energy density again, giving rise to a matter-dominated phase with $w\simeq
0$. If the inflaton potential is non-quadratic at the minimum, then similar to
the single-field case (see discussion near Eq. (6)), the equation of state
always approaches $1/3$, albeit the time-scales to approach radiation
domination can be shortened based on strength of the coupling to other fields
[158].

If the inflaton couples to vector bosons, the Universe can transition rapidly
to $w=1/3$ after inflation [159, 160, 161, 162, 163, 164, 165, 166]. For
example, if the inflaton is an axion-like field, then a coupling of the form
$\lambda\phi F\tilde{F}$ (where $F_{\mu\nu}$ is the gauge-boson field-strength
tensor) can lead to a virtually instantaneous transition to RD phase at the
end of inflation, for sufficiently large coupling $\lambda$ [167, 168].
Moreover, nearly all of the energy of the inflaton condensate can be
transferred to gauge fields in such processes, potentially avoiding a
primordial period of expansion with $w\simeq 0$.

In other regions of parameter space, for sufficiently small amplitudes of the
inflaton oscillations $\varphi$ and for weak couplings $\kappa$ and $g$,
resonances will remain subdominant, and the decay of the inflaton condensate
will occur mostly due to perturbative decays. The decay rates may be estimated
from Feynman diagrams for the tree-level couplings. Even for models in which
strong resonances occur at early times after inflation, such perturbative
decays can play an important role at late times, providing a mechanism to
complete the transfer of energy from the inflaton condensate into light decay
products. The rate of such perturbative decays can be enhanced in the case of
bosonic fields $\chi$. In those cases, Bose enhancement yields correction
factors to the tree-level decay rates that one would estimate in vacuum,
$\Gamma_{\chi}^{\rm vac}$, of the form $\Gamma_{\chi}\propto\Gamma_{\chi}^{\rm
vac}\,n_{\chi}$, where $n_{\chi}$ is the occupation number. Then the change of
the occupation number, $\dot{n}_{\chi}=\Gamma_{\chi}$, can yield exponential
growth rates for $n_{\chi}$ rather than typical linear rates of growth in
vacuum [116]. The perturbative decay of the $\phi$ condensate will be complete
when $\Gamma^{\rm vac}_{\chi}\sim H$, after which the energy will reside in
relativistic decay products and the Universe will become RD.

If the inflaton couples to fermions $\psi$, the effective mass
$m_{\psi}(\varphi(t))$ can become time-dependent at the end of inflation, much
as for bosonic fields $\chi$. This can yield a very different distribution of
fermionic decay products than would result from perturbative decays, although
the Pauli exclusion principle prevents the occupation number for any fermionic
mode to grow larger than 1 [169, 170, 171, 172, 173, 57]. In models in which
the inflaton couples to both bosonic and fermionic fields, perturbative decays
of the inflaton condensate into fermions can help complete the transfer of
energy out of the inflaton and avoid a prolonged phase of $w\simeq 0$.

Finally, we note that if significant dissipative effects due to particle
production are present during inflation (for example, during warm inflation
[174, 175]), and a significant thermal bath is already present when inflation
ends, the duration of transition to radiation domination from the end of
inflation might be shortened.

####  2.1.3 Future directions

The period immediately after the end of inflation can feature rich, nonlinear
and nonperturbative dynamics. Parametric resonance and tachyonic instabilities
can transfer energy efficiently from the inflaton condensate into higher-
momentum decay products. In both single-field and multifield models, the
efficient production of relativistic decay products can drive a transition in
the equation of state from $w\simeq-1$ during inflation to $w=1/3$ within a
few e-folds after the end of inflation. If only slow, perturbative processes
are involved, the transition can take considerably longer. Details of the
energy transfer depend on the model; in some cases, the rapid transition to RD
expansion can slide into a later phase of matter-dominated expansion (with
$w\simeq 0$) unless some mechanism exists (such as perturbative decays) to
completely drain the energy from the inflaton condensate. Although the details
of nonperturbative post-inflationary dynamics remain model dependent, recent
work has made progress towards more model-independent means of characterizing
this phase [176, 177, 102, 178].

For a given model, the full nonlinear dynamics of fields after inflation can
now be explored numerically using a number of codes which evolve scalar and
vector fields on a lattice [179, 180, 181, 182, 183, 149, 184, 185, 186, 187].
After such nonlinear evolution, however, the spectrum of decay products is
typically far from an equilibrium distribution. The processes and time-scales
over which such distributions of particles relax into thermal equilibrium
remain much less understood, and an important area for further research;
progress may be made by exploiting methods developed for the study of other
systems, such as quark-gluon plasma in the context of heavy ion collisions
[188]. Another interesting area of research concerns incorporating
gravitational effects more fully into our understanding of post-inflation
reheating [189, 190, 191, 192, 193, 194, 195, 196, 131, 197, 185].

###  2.2 Extra stages of inflation (Authors: T. Tenkanen & V. Vaskonen)

In this section we discuss scenarios in which the primary or primordial
inflationary phase was followed by a secondary or even multiple extra stages
of inflation. These additional inflationary stages may have been punctuated by
an era of either standard RD expansion or by some type of nonstandard
expansion; see Fig. 4 for illustration. Because the standard way of treating
the dynamics of the early Universe usually contain only one period of
inflation – the primordial one –, any additional stages of inflation can be
seen as a nonstandard addition regardless of how the Universe expanded between
the inflationary stages. The energy scales at which the additional
inflationary stages may have occurred are not particularly limited, except for
the usual requirements that the upper limit for the energy scale of inflation
is $\rho_{\rm tot}\simeq(10^{16}\,{\rm GeV})^{4}$ (see Sec. 3.3) and any
nonstandard phase must have ended prior to BBN at $T\sim 1$ MeV (see Sec.
4.1).

####  2.2.1 Multistep inflation

We begin by discussing scenarios of multistep inflation by which we refer to
models where (nonthermal) inflation occurs in at least two phenomenologically
different stages with distinct characteristics, as originally suggested in
Refs. [198, 199]. This type of scenarios can be generically divided into three
categories: those in which the same scalar field(s) that were responsible for
the primary inflation caused also the secondary stage of inflation (for
instance because the inflaton potential has two separate flat regions), so-
called hybrid inflation models in which inflation at its last stages was
driven not by the energy density of the inflaton field but by the vacuum
energy density induced by an interaction to another field, and those in which
another scalar field that was an energetically subdominant “spectator” field
during the primary phase of inflation either acquired a large vacuum
expectation value (vev) during the first inflationary stage or was kept frozen
in its initial value with a finite energy density and later on became the
dominant component, thus starting a new stage of inflation. Examples of the
first category above include scenarios studied recently in e.g. Refs. [200,
201, 202] (see also Refs. [203, 204] for recent studies on double inflation in
multifield models with a rapid turn in field-space), while scenarios belonging
to the second category were originally introduced in Refs. [205, 206] and
which have been studied in a large number of works since. However, generically
hybrid models predict a blue-tilted spectrum for primordial
perturbations666See Eq. (38) for a parameterization of the curvature power
spectrum. and are therefore now ruled out by observations [13].

A secondary stage of inflation started by a spectator field was originally
studied in Refs. [198, 199] and has been studied more recently in e.g. Refs.
[207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221].
Typically in scenarios where the secondary stage of inflation starts due to a
spectator field $\sigma$ acquiring a vev during the first stage of inflation,
the vev has to be very large for the field to become energetically dominant
while still in slow-roll to trigger a new period of inflation: for example,
for a quadratic spectator potential $V_{\sigma}\propto\sigma^{2}$ the
criterion is $|\sigma|\gtrsim M_{\rm P}/\sqrt{4\pi}$. A natural mechanism for
this, based on the stochastic evolution of light scalar fields in (quasi-) de
Sitter space was discussed in Ref. [216]. Another possibility is based on the
type of scenarios studied in e.g. Ref. [199], where the authors considered the
potential

$V=V_{0}+\lambda_{\phi}\phi^{4}\left[\ln\left(\frac{\phi^{2}}{f^{2}}\right)-\frac{1}{2}\right]+\lambda_{\sigma}\sigma^{4}\,,$
(7)

where $f$ is a mass scale related to the ultraviolet completion of the theory.
Here both fields are assumed to have a large vev initially,
$\phi_{i}\,,\sigma_{i}>M_{\rm P}$ and the field $\phi$ is assumed to drive the
first stage of inflation. If $\lambda_{\sigma}$ is not very small, the
stochastic quantum fluctuations of $\sigma$ acquired during the first stage of
inflation were smaller than its initial value, and the field remained
essentially frozen at $\sigma_{i}$. The first stage of inflation then ended
due to $\phi$ rolling down to $\phi\lesssim M_{\rm P}$, and the second stage
began shortly after when the field $\sigma$ became the energetically dominant
species. In practice, however, also a much simpler chaotic potential with a
suitable choice of parameters can permit similar dynamics.

Finally, we note that in addition to the scenarios discussed above, also the
recently proposed Transplanckian Censorship Conjecture [222] has motivated
studies on the occurrence of multiple stages of inflation, see e.g. Refs.
[223, 224] (however, see also Ref. [225]).

Figure 4: The black curve shows the evolution of the Hubble horizon $1/H$ as a
function of the scale factor $a$, whereas the gray dashed line shows, as an
example, a scale that re-enters horizon after the matter-radiation equality.

####  2.2.2 Thermal inflation

Next, we discuss scenarios where the secondary period of inflation is caused
by thermal effects. In this case the inflationary stage is known as thermal
inflation [226, 227], and it is is driven by a scalar field that at a high
temperature is far from its zero temperature vev. Part of the total energy
density of the Universe is then in the form of vacuum energy, and as
temperature drops, the vacuum energy can become the dominant energy density
component, therefore causing a period of inflation. Eventually, the thermal
inflation ends when the scalar field either tunnels or rolls down to a lower
energy minimum of its potential. Of particular interest is the former case,
where the vacuum energy dominance ends with a first-order phase transition
[228, 229, 230, 231] (see also Sec. 3.2). This happens by formation of bubbles
of the true vacuum which expand, finally turning the whole Universe into the
new phase. These bubbles nucleate sparsely and are initially small. They then
expand by many orders of magnitude and as they collide, most of the energy is
in very thin regions around the bubble walls that move almost at the speed of
light [232, 233, 234, 235]. After the bubbles have collided, the scalar field
oscillates around the true vacuum, potentially causing a short period of
matter-dominance before decaying and reheating the radiation bath. Similarly,
a short MD period can be caused in the case where the thermal inflation ends
with the scalar field rolling down to the new minimum.

A period of thermal inflation can be realized, for example, in classically
scale-invariant models or strongly-interacting theories [236, 237, 238, 239,
240, 241, 242, 235] 777We note that the early thermal inflation models
considered flat directions in supersymmetric and grand unified theories (see
e.g. Refs. [226, 227, 243, 244, 245, 246]).. In this case, the scalar
potential is of the form

$V\simeq
V_{0}+\frac{3g^{4}\phi^{4}}{4\pi^{2}}\left[\ln\left(\frac{\phi^{2}}{v_{\phi}^{2}}\right)-\frac{1}{2}\right]\,+g^{2}T^{\,2}\phi^{2}\,,$
(8)

where $g$ is a coupling constant and $v_{\phi}$ the vev of $\phi$. The scalar
field is stuck in a metastable phase that is separated from the true vacuum by
a potential energy barrier generated by purely thermal effects (the last term
in Eq. (8)). Therefore, a very long period of supercooling is possible,
because the potential energy barrier shrinks as the temperature decreases,
unlike in models where the potential energy barrier is dominated by nonthermal
terms. The potential energy barrier can also disappear at a nonzero
temperature in models that are not classically scale-invariant. In this case
the supercooling period is typically not very long.

