is subsequently reaches the observable value through dilution. As moduli
cosmologies naturally lead to considerable dilution, the second approach
appears the most promising within stringy models.

The second approach is normally associated to Affleck-Dine baryogenesis [94],
in which an Affleck-Dine field (a susy D-flat direction carrying baryon
number) acquires a vev during inflation. This field experiences a torque about
the origin from the trilinear A-terms, which result in an eventual non-zero
baryon number. As the Affleck-Dine field directly carries baryon number, the
Affleck-Dine mechanism can result in very large baryon numbers; an epoch of
dilution is therefore beneficial for reducing the baryon asymmetry to be
consistent with observed values. Given the presence of large dilution factors
and low reheating temperature, the Affleck-Dine mechanism is probably the most
appealing in a stringy context (although baryogenesis has not been extensively
studied in a stringy context). A treatment of Affleck-Dine baryogenesis in the
context of LVS is [95], while also see [96] and [97] for other stringy
approaches to baryogenesis.

The cosmology outlined in this paper offers novel and beneficial features in
terms of Affleck-Dine baryogenesis. During the long evolution to the minimum,
CCB minima may be present and indeed the preferred vacuum of the Standard
Model – thus triggering a vev for the Affleck-Dine field – even though CCB
vacua may be absent in the final minimum. This mitigates some of the
difficulties of Affleck-Dine model-building as it is no longer necessary for
the Affleck-Dine vev to be generated during inflation. Instead, the vev may be
generated during the transPlanckian post-inflationary evolution towards the
final minimum, in a period when the Standard Model couplings are very
different from their final values in today’s minimum.

The cosmology described in this paper offer a further clear general advantage
for baryogenesis. As the tracker approaches, overshoots and eventually settles
down in the minimum, the changing time-dependent couplings define an explicit
arrow of time when restricted to the Standard Model sector, thereby violating
Lorentz symmetry, CPT and ensuring that the 3rd Sakharov condition is
satisfied.

In addition to baryogenesis, a large dilution factor may also be relevant for
understanding the origin of the dark matter content of the universe. The
similarities in the baryon ($\Omega_{b}\sim 0.05$) and dark matter
($\Omega_{dm}\sim 0.25$) energy densities have long been striking. In cases
where the dark matter mass is comparable to the baryonic mass, the number
densities are also comparable, and in this case a universal dilution factor
from the moduli epoch could sit behind both the dark matter number density and
the baryon number density.

###  4.3 Primordial Legacies from Gravitational Waves

The holy grail of work in string cosmology are observational legacies that
offer distinctive signatures of stringy epochs or behaviour. There are several
difficulties in obtaining these. One is that of scales: in terms of Hubble
horizons, post-inflationary epochs correspond to scales that are much smaller
than CMB scales today. In terms of density perturbations, such galactic and
sub-galactic scales are highly contaminated by non-linear and baryonic
effects. The other is that of particle lifetimes: even if they once dominated
the universe, moduli and other stringy particles eventually decay and are no
longer present today.

For these reasons, one of the most promising approaches to such stringy
legacies are through relativistic particles that survive in the present
universe. Prominent examples include relic spectra of either axion-like
particles (for example, through the notion of a Cosmic Axion Background
arising from moduli decays [18, 19, 98, 99, 100, 21, 101, 102]; also see [103]
which considers an early population of Primordial Black Holes as a source of
relativistic axion-like particles) or gravitational waves through non-linear
structures (e.g. [104, 70, 105, 106]) or via axions (e.g. see the recent paper
[107]).

The cosmology described in this paper gives several mechanisms to produce and
enhance primordial gravitational wave signatures. The simplest is that of
direct enhancement through a kination epoch, where gravitational waves
represent one candidate for the radiation that grows relative to the kination
background, and in principle could constitute the radiation portion of the
tracker solution with gravitational waves an $\mathcal{O}(1)$ part of the
energy density of the universe. Such gravitational waves could be formed, for
example, from a high-tension ($G\mu_{then}\gg 10^{-7}$) cosmic string network
(bounds on the string tension today which restrict $G\mu_{today}\lesssim
10^{-7}$ are not applicable as the string tension will also evolve with the
evolution of the volume modulus). The difficulty is to ensure that this
spectrum is not diluted to unobservable levels during the period of moduli
domination preceding reheating: this may require the existence of ‘fast’
moduli decay channels as per the ideas of [27].

Other mechanisms exploit the rapid growth of small scale structure that is
present within kination, matter tracker and moduli domination epochs. In each
of these, sub-horizon density perturbations grow with powers of the scale
factor. If these go non-linear (and note the initial magnitude of these
perturbations can be significantly larger than those present on CMB scales)
they can form non-linear structures, and potentially primordial black holes,
at very early scales. If formed at any stage prior to moduli domination, the
density of such PBHs will increase relative to the background.

Indeed, PBHs are a candidate to source a matter tracker during the modulus
evolution and, as matter dominates over radiation, may come to form an
$\mathcal{O}(1)$ fraction of the universe during a long kination and tracker
epoch.

A primordial black hole of mass $M_{PBH}$ has a lifetime

$\tau_{Hawking}\sim\frac{M_{PBH}^{3}}{M_{P}^{4}},$

and typically forms with a horizon mass at time of formation,

$M_{PBH}\sim\frac{M_{P}^{2}}{H_{formation}}.$

With a standard modulus lifetime $\tau\sim\frac{M_{P}^{2}}{m_{\Phi}^{3}}$, it
follows that PBHs formed as above prior to the epoch of moduli domination,
$H>m_{\Phi}$, will evaporate prior to moduli decay.

This does not mean that such early PBHs would be observationally irrelevant.
There are two ways such early PBHs can lead to gravitational waves. One is
through their evaporation and direct emission of gravitons: as early PBHs can
survive deep into the moduli epoch, effects of dilution will be greatly
diminished. The other is through mergers of PBHs and emission of gravitational
waves from the merger. During the long epoch of moduli/matter domination, all
matter (including PBHs) will clump, which can (as with the present universe)
lead to mergers of black holes, albeit on much smaller scales.

The other clear origin of gravitational waves would be those emitted from
networks of cosmic (super)strings [108] (see [109] for an analysis also
focusing on kination epochs). One effect of the kination and tracker epochs is
that the tension of such cosmic strings may vary through the cosmic history:
in particular, tensions $G\mu\gg 10^{-7}$ would be possible at earlier stages
in this evolution. This motivates a largely unstudied scenario in which the
string tension varies appreciably during the cosmic evolution, emitting
gravitational waves as it does so.

In principle, the form of the gravitational wave spectrum could provide
evidence for the non-standard evolution of string cosmologies due to the
distinctive nature of the various epochs the system passes through; however,
we defer a detailed analysis of it to further work.

##  5 Conclusions and Open Directions

The epoch between the end of inflation and nucleosynthesis is poorly
constrained observationally and also one where string models strongly motivate
changes to the equations of state. In this paper we have performed a detailed
analysis of both the background cosmology and perturbations to this cosmology
for a string cosmology across this epoch. This analysis reveals many striking
features. These include the amplification of normally small effects (such as
background radiation or gravitational waves) during a long kination epoch, a
detailed understanding of perturbations during matter and radiation trackers,
and the phenomenological opportunities offered by changes in Standard Model
coupling constants during these eras.

Directions of future work in this area include:

  * •

In this paper, we have studied perturbations separately in each of the
different eras the string cosmology passes through. It would be good to extend
this to a single unified analytical or numerical treatment which matches the
perturbations across all of these epochs and carries the cosmology all the way
from the end of inflation to match into BBN and $\Lambda$CDM. Such a treatment
would also require an understanding of the initial spectrum of perturbations
within the radiation (or matter) seed that catches up with the kinating scalar
to bring the cosmology onto the tracker solutions.

  * •

In the approach to the tracker, we have assumed that the seed radiation/matter
is separate to the rolling scalar field, and that self-perturbations in the
scalar field remain subdominant. In Eq. (3.31), we saw that scalar field self-
perturbations do indeed grow relative to the background. With the analytic
perturbative techniques of this paper, we are unable to study what happens if
these perturbations reach the level of the background with an $\mathcal{O}(1)$
density contrast. However, it is a well-defined problem in numerical GR, which
should be accessible with tools such as [110], to determine the end-point of a
kinating scalar with self-perturbations on an exponential potential. What
happens? Do the self-perturbations end up forming black holes? Does the system
end up in a tracker in which the self-perturbations of the scalar field can
act as radiation?

  * •

The development and analysis of vanilla test particle physics models for the
scenarios described in sections 4.1 and 4.2, in which the effects of explicit
time dependence of MSSM couplings on the vacuum structure of the Standard
Model is studied. In particular, this would enable a more quantitative study
of the prospects of baryogenesis in this fashion.

  * •

A detailed study of the gravitational wave spectrum this non-standard
cosmology gives rise to, arising either from the evolution of cosmic string
networks or through formation and decay/merger of primordial black holes: with
the aim of determining any unusual spectral features that could give
distinctive signatures of the stringy cosmological evolution, for example the
correlation between gravitational wave production and cosmic microwave
background fluctuations induced by a network of cosmic superstrings [111,
112].

In summary, the era between inflation and BBN offers one of the most promising
opportunities for string cosmology in terms of chances to connect string
theory to observations.

## Acknowledgments

FA is supported by the Clarendon Scholarship in partnership with the Scatcherd
European Scholarship, Saven European Scholarship, and the Hertford College
Peter Howard Scholarship. JC acknowledges support from the STFC consolidated
grants ST/T000864/1 and ST/X000761/1. MM is supported by the St John’s College
Graduate Scholarship in partnership with STFC. The research of FR is partly
supported by the Dutch Research Council (NWO) via a Start-Up grant and a Vici
grant. EJC is supported by STFC Consolidated Grants [ST/T000732/1 and
ST/X000672/1] and by a Leverhulme Research Fellowship [RF- 2021 312]. For the
purpose of Open Access, the authors have applied a CC BY public copyright
licence to any Author Accepted Manuscript version arising from this
submission. We thank Michele Cicoli, Anish Ghoshal, Eugene Lim, Swagat Mishra,
Sirui Ning, Fernando Quevedo and Gonzalo Villa for discussions. EJC would like
to thank the members of Oxford’s Rudolf Peierls Centre for Theoretical Physics
for their kind hospitality during the period this work was being undertaken.