An interesting consequence of thermal inflation ending to a first-order phase
transition is the formation of a stochastic gravitational wave background in
bubble collisions (see Sec. 4.4). In addition, the collisions of very
energetic bubble walls may lead to copious formation of PBHs [247, 248, 249]
(see also Sec. 3.6). Alternatively, in the case that the thermal inflation
ends by the field rolling down its potential PBHs can form from large density
perturbations generated when the thermal inflation potential turns from convex
to concave at the high-temperature minimum [250].

Finally, a secondary nonthermal stage of inflation can be caused by symmetry
restoration by nonthermal effects as the particles produced in preheating
after inflation can for a long time remain out of thermal equilibrium [251,
252, 253]. Also this inflationary stage ends with a symmetry breaking phase
transition that can be of first order [254, 255].

A common feature of all these scenarios is that if the secondary stage of
inflation lasted for a sufficiently long time, this becomes the “primordial”
inflation whose imprints on the structure of the Universe at different scales
are the only ones we can observe. On the other hand, if the secondary phase
only lasted for a short time, the initial conditions for large and small
structure formation decouple so that the first stage determines the spectrum
on large scales and the second the spectrum on small scales [199]. In addition
to providing initial conditions for large scale structure formation, the
primary stage of inflation provides initial conditions also for all the stages
that follow it. This has been argued to alleviate the problems some
inflationary models face with initial conditions [200].

###  2.3 Heavy particles and dark sectors (Authors: N. Bernal & J. Unwin)

In this section we discuss how nonstandard cosmological histories may arise
from different particle physics scenarios including heavy particles or dark
sectors.

####  2.3.1 Nonstandard cosmology from heavy particles

We start with the intuitive idea that a long lived heavy particle $\phi$ could
source an EMD epoch. The state $\phi$ could initially be in thermal
equilibrium with the SM states, or could be out-of-equilibrium at all times.
Once $\phi$ becomes nonrelativistic its contribution $\rho_{\phi}$ to the
energy density of the Universe will grow relative to the contribution due to
SM radiation $\rho_{R}$. Their energy densities evolve with the scale factor
according to $\rho_{\phi}\propto a^{-3}$ while $\rho_{R}\propto a^{-4}$. Thus,
even if $\phi$ initially contributes a small component to the total energy
density, then provided it is sufficiently long-lived it will grow to dominate
the energy density of the Universe and lead to an EMD epoch. These ideas first
arose in the context supersymmetry and string theory, see e.g. Refs. [42, 256,
257, 258, 259], and the particular case of string moduli fields will be
discussed in Sec. 2.4.

More generally, any massive meta-stable state could potentially come to
dominate the energy density of the early Universe. Specifically, in the early
Universe the evolution of $\rho_{\phi}$ and the SM entropy density $s$ are
governed by the following coupled Boltzmann equations [260]

$\displaystyle\dot{\rho}_{\phi}+3H\rho_{\phi}=-\Gamma_{\phi}\,\rho_{\phi}\,,$
(9) $\displaystyle\dot{s}+3Hs=+\frac{\Gamma_{\phi}\,\rho_{\phi}}{T}\,,$ (10)

where $\Gamma_{\phi}$ is the $\phi$ decay width, $H$ is the Hubble parameter,
and $T$ is the temperature of the SM photons. Notably, the evolution of $T$
can be tracked via Eq. (10), since the entropy density
$s(T)\equiv(\rho_{R}+p_{R})/T=2\pi^{2}\,g_{*,s}(T)\,T^{3}/45$. Additionally,
under the assumption that the SM bath maintains internal equilibrium at all
times in the early Universe, its temperature dependence follows from its
energy density $\rho_{R}(T)=\pi^{2}\,g_{*}(T)\,T^{4}/30$. Here $g_{*}(T)$ and
$g_{*,s}(T)$ correspond to the effective number of relativistic degrees of
freedom for SM energy and entropy densities, respectively. The Hubble
expansion rate $H$ is defined by
$H^{2}=8\pi(\rho_{\phi}+\rho_{R})/(3M_{P}^{2})$. While $\Gamma_{\phi}\ll H$
(such that the radiation bath is mainly composed of particles that are not
produced by decays) the entropy is separately conserved in each sector and the
scale factor and temperature scale as $T\propto a^{-1}$ (up to variations of
$g_{*,s}$). The expansion rate will be different depending on the fractional
distribution of the total energy density. Specifically, during the era of MD
one has $H\propto T^{3/2}$, whereas for the RD case $H\propto T^{2}$. Also
note that once $\phi$ decays, the entropy conservation in the bath no longer
holds (which corresponds to when the energy density in the relativistic decay
products of $\phi$ becomes significant), and the evolution switches to
$T\propto a^{-3/8}$ and thus $H\propto T^{4}$ [38, 261] until the Universe
transitions to the RD era.

Going beyond an EMD epoch, one can envisage instead that the Universe may have
been dominated at one stage by a particle species with a general equation of
state $w$, such that $\rho_{\phi}\propto a^{-3(1+w)}$ [113]. In this case the
Boltzmann equation (9) depends on $w$ and (assuming the scalar field is
decaying at zero-momentum) is instead given by888If $\phi$ is coherently
oscillating in a potential $V(\phi)\propto\phi^{n}$, the right hand side
should read $-\frac{2n}{2+n}\Gamma_{\phi}\,\rho_{\phi}$ [113].

$\dot{\rho}_{\phi}+3\,(1+w)H\rho_{\phi}=-\Gamma_{\phi}\,\rho_{\phi}\,.$ (11)

Thus, while the $\phi$ field dominates the energy density of the Universe the
Hubble rate scales as $H\propto a^{-\frac{3}{2}(1+w)}\propto
T^{\frac{3}{2}(1+w)}$.

As in the MD case, once the energy density in the relativistic decay products
of $\phi$ becomes significant this evolution switches to $H\propto T^{4}$
(regardless of the value of $w$ [262]). Simple cases corresponding to MD and
RD expansion are found for $w=0$ and $w=1/3$, respectively. Examples with more
general $w$ can occur for oscillating scalar fields with certain potentials
(e.g. periodic [263, 264] or special constructions [265, 266]), in brane world
cosmologies [267, 268], and for scalar-tensor theories [269, 270]. An
especially well motivated case is that of kination [271, 272, 273] in which
$\phi$ is a “fast-rolling” field whose kinetic term $\dot{\phi}$ dictates the
expansion rate of the post-inflation Universe, implying an equation of state
$w\approx 1$. In most models of interest $|w|\leq 1$, with $w=-1$
corresponding to dark energy or quintessence domination [274, 275]. For models
with $w>1$ there are concerns about superluminal propagation, although it has
been claimed this may be resolved in certain cases [265]. Importantly, for
$w>1/3$ the energy density $\rho_{\phi}$ gets diluted faster than radiation,
and if $\rho_{\phi}\ll\rho_{R}$ at $T\sim 1$ MeV, the late decay of $\phi$
typically has a negligible impact.

####  2.3.2 The transition to radiation domination

There are two main possibilities through which a period of nonstandard
cosmology can come to an end in the case that it is sourced by a population of
heavy state $\phi$ or similar physics. Firstly, if the energy density
$\rho_{\phi}$ redshifts faster than radiation ($w>1/3$) then after some
duration the SM radiation will dominate the energy density, leading to a
smooth transition to a RD Universe. Kination ($w=1$) is the prime example of
this scenario. Alternatively, the period of nonstandard cosmology could end
due to the decays of $\phi$ to lighter degrees of freedom which are produced
relativistically. This leads to a rather more abrupt exit from the era of
nonstandard expansion which coincides with $3H\sim\Gamma_{\phi}$. For
scenarios with $w<1/3$ (such as an EMD epoch) decays of $\phi$ provide the
principle route to a RD Universe prior to BBN.

A common model assumption is that the $\phi$ states can decay to SM particles
(and potentially also states beyond the SM). Note that if the decay products
of $\phi$ are produced with relativistic momenta, then when the population of
$\phi$ decay at $H\sim\Gamma_{\phi}$ this causes the Universe to transition
from a $\phi$-sourced era of nonstandard cosmology to a RD Universe.
Importantly, for successful BBN the temperature at the end of the
$\rho_{\phi}$ dominated phase must satisfy $T_{\rm reh}\gtrsim\mathcal{O}(1)$
MeV (see Sec. 4.1), where $T_{\rm reh}$ is determined by the decay rate,
$\Gamma_{\phi}=3H(T=T_{\rm reh})$, giving

$T_{\rm reh}^{4}\equiv\frac{5}{4\pi^{3}\,g_{*}(T_{\rm
reh})}\,M_{P}^{2}\,\Gamma_{\phi}^{2}\,.$ (12)

Note that if $T_{\rm reh}$ is below the electroweak phase transition
temperature, one must address questions regarding the origin of the baryon
asymmetry; for discussion and potential resolutions see e.g. Refs. [276, 277,
278, 279] and Sec. 3.2.

This transition to a RD Universe can have a number of important consequences,
most prominently the dilution of frozen-out cosmological abundances and the
nonthermal production of states, which we discuss next. For a particle species
$X$ which is decoupled from the thermal bath its comoving number density
$Y\equiv n_{X}/s$ is typically expected to be constant. However, the decays of
$\phi$ can lead to changes in the comoving number density of decoupled
species. The first possibility is that $n_{X}$ is increased due to
(nonthermal) production of $X$ via direct $\phi$ decays to these $X$ states.
In particular, it has been observed that by adjusting the $\phi$ branching
fraction to DM one can reproduce the observed relic density in a large range
of scenarios [42, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291,
292].

Alternatively, if $\phi$ decays dominantly to SM states then this leads to
heating of the thermal bath relative to the decoupled species, breaking the
common scaling of $n$ and $s$ [38, 260, 261, 293]. This scenario is often
referred to as an “entropy injection” (or “entropy dump”) since in the
instantaneous $\phi$ decay approximation it appears to imply a jump in the
entropy density, although in actuality the temperature of the bath simply
cools more slowly due to $\phi$ decays and at no point does the temperature
increase [38]. Notably, similar to nonthermal production one can adjust the
magnitude of dilution due to entropy injections to obtain the observed DM
relic abundance in many scenarios. Indeed, this is the premise behind Flooded
DM [278] and is also used in related scenarios which use entropy production to
adjust the DM abundance after thermal freeze-out [38, 294, 295, 296, 297, 298,
299, 300, 301, 302]. A dedicated discussion on the consequences of nonstandard
expansion phases on DM models is presented in Sec. 3.1

The magnitude of the entropy injection during the transition to RD epoch due
to $\phi$ decays depends on the contribution to the total energy density from
the $\phi$ states relative to the SM radiation. Assuming the instantaneous
decay approximation, in which all $\phi$ states decay simultaneously at
$T=T_{\rm reh}$, the entropy prior and after decays are related via

$\frac{s_{\rm before}}{s_{\rm after}}=\left[\frac{\rho_{R}(T_{\rm
reh})}{\rho_{\phi}(T_{\rm reh})}\right]^{3/4},$ (13)

up to variations of the relativistic degrees of freedom. Instantaneous $\phi$
decays is a reasonable approximation provided no physical processes of
interest (e.g. freeze-out) occur near the point of $\phi$ decays [38]. Since
it is assumed that below some critical temperature $T_{*}$ the $\phi$ energy
density starts evolving as $a^{-3(1+w)}$, whereas the radiation continues to
evolve as $a^{-4}$, over time the $\rho_{\phi}$ grows relative to $\rho_{R}$.
Thus to calculate the entropy injection one needs to track both the initial
contributions to the energy density at $T=T_{*}$ and the duration from this
point to $T=T_{\rm reh}$. Specifically, these energy densities can be followed
from inspection of the Friedmann equation (for $H(T_{*})>H(T)>\Gamma_{\phi}$)

$H(a)^{2}=H(a_{*})^{2}\left[\mathcal{R}_{\phi}\left(\frac{1}{\Delta
a}\right)^{3(1+w)}+\mathcal{R}_{R}\left(\frac{1}{\Delta a}\right)^{4}\right],$
(14)

where $\Delta a(T)\equiv a(T)/a(T_{*})$ and
$\mathcal{R}_{i}\equiv\rho_{i}/(\rho_{\phi}+\rho_{R})\big{|}_{T=T_{*}}$.

Provided that $\phi$ is sufficiently long lived that it becomes the dominant
energy density component, then between $T=T_{*}$ and $T=T_{\rm reh}$ the scale
factor grows by the following factor

$\Delta a_{\Gamma}\equiv\Delta a(T_{\rm
reh})\simeq\left[\frac{H(a_{*})^{2}\,\mathcal{R}_{\phi}}{\Gamma_{\phi}^{2}}\right]^{\frac{1}{3(1+w)}}.$
(15)

If one then assumes that prior to $T=T_{*}$ the $\phi$ are relativistic then
$\rho_{\phi}(T_{*})=\rho_{R}(T_{*})\,\mathcal{R}_{\phi}/\mathcal{R}_{R}$.
Further supposing all the energy in $\phi$ is transferred to radiation when it
decays, then it follows that the magnitude of the entropy dump is

$\Delta s=\left[\frac{\rho_{R}(T_{\rm reh})}{\rho_{\phi}(T_{\rm
reh})}\right]^{3/4}=\left[\frac{\rho_{R}(T_{*})}{\rho_{\phi}(T_{*})}\frac{\Delta
a_{\Gamma}^{-4}}{\Delta
a_{\Gamma}^{-3(1+w)}}\right]^{3/4}=\left[\frac{\mathcal{R}_{R}}{\mathcal{R}_{\phi}}\,\frac{1}{\Delta
a_{\Gamma}^{1-3w}}\right]^{3/4}\simeq\frac{\mathcal{R}_{R}^{3/4}}{\mathcal{R}_{\phi}^{\frac{1}{1+w}}}\,\left[\frac{T_{\rm
reh}}{T_{*}}\right]^{\frac{1-3w}{1+w}}.$ (16)

This ratio is important since it implies that for decoupled particle species
which are not significantly produced due to $\phi$ decays, then the associated
comoving number density will be reduced by a (potentially large) factor
$\Delta s$ as the states are diluted relative to the SM radiation. Note that
this dilution also applies to particle asymmetries [303]. In particular, it
dilutes any pre-existing baryon asymmetry in the Universe. This is discussed
in detail in Sec. 3.2.

We highlight that in determining the magnitude of the entropy injection the
temperature of the visible sector bath at which $\rho_{\phi}$ starts evolving
as $a^{-3(1+w)}$, denoted $T_{*}$, is a critical parameter. In the simple case
where $\phi$ is initially in thermal contact with the SM prior to decoupling
and then becomes nonrelativistic, it is reasonable to identify $T_{*}\sim
m_{\phi}$. However, if the $\phi$ decouple very early, or are never in thermal
contact, the $\phi$ could well be thermally distributed with a different
temperature and $T_{*}$ is a free parameter. We shall discuss establishing
temperature difference between sectors shortly. Another important example is
the case in which $\phi$ is a bosonic field $a$ oscillating in its potential,
such that the average equation of state is $w\simeq 0$ (matter-like), e.g. an
axion-like particle (ALP) or any scalar field oscillating in a quadratic
potential. The ALP case is distinct from the heavy particle scenario since $a$
becomes matter-like once it begins to oscillate around its minimum which
occurs for ALP mass $m_{a}\sim H$ [304, 305], corresponding to
$T_{*}\sim\sqrt{3m_{a}\,M_{\rm P}}$, for a simple quadratic potential
$V(a)\supset m_{a}^{2}\,a^{2}$.

####  2.3.3 Nonstandard cosmology beyond heavy particles

While our discussion thus far has been in the context of heavy elementary
particles, the state responsible for determining the equation of state of the
Universe need not be elementary and could be composite states, nonperturbative
states [54, 306], or other forms of matter. For instance, a natural proposal
is that a period of MD could arise if the energy density of the Universe is
mainly composed of PBHs which subsequently evaporate. Requiring that the
Universe transitions to RD epoch prior to BBN implies an upper mass bound on
the PBHs, around $6\times 10^{5}$ kg [307]. Additionally, it was recently
pointed out that PBHs could be very efficiently produced from the preheating
instability so that they dominate the energy density of the Universe, and
therefore the reheating no longer occurs because of inflaton decays, but
rather due to evaporation of the PBHs [197, 308]. PBHs are discussed further
in Sec. 3.6.

Dark sectors999It is common to refer the set of states which are charged under
the SM gauge groups, or otherwise with significant couplings to SM states, as
the visible sector. In contrast to this a dark sector (or hidden sector)
typically refers to a set of states which are neutral singlets under the SM
gauge groups and with weak direct coupling to SM states. offer another
interesting alternative through which to introduce nonstandard elements to
cosmology. Although the dark and visible sectors could be in equilibrium in
the early Universe, this need not necessarily be the case and in the converse
scenario there can potentially be significant temperature differences between
the two sectors. Such temperature differences can arise due to asymmetric
inflationary reheating of sectors [309, 310, 311]. Moreover, even in the case
that the visible and dark sector are initially decoupled with similar
temperatures, subsequently evolution can lead to temperature differences.
Examples in which decoupled dark sectors dynamically develop different
temperatures include self-heating via semi-annihilation processes [312] or
self-interactions [313, 314, 315], also via number-depleting annihilations via
3-to-2 [313, 316, 317] or 4-to-2 [318, 319, 320, 321] processes, and in co-
decaying DM models [322].

To appreciate (one reason) why temperature differences between sectors can be
important, consider the case in which the temperature of $\phi$ states
$T_{\phi}$ is significantly lower than that of the SM photon temperature $T$.
Then the $\phi$ become nonrelativistic at $T_{\phi}\sim m_{\phi}$ which could
be much earlier than the point at which $T\sim m_{\phi}$. Thus the states
$\phi$ need not necessarily be extremely heavy, relative to SM states, in
order to give rise to an EMD era. Furthermore, the temperature of the visible
sector at which $\phi$ becomes nonrelativistic corresponds to $T_{*}$ in Eq.
(16) and, notably, if $T_{*}$ can be treated as a (largely) free parameter
this permits a large range in the magnitude of the entropy injection due to
$\phi$ decays, and increases the potential impact of dilution on decoupled
particle species.

###  2.4 Moduli fields (Authors: K. Sinha & S. Watson)

String theories and M-theory compactifications yield a large number of scalar
fields, or moduli. A well-studied example is Calabi-Yau compactification where
these moduli correspond to the complex structure and Kähler, also below
deformations of the internal manifold [323, 324]. Generating masses for moduli
has been a central program in string phenomenology and cosmology [325, 326,
327, 328, 329]. This is necessitated by several considerations: massless
scalars lead to long range interactions which are severely constrained by
“fifth force” experiments. Moreover, if the couplings of moduli to different
sectors of matter fields are non-universal, the corresponding long-range
forces would violate the equivalence principle. An additional question is that
of predictability: the vacuum expectation values of moduli determine the
parameters of the low-energy theory and thus should be fixed. Moduli
stabilization has been a central program in string theory over several decades
and the construction of the effective potential draws upon many ingredients
that are central to string theory: branes, fluxes, and strong dynamics in the
hidden sector. Explicit constructions are still being actively discussed by
the community [330, 331, 332, 333, 334].

From the perspective of low-energy phenomenology, an important question about
moduli stabilization is the resulting mass $m_{\sigma}$ of the modulus
$\sigma$. The physics of moduli should decouple at scales $E\ll m_{\sigma}$.
Nevertheless, decoupled moduli can still impact the pattern of the low-energy
particle spectrum in many situations, for example when they are responsible
for supersymmetry breaking [335, 336, 337, 338]. On the other hand, the impact
of light moduli on the evolution of the early Universe can also lead to many
different observational predictions [283, 339, 340, 341]. In the early
Universe, moduli are typically displaced from the minimum of their effective
potential and undergo coherent oscillations that can mimic a MD era [341,
284]. Since moduli couple to other fields gravitationally, their decay width
is given parametrically as $\Gamma_{\sigma}\sim m_{\sigma}^{3}/M^{2}_{\rm P}$.
Depending on the mass $m_{\sigma}$, moduli can decay late enough to affect the
successes of BBN. This is the so-called cosmological moduli problem, which
first appeared as the Polonyi problem in early versions of spontaneously
broken supergravity. Stated most generally, the problem is that supersymmetry
breaking in the hidden sector with gravity mediation often contains scalar
fields that have weak scale masses and couple gravitationally. After
inflation, these fields behave like nonrelativistic matter and dominate the
energy density of the Universe until after BBN, spoiling its successful
predictions [342, 343, 257, 344].

The cosmological moduli problem may actually be more of an opportunity if the
masses of the moduli result in decays prior to BBN. Indeed, this situation
leads to new cosmological predictions and an alternative to a standard thermal
history. That is, if a modulus is massive enough to decay before BBN, but not
so massive as to decouple entirely from low energy physics (a situation that
commonly occurs in UV complete examples), its effects on DM, baryogenesis, and
structure formation can lead to novel and interesting predictions [341]. The
late decay of moduli can be viewed as a second reheating of the Universe
(after the first, inflationary one), with the production of relativistic SM
and other light (beyond the SM) particles. Depending on the masses and
couplings of the modulus to other fields, such as DM and baryons, this
framework can lead to new expectations for post-inflationary cosmology prior
to BBN.

The critical question then is: what is the mass $m_{\sigma}$ of a modulus in
string theory, or, framed another way, what is the behaviour of the curvature
of the effective potential near its minimum? Given that the scales relevant in
string theory are the Planck scale, the string tension, and the typical size
of the extra dimensions (the Kaluza-Klein scale), one would expect a priori
that $m_{\sigma}$ is determined by a combination of these scales and that
there is no obvious role of a low-energy scale in the stabilization mechanism.
If that were the case, moduli would be too massive and decay too early to
leave discernible imprints on pre-BBN cosmology.

In explicitly constructed examples of moduli stablization, a common result is
that some moduli remain parametrically lighter than the string scale. This can
be understood at the level of effective supergravity, since the key is the
connection between moduli stabilization and supersymmetry breaking. The
effective potential determines the moduli F-terms dynamically. This, combined
with the Kähler metric of the matter sector, fixes the strength of gravity
mediation. More concretely, the scalar potential in the supergravity limit is
given by

$V=e^{K/M^{2}_{\rm
P}}\left(\sum_{\sigma}\left|D_{\sigma}W\right|^{2}-{\frac{3}{M^{2}_{\rm
P}}|W|^{2}}\right)\,\,,$ (17)

where $W(\sigma)$ and $K(\sigma)$ are the superpotential and Kähler potential,
respectively. The gravitino mass is defined as $m^{2}_{3/2}=\langle
e^{K/M^{2}_{\rm P}}|W|^{2}/M^{4}_{\rm P}\rangle$. At the extremum,
$\partial_{\sigma}V=0$ and $\partial_{\sigma\sigma}V\,>\,0$. Moreover, an
almost vanishing cosmological constant requires $V\simeq 0$. Then, both terms
in Eq. (17) are forced to be $\sim m^{2}_{3/2}$. Since the modulus mass matrix
is positive definite, the smallest eigenvalue should then be
$\mathcal{O}(m_{3/2})$. For a model with a single modulus with nonvanishing
F-term, the trivial conclusion is
$\partial_{\sigma\sigma}V\,\sim\,m^{2}_{3/2}$.