##  Appendix A Additional Details on the Background Cosmology

This section contains extra details on the background cosmology.

###  A.1 Oscillations about the Tracker Solution

Here we give further details about the oscillations of the background about
the tracker solution, as the system settles down into the tracker (i.e.
section 2.4).

The result of Eq. (2.60) can be rewritten in a simpler form as

$\begin{split}x(N)&=x_{0}+e^{\frac{3}{4}(\gamma-2)N}\eta_{1}\sin(\omega
N+\tilde{\alpha}_{1}),\\\
y(N)&=y_{0}+e^{\frac{3}{4}(\gamma-2)N}\eta_{2}\sin(\omega
N+\tilde{\alpha}_{2}),\end{split}$ (A.1)

where $(x_{0},y_{0})$ are the tracker values. Here

$\begin{split}\eta_{1}=\sqrt{c_{1}^{2}+\frac{1}{4\omega^{2}}[2\beta
c_{2}+(\alpha-\delta)c_{1}]^{2}},\\\
\eta_{2}=\sqrt{c_{2}^{2}+\frac{1}{4\omega^{2}}[2\gamma
c_{1}-(\alpha-\delta)c_{2}]^{2}}.\end{split}$ (A.2)

Meanwhile the angular shifts are given by

$\tan\alpha_{1}=\frac{2\omega c_{1}}{2\beta c_{2}+(\alpha-\delta)c_{1}},\ \ \
\ \ \ \ \tan\alpha_{2}=\frac{2\omega c_{2}}{2\gamma
c_{1}-(\alpha-\delta)c_{2}}.$ (A.3)

From $x(N)$ we can also get the expression for the scalar field as

$\phi(N)=\phi_{0}+\sqrt{6}M_{P}x_{0}N+\frac{\sqrt{6}M_{P}\eta_{1}}{\sqrt{\omega^{2}+\frac{9}{16}(\gamma-2)^{2}}}e^{\frac{3}{4}(\gamma-2)N}\sin(\omega
N+\tilde{\alpha}_{1}+\alpha^{\prime}),$ (A.4)

where

$\tan\alpha^{\prime}=-\frac{4}{3}\frac{\omega}{\gamma-2}.$ (A.5)

Lastly, we can also calculate how the Hubble constant behaves during these
tracker oscillations by expanding its equation around the tracker solution
$H=H_{0}e^{-3\gamma N/2}+\delta H$, then at linear order this gives

$\delta H^{\prime}=-\frac{3}{2}\gamma\delta
H-3\sqrt{\frac{3}{2}}\frac{\gamma}{\lambda}H_{0}e^{-3\gamma
N/2}[(2-\gamma)(x(N)-x_{0})-\sqrt{(2-\gamma)\gamma}(y(N)-y_{0})].$ (A.6)

Inserting our expressions for the perturbations yields a differential equation
of the form

$\delta H^{\prime}=-\frac{3}{2}\gamma\delta
H-3\sqrt{\frac{3}{2}}\frac{\gamma}{\lambda}H_{0}e^{-3(\gamma+2)N/4}[(2-\gamma)\eta_{1}\sin(\omega
N+\tilde{\alpha}_{1})-\sqrt{(2-\gamma)\gamma}\eta_{2}\sin(\omega
N+\tilde{\alpha}_{2})].$ (A.7)

This can be solved directly to give

$\delta
H=H\bigg{[}c_{3}+\frac{12}{9(\gamma-2)^{2}+16\omega^{2}}\sqrt{\frac{3}{2}}\frac{\gamma}{\lambda}H_{0}e^{-3(\gamma+2)N/4}h(N)\bigg{]},$
(A.8)

where $c_{3}$ is an integration constant and $h(N)$ is a trigonometric
function given by

$\begin{split}h(N)&=3(2-\gamma)[(2-\gamma)\eta_{1}\sin(\omega
N+\tilde{\alpha}_{1})-\sqrt{(2-\gamma)\gamma}\eta_{2}\sin(\omega
N+\tilde{\alpha}_{2})]\\\ &\ \ \ \ \ \ \ \
+4\omega[(2-\gamma)\eta_{1}\cos(\omega
N+\tilde{\alpha}_{1})-\sqrt{(2-\gamma)\gamma}\eta_{2}\cos(\omega
N+\tilde{\alpha}_{2})].\end{split}$ (A.9)

Through appropriate trigonometric manipulations the Hubble constant can be re-
expressed in a simpler form which schematically behaves as

$\delta H=H\bigg{[}c_{3}+A^{\prime}e^{\frac{3}{4}(\gamma-2)N}\sin(\omega
N+\alpha^{\prime})\bigg{]},$ (A.10)

for some coefficients $A^{\prime}$ and $\alpha^{\prime}$.

###  A.2 More Details on Reheating

Here we give, for completeness, an account of reheating that goes beyond the
instantaneous decay approximation.

In an initially completely matter dominated universe where the matter
particles start to decay into pairs of photons at time $t_{0}$, the total
number of matter particles in a volume $V$ is given by
$N=N_{0}e^{-\alpha(t-t_{0})}$, so then the energy density of the radiation
component is

$\rho_{\gamma}=\frac{N_{\gamma}}{V}\langle
E_{\gamma}(t)\rangle=\frac{2N_{0}(1-e^{-\alpha(t-t_{0})})}{V_{0}}\frac{a_{0}^{3}}{a^{3}}\langle
E_{\gamma}(t)\rangle,$ (A.11)

where $\langle E_{\gamma}(t)\rangle$ is the average energy of the photon bath
at a particular time.

The energy of a photon can be related to the scale factor through
$E_{\gamma}(\lambda)=2\pi/\lambda=ma_{c}/2a$ where $a_{c}$ is the scale factor
at which the photon was created. The average energy density is then given by

$\langle
E_{\gamma}(t)\rangle=\frac{1}{n_{\gamma,\text{tot}}}\int_{4\pi/m}^{\frac{4\pi
a(t)}{ma(t_{0})}}d\lambda\ \tilde{n}(t,\lambda)E_{\gamma}(\lambda),$ (A.12)

where $\tilde{n}_{\gamma}(t,\lambda)d\lambda$ is the total number of photons
per unit volume in the wavelength interval $d\lambda$, while the total number
density is

$n_{\gamma,\text{tot}}=\int_{4\pi/m}^{\frac{4\pi a(t)}{ma(t_{0})}}d\lambda\
\tilde{n}_{\gamma}(t,\lambda)=2n_{0}(1-e^{-\alpha(t-t_{0})}),$ (A.13)

where $n_{0}=N_{0}/V_{0}$ is the initial number density of the matter
particles.

To rewrite the average energy in a more useful form, consider photons
originally created with a wavelength of $\lambda=4\pi/m$. Over time the change
in their energy is due to the redshifting of the wavelength with

$d\lambda=\frac{4\pi}{m}\frac{a(t)}{a_{c}(t_{c})}-\frac{4\pi}{m}\frac{a(t)}{a_{c}(t_{c}+dt)}\approx\frac{4\pi}{m}\frac{aH_{c}}{a_{c}}dt,$
(A.14)

providing a relation between $d\lambda$ and $dt$. Since
$\dot{n}_{\gamma}dt|_{c}$ photons are initially created in a time interval
$dt$, then this can be related to the number density per unit wavelength
interval as

$\tilde{n}_{\gamma}d\lambda=\dot{n}_{\gamma}(a_{c})\frac{m}{4\pi
H_{c}}\frac{a_{c}d\lambda}{a(t)}.$ (A.15)

This formula states that the number density over a wavelength interval is
given by the number of photons created per unit time, together with a
$dt\rightarrow d\lambda$ conversion factor, along with stretching of the
wavelength interval by a factor of $a_{c}/a$.

The average energy of the radiation bath can now be written through a change
of integration variables $da_{c}/d\lambda=-ma_{c}^{2}/4\pi a$ as

$\langle
E_{\gamma}(t)\rangle=\frac{1}{n_{\gamma,\text{tot}}}\frac{m}{2a(t)}\int_{a(t_{0})}^{a(t)}da_{c}\frac{\dot{n}_{\gamma}(a_{c})}{H_{c}}=\frac{1}{n_{\gamma,\text{tot}}}\frac{m}{2a(t)}\int_{t_{0}}^{t}dt_{c}\
a(t_{c})\dot{n}_{\gamma}(t_{c}).$ (A.16)

Writing the production rate as

$\dot{n}_{\gamma}(t_{c})=2\alpha n_{0}e^{-\alpha(t_{c}-t_{0})}.$ (A.17)

then the Hubble equation can be written as

$H^{2}=\frac{H_{0}^{2}a_{0}^{3}}{a^{3}}e^{\alpha t_{0}}\bigg{(}e^{-\alpha
t}+\frac{\alpha}{a}\int_{t_{0}}^{t}dt_{c}a(t_{c})e^{-\alpha t_{c}}\bigg{)},$
(A.18)

where $H_{0}$ is the initial Hubble constant at $t_{0}$. This can be
differentiated to give a differential equation for $a(t)$

$\ddot{a}+\frac{\dot{a}^{2}}{a}=\frac{H_{0}^{2}a_{0}^{3}}{2a^{2}}e^{-\alpha(t-t_{0})}.$
(A.19)

##  Appendix B Cosmological perturbations for general equation-of-state
parameters

###  B.1 Large-scale evolution of gravitational potential through transitions
between cosmic fluids

In this Appendix, we derive a general expression for the large-scale evolution
of gravitational potential through transitions between cosmic fluids
characterized by parameters $\gamma_{1}$ and $\gamma_{2}$. We consider the
large-scale Einstein equation

$-3\mathcal{H}(\Phi^{\prime}+\mathcal{H}\Psi)=4\pi
Ga^{2}(\rho_{1}\delta_{1}+\rho_{2}\delta_{2}).$ (B.1)