This set of conditions is restrictive enough to yield $m_{\sigma}\sim
m_{3/2}$. Indeed, taking into account a redefinition to obtain canonical
kinetic terms, one obtains [341]

$m_{\sigma}\,\sim\,\langle\sigma\rangle\,m_{3/2}\,\,\sim\,\,\mathcal{O}(1-100)\,\times\,m_{3/2}.$
(18)

We note that to remain in the supergravity limit, stabilization schemes
typically require $\langle\sigma\rangle\,\gtrsim\mathcal{O}(1)$ in Planck
units. The exact hierarchy between the mass of the modulus and the gravitino
mass, however, depends on the specific stabilization scheme. For example, in
M-theory compactifications the pre-factor is $\mathcal{O}(1)$, while for type
IIB Kachru-Kallosh-Linde-Trivedi (KKLT) compactifications the pre-factor is
$\mathcal{O}(10-100)$.

The corresponding reheat temperature is given by $T_{\rm reh}\sim
g_{*}^{-1/4}\sqrt{\Gamma_{\sigma}\,M_{\rm P}}$ or

$T_{\rm
reh}=c^{1/2}\left(\frac{10.75}{g_{*}}\right)^{1/4}\left(\frac{m_{\sigma}}{50\,{\rm
TeV}}\right)^{3/2}\,{\rm MeV}\,,$ (19)

where $g_{*}$ is the number of relativistic degrees of freedom at $T_{\rm
reh}$, and $c$ is a model-dependent $\mathcal{O}(0.1-1)$ number. It is
intriguing that the lower bound on the modulus mass required to avoid the
cosmological moduli problem corresponds to the gravitino mass scale required
to address the electroweak hierarchy problem.

Although the appeal to supergravity shown above is suggestive, the expectation
of light moduli at the TeV scale does not necessarily depend on having low-
scale supersymmetry as a solution to the hierarchy problem. Statistical
arguments in the type IIB string landscape indicate that the distribution of
moduli masses is logarithmic in vacua with stabilized Kähler moduli [345].
Since explicit compactifications feature moduli in the mass range we have
discussed, one can expect a preponderance of nearby vacua with similar moduli
masses. To our knowledge such statistical arguments have not been advanced for
M-theory compactifications, but similar results can be expected to hold in
those cases as well. Indeed, there are indications from recent work that very
general features of string compactifications lead to the existence of moduli
with masses exponentially suppressed compared to the Planck scale. In explicit
constructions, the moduli masses arise from exponentially suppressed
contributions to the superpotential coming from non-perturbative effects (in
KKLT, such contributions are essential to obtain a small value of the Gukov-
Vafa-Witten flux superpotential $W_{0}$ [346]; in Large Volume Scenarios, they
occur in the superpotential term for the Kähler moduli).

Another interesting result from these models is that the reheat temperature is
deeply connected to fundamental physics and is not a free parameter. Since
moduli are ubiquitous in string theory, a nonstandard cosmological history
thus appears to be a robust possibility and perhaps an inevitable one. Of
course, connecting the physics of moduli to a consistent nonstandard
cosmological history presents many challenges. We now turn to a discussion of
some of these issues.

One class of challenges lies in consistent embeddings of the matter sector
into a given string compactification and the interactions of various particles
with moduli. For example, the moduli-induced gravitino problem [347, 348, 283]
arises because gravitinos produced from moduli themselves subsequently decay
and produce DM in supersymmetric models. This tends to overclose the Universe
unless the decay of the modulus to the gravitino is suppressed due to
branching or kinematics. Whether or not such blocking is possible is a model-
dependent question. Similarly, the moduli-induced dark radiation problem
arises from the fact that moduli are either accompanied by light pseudoscalars
in typical string compactifications, or themselves decay to such axion-like-
particles [349, 350, 351]. The existence of such relativistic species in the
dark sector is subject to $\Delta N_{\rm eff}$ constraints from cosmological
observations, in turn imposing constraints on the moduli physics. Another
class of challenges lies in various features of the low-energy effective
potential of the modulus. An example of this is the “overshoot problem”: since
moduli are displaced from their minima during inflation, this displacement
should not be such that a modulus overcomes the barrier protecting the local
low-energy minimum. This problem is again model-dependent and depends on the
stabilization mechanism, and may suggest a correlation of the scales of
supersymmetry breaking (the barrier height) and inflation [352, 353]. Many of
these issues require further theoretical investigation. Regardless, we have
seen that a nonstandard cosmological history is a robust prediction of
string/M-theoretic approaches to beyond-SM physics.

There can be a number of effects of a decaying modulus on early Universe
cosmology: nonthermal production of DM [354, 348], late-time baryogenesis
[355, 277, 356], effects on the CMB [340] and LSS [357], and implications for
the matter power spectrum [45], effective number of neutrino species [349],
and isocurvature constraints. We briefly discuss some of these topics, leaving
a more detailed discussion for other authors in this review.

Gravitational effects: matter power spectrum and CMB: We first discuss the
effect of moduli on the DM halo. In a RD Universe, DM perturbations
$\delta\rho/\rho$ grow logarithmically with the scale factor. In a matter-
dominated Universe, on the other hand, they grow linearly:
$\delta\rho/\rho\sim a(t)\sim t^{2/3}$. This implies that while in standard
cosmology the growth of structure only becomes significant after matter-
radiation equality, an early MD era with cosmological moduli results in an
early period of growth and a new scale at smaller wavelengths in the matter
power spectrum. For $T_{\rm reh}$ near the BBN limit, the result may be the
formation of Earth-sized, ultracompact microhalos [45, 43, 358]. If the DM
decouples both thermally and kinetically prior to the subsequent onset of RD
epoch, this enhancement of the small-scale matter power spectrum may survive,
with profound consequences for WIMP detection [358, 359, 360] and axion
cosmology [48]. For a more detailed discussion on effects on the matter power
spectrum, see Sec. 3.5 and Sec. 4.2.

We next turn to possible effects of moduli on the CMB [340]. The main point
here is that the equation of state (e.g. thermal $w=1/3$ or nonthermal $w=0$)
in the aftermath of inflation determines the expansion rate of the Universe.
Thus, the rate at which primordial perturbations re-enter the Hubble horizon
is impacted by an era when moduli dominate the energy density. The total
number of e-folds is given by $N_{k}={\rm ln}(a_{\rm end}/a_{k})$, where
$a_{\rm end}$ and $a_{k}$ are the scale factors at the end of inflation and
when the pivot scale exited the horizon, respectively. It is dependent on,
apart from the inflationary potential, the value of the energy density at the
end of inflation and the energy density at which the Universe is assumed to
become thermalized. The energy densities in turn depend on the mass and
reheating temperature of the modulus. Thus, the properties of the modulus
sector impact inflationary observables through $N_{k}$ and conversely, CMB
observations can be used to constrain the modulus sector. For a more detailed
discussion on inflationary observables, see Sec. 3.3.

Nonthermal DM: Moduli-dominated cosmological histories have a major impact on
the standard thermal freeze-out paradigm of DM. The reason is that the decay
of the modulus dilutes any previously existing population of DM by a factor
$\Omega_{\rm CDM}\rightarrow\Omega_{\rm CDM}(T_{\rm reh}/T_{f})^{3}$, where
$T_{f}$ is the DM freeze-out temperature. DM is produced during the modulus
reheating process at low temperatures $\sim T_{\rm reh}$. If the number
density of DM particles produced exceeds the critical value
$n^{c}_{x}=(H/\langle\sigma_{x}v\rangle)_{T=T_{\rm reh}}$, then the particles
quickly annihilate down to this value, which is an attractor. The result is a
parametric enhancement of the relic density $\Omega^{NT}_{\rm
DM}\,=\,\Omega^{T}_{\rm DM}\left(T_{f}/T_{\rm reh}\right)$. If, on the other
hand, the number density of DM particles produced from modulus decay is lower
than the attractor value, no further annihilation takes place and the final
relic density is simply given by $n_{\rm DM}\sim n_{\sigma}\times{\rm
Br}_{\sigma\rightarrow{\rm DM}}$, where ${\rm Br}_{\sigma\rightarrow{\rm DM}}$
is the branching ratio for modulus decay to DM and $n_{\sigma}$ is the number
density of the scalar condensate. Due to the extra freedom in choosing $T_{\rm
reh}$ in the first case and ${\rm Br}_{\sigma\rightarrow{\rm DM}}$ and
$n_{\sigma}$ in the second, a much wider range of DM masses and annihilation
cross-sections can satisfy the observed relic density constraint compared to
the thermal freeze-out scenario. For a more detailed discussion on DM, see
Sec. 2.3 and Sec. 3.1.

Baryogenesis: Just as in the case of DM, the decay of a modulus dilutes baryon
asymmetry existing from any previous era and has to be regenerated. This turns
out to be quite natural. The decay of the modulus provides the non-equilibrium
condition required for baryogenesis and a typical value for the dilution
factor101010The dilution factor is defined as $Y\equiv n_{\sigma}/s=3T_{\rm
reh}/(4m_{\sigma})$, where the last expression can be obtained directly from
the Boltzmann equation for $n_{\sigma}$. is $Y\sim 10^{-8}$. If the modulus
decays to a field that has $CP$\- and $B$\- violating couplings to SM fields,
baryogenesis can occur through usual interference between tree and loop-level
diagrams. These interference terms are usually suppressed by factors like
$(1/4\pi)$, so that a net factor of $\sim 10^{-10}$ for the baryon asymmetry
can be obtained with $\mathcal{O}(1)$ Yukawa couplings [355, 277]. For a more
detailed discussion on baryogenesis, see Sec. 3.2.

###  2.5 Post-BBN modifications (Authors: T. Karwal & T. L. Smith)

In this section, we discuss causes of nonstandard, post-BBN expansion history.
We outline examples of modifications to the properties of the standard
$\Lambda$CDM components, and scenarios which introduce new components
altogether.

These two categories can be further refined into modifications to the
background and/or perturbative evolution. The conservation of stress-energy
dictates how all minimally coupled components evolve in both the (homogeneous
and isotropic) background as well as at linear order in their inhomogeneous
perturbations (see e.g. Ref. [361]):

$\displaystyle\dot{\bar{\rho}}$ $\displaystyle=$
$\displaystyle-3\frac{\dot{a}}{a}\bar{\rho}(1+w)\,,$ (20)
$\displaystyle\dot{\delta}$ $\displaystyle=$
$\displaystyle-(1+w)(\theta+3\dot{\phi})-3\frac{\dot{a}}{a}\left(c_{s}^{2}-w\right)\delta\,,$
(21) $\displaystyle\dot{\theta}$ $\displaystyle=$
$\displaystyle-\frac{\dot{a}}{a}(1-3w)\theta-\frac{\dot{w}}{1+w}\theta+\frac{c_{s}^{2}}{1+w}k^{2}\delta-k^{2}\Sigma+k^{2}\psi\,,$
(22)

where dots indicate derivatives with respect to conformal time, $\bar{\rho}$
is the fluid’s homogeneous density, $w\equiv\bar{P}/\bar{\rho}$ is its
equation of state parameter, $\delta\equiv\delta\rho/\bar{\rho}$ is its
density contrast, $\theta$ is the divergence of its velocity perturbation,
$c_{s}^{2}\equiv\delta P/\delta\rho$ is its sound speed, $\phi$ and $\psi$ are
the temporal and spatial gravitational metrics in conformal Newtonian gauge,
and $\Sigma$ is the fluid’s anisotropic stress.