Define

$y\equiv\dfrac{\rho_{1}}{\rho_{2}}\sim a^{-3(\gamma_{1}-\gamma_{2})},$ (B.2)

and assume adiabaticity

$\displaystyle\dfrac{\delta_{1}}{\gamma_{1}}=\dfrac{\delta_{2}}{\gamma_{2}}$
(B.3)

to obtain

$-3\mathcal{H}(\Phi^{\prime}+\mathcal{H}\Phi)=4\pi
Ga^{2}\rho_{2}\delta_{2}\left(1+\dfrac{1}{y}\dfrac{\gamma_{1}}{\gamma_{2}}\right)=\dfrac{3}{2}\mathcal{H}^{2}\dfrac{y}{y+1}\delta_{2}\left(1+\dfrac{1}{y}\dfrac{\gamma_{1}}{\gamma_{2}}\right),$
(B.4)

where we substituted the Friedman equation $8\pi
G\rho_{2}/3=(\mathcal{H}^{2}y)/(a^{2}(y+1))$. This turns into

$-3(3(\gamma_{1}-\gamma_{2})y\dfrac{d\Phi}{dy}+\Phi)=\dfrac{3}{2}\dfrac{y}{y+1}\left(1+\dfrac{1}{y}\dfrac{\gamma_{1}}{\gamma_{2}}\right)\delta_{2}$
(B.5)

and with $\delta_{2}^{\prime}\approx 3\gamma_{2}\Phi^{\prime}$,

$\displaystyle-\left(\gamma_{1}\left(9\gamma_{1}-6\gamma_{2}+2\right)+\left(6\gamma_{1}-3\gamma_{2}+2\right)\gamma_{2}y^{2}+2\left(6\gamma_{1}^{2}-3\gamma_{2}\gamma_{1}+\gamma_{1}+\gamma_{2}\right)y\right)\Phi^{\prime}(y)$
(B.6)
$\displaystyle-6\left(\gamma_{1}-\gamma_{2}\right)y(y+1)\left(\gamma_{1}+\gamma_{2}y\right)\Phi^{\prime\prime}(y)+2\left(\gamma_{2}-\gamma_{1}\right)\Phi(y)=0.$
(B.7)

The general solution is

$\displaystyle\Phi(y)=\frac{2\left(\gamma_{1}-\gamma_{2}\right)\left(3\gamma_{1}-2\sqrt{y+1}\,_{2}F_{1}\left(\frac{1}{2},\frac{3\gamma_{1}+2}{6\gamma_{1}-6\gamma_{2}};\frac{3\gamma_{1}+2}{6\gamma_{1}-6\gamma_{2}}+1;-y\right)+2\right)}{3\gamma_{1}+2}$
(B.8)

Then

$\displaystyle\lim_{y\rightarrow
0}\Phi(y)=\frac{6\gamma_{1}\left(\gamma_{1}-\gamma_{2}\right)}{3\gamma_{1}+2}$
(B.9)

and

$\displaystyle\sqrt{y+1}\,_{2}F_{1}\left(\frac{1}{2},b;b+1,-y\right)=b\sqrt{y+1}\int^{1}_{0}\dfrac{t^{b-1}}{(1+ty)^{1/2}}\rightarrow
b\int^{1}_{0}t^{b-3/2}=\dfrac{b}{b-1/2}$ (B.10)

as $y\rightarrow\infty$ so that

$\displaystyle\lim_{y\rightarrow\infty}\Phi(y)=\frac{6\left(\gamma_{1}-\gamma_{2}\right)\gamma_{2}}{3\gamma_{2}+2}.$
(B.11)

Hence

$\displaystyle\lim_{y\rightarrow\infty}\Phi(y)/\Phi(0)=\frac{\gamma_{2}\left(3\gamma_{1}+2\right)}{\gamma_{1}\left(3\gamma_{2}+2\right)}.$
(B.12)

As a different way to check this result, during an epoch dominated by a single
fluid with EoS parameter $\gamma_{i}$, the Bardeen variable $\Phi$ and
curvature perturbation in the super-horizon limit are related by [5]

$\mathcal{R}\simeq\frac{3\gamma_{i}+2}{3\gamma_{i}}\Phi\quad\quad\quad\text{for}\quad\quad
k\eta\ll 1.$ (B.13)

Assuming adiabaticity, the curvature pertubation on super-horizon scales is
conserved, and one can deduce that in the transition between epochs dominated
by different fluids $\Phi$ drops exactly as predicted by (B.12).

###  B.2 Cosmological perturbations for $\gamma$-trackers

In this section, we discuss perturbations in tracker regimes, characterized by
an equation-of-state parameter $\gamma$. For super-horizon limits (long
wavelength modes), the discussion is the same for all values of $\gamma$,
while for the sub-horizon limits (short wavelength modes), we need to
distinguish between the cases $\gamma>1$ and $\gamma<1$. The limiting case
$\gamma=1$ is described in the main text.

For a $\gamma$-tracker, the scale factor goes like

$\displaystyle a(\eta)=a_{0}\eta^{\frac{2}{3\gamma-2}},$ (B.14)

and the evolution of the scalar field is described by

$\displaystyle\phi=\phi_{t}+\frac{6\gamma}{(3\gamma-2)\lambda}\log\left(\eta\right),$
(B.15)

and the scalar potential $V$ is expressed as

$\displaystyle
V=V_{0}e^{-\lambda\phi}=-\frac{18(\gamma-2)\gamma\eta^{\frac{6\gamma}{2-3\gamma}}}{(2-3\gamma)^{2}\lambda^{2}}.$
(B.16)

The Einstein equations, within a tracker background, take the form

$\nabla^{2}\Phi-3\mathcal{H}\left(\Phi^{\prime}+\mathcal{H}\Phi\right)=\frac{a^{2}}{2M_{P}^{2}}\left(\delta\rho_{\phi}+\delta\rho_{\gamma}\right)$
(B.17)
$\Phi^{\prime\prime}+3\mathcal{H}\Phi^{\prime}+\left(2\mathcal{H}^{\prime}+\mathcal{H}^{2}\right)\Phi=\frac{a^{2}}{2M_{P}^{2}}\left(\delta
P_{\phi}+(\gamma-1)\delta\rho_{\gamma}\right),$ (B.18)

where $\delta\rho_{\phi}$ and $\delta P_{\phi}$ are given by (3.60), (3.61):

$\displaystyle\delta\rho_{\phi}$
$\displaystyle=\frac{1}{a^{2}}\left(\phi^{\prime}\delta\phi^{\prime}-\Phi\phi^{\prime
2}\right)+\frac{dV(\phi)}{d\phi}\delta\phi$ (B.19)
$\displaystyle=\frac{6\gamma\eta^{\frac{6\gamma}{2-3\gamma}}\left((3\gamma-2)\eta\lambda\delta\phi^{\prime}(\eta)+3(\gamma-2)\lambda\delta\phi(\eta)-6\gamma\Phi(\eta)\right)}{(2-3\gamma)^{2}\lambda^{2}},$
$\displaystyle\delta P_{\phi}$
$\displaystyle=\frac{1}{a^{2}}\left(\phi^{\prime}\delta\phi^{\prime}-\Phi\phi^{\prime
2}\right)-\frac{dV(\phi)}{d\phi}\delta\phi$ (B.20)
$\displaystyle=-\frac{6\gamma\eta^{\frac{6\gamma}{2-3\gamma}}\left((2-3\gamma)\eta\lambda\delta\phi^{\prime}(\eta)+3(\gamma-2)\lambda\delta\phi(\eta)+6\gamma\Phi(\eta)\right)}{(2-3\gamma)^{2}\lambda^{2}}.$

By combining the Einstein equations, we can eliminate the dependence on
$\delta\rho_{\gamma}$, and we also consider the Klein-Gordon equation (3.62),
to obtain a system of second order equations for $\Phi(\eta)$ and
$\delta\phi(\eta)$,

$\begin{cases}\Phi^{\prime\prime}(\eta)+\frac{6\gamma\Phi^{\prime}(\eta)}{\left(3\gamma-2\right)\eta}+\Phi(\eta)\left((\gamma-1)k^{2}-\frac{18(\gamma-2)\gamma^{2}}{(3\gamma-2)^{2}\eta^{2}\lambda^{2}}\right)=-\frac{3(\gamma-2)\gamma\left((3\gamma-2)\eta\delta\phi^{\prime}(\eta)+3\gamma\delta\phi(\eta)\right)}{(3\gamma-2)^{2}\eta^{2}\lambda}.\\\\[14.22636pt]
\delta\phi^{\prime\prime}(\eta)+\frac{4\delta\phi^{\prime}(\eta)}{\left(3\gamma-2\right)\eta}+\delta\phi(\eta)\left(k^{2}-\frac{18(\gamma-2)\gamma}{(3\gamma-2)^{2}\eta^{2}}\right)=\frac{12\gamma\left(2(3\gamma-2)\eta\Phi^{\prime}(\eta)-3(\gamma-2)\Phi(\eta)\right)}{(3\gamma-2)^{2}\eta^{2}\lambda}.\\\
\end{cases}$ (B.21)

####  B.2.1 Long wavelength modes

In the limit $k\rightarrow 0$, the system (B.21) can be solved analytically.
When $\lambda^{2}>24\gamma^{2}/(9\gamma-2)$, which is the condition for the
tracker solution to be a stable spiral [14], the solution is given by

$\displaystyle\Phi(\eta)=c_{1}+c_{2}\eta^{\frac{3\gamma+2}{2-3\gamma}}+\gamma\eta^{\frac{3(\gamma-2)}{2(3\gamma-2)}}\left(c_{3}\cos\left(\omega\log\eta\right)+c_{4}\sin\left(\omega\log\eta\right)\right),$
(B.22) $\displaystyle\delta\phi(\eta)=\frac{2c_{1}}{\lambda}-3\gamma
c_{2}\eta^{\frac{3\gamma+2}{2-3\gamma}}+\frac{4(3\gamma-2)\lambda\omega}{9(\gamma-2)}\eta^{\frac{3(\gamma-2)}{2(3\gamma-2)}}\left(c_{3}\cos\left(\omega\log\eta\right)+c_{4}\sin\left(\omega\log\eta\right)\right),$
(B.23)

where we defined

$\displaystyle\omega=\frac{3\sqrt{-(3\gamma-2)^{2}(\gamma-2)\left((9\gamma-2)\lambda^{2}-24\gamma^{2}\right)}}{2(3\gamma-2)^{2}\lambda}.$
(B.24)

When $\lambda^{2}<24\gamma^{2}/(9\gamma-2)$, on the other hand, all solutions
are power-law

$\displaystyle\Phi(\eta)=c_{1}+c_{2}\eta^{\frac{3\gamma+2}{2-3\gamma}}+\gamma\eta^{\frac{3(\gamma-2)}{2(3\gamma-2)}}\left(c_{3}\eta^{\omega}+c_{4}\eta^{-\omega}\right),$
(B.25) $\displaystyle\delta\phi(\eta)=\frac{2c_{1}}{\lambda}-3\gamma
c_{2}\eta^{\frac{3\gamma+2}{2-3\gamma}}+\frac{4(3\gamma-2)\lambda\omega}{9(\gamma-2)}\eta^{\frac{3(\gamma-2)}{2(3\gamma-2)}}\left(c_{3}\eta^{\omega}+c_{4}\eta^{-\omega}\right),$
(B.26)

where

$\displaystyle\omega=\frac{3\sqrt{(3\gamma-2)^{2}(\gamma-2)\left((9\gamma-2)\lambda^{2}-24\gamma^{2}\right)}}{2(3\gamma-2)^{2}\lambda}.$
(B.27)

We note that a special case of (B.25), (B.26) for $\gamma=0$ with
$\Phi(\eta)=c_{1}+c_{2}\eta$ and
$\delta\phi(\eta)=(2\lambda/3)c_{3}\eta^{3}+(2c_{1}/\lambda+2\lambda
c_{4}/3)$.

The solutions corresponding to the coefficients $c_{3}$ and $c_{4}$ are non-
adiabatic, and for $\gamma>2/3$, these are decaying.

####  B.2.2 Small wavelength modes for trackers with $\gamma>1$

We consider an Ansatz solution to Eqs. (B.21) of the form

$\displaystyle\delta\phi(\eta)=(k\eta)^{\frac{-2}{3\gamma-2}}f(k\eta),\quad\Phi(\eta)=(k\eta)^{\frac{-3\gamma}{3\gamma-2}}g(k\eta),$
(B.28)

where $f(k\eta)$ and $g(k\eta)$ are assumed to be trigonometric functions.
Plugging this Ansatz into the equation for the scalar perturbations, and
keeping the leading terms as $k\eta\gg 1$, we obtain

$\displaystyle f^{\prime\prime}(k\eta)+f(k\eta)=0,$ (B.29)

so that we can take

$\displaystyle\delta\phi(\eta)=A_{1}(k\eta)^{\frac{-2}{3\gamma-2}}\cos\left(k\eta\right).$
(B.30)

The leading terms to both $\delta\phi$ and $\delta\phi^{\prime}$ go as $1/a$,
and hence the intrinsic kinetic energy in the scalar field perturbations
$a^{-2}\delta\phi^{\prime}(\eta)^{2}\sim a^{-4}$ will behave like radiation in
all trackers.
Similarly, keeping the leading terms in the equation for the gravitational
potential assuming that $\gamma\neq 1$, we obtain

$\displaystyle(2-3\gamma)^{2}\eta
k\lambda\left(g^{\prime\prime}(k\eta)+(\gamma-1)g(k\eta)\right)=3A_{1}(\gamma-2)\gamma\sin(k\eta)$
(B.31)

which is solved by

$\displaystyle
g(k\eta)=A_{1}\frac{3\gamma\sin(k\eta)}{(3\gamma-2)\lambda}+A_{2}\cos\left(k\sqrt{\gamma-1}\eta\right),$
(B.32)

as long as $\gamma>1$. We thus conclude that

$\displaystyle\Phi(\eta)=(k\eta)^{\frac{-3\gamma}{3\gamma-2}}\left[A_{1}\frac{3\gamma\sin(k\eta)}{(3\gamma-2)\lambda}+A_{2}\cos\left(k\sqrt{\gamma-1}\eta\right)\right],$
(B.33)

consists of a superposition of both modes characteristic to the fluid, and
those following the scalar field.

##### Behaviour of the energy density of the scalar perturbations

To understand how the scalar perturbations redshift, we consider the solution
to the scalar equation
$\phi^{\prime\prime}+2\mathcal{H}\phi^{\prime}+k^{2}\phi=-a^{2}V^{\prime}(\phi)$,

$\displaystyle\phi(\eta)=\phi_{t}+\frac{6\gamma}{(3\gamma-2)\lambda}\log\left(\eta\right)+\sum_{k}a_{\pm
k}\frac{e^{\pm ik\eta}}{(k\eta)^{\frac{2}{3\gamma-2}}},$ (B.34)

and for sub-horizon scales we have

$\displaystyle\phi^{\prime}(\eta)=\frac{6\gamma}{(3\gamma-2)\lambda}\frac{1}{\eta}+\sum_{k}(\pm
ik)a_{\pm k}\frac{e^{\pm ik\eta}}{(k\eta)^{\frac{2}{3\gamma-2}}},$ (B.35)

and we see that both the velocity-term and the Fourier modes have the same
dependence on $\eta$. Hence the leading contribution to
$a^{-2}\phi^{\prime}(\eta)^{2}$ will scale like $a^{-3\gamma}$, while the
Fourier modes go as $a^{-4}$, and the cross-term mixes these behaviors:
$a^{-\frac{4+3\gamma}{2}}$.The contributions to the energy density of the
scalar perturbations
$\delta\rho_{\phi}=a^{-2}\left(\phi^{\prime}\delta\phi^{\prime}-\Phi\phi^{\prime
2}\right)+\left(dV(\phi)/d\phi\right)\delta\phi$ go like

$\displaystyle\frac{1}{a^{2}}\left(\phi^{\prime}\delta\phi^{\prime}\right)\sim
a^{-\frac{4+3\gamma}{2}},\quad\frac{1}{a^{2}}\left(\Phi\phi^{\prime
2}\right)\sim a^{-\frac{9\gamma}{2}},\quad\frac{dV(\phi)}{d\phi}\delta\phi\sim
a^{-1-3\gamma},$ (B.36)

and we see that the first contribution matches with the behavior of the cross-
term, and this contribution is the leading one (as $\gamma>1$).

####  B.2.3 Small wavelength modes for trackers with $0<\gamma<1$

Here, we consider the Ansatz

$\displaystyle\Phi(\eta)=f(k\eta)e^{\sqrt{1-\gamma}k\eta},\quad\delta\phi(\eta)=g(k\eta)e^{\sqrt{1-\gamma}k\eta},$
(B.37)

where we assume that

$\displaystyle f(k\eta)\gg
f^{\prime}(k\eta),f^{\prime\prime}(k\eta),g(k\eta),g^{\prime}(k\eta),g^{\prime\prime}(k\eta)$
(B.38)

in the limit for large $k\eta$. In this limit the equation for the
gravitational potential decouples from the scalar perturbations, leading to

$\displaystyle
f(k\eta)=\left(\sqrt{1-\gamma}(3\gamma-2)k\eta\right)^{\frac{3\gamma}{2-3\gamma}}c_{1}.$
(B.39)

Substituting this in the equation for the scalar perturbations, we obtain

$\displaystyle g(k\eta)$
$\displaystyle=-\frac{24\gamma\left(\left(\sqrt{1-\gamma}(3\gamma-2)k\eta\right)^{\frac{2}{2-3\gamma}}-3\gamma\left(\sqrt{1-\gamma}(3\gamma-2)k\eta\right)^{\frac{3\gamma}{2-3\gamma}}\right)}{(2-3\gamma)^{2}(\gamma-2)\lambda
k\eta^{2}}c_{1}$ (B.40)
$\displaystyle+e^{-\sqrt{1-\gamma}(k\eta)}(c_{2}\cos(k\eta)+c_{3}\sin(k\eta)).$

####  B.2.4 Small wavelength modes for trackers with $\gamma=0$

For this value of $\gamma$, the equations (B.21) decouple, and we can solve
without the need to go to an asymptotic limit,

$\displaystyle\Phi(\eta)=c_{1}e^{k\eta}+c_{2}e^{-k\eta},$ (B.41)
$\displaystyle\delta\phi(\eta)=(c_{3}k\eta+c_{4})\cos(k\eta)+(c_{4}k\eta-
c_{3})\sin(k\eta).$ (B.42)

####  B.2.5 Perturbations of the volume axion

The equation for the axion perturbations is

$\displaystyle\chi_{k}^{\prime\prime}(\eta)+\left[\frac{2a^{\prime}}{a}+\frac{f^{\prime}}{f}\right]\chi_{k}^{\prime}(\eta)+k^{2}\chi_{k}(\eta)=0,$
(B.43)

where in a tracker background

$\displaystyle a(\eta)=\eta^{\frac{2}{3\gamma-2}},\quad
f(\eta)=\exp\left(\sqrt{\frac{8}{3}\frac{\Phi}{M_{p}}}\right)\sim\eta^{-\frac{4\sqrt{6}\gamma}{(3\gamma-2)\lambda}},$
(B.44)

so that this equation takes the form

$\displaystyle
k^{2}\chi_{k}(\eta)+\chi_{k}^{\prime\prime}(\eta)+\frac{4\left(\lambda-\sqrt{6}\gamma\right)\chi_{k}^{\prime}(\eta)}{(3\gamma-2)\eta\lambda}=0.$
(B.45)

The solutions are

$\displaystyle\chi_{k}(\eta)=\eta^{-\frac{3\gamma\lambda+4\sqrt{6}\gamma-6\lambda}{4\lambda-6\gamma\lambda}}\left(c_{1}J_{\alpha}(k\eta)+c_{2}Y_{\alpha}(k\eta)\right),$
(B.46)

with

$\displaystyle\alpha=-\frac{\sqrt{96\gamma^{2}+9(\gamma-2)^{2}\lambda^{2}+24\sqrt{6}(\gamma-2)\gamma\lambda}}{2(3\gamma-2)\lambda}.$
(B.47)

The kinetic energy scales then scales like radiation

$\displaystyle\rho=\dfrac{f(\eta)\chi_{k}^{\prime}(\eta)^{2}}{a^{(}\eta)}\sim
a^{-4}.$ (B.48)