The simplest modifications to the $\Lambda$CDM expansion history can be
characterised as changes to $w$ in Eq. (20). Models which do this can
generically be characterized by the way in which the equation of state
parameter, $w$, changes with time. For example, quintessence models which
promote the cosmological “constant” to a dynamical quantity predict some time
variation in the dark energy equation of state, $w_{\rm de}$, affecting the
late-time expansion history relative to “$\Lambda$”-CDM. Particular
quintessence models have the property that $w_{\rm de}$ tracks the matter
equation of state parameter, $w_{\rm m}$ [362, 363, 364, 365]. These “tracker”
models are particularly interesting as potential solutions to the coincidence
problem of dark energy. Such tracker dark energies can arise from various
mechanisms including scalar fields with particular potentials [363],
quintessence and k-essence models [366, 367, 368] and alternative models of
gravity [365]. Models which propose additional forms of energy density, beyond
what is contained in $\Lambda$CDM, can also modify the post-BBN expansion
history. A universe which contains a collection of scalar fields may undergo
periodic epochs of anomalous expansion (see e.g. Refs. [369, 370]). Some
models of minimally coupled “early” dark energy (EDE) envision a scalar field
which is fixed at some point in its potential by Hubble friction which becomes
dynamical when the Hubble parameter drops below some critical value around the
matter-radiation equality [371, 372]. The fractional contribution of this
field to the total energy density briefly increases and then decreases, as
long as it dilutes faster than matter. Other models propose a nonminimal
coupling to other components (such as the neutrino sector) which then produces
a “kick” to the scalar field, again leading to a brief increase in the total
energy density of the Universe [373]. Due to its success at providing a
compelling resolution to the Hubble tension [374], numerous particle models
have now been proposed to comprise of such EDEs, including axion-like [375,
376, 377, 372] and power-law scalar fields [378] with nonstandard kinetic
terms [379], and first-order phase transitions in the dark sector [380].

Several models of nonstandard expansion also arise due to interactions between
the different components that fill the Universe. Conservation of stress-energy
in the presence of exchange of energy-momentum between different components
($X$ and $Y$) can be written as (see e.g. Ref. [381])

$\nabla_{\mu}T^{\mu}_{X\nu}=Q_{\nu}\,,\ \
\nabla_{\mu}T^{\mu}_{Y\nu}=-Q_{\nu},$ (23)

where $Q_{\nu}$ is the energy-momentum flux exchanged between the two
components. The form of $Q_{\nu}$ depends on the specific physics of the
coupling, but a natural choice is to write the energy-momentum transfer as a
linear combination of the four-velocities [382]

$Q_{\nu}=\zeta_{X}u_{X\nu}+\zeta_{Y}u_{Y\nu}\,,$ (24)

where $u_{(X,Y)\nu}$ is the four-velocity of the components. Models with
interactions of this form include interactions between dark matter and dark
energy [383], developed in order to address the coincidence problem [384,
385]. A similar inter-species interaction is DM decaying into dark radiation
[386, 387], invoked to explain the $\sigma_{8}$ and Hubble tensions. While
there are stringent bounds on all DM behaving this way, a fraction of DM
annihilating into dark radiation is permitted by cosmological data. Depending
on the rate $\Gamma$ of decay of the DM, when the expansion rate
$H\sim\Gamma$, all the interacting DM decays into dark radiation over one
cosmic time scale, which rapidly dilutes away. If this event occurs during MD,
the expansion of the Universe is suddenly slowed by a factor roughly
proportional to the square-root of the fraction of decaying DM, and then
follows the usual dilution of matter density again. Close to matter-radiation
equality or matter-dark-energy equality, such an event can shift the redshift
of equality. At other times, when matter is subdominant, the effects of
decaying DM on the background evolution of the Universe are minimised, with
the principal effects being confined to the evolution of perturbations.

Given the form of the energy-momentum exchange rate in Eq. (24), at the
background level we have

$\displaystyle\dot{\bar{\rho}}_{X}+3H\bar{\rho}_{X}(1+w_{X})$ $\displaystyle=$
$\displaystyle Q_{0}\,,$ (25)
$\displaystyle\dot{\bar{\rho}}_{Y}+3H\bar{\rho}_{Y}(1+w_{Y})$ $\displaystyle=$
$\displaystyle-Q_{0}\,,$

which generalize Eq. (11). The impact of this interaction on the expansion
rate can be more simply understood once we realize that the sum of the
continuity equations for the interacting components is, itself, conserved.
Therefore, at the background level, the two components behave as a single,
effective, component with an effective equation of state parameter $w_{\rm
eff}\equiv(\bar{\rho}_{X}w_{X}+\bar{\rho}_{Y}w_{Y})/(\bar{\rho}_{X}+\bar{\rho}_{Y})$.
In the noninteracting case, $Q_{0}=0$, for constant equation of state
parameters, we know that $\rho_{X,Y}\propto a^{-3(1+w_{X,Y})}$ so that, taking
$w_{X}>w_{Y}$, $w_{\rm eff}$ monotonically transitions from $w_{X}$ to $w_{Y}$
as the Universe expands. In the interacting case, $Q_{0}\neq 0$, as energy
density flows from one component to the other, $w_{\rm eff}$ will deviate from
a monotonic evolution, leading to a change in the expansion history. Note that
it is possible to have $w_{X}<-1$ (for example, a scalar field with a non-
canonical kinetic term, e.g., Ref. [388]) in which case $w_{\rm eff}<-1$. In
this case we can still model the background and perturbative dynamics as an
effective fluid [389]. A similar effect is seen when working out the dynamics
in some modified gravity theories. For example, we can write any scalar-tensor
gravity theory in the Einstein frame, in which the modified expansion is due
to a nonminimal coupling between the scalar field and other components [390].
In addition to this, an agnostic treatment of the microscopic dynamics of dark
energy makes it indistinguishable from modified gravity [391]. This effective
treatment of modified gravity and nonstandard dark energy interactions is the
main focus of the “parameterized post-Friedmann” formalism [392].

Finally, DM-baryon interactions [393, 394, 395] have been recently considered
as a potential explanation of the anomalous measurements from EDGES [37]. In
these models the DM-baryon cross-section behaves as $\sigma=\sigma_{0}v^{n}$,
where $v$ is the relative velocity bewteen the scattering particles. This
velocity dependence may arise in several natural interactions, such as
electric- or magnetic-dipole interactions through light mediators ($n=-2$) or
Coulomb-like interactions or Yukawa interactions through light mediators
($n=-4$).

It is important to note that when the background equations are modified, in
order to maintain energy-momentum conservation at the linear level, the
perturbation equations must also be modified. However, there are interactions
that preserve the background equations and just modify the evolution of the
linear perturbations. Thomson scattering between photons and baryons and any
self-interactions are examples of such mechanisms. In the remainder of this
section we will consider the effects of scattering so that the number of each
constituent remains unchanged.

For a self-interacting relativistic component [396, 397], the linear
conservation equations [Eqs. (21) and (22)], remain unchanged. The
modifications enter through the evolution of the component’s anisotropic
stress, $\Sigma$. The evolution of $\Sigma$ is governed by the Boltzmann
equation which, in turn, is sourced by a collision term. In Refs. [398, 399]
it was shown that, in the case of neutrino self-interactions mediated by a
massive scalar, the collision integral can be approximated to be of the form
$\mathcal{C}[\mathcal{F}]=\mathcal{F}/\dot{\tau}$, where $\mathcal{F}$ is the
component’s distribution function and $\dot{\tau}$ is the rate at which the
component’s opacity changes. In the absence of a collision term, the equations
of motion for the anisotropic stress $\Sigma$ cause a decrease over time of
the density contrast and velocity perturbation (i.e. “free-streaming”); self-
interactions disrupt this and lead to enhanced clustering for that component.
On the other hand, self-interacting DM [400, 401] has been considered to
alleviate a variety of small-scale structure “problems”. However, the typical
interaction cross-sections that are considered are only active in collapsed,
nonlinear, structures and hence have little effect on the linear physics
probed by the CMB and large-scale structure observations.

Models which excite initial conditions beyond the standard adiabatic initial
conditions also modify the late-time evolution of perturbations. Any model of
inflation that involves more than one dynamically important scalar field, such
as the curvaton scenario [402] or axion DM [403], produce nonadiabatic initial
conditions, which will be discussed in more detail in Sec. 3.4 (see also Ref.
[404]). As a specific example, we mention compensated isocurvature
perturbations which balance an overdensity of baryons with an underdensity of
cold DM and vice-versa (see e.g. Refs. [405, 406]). This scenario leaves the
local matter density perturbations unchanged but affects the plasma sound
speed $c_{s}^{2}$, leading to a smoothing of primordial perturbations. This
has been suggested as a solution to the $A_{\rm lens}$ anomaly [407, 408] and
may partially alleviate the Hubble tension through large-scale fluctuations in
$\Omega_{\rm b}$ [409].

Finally, models which promote “constants” of nature to dynamical fields may
also cause nonstandard cosmological evolution. Scalar-tensor theories of
gravity, mentioned before, are an example of this, leading to a dynamical
gravitational coupling, $G(\vec{x},t)$. Other intriguing examples are models
of a dynamical fine-structure constant [410, 411]. These models predict a time
and spatial variation in the strength of electromagnetic interactions (such as
Thomson scattering) which has a unique impact on the structure of the observed
CMB anisotropies (see e.g. Ref. [412, 413]).

##  3 Consequences of nonstandard expansion phases

In this section, we will discuss several consequences of nonstandard expansion
eras, paying particular attention to models of dark matter, early Universe
phase transitions and baryogenesis, models of inflation, and generation of the
primordial curvature perturbations, as well as microhalos and PBHs.

###  3.1 Consequences for dark matter (Authors: A. Berlin, D. Hooper & G.
Krnjaic)

If the early Universe was not consistently dominated by radiation as is
typically assumed, the phenomenology of DM production and detection could be
substantially altered from standard expectations. In particular, as discussed
in Sec. 2.3, if the visible sector (which contains the SM) is supplemented by
a decoupled hidden (or dark) sector (which contains the DM), these two sectors
can be produced independently during post-inflation reheating and maintain
separate temperatures throughout cosmological evolution. Within the hidden
sector, the DM could annihilate to lighter hidden sector states which
ultimately decay into SM particles. If these unstable states are sufficiently
heavy and long-lived, they will come to dominate the energy density of the
Universe before decaying to SM particles, thereby diluting the abundances of
all relics, including that of the DM [298, 296, 297]. This makes it possible
to circumvent the well-known $\sim$100 TeV upper limit on the mass of the DM,
based on partial wave unitarity [414] (for related scenarios, see Refs. [415,
416, 417, 341, 418, 295, 278, 419, 227, 420, 421, 422, 39, 261, 282, 423, 284,
424, 288, 265, 425, 426, 289, 301, 300, 427, 428, 429, 430, 431, 432, 433,
302, 434, 435, 436, 437, 438, 439, 290]; for alternative “freeze-in”
production of DM in scenarios with a nonstandard expansion phase, see Refs.
[440, 441, 425, 442, 443, 426, 289, 301, 444, 262, 292, 445, 446, 447, 448]).

Long lifetimes are straightforwardly realized if the decaying particle is the
lightest hidden sector state. In fact, if the hidden and visible sectors are
decoupled, the lightest hidden sector state will automatically be long-lived,
since its width relies on a coupling that is too small to sustain thermal
equilibrium between the two sectors. This picture is relatively universal, and
can be found within any model in which the DM freezes out through
annihilations in a heavy and highly decoupled hidden sector that is populated
after inflation. In contrast to scenarios in which an additional out-of-
equilibrium decay is invoked solely to dilute the initial cosmological
abundances of various species, dilutions of this kind are an inevitable
consequence of thermal decoupling.

In this framework, the DM coupling to the decaying particle is essential for
generating the relic abundance. Initially the DM and lighter long-lived state
are in chemical equilibrium until the DM freezes out by annihilating to the
lighter state when the dark sector temperature falls below the DM mass.
However, unlike in conventional freeze-out, there is no thermal contact with
the SM and there is a substantial entropy transfer when the lighter unstable
particle eventually dominates the energy density and then heats the SM
radiation bath upon decay.