##  Appendix C Entropy formalism

We can alternatively express the second-order equations (3.66) and (3.67) as a
set of four first-order equations, employing the formalism introduced in [60].
In this approach we keep the gravitational potential $\Phi$ as one of the
variables. Additionally, we introduce the gauge-invariant curvature
perturbation $\mathcal{R}$, along with the gauge-invariant entropy
perturbations $\mathcal{S}$ and $\Gamma$. For convenience we drop the
subscript $k$ in what follows for $\Phi,\mathcal{R},\mathcal{S}$ and $\Gamma$,
but it is important to remember that the quantities are all being evaluated
for a particular $k$ in Fourier space. $S$, denotes the relative entropy
perturbation and is defined as

$\displaystyle\mathcal{S}=\dfrac{3\mathcal{H}\gamma_{r}\gamma_{\phi}\Omega_{r}}{\gamma}\left(\dfrac{\delta\rho_{\phi}}{\rho_{\phi}^{\prime}}-\dfrac{\delta\rho_{r}}{\rho_{r}^{\prime}}\right)=\Omega_{r}\frac{\gamma_{\phi}\delta_{r}-\gamma_{r}\delta_{\phi}}{\gamma},$
(C.1)

where $\gamma=\Omega_{\phi}\gamma_{\phi}+\Omega_{r}\gamma_{r}$, where the
equation of state $\gamma_{r}=1+w_{r}$ etc…, and the intrinsic entropy
perturbation of the scalar field is

$\displaystyle\Gamma=\dfrac{3\mathcal{H}\gamma_{\phi}c_{\phi}^{2}}{1-c_{\phi}^{2}}\left(\dfrac{\delta\rho_{\phi}}{\rho_{\phi}^{\prime}}-\dfrac{\delta
P_{\phi}}{P_{\phi}^{\prime}}\right),$ (C.2)

where the adiabatic sound speed is
$c_{\phi}^{2}=P_{\phi}^{\prime}/\rho_{\phi}^{\prime}$, in terms of the
background scalar field pressure and energy density. The resulting first-order
system in terms of these variables is then expressed as [60]

$\displaystyle\dfrac{d\mathbf{v}}{dN}=\left(M_{0}+k^{2}\mathcal{H}^{-2}M_{1}+k^{4}\mathcal{H}^{-4}M_{2}\right)\mathbf{v},\quad\mathbf{v}=\left(\Phi,\mathcal{R},\mathcal{S},\Gamma\right)^{T},$
(C.3)

with $N=\log(a)$, and the matrices are defined as follows

$\displaystyle
M_{0}=\left(\begin{array}[]{cccc}-\frac{3\gamma}{2}-1&\frac{3\gamma}{2}&0&0\\\
0&0&\frac{\Omega_{\phi}\left(\omega_{r}-c_{\phi}^{2}\right)}{\gamma}&\frac{\left(1-c_{\phi}^{2}\right)\Omega_{r}}{\gamma}\\\
0&0&\frac{3\gamma_{r}\Omega_{r}\left(\omega_{r}-c_{\phi}^{2}\right)}{\gamma}+3\left(\omega_{\phi}-\omega_{r}\right)&\frac{3\left(1-c_{\phi}^{2}\right)\gamma_{r}\Omega_{r}}{\gamma}\\\
0&0&-\frac{3\gamma}{2}&3\left(\omega_{\phi}-\frac{\gamma}{2}\right)\\\
\end{array}\right),$ (C.4) $\displaystyle
M_{1}=\left(\begin{array}[]{cccc}0&0&0&0\\\ -\frac{2c^{2}}{3\gamma}&0&0&0\\\
0&0&\frac{1}{3}&\frac{1}{3}\\\ 0&-\gamma_{\phi}&-\frac{1}{3}&-\frac{1}{3}\\\
\end{array}\right),\quad M_{2}=\left(\begin{array}[]{cccc}0&0&0&0\\\
0&0&0&0\\\ \frac{2\gamma_{\phi}}{9\gamma}&0&0&0\\\
-\frac{2\gamma_{\phi}}{9\gamma}&0&0&0\\\ \end{array}\right),$ (C.5)

with

$\displaystyle
c^{2}=\dfrac{\left(\rho_{\phi}+P_{\phi}\right)c_{\phi}^{2}+\left(\rho_{r}+P_{r}\right)c_{r}^{2}}{\left(\rho_{\phi}+P_{\phi}\right)+\left(\rho_{r}+P_{r}\right)}.$
(C.6)

This formalism is mainly useful for efficiently solving superhorizon limits
($k=0$) and identifying which are the adiabatic modes. We illustrate this in a
couple of examples.

###  C.1 Example 1: Transition from kination to fluid domination

We consider the transition from kination to some fluid with parameter $\gamma$
and introduce the variable

$\displaystyle
y=\dfrac{\rho_{\gamma}}{\rho_{\phi}}=\dfrac{a^{3(2-\gamma)}}{a^{3(2-\gamma)}_{eq}}.$
(C.7)

and consider the system

$\displaystyle
3(2-\gamma)y\dfrac{d\mathbf{v}}{dy}=\dfrac{d\mathbf{v}}{dN}=\left(M_{0}+k^{2}\mathcal{H}^{-2}M_{1}+k^{4}\mathcal{H}^{-4}M_{2}\right)\mathbf{v}\equiv\mathcal{M}\mathbf{v},\quad\mathbf{v}=\left(\Phi,\mathcal{R},\mathcal{S},\Gamma\right)^{T}.$
(C.8)

The matrix $M_{0}$ governs the large-scale evolution ($k=0$), and because

$\displaystyle\Omega_{\phi}(y)=\frac{1}{1+y},\quad\Omega_{r}(y)=\frac{y}{1+y},$
(C.9)

this is given by

$\displaystyle M_{0}=\left(\begin{array}[]{cccc}-\frac{3(\gamma
y+2)}{2(y+1)}-1&\frac{3\gamma y+6}{2y+2}&0&0\\\ 0&0&\frac{\gamma-2}{\gamma
y+2}&0\\\ 0&0&-\frac{6(\gamma-2)}{\gamma y+2}&0\\\ 0&0&-\frac{3(\gamma
y+2)}{2(y+1)}&-\frac{3(\gamma-2)y}{2(y+1)}\\\ \end{array}\right).$ (C.10)

This system has an exact solution equal to

$\displaystyle S(y)=\frac{c_{1}y}{\gamma
y+2},\quad\Gamma(y)=\sqrt{2}c_{2}\sqrt{y+1}-\frac{c_{1}}{\gamma-2},\quad\mathcal{R}(y)=\frac{c_{1}}{6\gamma+3\gamma^{2}y}+c_{3},$
(C.11) $\displaystyle\Phi(y)$
$\displaystyle=\frac{\sqrt{y+1}}{8\gamma}\Big{(}3\gamma^{2}c_{3}\,{}_{2}F_{1}\left(\frac{1}{2},\frac{4}{6-3\gamma};1+\frac{4}{6-3\gamma};-y\right)$
(C.12) $\displaystyle+(c_{1}-3(\gamma-2)\gamma
c_{3})\,_{2}F_{1}\left(\frac{3}{2},\frac{4}{6-3\gamma};1+\frac{4}{6-3\gamma};-y\right)+8\gamma
c_{4}y^{\frac{4}{3(\gamma-2)}}\Big{)}.$

For adiabatic solutions that remain well-defines as $y\rightarrow 0$, we
require that $c_{1}=c_{2}=c_{4}=0.$ For a transition to radiation domination
($\gamma=4/3$), we then obtain

$\displaystyle\Phi(y)=\frac{c_{3}\left(2y^{2}+y-2\sqrt{y+1}+2\right)}{3y^{2}},$
(C.13)

while for a transition to matter domination ($\gamma=1$), we have

$\displaystyle\Phi(y)=c_{3}-\frac{1}{4}c_{3}(y+1)\,_{2}F_{1}\left(1,\frac{11}{6};\frac{7}{3};-y\right).$
(C.14)

###  C.2 Example: $\gamma$-tracker

The behavior of long-wavelength perturbations within the tracker regime has
been discussed in [59] and [60]. These modes can be characterized by the
following matrix

$\displaystyle
M_{0}=\left(\begin{array}[]{cccc}-\frac{3\gamma}{2}-1&\frac{3\gamma}{2}&0&0\\\
0&0&0&\frac{3(2-\gamma)}{\lambda^{2}}\\\
0&0&0&3(2-\gamma)\left(1-\frac{3\gamma}{\lambda^{2}}\right)\\\
0&0&-\frac{3\gamma}{2}&3\left(\frac{\gamma}{2}-1\right)\\\
\end{array}.\right)$ (C.15)

This matrix has constant coefficients, allowing us to express the solution as

$\displaystyle\mathbf{v}=\sum_{i}c_{i}\mathbf{v}_{i}a^{\lambda_{i}},$ (C.16)

where $\mathbf{v}_{i}$ and $\lambda_{i}$ are the eigenvectors and eigenvalues
of $M_{0}$. The eigenvalues are given by

$\displaystyle\lambda_{1},\lambda_{2},\lambda_{3,4}=\left\\{0,-\frac{3\gamma}{2}-1,\frac{3\left((\gamma-2)\pm\sqrt{(\gamma-2)\left(-24\gamma^{2}+9\gamma\lambda^{2}-2\lambda^{2}\right)}\lambda\right)}{4\lambda}\right\\}.$
(C.17)

The first two modes are adiabatic ($S(a)=\Gamma(a)=0$), with eigenvectors
equal to

$\displaystyle\mathbf{v}_{1}=\left(\frac{3\gamma}{3\gamma+2},1,0,0\right),\quad\mathbf{v}_{2}=(1,0,0,0).$
(C.18)

The other two modes are non-adiabatic and the expressions for their
eigenvectors are long and not very illuminating. When
$\lambda^{2}>24\gamma^{2}/(9\gamma-2)$, the real part of these eigenvalues are
negative, and so show an oscillatory behaviour.