Although this class of scenarios can be realized within the context of a wide
of range of hidden sector models, we will consider a simple vector portal
model as an illustrative example. For our DM candidate, we introduce a stable
Dirac fermion, $X$, which has unit charge under a spontaneously broken
$U(1)_{X}$ gauge symmetry, corresponding to the massive gauge boson,
$Z^{\prime}$. The hidden sector Lagrangian contains

${\cal L}\supset-\frac{\epsilon}{2}B^{\mu\nu}Z^{\prime}_{\mu\nu}+g_{\rm
DM}Z^{\prime}_{\mu}\overline{X}\gamma^{\mu}X\,,$ (26)

where $Z^{\prime}_{\mu\nu}$ and $B_{\mu\nu}$ are the $U(1)_{X}$ and
hypercharge field strengths, respectively, and $\epsilon$ quantifies the
degree of kinetic mixing between them [449, 450]. If $\epsilon$ is small (a
choice which is technically natural), and the $Z^{\prime}$ is the lightest
hidden sector particle, it will be long-lived, leading it to dominate the
energy density of the Universe before decaying. The DM undergoes the process
of thermal freeze-out entirely within the hidden sector, dictated by the
cross-section for $X\bar{X}\rightarrow Z^{\prime}Z^{\prime}$. For $m_{X}\gg
m_{Z^{\prime}}$ this cross-section is given by $\sigma
v\simeq\pi\alpha^{2}_{X}/m^{2}_{X}$, where $\alpha_{X}\equiv g_{\rm
DM}^{2}/4\pi$.

As an initial condition, we take the hidden and visible sectors to be
described by separate thermal distributions, with temperatures of $T_{\rm h}$
and $T$, respectively. The ratio of these temperatures, $\xi_{\rm
inf}\equiv(T_{\rm h}/T)_{\rm inf}$, is determined by the physics of inflation,
including the sectors’ respective couplings to the inflaton [451, 452]. From
entropy conservation in each sector, we can calculate the time evolution of
$\xi$ (prior to the decays of $Z^{\prime}$)

$\frac{s_{h}}{s}=\frac{g^{h}_{*,s}}{g_{*,s}}\,\xi^{3}={\rm
constant}\qquad\longrightarrow\qquad\xi=\xi_{\rm
inf}\,\bigg{(}\frac{g^{h}_{*,s,{\rm
inf}}}{g^{h}_{*,s}}\bigg{)}^{1/3}\,\bigg{(}\frac{g_{*,s}}{g_{*,s,{\rm
inf}}}\bigg{)}^{1/3}\,,$ (27)

where $s$ and $s_{h}$ are the hidden and visible sector entropy densities,
$g_{*,s}$ and $g_{*,s}^{h}$ are the numbers of effective relativistic entropy
degrees of freedom in the visible and hidden sectors. If the SM temperature is
well above the electroweak scale, $g_{*,s}\simeq g_{*,s,{\rm inf}}$. As the
temperature of the hidden sector falls below $m_{X}$, $g^{h}_{*,s}$ decreases
from $g_{Z^{\prime}}+(7/8)g_{X}$ to $g_{Z^{\prime}}$, bringing $\xi$ from
$\xi_{\rm inf}$ to $(13/6)^{1/3}\,\xi_{\rm inf}\approx 1.3\,\xi_{\rm inf}$,
for $m_{Z^{\prime}}\ll m_{X}$.

As the Universe expands, $X$ will eventually freeze-out of chemical
equilibrium, yielding a nonnegligible relic abundance. The evolution of the
number density of $X$ (plus $\bar{X}$), $n_{X}$, is described by the Boltzmann
equation

$\dot{n}_{X}+3Hn_{X}=-\frac{1}{2}\langle\sigma
v\rangle~{}\left(n_{X}^{2}-\frac{n^{2}_{Z^{\prime}}}{\,n^{2}_{Z^{\prime},{\text{eq}}}}n_{X,{\text{eq}}}^{2}\right)\,,$
(28)

where $\langle\sigma v\rangle$ is the thermally averaged cross-section for the
process $X\,\bar{X}\rightarrow Z^{\prime}\,Z^{\prime}$, and
$H=[8\pi\,(\rho_{R}+\rho_{h})/3\,M_{\rm P}^{2}]^{1/2}$ describes the expansion
rate of the Universe in terms of the energy densities in the visible and
hidden sectors.

In the case that $n_{Z^{\prime}}$ remains close to its equilibrium value
during the freeze-out of $X$, the Boltzmann equation can be solved semi-
analytically. In this case, the thermal relic abundance of $X$ (plus
$\bar{X}$) is given by,

$\Omega_{X}h^{2}\approx 1.7\times
10^{-10}~{}\frac{x_{f}\sqrt{g_{\text{eff}}}}{g_{*,s}}~{}\left(\frac{\sigma
v_{X\bar{X}\rightarrow
Z^{\prime}Z^{\prime}}}{\text{GeV}^{-2}}\right)^{-1}\approx 1.6\times
10^{4}\bigg{(}\frac{x_{f}}{30}\bigg{)}\bigg{(}\frac{0.1}{\alpha_{X}}\bigg{)}^{2}\bigg{(}\frac{m_{X}}{{\rm
PeV}}\bigg{)}^{2}\bigg{(}\frac{\sqrt{g_{\rm eff}}/g_{*}}{0.1}\\!\bigg{)}\,,$
(29)

where $g_{\rm eff}\equiv g_{*,s}+g^{h}_{*,s}\,\xi^{4}$ at freeze-out. $x_{f}$,
which is defined as the mass of $X$ divided by the SM temperature at freeze-
out, is found to be $\sim 20\times\xi$ over a wide range of parameters. From
Eq. (29), it is clear that a PeV-scale DM candidate with perturbative
couplings will initially freeze-out with an abundance that exceeds the
observed DM density ($\Omega_{X}h^{2}\gg\Omega_{\rm DM}h^{2}\simeq 0.12$). It
has long been appreciated, however, that this conclusion can be circumvented
if the Universe departed from the standard RD picture after DM freeze-out
[416, 417, 341, 418, 295, 278, 295, 419, 227, 420, 421, 453, 454, 455]. More
specifically, as the Universe expands, the remaining $Z^{\prime}$s will become
nonrelativistic and quickly come to dominate the energy density of the
Universe when $\rho_{Z^{\prime}}=0.0074\,g_{*,s}\xi_{\rm
inf}^{3}\,m_{Z^{\prime}}T_{\rm dom}^{3}>(\pi^{2}/30)g_{*}T_{\rm dom}^{4}$,
which occurs at a visible sector temperature

$T_{\rm dom}\sim 1\,{\rm TeV}\,\times\,\xi_{\rm
inf}^{3}\left(\frac{m_{Z^{\prime}}}{50\,\rm TeV}\right)\,.$ (30)

This expression is valid so long as the $Z^{\prime}$ population departs from
chemical equilibrium while relativistic. When the $Z^{\prime}$s ultimately
decay (see Fig. 5), they will deposit energy and entropy into the visible
sector, potentially diluting the DM abundance to acceptable levels. In the
left frame of Fig. 6, we show the evolution of the energy densities in the
visible and hidden sectors, for a representative choice of parameters in this
model.

Figure 5:  A schematic diagram of the processes being considered in Sec. 3.1.
Here $X$, the DM candidate, annihilates (possibly through intermediate
processes) into pairs of metastable hidden sector particles, $Y$. If the
hidden sector is heavy and extremely decoupled from the visible sector (which
contains the SM), then $Y$ will be long-lived, and may eventually dominate the
Universe’s energy density. Upon its decay into SM particles, $Y$ reheats the
visible Universe and dilutes all particle abundances, including the relic
density of $X$.

Figure 6: Left: The evolution of the energy densities of DM (blue solid), of
$Z^{\prime}$s (yellow dashed), and in the visible sector (orange dot-dashed),
as a function of the visible sector temperature. Upon becoming
nonrelativistic, the $Z^{\prime}$s quickly come to dominate the energy density
of the Universe. When they decay, they heat the SM bath and dilute the $X$
abundance. Right: The black contours represent the regions of the
$m_{X}-\epsilon$ plane in which the DM density is equal to the measured
cosmological abundance, for three values of the hidden sector interaction
strength, $\alpha_{X}$, and for $m_{Z^{\prime}}=m_{X}/20$ and $\xi_{\rm
inf}\equiv(T_{h}/T)_{\rm inf}=1$. The blue region is excluded by measurements
of the light element abundances. In and above the orange and yellow regions,
the hidden and visible sectors are in kinetic equilibrium during DM freeze-
out, or the $Z^{\prime}$ population decays before the freeze-out of $X$,
respectively. In and above the brown region, the $Z^{\prime}$ population never
dominates the energy density of the Universe, and thus does not significantly
dilute the DM relic abundance. In contrast to the case of a standard thermal
relic, DM from a decoupled sector can be as heavy as $\sim$1-10 PeV without
being overproduced in the early Universe.

To calculate the impact of this effect, we integrate over the $Z^{\prime}$
decay rate to find the factor by which the relics will be diluted [15]

$\frac{s_{f}}{s_{i}}\approx
680\times\bigg{(}\\!\frac{10^{-10}}{\epsilon}\\!\bigg{)}\bigg{(}\frac{m_{Z^{\prime}}}{100\,{\rm
TeV}}\bigg{)}^{1/2}\bigg{(}\frac{\langle
g^{1/3}_{*,s}\rangle^{3}}{100}\bigg{)}^{1/4}\,\xi_{\rm inf}^{3}\,,$ (31)

where $\langle g_{*,s}\rangle$ denotes the time-averaged value over the period
of decay. Combining this with Eq. (29), we find that the final DM relic
abundance is given by

$\Omega_{X}h^{2}\approx\frac{0.12}{\xi_{\rm
inf}^{3}}\,\bigg{(}\frac{\epsilon}{10^{-13}}\bigg{)}\,\bigg{(}\frac{0.045}{\alpha_{X}}\bigg{)}^{2}\,\bigg{(}\frac{m_{X}}{{\rm
PeV}}\bigg{)}^{2}\,\bigg{(}\frac{100\,{\rm
TeV}}{m_{Z^{\prime}}}\bigg{)}^{1/2}\,\bigg{(}\frac{x_{f}}{30}\bigg{)}\bigg{(}\frac{\sqrt{g_{\rm
eff}}/g_{*,s}}{0.1}\bigg{)}\bigg{(}\frac{100}{\langle
g^{1/3}_{*,s}\rangle^{3}}\bigg{)}^{1/4}\,.$ (32)

In the right panel of Fig. 6 we plot some of the phenomenological features of
this model as a function of the DM mass and the degree of kinetic mixing
between the $Z^{\prime}$ and SM hypercharge. The black contours denote the
regions where the DM density is equal to the measured cosmological abundance,
for three values of the hidden sector interaction strength, $\alpha_{X}$.
Below the brown region, $Z^{\prime}$ decays deposit significant entropy into
the visible sector, reducing the final $X$ abundance. To assure consistency
with BBN (see Sec. 4.1 for details), we require that the temperature of the
Universe exceeds 10 MeV after the decays of the $Z^{\prime}$ population,
resulting in the following constraint: $\epsilon\gtrsim 2\times
10^{-13}\times(100\,{\rm TeV}/m_{Z^{\prime}})^{1/2}\,(g_{*,s}/10)^{1/4}$.