####  C.2.1 Radiation tracker

For the radiation tracker, with $\gamma=4/3$, this looks

$\displaystyle M_{0}=\left(\begin{array}[]{cccc}-3&2&0&0\\\
0&0&0&\frac{2}{\lambda^{2}}\\\ 0&0&0&2\left(1-\frac{4}{\lambda^{2}}\right)\\\
0&0&-2&-1\\\ \end{array}\right),$ (C.19)

and we can express the solution as

$\displaystyle\mathbf{v}=\sum_{i}c_{i}\mathbf{v}_{i}a^{\lambda_{i}},\quad\mathbf{v}=\left(\Phi,\mathcal{R},\mathcal{S},\Gamma\right)^{T}$
(C.20)

where $\mathbf{v}_{i}$ are the eigenvectors of $M_{0}$, given by

$\displaystyle\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3,4}=\left\\{(2,3,0,0),(1,0,0,0),\left(-\frac{4}{5\lambda^{2}\pm
i\lambda\nu-16},-\frac{4}{\lambda^{2}\pm i\lambda\nu},\frac{-\lambda\pm
i\nu}{4\lambda},1\right)\right\\},$ (C.21)

and $\mathbf{v}_{3,4}$ refers to the plus and minus solutions in Eq. (C.21)
respectively, with $\nu=\sqrt{64-15\lambda^{2}}$ and the values for the
eigenvalues $\lambda_{i}$ are given by

$\displaystyle\lambda_{1},\lambda_{2},\lambda_{3,4}=\left\\{0,-3,\frac{-1\pm\sqrt{\frac{64}{\lambda^{2}}-15}}{2}\right\\}.$
(C.22)

The first two modes are a leading $\lambda_{1}=0$ constant and a subleading
$\lambda_{2}=-3$ decaying mode, both of which are adiabatic ($\mathcal{S}=0$,
$\Gamma=0$). The last two modes are non-adiabatic and decaying (given that
$\lvert\lambda\rvert>2=\sqrt{3\gamma}$), and these eigenvalues have an
imaginary part when $\lambda>\sqrt{64/15}\approx 2.07$.

####  C.2.2 Matter tracker

The discussion of super-horizon modes is qualitatively the same as for the
radiation tracker. The large-scale matrix in the matter tracker with
$\gamma=1$ is given by,

$\displaystyle
M_{0}=\left(\begin{array}[]{cccc}-\frac{5}{2}&\frac{3}{2}&0&0\\\
0&0&0&\frac{3}{\lambda^{2}}\\\ 0&0&0&3\left(1-\frac{3}{\lambda^{2}}\right)\\\
0&0&-\frac{3}{2}&-\frac{3}{2}\\\ \end{array}\right).$ (C.23)

We express the general solution as

$\displaystyle\mathbf{v}=\sum_{i}c_{i}\mathbf{v}_{i}a^{\lambda_{i}},$ (C.24)

where $\mathbf{v}_{i}$ and $\lambda_{i}$ are the eigenvectors and eigenvalues
of $M_{0}$. We find that

$\displaystyle\lambda_{1},\lambda_{2},\lambda_{3,4}=\left\\{0,-\frac{5}{2},\frac{3\left(\lambda\pm\sqrt{24-7\lambda^{2}}\right)}{4\lambda}\right\\},$
(C.25)

and the respective eigenvectors are given by

$\displaystyle\mathbf{v}_{1}=\left(\frac{3}{5},1,0,0\right),\quad\mathbf{v}_{2}=(1,0,0,0),\quad\mathbf{v}_{3,4}=\left(\frac{6}{-7\lambda^{2}\pm
i\lambda\nu+18},\frac{4}{\lambda(-\lambda\pm i\nu)},-\frac{\lambda\pm
i\nu}{2\lambda},1\right)$ (C.26)

The last two eigenvalues represent non-adiabatic modes. These modes are
subdominant compared to the first mode as they exhibit decay over time as long
as $\lvert\lambda\rvert>\sqrt{3}=\sqrt{3\gamma}$, which is a necessary
condition for the tracker to exist.

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Euler-type integrals for the generalized hypergeometric matrix function

Ankit Pala, Kiran Kumarib
<EMAIL_ADDRESS><EMAIL_ADDRESS>
aDepartment of Mathematics, School of Advanced Sciences & Languages

VIT Bhopal University, Sehore-466 114, Madhya Pradesh, India.

bDepartment of Mathematics and Humanities,

Sardar Vallabhbhai National Institute of Technology, Surat-395 007, Gujarat,
India.

Abstract: In this paper, we investigate the Euler-type integral
representations for the generalized hypergeometric matrix function and develop
some transformations in terms of hypergeometric matrix functions. Furthermore,
unit and half arguments have been provided for several particular cases.
Keywords: Generalized hypergeometric matrix function, Matrix functional
calculus, Gamma and Beta matrix function.
AMS Subject Classification (2010): 15A15, 33C05, 33C20, 44A99.

##  1 Introduction and Preliminaries

In recent years, mathematical analysis has paid a lot of attention to the
theory of special matrix functions and polynomials. The matrix analogue of
hypergeometric functions play a significant role in the area of applied and
pure analysis. These functions appears in the study of statistics [3],
probability theory [13] and Lie theory [10]. The matrix analogue of the Gauss
hypergeometric function was introduced by Jódar and Cortés [11]. They studied
the integral representation and matrix differential equation. Later, Abdalla
[1, 2], Dwivedi and Sahai [6, 7] studied the various properties like integral
and differential relations, finite sum formulas, generating functions of
special matrix functions. Particularly, these functions play a vital role in
solving numerous problems of mathematical physics, engineering and
mathematical sciences.

Throughout this work, we consider the complex space $\mathbf{C}^{R\times R}$
of complex matrices of common order $R$. For any matrix
$P\in\mathbf{C}^{R\times R}$, $\sigma(P)$ is the spectrum of $P$ and

$\displaystyle a(P)=\max\left\\{\Re(z):z\in\sigma(P)\right\\},\quad
b(P)=\min\left\\{\Re(z):z\in\sigma(P)\right\\},$ (1.1)

where $a(P)$ is the spectral abscissa of $P$ and $b(P)=-a(-P)$. A Hermitian
matrix $P$ in $\mathbf{C}^{R\times R}$ is a positive stable matrix if
$\Re(\lambda)>0$ for all $\lambda\in\sigma(P)$, where $\sigma(P)$ is the set
of all eigenvalues of $P$ or spectrum of $A$ and its two-norm is given by

$\displaystyle||P||=\sup_{x\neq
0}\frac{||Px||_{2}}{||x||_{2}}=\max\left\\{\sqrt{\lambda}:\lambda\in\sigma(P^{*}P)\right\\},$

where for any vector $x\in\mathbf{C}^{r}$, $||x||_{2}=(x^{*}x)^{1/2}$ is the
Euclidean norm of $x$ and $P^{*}$ denotes the conjugate transpose of $P$. $I$
and $\mathbf{0}$ stands for the identity matrix and null matrix in
$\mathbf{C}^{R\times R}$, respectively. Taking into account the Schür
decomposition of a matrix $P$ [8], we have

$\displaystyle\left\|e^{Pt}\right\|\leq
e^{ta(P)}\sum_{u=0}^{r-1}\frac{\left(\left\|P\right\|r^{1/2}t\right)^{u}}{u!}\quad(t\geq
0),$

which yields

$\displaystyle\left\|t^{P}\right\|\leq\left\|e^{P\ln t}\right\|\leq
t^{a(P)}\sum_{u=0}^{r-1}\frac{\left(\left\|P\right\|r^{1/2}\ln
t\right)^{u}}{u!}\quad(t\geq 1).$

If $f(z)$ and $g(z)$ are holomorphic functions of the complex variable $z$,
which are defined in an open set $\Omega$ of the complex plane, and $P$ is a
matrix in $\mathbf{C}^{R\times R}$ with $\sigma(P)\subset\Omega$, then from
the properties of the matrix functional calculus [5], it follows that

$\displaystyle f(P)g(P)=g(P)f(P).$

Furthermore, if $Q\in\mathbf{C}^{R\times R}$ with $\sigma(Q)\subset\Omega$,
and if $PQ=QP$, then

$\displaystyle f(P)g(Q)=g(Q)f(P).$ (1.2)

The logarithmic norm of a matrix $P\in\mathbf{C}^{R\times R}$ is defined as
(see [4, 9]),

$\displaystyle\mu(P)=\lim_{k\to
0}\frac{\left\|I+kP\right\|-1}{k}=\max\left\\{z\ |\
z\in\sigma\left(\frac{P+P^{*}}{2}\right)\right\\}$ (1.3)

Let the number $\tilde{\mu}(P)$ such that

$\displaystyle\tilde{\mu}(P)=-\mu(-P)=\min\left\\{z\ |\
z\in\sigma\left(\frac{P+P^{*}}{2}\right)\right\\}.$ (1.4)

The reciprocal gamma function $\Gamma^{-1}(z)=1/\Gamma(z)$ is an entire
function of the complex variable $z$. The image of $\Gamma^{-1}(z)$ acting on
$P$, denoted by $\Gamma^{-1}(P)$, is a well defined matrix. If $P+nI$ is
invertible for all integers $n\geq 0$, then the reciprocal gamma function is
defined as (see [12]),

$\displaystyle\Gamma^{-1}(P)=P(P+I)\dots(P+(n-1)I)\ \Gamma^{-1}(P+nI),\ \
n\geq 1.$ (1.5)

By applications of the matrix functional calculus, the Pochhammer symbol [12]
for $P\in\mathbf{C}^{R\times R}$ is given by

$\displaystyle(P)_{m}=\begin{cases}I,\;\;\;\;\;\;\;\;\;\;\hskip
113.81102pt\text{if}\;m=0\\\ P(P+I)\dots(P+(m-1)I),\hskip
22.76228pt\text{if}\;m\geq 1,\end{cases}$ (1.6)

which gives

$\displaystyle(P)_{m}=\Gamma^{-1}(P)\ \Gamma(P+mI),\quad m\geq 1.$ (1.7)

If $P\in\mathbf{C}^{R\times R}$ is a positive stable matrix and $m\geq 1$ is
an integer, then the gamma matrix function can be represented in the following
limit form as [11]:

$\displaystyle\Gamma(P)=\lim_{m\to\infty}(m-1)!\ (P)_{m}^{-1}\ m^{P}.$ (1.8)