Viable phenomenology can be obtained within this class of models for a wide
range of portal couplings, spanning many orders of magnitude. Depending on the
degree of kinetic mixing, the hidden and visible sectors may have been
entirely decoupled from one another, or kept in kinetic equilibrium through
interactions of the type $\gamma f\leftrightarrow Z^{\prime}f$.
Quantitatively, we find that the rate for these processes exceed that of
Hubble expansion if: $\epsilon\gtrsim 10^{-7}\times(T/10\,{\rm GeV})^{1/2}$
(shown as the orange region in the right panel of Fig. 6). Thus for smaller
values of $\epsilon$, the hidden sector will not reach equilibrium with the
visible sector and will remain decoupled. Furthermore, in the yellow regions
of the right panel of Fig. 6, the $Z^{\prime}$ population decays prior to the
freeze-out of $X$.

Many hidden sector DM models are difficult to constraint or otherwise test. In
particular, the feeble couplings between the hidden and SM sectors makes the
prospects for DM searches at colliders and at direct detection experiments
quite bleak (see, however, Ref. [456]). This is less true for indirect
searches. In particular, during the EMD epoch that is predicted in many of
these scenarios, density perturbations grow more quickly than otherwise
expected, leading to a large abundance of sub-earth-mass DM microhalos (see
Sec. 3.5 for details). Since the DM does not couple directly to the SM, the
minimum halo mass is much smaller than predicted for weakly interacting DM,
and the smallest halos could form during the RD era. In Ref. [360], a
calculation was performed of the evolution of density perturbations within the
context of such hidden sector models, using a series of $N$-body simulations
to determine the outcome of nonlinear collapse during a RD era. The resulting
microhalos were found to be extremely dense, leading to very high rates of DM
annihilation and to large indirect detection signals that resemble those
ordinarily predicted for decaying DM. Existing measurements of the high-
latitude gamma-ray background rule out a wide range of parameter space within
this class of models. The scenarios that are most difficult to constrain are
those that feature a very long EMD epoch; if microhalos form prior to the
decay of the unstable hidden sector matter, the destruction of these
microhalos effectively heats the DM, suppressing the later formation of
microhalos. This gravitational heating is discussed in more detail in Sec.
3.5.

Although we have only considered one representative scenario here, our
treatment is without loss of essential generality. As long as the dark sector
freeze out proceeds through 2 $\to$ 2 annihilation between the DM and the
unstable long-lived species, the relic abundance will have the same parametric
scaling as outlined above regardless of the spins of these species or the
Lorentz structures of their interactions; variations relative to our benchmark
case only modify these results by order-one amounts [298, 296].

###  3.2 Phase transitions and baryon asymmetry (Authors: R. Allahverdi & M.
Lewicki)

####  3.2.1 Phase transition dynamics

Let us begin with discussion of the modification of dynamics of a first order
phase transition due to modified expansion history. While the modification
itself can seem to be rather drastic, phase transitions prove to be largely
insensitive. This is simply because the dynamics of a phase transition are
largely dictated by the dynamics of fields undergoing the transition. For
cosmological applications, in the most common case of a thermally driven
transition of a scalar field, decay rate per unit volume is given by [231,
230, 228, 229]

$\Gamma(T)\approx T^{4}\exp\left(-\frac{S_{3}(T)}{T}\right),\quad
S_{3}=4\pi\int{\rm d}rr^{2}\left[\frac{1}{2}\left(\frac{{\rm d}\phi}{{\rm
d}r}\right)^{2}+V(\phi,T)\right],$ (33)

where the action $S_{3}(T)$ depends on the potential of the field in question
including thermal corrections. Expansion history comes into play when we find
the temperature at which bubbles begin nucleating defined by

$\int_{0}^{t_{n}}{\rm
d}t\frac{\Gamma(t)}{H(t)^{3}}=\int_{T_{n}}^{\infty}\frac{{\rm
d}T}{T}\frac{\Gamma(T)}{H(T)^{4}}=1,$ (34)

where we assumed adiabatic expansion, that is ${\rm d}t=-{\rm d}T/(T\,H(T))$.
Given the exponential dependence of the decay rate we can see that modified
expansion history will impact the resulting temperature only logarithmically
(see e.g. Ref. [457]). Thus, phase transitions are rather insensitive to
expansion history and their course is instead linked to the (subdominant)
radiation sector the fields in question are coupled to.

With that said, it is of course true that possible observable consequences of
such a transition can still be impacted a lot more severely due to the
modified expansion. We will discuss these modification in terms of electroweak
baryogenesis in the remainder of this section and the gravitational wave
background in Sec. 4.4.

####  3.2.2 Electroweak baryogenesis

Let us now turn to electroweak baryogenesis [458, 459, 460, 461] in which the
observed baryon asymmetry of the Universe is a result of a strong electroweak
phase transition. This requires a modification of the SM to make the
electroweak phase transition first order and introduce extra sources of
$CP$-violation. Our goal, however, is not to discuss any specific modification
of SM aimed at satisfying these conditions but rather to discuss how they can
be modified in nonstandard expansion histories.

We will focus on the main constraint typically put on the transition itself,
that is the vacuum expectation value of the field over the temperature has to
be large enough $v/T\gtrsim 1$. This comes from the requirement that the
electroweak sphalerons providing required baryon number violation are
suppressed after the transition to avoid washing out the asymmetry as the
system goes back to thermal equilibrium. To be more specific, the sphaleron
processes in question are SM $SU(2)$ gauge interactions and as a result are
automatically suppressed once this symmetry is broken. The simplest criterion
for their necessary decoupling translates to the sphaleron rate being smaller
than the Hubble rate [462, 463]

$\Gamma_{\rm
Sph}=T^{4}\mathcal{B}_{0}\frac{g}{4\pi}\left(\frac{v}{T}\right)^{7}\exp\left(-\frac{4\pi}{g}\frac{v}{T}\right)\lesssim
H(T),$ (35)

where $v$ is the vev of the Higgs field after the transition. Calculation of
the prefactor $\mathcal{B}_{0}$ is difficult and leads to a small spread of
values used in the literature for the final bound on $v/T$ [464, 465, 466].
Since our interest is mainly in the modification of this result in a
nonstandard thermal history, we will simply use the value of $\mathcal{B}_{0}$
which leads to the standard bound $v/T\leq 1$ when the expansion is driven by
SM radiation $H(T)=H_{\rm R}(T)\propto T^{2}$. The modification of the
required $v/T$ is shown in Fig. 7 assuming the transition occurs at the
typical temperature for electroweak symmetry breaking $T\approx 100$ GeV.
While lowering the value of $v/T$ required to avoid wash-out by sphalerons
might not seem like a huge modification, it has a significant effect on the
parameter space of specific models [467, 468].

Figure 7: Left panel: vev of the field over the temperature required to
successfully decouple the electroweak sphaleron and avoid wash-out of the
baryon asymmetry after the transition as a function of the Hubble rate during
the transition normalized to the contribution to the expansion rate due to the
subdominant radiation component. Right panel: Dilution of the baryon asymmetry
upon reheating in models with expansion dominated by a component redshifting
slower than radiation $(w<1/3)$ on the example of an EMD epoch with $w=0$. The
result is shown as a function of the Hubble rate at the generation of the
asymmetry normalized to the expansion rate at reheating temperature at which
RD era resumes.

As discussed in Sec. 2, a particularly well-motivated example of a nonstandard
thermal history involves an EMD epoch. The EMD epoch ends upon completion of
the decay and the Universe enters a RD phase with temperature $T_{\rm reh}$.
Electroweak baryogenesis in such a scenario with $1\,{\rm GeV}\lesssim T_{\rm
reh}\ll 100$ GeV has been studied in Ref. [276]. There it was found that a
faster expansion rate at the same temperature results in a bound $v/T\gtrsim
0.77~{}(0.92)$ for $T_{\rm reh}=1~{}(10)$ GeV. Note that a modified expansion
rate affects the out-of-equilibrium condition necessary for generation of
baryon asymmetry (i.e. the third Sakharov condition [469]), as well as the
efficiency of wash-out processes that could erase the asymmetry, in any
baryogenesis scenario.

####  3.2.3 Constraints on baryogenesis from dilution by reheating

In general, a nonstandard thermal history can also affect baryogenesis by
diluting the generated baryon asymmetry. This is due to entropy release during
transition to the final RD phase. This can be readily seen by noting that the
energy density of a component with equation of state $w<1/3$ is redshifted
more slowly than that of radiation. Thus its dominance keeps growing and there
can be no transition to RD era unless it decays to SM radiation. This
reheating process is accompanied by (a typically significant) increase in the
entropy of the Universe.

To demonstrate the effect of reheating on baryon asymmetry, we consider a
nonstandard thermal history that involves a phase with equation of state
$w<1/3$ that is preceded and followed by RD era. Let us denote the Hubble rate
at the onset of the nonstandard expansion phase or generation of the baryon
asymmetry (whichever happened later) by $H$ and at the time of return to RD
era by $H_{\rm reh}$. The number density of any quantity produced at $H>H_{\rm
reh}$ is redshifted by a factor $(a_{\rm reh}/a)^{3}=(H/H_{\rm
reh})^{2/(1+w)}$ between $H$ and $H_{\rm reh}$. Applying this to the
normalized baryon asymmetry $n_{\rm B}/s$, where $n_{\rm B}$ is the number
density of baryon asymmetry, the maximum dilution factor during a nonstandard
expansion phase with $w<1/3$ is found to be

$d_{\rm max}\sim\left({H\over H_{\rm reh}}\right)^{(1-3w)/2(1+w)}\,.$ (36)

Fig. 7 shows the dilution factor in the case $w=0$ as a function the Hubble
rate normalised to the Hubble rate at reheating $H_{\rm reh}$.

To put things in perspective, let us consider an EMD epoch ($w=0$) driven by
some string modulus $\phi$ with mass $m_{\phi}$. We typically have $H\sim
m_{\phi}$ at the onset of the EMD epoch and $H_{\rm reh}\sim
m^{3}_{\phi}/M^{2}_{\rm P}$ in this case (see Sec. 2.4), which results in
$d_{\rm max}={\mathcal{O}}(M_{\rm P}/m_{\phi})$. Considering that
$m_{\phi}\gtrsim{\cal O}(100~{}{\rm GeV})$ is needed in order to to reheat the
Universe before the onset of BBN (i.e. $T_{\rm reh}\gtrsim\mathcal{O}(1)$
MeV), we find $d_{\rm max}\lesssim 10^{13}-10^{14}$. Such a large dilution can
render any previously generated baryon asymmetry totally insignificant.

Therefore, to obtain the observed value $(n_{\rm B}/s)\simeq 0.9\times
10^{-10}$, we are left with two general options. First, to generate an
initially large baryon asymmetry that matches the correct value after
dilution. Second, to generate the correct baryon asymmetry after the
transition to the final RD phase thereby avoiding dilution by reheating. Let
us discuss these possibilities and specific baryogenesis mechanisms that can
work in each case in some detail.

Affleck-Dine (AD) baryogenesis [276, 470] (for reviews, see Refs. [471, 472,
473]) is a mechanism that can naturally fit the first option. This scenario
utilizes flat directions in the scalar potential of supersymmetric extensions
of the SM. AD baryogenesis is known to be able to generate large values of
baryon asymmetry, even ${\cal O}(1)$, in a standard thermal history. An EMD
epoch (driven, for example, by string theory moduli) can be therefore helpful
in this regard by regulating the baryon asymmetry and bringing it down to the
observed value [474, 475, 476, 356]. A viable string embedding of AD
baryogenesis has been discussed in Ref. [477] where the volume modulus in type
IIB string models drives an EMD phase resulting in the observed baryon
asymmetry for natural values of the underlying parameters. We note that
dilution by late reheating can also help with other models of baryogenesis
that generate a too large baryon asymmetry in scenarios with a standard
thermal history.