Let $P$ and $Q$ be two positive stable matrices in $\mathbf{C}^{R\times R}$.
The gamma matrix function $\Gamma(P)$ and the beta matrix function $B(P,Q)$
have been defined in [11, 12], as follows

$\displaystyle\Gamma(P)=\int_{0}^{\infty}e^{-t}\ t^{P-1}dt;\quad
t^{P-1}=\exp((P-I)\ln t),$ (1.9)

and

$\displaystyle B(P,Q)=\int_{0}^{1}t^{P-1}\ (1-t)^{Q-1}dt.$ (1.10)

Let P and Q be commuting matrices in $\mathbf{C}^{R\times R}$ such that the
matrices $P+nI,Q+nI$ and $P+Q+nI$ are invertible for every integer $n\geq 0$,
then according to [12], we have

$\displaystyle B(P,Q)=\Gamma(P)\Gamma(Q)\left[\Gamma(P+Q)\right]^{-1}.$ (1.11)

##  2 Main results

Dwivedi and Sahai [6] introduced a natural generalization of hypergeometric
matrix function called generalized hypergeometric matrix function and defined
by

$\displaystyle{}_{p}F_{q}\left(P_{1},\dots,P_{p};Q_{1},\dots,Q_{q};z\right)=\sum_{m=0}^{\infty}\frac{(P_{1})_{m}\dots(P_{p})_{m}(Q_{1})_{m}^{-1}\dots(Q_{q})_{m}^{-1}}{m!}z^{m},$
(2.1)

where $P_{i},Q_{j}\in\mathbf{C}^{R\times R},1\leq i\leq p,1\leq j\leq q$, such
that $Q_{j}+mI,1\leq j\leq q$ are invertible for all integers $m\geq 0$.

In the present work, we consider an Euler-type integral and present some
integral representations of ${}_{3}F_{2}(\cdot)$ matrix function using the
suitable adjustment of matrix parameters.

######  Theorem 2.1.

Let $P,Q$ and $R$ be commuting matrices in $\mathbf{C}^{R\times R}$ such that
$Q,R$ and $R-Q$ are positive stable. Then, for $|z|<1$, we have the following
integral representation:

$\displaystyle{}_{3}F_{2}\left(P,\frac{Q}{2},\frac{Q+I}{2};\frac{R}{2},\frac{R+I}{2};z\right)=\frac{\Gamma(R)}{\Gamma(Q)\Gamma(R-Q)}\int_{0}^{1}u^{Q-I}(1-u)^{R-Q-I}(1-zu^{2})^{-P}du.$
(2.2)

###### Proof.

From the left-hand side of (2.2), we find that

$\displaystyle{}_{3}F_{2}\left(P,\frac{Q}{2},\frac{Q+I}{2};\frac{R}{2},\frac{R+I}{2};z\right)=\sum_{m=0}^{\infty}\frac{\left(P\right)_{m}\left(\frac{Q}{2}\right)_{m}\left(\frac{Q+I}{2}\right)_{m}}{\left(\frac{R}{2}\right)_{m}\left(\frac{R+I}{2}\right)_{m}}\frac{z^{m}}{m!}.$
(2.3)

Using the relation

$\displaystyle\left(P\right)_{2m}=2^{2m}\left(\frac{P}{2}\right)_{m}\left(\frac{P+I}{2}\right)_{m},$

equation (2.3) becomes

$\displaystyle{}_{3}F_{2}\left(P,\frac{Q}{2},\frac{Q+I}{2};\frac{R}{2},\frac{R+I}{2};z\right)$
$\displaystyle=\sum_{m=0}^{\infty}\frac{\left(P\right)_{m}\left(Q\right)_{2m}}{\left(R\right)_{2m}}\frac{z^{m}}{m!}$
$\displaystyle=\frac{\Gamma(R)}{\Gamma(Q)\Gamma(R-Q)}\int_{0}^{1}u^{Q-I}(1-u)^{R-Q-I}\sum_{m=0}^{\infty}\frac{\left(P\right)_{m}(zu^{2})^{m}}{m!}\
du$
$\displaystyle=\frac{\Gamma(R)}{\Gamma(Q)\Gamma(R-Q)}\int_{0}^{1}u^{Q-I}(1-u)^{R-Q-I}(1-zu^{2})^{-P}du.$

This completes the proof of Theorem 2.1. ∎

######  Corollary 2.1.

Under the conditions stated in Theorem 2.1, the following integral relation
holds true:

$\displaystyle{}_{3}F_{2}\left(-kI,\frac{Q}{2},\frac{Q+I}{2};\frac{R}{2},\frac{R+I}{2};z\right)=\frac{\Gamma(R)}{\Gamma(Q)\Gamma(R-Q)}\int_{0}^{1}u^{Q-I}(1-u)^{R-Q-I}(1-zu^{2})^{kI}du.$
(2.4)

######  Theorem 2.2.

Let $P,Q$ and $R$ be commuting matrices in $\mathbf{C}^{R\times R}$ such that
$R,R-P,R-Q$ and $R-Q-P$ are positive stable. Then the following integral
representation holds true:

$\displaystyle{}_{3}F_{2}\left(P,\frac{Q}{2},\frac{Q+I}{2};\frac{R}{2},\frac{R+I}{2};1\right)=\frac{\Gamma(R)\Gamma(R-Q-P)}{\Gamma(R-P)\Gamma(R-Q)}\
{}_{2}F_{1}\left(P,Q;R-P;-1\right).$ (2.5)

###### Proof.

Substituting $z=1$ in (2.2), it becomes

$\displaystyle{}_{3}F_{2}\left(P,\frac{Q}{2},\frac{Q+I}{2};\frac{R}{2},\frac{R+I}{2};1\right)$
$\displaystyle=\frac{\Gamma(R)}{\Gamma(Q)\Gamma(R-Q)}\int_{0}^{1}u^{Q-I}(1-u)^{R-Q-
P-I}(1+u)^{-P}du$
$\displaystyle=\frac{\Gamma(R)}{\Gamma(Q)\Gamma(R-Q)}\sum_{m=0}^{\infty}{-P\choose
m}\int_{0}^{1}u^{Q+(m-1)I}(1-u)^{R-Q-P-I}\ du$
$\displaystyle=\frac{\Gamma(R)}{\Gamma(Q)\Gamma(R-Q)}\sum_{m=0}^{\infty}{-P\choose
m}\frac{\Gamma(Q+mI)\Gamma(R-Q-P)}{\Gamma(R-P+mI)}$
$\displaystyle=\frac{\Gamma(R)\Gamma(R-Q-P)}{\Gamma(R-P)\Gamma(R-Q)}\sum_{m=0}^{\infty}\frac{(P)_{m}(Q)_{m}}{(R-P)_{m}}\frac{(-1)^{m}}{m!}$
$\displaystyle=\frac{\Gamma(R)\Gamma(R-Q-P)}{\Gamma(R-P)\Gamma(R-Q)}\
{}_{2}F_{1}\left(P,Q;R-P;-1\right),$

which completes the proof of Theorem 2.2. ∎

######  Corollary 2.2.

Under the conditions stated in Theorem 2.2, the following integral relation
holds true:

$\displaystyle{}_{3}F_{2}\left(-nI,\frac{Q}{2},\frac{Q+I}{2};\frac{R}{2},\frac{R+I}{2};1\right)=\frac{\left(R-Q\right)_{n}}{\left(R\right)_{n}}\
{}_{2}F_{1}\left(-nI,Q;R+nI;-1\right).$

######  Theorem 2.3.

Let $P,Q$ and $R$ be commuting matrices in $\mathbf{C}^{R\times R}$ such that
$Q,R$ and $R-Q$ are positive stable. Then the following integral
representation holds true:

$\displaystyle{}_{3}F_{2}\left(P,\frac{Q}{2},\frac{Q+I}{2};\frac{R}{2},\frac{R+I}{2};\frac{1}{2}\right)=2^{P}\sum_{m=0}^{\infty}{-P\choose
m}\frac{(R-Q)_{m}}{(R)_{m}}\times{}_{2}F_{1}\left(-mI,Q;R+mI;-1\right).$ (2.6)

###### Proof.

Substituting $z=\frac{1}{2}$ in Theorem 2.1, we obtain

$\displaystyle{}_{3}F_{2}\left(P,\frac{Q}{2},\frac{Q+I}{2};\frac{R}{2},\frac{R+I}{2};\frac{1}{2}\right)$
$\displaystyle\quad=\frac{\Gamma(R)}{\Gamma(Q)\Gamma(R-Q)}\int_{0}^{1}u^{Q-I}(1-u)^{R-Q-I}\left(1-\frac{1}{2}u^{2}\right)^{-P}du$
$\displaystyle\quad=\frac{2^{P}\
\Gamma(R)}{\Gamma(Q)\Gamma(R-Q)}\int_{0}^{1}u^{Q-I}(1-u)^{R-Q-I}\left(2-u^{2}\right)^{-P}du$
$\displaystyle\quad=\frac{2^{P}\
\Gamma(R)}{\Gamma(Q)\Gamma(R-Q)}\sum_{m=0}^{\infty}{-P\choose
m}\int_{0}^{1}u^{Q-I}(1-u)^{R-Q+(m-1)I}\left(1+u\right)^{m}du$
$\displaystyle\quad=\frac{2^{P}\
\Gamma(R)}{\Gamma(Q)\Gamma(R-Q)}\sum_{m=0}^{\infty}\sum_{k=0}^{m}{-P\choose
m}{m\choose k}\int_{0}^{1}u^{Q+(k-1)I}(1-u)^{R-Q+(m-1)I}du$
$\displaystyle\quad=2^{P}\sum_{m=0}^{\infty}\sum_{k=0}^{m}{-P\choose
m}{m\choose k}\frac{(Q)_{k}(R-Q)_{m}}{(R)_{k+m}}.$

Using the transformation $(R)_{k+m}=(R)_{m}(R+m)_{k}$, we arrive at

$\displaystyle{}_{3}F_{2}\left(P,\frac{Q}{2},\frac{Q+I}{2};\frac{R}{2},\frac{R+I}{2};\frac{1}{2}\right)$
$\displaystyle\quad=2^{P}\sum_{m=0}^{\infty}{-P\choose
m}\frac{(R-Q)_{m}}{(R)_{m}}\sum_{k=0}^{m}{m\choose
k}\frac{(Q)_{k}}{(R+m)_{k}}$
$\displaystyle\quad=2^{P}\sum_{m=0}^{\infty}{-P\choose
m}\frac{(R-Q)_{m}}{(R)_{m}}\times{}_{2}F_{1}\left(-mI,Q;R+mI;-1\right).$

This completes the proof of Theorem 2.3. ∎

######  Corollary 2.3.

Under the conditions stated in Theorem 2.3, the following integral relation
holds true:

$\displaystyle{}_{3}F_{2}\left(P,\frac{Q}{2},\frac{Q+I}{2};\frac{R}{2},\frac{R+I}{2};\frac{1}{2}\right)=2^{P}\sum_{m=0}^{\infty}{-P\choose
m}\
{}_{3}F_{2}\left(-mI,\frac{Q}{2},\frac{Q+I}{2};\frac{R}{2},\frac{R+I}{2};1\right)$
(2.7)

Now we generalize the Theorem 2.1 in the following form using the suitable
adjustment of argument in the ${}_{3}F_{2}(\cdot)$ matrix function.