The second route is to generate baryon asymmetry upon completion of transition
to a RD Universe when the Hubble rate is $H_{\rm reh}$. Many scenarios with a
nonstandard thermal history result in $T_{\rm reh}\ll 100$ GeV. In particular,
in cases where an EMD epoch is responsible for nonthermal production of weak-
scale DM, we may have $T_{\rm reh}\lesssim{\cal O}(1~{}{\rm GeV})$. This
severely constrains viable baryogenesis mechanisms that can work. For example,
it rules out leptogenesis [477] (for reviews, see Refs. [478, 479]) as
sphalerons are out-of-equilibrium at these temperatures and hence unable to
convert the generated lepton asymmetry to baryon asymmetry. Thermal
baryogenesis is essentially ruled out as a possibility too because it will
require $C$-, $CP$-, and $B$-violating interactions at such low scales.

Therefore, nonthermal post-sphaleron baryogenesis seems to be the available
possibility for generating the observed baryon asymmetry at late times. An
explicit model is discussed in Ref. [277] in the context of EMD epoch driven
by some string modulus $\sigma$. This model includes new particles $N$ with a
mass $m_{N}\gg T_{\rm reh}$ that have $C$-, $CP$-, and $B$-violating
interactions with the SM particles. $N$ quanta are produced in $\sigma$ decay,
and their subsequent decay to the SM particles generates a baryon asymmetry
given by

${n_{\rm B}\over s}=Y_{\sigma}~{}{\rm Br}_{N}~{}\epsilon\,,\hskip
28.45274ptY_{\sigma}\equiv{3T_{\rm reh}\over 4m_{\sigma}}.$ (37)

Here $Y_{\sigma}$ is the yield from $\sigma$ decay, ${\rm Br}_{N}$ is the
branching fraction for production of $N$ from $\sigma$ decay and $\epsilon$ is
the asymmetry generated per $N$ decay. One point to keep in mind is that
$Y_{\sigma}$ is typically very small. For example, a modulus with mass
$m_{\sigma}\sim 100-1000$ TeV gives $Y_{\sigma}\lesssim{\cal O}(10^{-6})$.
Therefore, ${\rm Br}_{N}\epsilon$ cannot be too small in order to generate the
correct value of $n_{\rm B}/s$. This implies an $\epsilon$ usually larger than
that in thermal scenarios.

This nonthermal scenario may also address the baryon-DM coincidence problem
(for a review, see Ref. [480]) if $\sigma$ decay is the common origin of
baryogenesis and DM [355, 481]. In addition, if $N$ particles are not much
heavier than $\mathcal{O}(1)\,{\rm TeV}$, they could be produced at the LHC.
The $B$-violating interactions of $N$ can also lead to observable low-energy
phenomena like neutron-antineutron oscillations. Such a TeV scale baryogenesis
scenario may therefore be tested by complementary probes at the cosmology,
high energy, and low energy frontiers [482].

So far, our quantitative discussion of the effects of a nonstandard thermal
history on baryogenesis has mainly focused on EMD epoch as an important and
well-motivated case. However, the main considerations regarding the out-of-
equilibrium condition to generate and preserve baryon asymmetry, and the
entropy release during transition to the final RD phase are valid for any
scenario with a modified expansion rate. Here, we mention in passing a number
of studies on baryogenesis in more general nonstandard thermal histories. TeV
scale leptogenesis in nonstandard thermal histories that involve a) an EMD
epoch, and b) a phase with equation of state $w>1/3$ has been discussed in
Ref. [429]. An important example of a scenario with $w>1/3$ is kination where
$w=1$. Ref. [483] considers AD baryogenesis in the context of a model that
involves a post-inflationary kination phase. The effect of kination on the
gravitational leptogenesis scenario has been investigated in Ref. [484].
Gravitational leptogenesis for a general equation of state during inflationary
reheating has been studied in Ref. [57]. It has been shown in Ref. [485] that
nonstandard cosmologies from scalar-tensor theories can open new paths to low-
scale leptogenesis and thereby allow direct experimental tests of the
scenario. Finally, it is possible to have more complicated situations that
involve various sectors with different temperatures. A specific example of
such a scenario is given in Ref. [279] that considers thermal leptogenesis in
the presence of a dark sector whose temperature is higher than that of SM
radiation.

###  3.3 Consequences for models of inflation (Authors: K. Freese & T.
Tenkanen)

Cosmic inflation can be regarded as a successful paradigm for two reasons.
First, if inflation lasted long enough so that during it the Universe expanded
at least by a factor111111Assuming high scale inflation and standard expansion
history. In nonstandard cosmologies, this number can range from $\sim e^{18}$
to $e^{68}$ [20, 21, 22]. of $\sim e^{60}\sim 10^{26}$, inflation provides a
plausible explanation for the classic horizon, monopole, and flatness problems
(see e.g. Ref. [486]). Second, inflation provides a natural way to generate
the initial metric perturbations and explain their super-horizon correlations
at the time of CMB decoupling. These correlations are usually described in
terms of the scalar curvature power spectrum, which at least at the largest
physical distance scales is measured to have a power-law form [13]

$\mathcal{P}_{\zeta}(k)=\mathcal{A}\left(\frac{k}{k_{*}}\right)^{n_{s}-1}\,,$
(38)

to a high precision. Here $\zeta$ denotes the curvature perturbation and $k$
the wavenumber in a Fourier decomposition. The spectrum (38) has the observed
amplitude $\mathcal{A}\simeq 2.1\times 10^{-9}$ and spectral tilt $n_{s}\simeq
0.965$ at the pivot scale $k_{*}=0.05\,{\rm Mpc}^{-1}$ [13]. The data are
consistent with no running, ${\rm d}n_{s}/{\rm d\,ln}k$, or running of the
running of the spectral tilt, ${\rm d}^{2}n_{s}/{\rm d(ln}k)^{2}$ [13], and
therefore we will neglect them here for simplicity. While in principle solving
the classic horizon, monopole, and flatness problems only requires exponential
expansion at early times, any successful model of inflation also has to
predict the measured values for observables, or be within the limits on them.
In addition to depending on the underlying particle physics scenario, these
predictions usually depend also on the post-inflationary expansion history,
which is the topic of this section.

In the following, we will assume, for simplicity, that inflation was driven by
a single scalar field $\phi$, which at early times dominated the total energy
density of the Universe and drove inflation by rolling down in its potential
$V$. In the slow-roll approximation, $|\ddot{\phi}|\ll
3H|\dot{\phi}|\,,\dot{\phi}^{2}\ll V$, where $H\equiv\dot{a}/a$ is the Hubble
parameter and the dot denotes derivative with respect to cosmic time, the
inflationary dynamics can be characterized by the slow-roll parameters

$\epsilon\equiv\frac{M_{\rm
P}^{2}}{16\pi}\left(\frac{V^{\prime}}{V}\right)^{2}\,,\quad\eta\equiv\frac{M_{\rm
P}^{2}}{8\pi}\frac{V^{\prime\prime}}{V}\,,$ (39)

where the prime denotes derivative with respect to $\phi$. In slow-roll
$\epsilon\,,|\eta|\ll 1$, and inflation ends when $\epsilon=1$. In this case,
then, the amplitude of the curvature power spectrum can be expressed in terms
of the slow-roll parameters as [487]

$\mathcal{A}=\frac{1}{24\pi^{2}M_{\rm
P}^{4}}\frac{V(\phi_{*})}{\epsilon(\phi_{*})},$ (40)

where $\phi_{*}$ denotes the field value at the time when the scale $k_{*}$
exited the horizon. The leading order expression for the spectral tilt is

$n_{s}-1\equiv\left.\frac{{\rm d\,ln}\mathcal{P}_{\zeta}(k)}{{\rm
d\,ln}k}\right|_{k=k_{*}}\simeq-6\epsilon(\phi_{*})+2\eta(\phi_{*})\,.$ (41)

In addition to generating scalar curvature perturbations, fluctuations of the
inflaton field also generate tensor perturbations, i.e. primordial
gravitational waves. The observational constraints are usually expressed in
terms of the tensor-to-scalar ratio

$r\equiv\frac{\mathcal{P}_{T}}{\mathcal{P}_{\zeta}}\simeq 16\epsilon\,,$ (42)

where $\mathcal{P}_{T}$ is the tensor power spectrum (see e.g. Ref. [486]) and
the last expression applies at the leading order in slow-roll parameters. As
primordial tensor perturbations have not been detected, observations of the
CMB place an upper limit on tensor-to-scalar ratio, $r<0.06$ at the pivot
scale $k_{*}=0.05\,{\rm Mpc}^{-1}$ [488].

As can be seen by substituting the slow-roll parameters in Eq. (39) into Eqs.
(40), (41), and (42), the inflationary observables depend solely on the shape
of the inflaton potential. Constraints on the observables therefore limit
different scalar field potentials, i.e. different models, at the energy scale
where inflation occurred. However, judging whether a given model complies with
observations is not entirely clear-cut. A given potential may predict wrong
values for observables at some energy scale but get them exactly right if the
distance scales where measurements are made exited the horizon when the
inflaton was rolling in some other part of the potential where its shape was
different and which therefore generated different values for observables. The
time when a given scale must have exited the horizon is determined by the
overall energy scale of inflation and the post-inflationary evolution of
scales. Therefore, details of the entire post-inflationary expansion history
and, in particular, any nonstandard period of expansion will affect
inflationary model selection, as we will demonstrate below.

It is convenient to characterize the observables in terms of the number of
e-folds between the horizon exit of the pivot scale and the end of inflation,
$N\equiv\ln(a_{\rm end}/a_{k})$ where $a_{\rm end}$ ($a_{k}$) is the value of
the scale factor at the end of inflation (when the scale $k$ exited the
horizon), which – together with the parameters that characterize the post-
inflationary evolution – relate the pivot scale to the present Hubble scale
$H_{0}$ as

$\frac{k}{a_{0}H_{0}}=\frac{a_{k}H_{k}}{a_{0}H_{0}}=e^{-N}\frac{a_{\rm
end}}{a_{\rm RD}}\frac{a_{\rm RD}}{a_{0}}\frac{H_{k}}{H_{0}}\,,$ (43)

where $H_{k}$ is the Hubble scale at $a_{k}$, $a_{\rm RD}$ is the scale factor
at the time when the nonstandard period ended and the Universe became RD, and
$a_{0}$ is the scale factor at present. Here we have assumed that there is
exactly one nonstandard phase between the end of inflation and the subsequent
RD period. Note that the nonstandard expansion period could have been caused
by, for example, a kinetic energy-dominated kination phase [273, 462] or
another period of inflation (see Sec. 2.2) instead of conventional reheating
discussed in Sec. 2.1. Including another nonstandard expansion phase(s) that
punctuated the subsequent RD era(s) is straightforward [20] but here we leave
them out for simplicity. Assuming that the nonstandard phase can be
characterized with a single effective equation of state parameter $w$, so that
the total energy density scaled as $\rho\propto a^{-3(1+w)}$ during the
nonstandard expansion epoch, we obtain

$N\simeq
57+\frac{1}{2}\ln\left(\frac{r}{0.1}\right)+\frac{1}{3(1+w)}\ln\left(\frac{\rho_{\rm
RD}}{\rho_{\rm end}}\right)-\ln\left(\frac{\rho_{\rm RD}^{1/4}}{10^{16}\,{\rm
GeV}}\right)-\ln\left(\frac{k}{0.05\,{\rm Mpc}^{-1}}\right)\,,$ (44)

where $\rho_{\rm RD}$ is the total energy density at the time the standard RD
epoch began, $\rho_{\rm end}=\linebreak 3/(8\pi)H^{2}(a_{\rm end})M_{\rm
P}^{2}$ is the total energy density at the end of inflation, and we have fixed
$a_{0}=1$, used $T_{0}=2.725\,{\rm K}$, and assumed, for simplicity, that the
transitions between different epochs were instantaneous121212Note that our
treatment does not account for non-instantaneous transitions between epochs
reminiscent of e.g. nonperturbative effects encountered in preheating (see