######  Theorem 2.4.

Let $P,Q$ and $R$ be commuting matrices in $\mathbf{C}^{R\times R}$ such that
$Q,R$ and $R-Q$ are positive stable and
$w\in\mathbf{R}\backslash\left\\{0,-1\right\\}$. Then the following integral
representation holds true:

$\displaystyle{}_{3}F_{2}\left(P,\frac{Q}{2},\frac{Q+I}{2};\frac{R}{2},\frac{R+I}{2};\frac{1}{w+1}\right)$
$\displaystyle\qquad\qquad\qquad=\left(\frac{w+1}{w}\right)^{P}\sum_{m=0}^{\infty}{-P\choose
m}w^{-m}{}_{3}F_{2}\left(-mI,\frac{Q}{2},\frac{Q+I}{2};\frac{R}{2},\frac{R+I}{2};1\right).$
(2.8)

###### Proof.

Substituting $z=\frac{1}{w+1}$ in Theorem 2.1, we have

$\displaystyle{}_{3}F_{2}\left(P,\frac{Q}{2},\frac{Q+I}{2};\frac{R}{2},\frac{R+I}{2};\frac{1}{w+1}\right)$
$\displaystyle\quad=\frac{\Gamma(R)}{\Gamma(Q)\Gamma(R-Q)}\int_{0}^{1}u^{Q-I}(1-u)^{R-Q-I}\left(1-\frac{u^{2}}{w+1}\right)^{-P}du$
$\displaystyle\quad=\frac{(w+1)^{P}\
\Gamma(R)}{\Gamma(Q)\Gamma(R-Q)}\int_{0}^{1}u^{Q-I}(1-u)^{R-Q-I}\left(w+1-u^{2}\right)^{-P}du$
$\displaystyle\quad=\left(\frac{w+1}{w}\right)^{P}\frac{\Gamma(R)}{\Gamma(Q)\Gamma(R-Q)}\int_{0}^{1}u^{Q-I}(1-u)^{R-Q-I}\left(1+\frac{1-u^{2}}{w}\right)^{-P}du$
$\displaystyle\quad=\left(\frac{w+1}{w}\right)^{P}\frac{\Gamma(R)}{\Gamma(Q)\Gamma(R-Q)}\int_{0}^{1}u^{Q-I}(1-u)^{R-Q-I}\sum_{m=0}^{\infty}{-P\choose
m}\left(\frac{1-u^{2}}{w}\right)^{m}du$
$\displaystyle\quad=\left(\frac{w+1}{w}\right)^{P}\frac{\Gamma(R)}{\Gamma(Q)\Gamma(R-Q)}\sum_{m=0}^{\infty}{-P\choose
m}w^{-m}\int_{0}^{1}u^{Q-I}(1-u)^{R-Q+(m-1)I}\left(1+u\right)^{m}du$
$\displaystyle\quad=\left(\frac{w+1}{w}\right)^{P}\frac{\Gamma(R)}{\Gamma(Q)\Gamma(R-Q)}\sum_{m=0}^{\infty}\sum_{k=0}^{m}{-P\choose
m}{m\choose k}w^{-m}\int_{0}^{1}u^{Q+(k-1)I}(1-u)^{R-Q+(m-1)I}du$
$\displaystyle\quad=\left(\frac{w+1}{w}\right)^{P}\sum_{m=0}^{\infty}\sum_{k=0}^{m}{-P\choose
m}{m\choose k}w^{-m}\frac{(Q)_{k}(R-Q)_{m}}{(R)_{m}(R+m)_{k}}$
$\displaystyle\quad=\left(\frac{w+1}{w}\right)^{P}\sum_{m=0}^{\infty}{-P\choose
m}w^{-m}\frac{(R-Q)_{m}}{(R)_{m}}\sum_{k=0}^{m}\frac{(-1)^{k}(-m)_{k}}{k!}\frac{(Q)_{k}}{(R+m)_{k}}$
$\displaystyle\quad=\left(\frac{w+1}{w}\right)^{P}\sum_{m=0}^{\infty}{-P\choose
m}w^{-m}\frac{(R-Q)_{m}}{(R)_{m}}{}_{2}F_{1}\left(-mI,Q;R+mI;-1\right).$

Using Corollary 2.2, this yields the right hand side of Theorem 2.4. ∎

######  Remark 2.1.

It is interesting to observe that for $w=1$, Theorem 2.4 reduces to Theorem
2.3 and Corollary 2.3 and for $w=-2$, it reduces to the following result
asserted by Corollary 2.4.

######  Corollary 2.4.

Under the conditions stated in Theorem 2.4, the following integral relation
holds true:

$\displaystyle{}_{3}F_{2}\left(P,\frac{Q}{2},\frac{Q+I}{2};\frac{R}{2},\frac{R+I}{2};-1\right)=\left(\frac{1}{2}\right)^{P}\sum_{m=0}^{\infty}{-P\choose
m}(-2)^{-m}{}_{3}F_{2}\left(-mI,\frac{Q}{2},\frac{Q+I}{2};\frac{R}{2},\frac{R+I}{2};1\right).$

Next we generalize the Theorem 2.1 in the obvious way using the suitable
adjustment of matrix parameters by introducing the sequence of $q$ parameters
in the ${}_{3}F_{2}(\cdot)$ matrix function.

######  Theorem 2.5.

Let $P,Q$ and $R$ be commuting matrices in $\mathbf{C}^{R\times R}$ such that
$Q,R$ and $R-Q$ are positive stable. Then, for $|z|<1$, the following integral
representation holds true:

$\displaystyle{}_{q+1}F_{q}\left(P,\frac{Q}{q},\frac{Q+I}{q},\dots,\frac{Q+(q-1)I}{q};\frac{R}{q},\frac{R+I}{q},\dots,\frac{R+(q-1)I}{q};z\right)$
$\displaystyle\qquad\qquad\qquad\qquad\quad=\frac{\Gamma(R)}{\Gamma(Q)\Gamma(R-Q)}\int_{0}^{1}u^{Q-I}(1-u)^{R-Q-I}(1-zu^{q})^{-P}du.$
(2.9)

###### Proof.

Using the following relation:

$\displaystyle\left(P\right)_{mn}=m^{mn}\left(\frac{P}{m}\right)_{n}\left(\frac{P+I}{m}\right)_{n}\dots\left(\frac{P+(m-1)I}{m}\right)_{n},$

we can easily proceed for the proof similar to Theorem 2.1. ∎

Now for $q=3$ and $z=1$, Theorem 2.5 leads to the following result asserted by
Theorem 2.6.

######  Theorem 2.6.

Let $P,Q$ and $R$ be commuting matrices in $\mathbf{C}^{R\times R}$ such that
$R,R-P,R-Q$ and $R-Q-P$ are positive stable. Then the following integral
representation holds true:

$\displaystyle{}_{4}F_{3}\left(P,\frac{Q}{3},\frac{Q+I}{3},\frac{Q+2I}{3};\frac{R}{3},\frac{R+I}{3},\frac{R+2I}{3};1\right)$
$\displaystyle\qquad=\frac{\Gamma(R)\Gamma(R-Q-P)}{\Gamma(R-P)\Gamma(R-Q)}\sum_{m=0}^{\infty}\frac{(-1)^{m}(P)_{m}(Q)_{m}}{m!(R-P)_{m}}\
{}_{2}F_{1}\left(-mI,Q+mI;R-P+mI;-1\right).$ (2.10)

###### Proof.

Substituting $q=3$ and $z=1$ in Theorem 2.5, we have

$\displaystyle{}_{4}F_{3}\left(P,\frac{Q}{3},\frac{Q+I}{3},\frac{Q+2I}{3};\frac{R}{3},\frac{R+I}{3},\frac{R+2I}{3};1\right)$
$\displaystyle\quad=\frac{\Gamma(R)}{\Gamma(Q)\Gamma(R-Q)}\int_{0}^{1}u^{Q-I}(1-u)^{R-Q-I}(1-u^{3})^{-P}du$
$\displaystyle\quad=\frac{\Gamma(R)}{\Gamma(Q)\Gamma(R-Q)}\int_{0}^{1}u^{Q-I}(1-u)^{R-Q-
P-I}(1+u+u^{2})^{-P}du$
$\displaystyle\quad=\frac{\Gamma(R)}{\Gamma(Q)\Gamma(R-Q)}\sum_{m=0}^{\infty}\frac{(P)_{m}(-1)^{m}}{m!}\int_{0}^{1}u^{Q+(m-1)I}(1-u)^{R-Q-
P-I}(1+u)^{m}du$
$\displaystyle\quad=\frac{\Gamma(R)}{\Gamma(Q)\Gamma(R-Q)}\sum_{m=0}^{\infty}\sum_{k=0}^{m}\frac{(P)_{m}(-1)^{m}}{m!}{m\choose
k}\int_{0}^{1}u^{Q+(m+k-1)I}(1-u)^{R-Q-P-I}du$
$\displaystyle\quad=\frac{\Gamma(R)\Gamma(R-Q-P)}{\Gamma(R-P)\Gamma(R-Q)}\sum_{m=0}^{\infty}\sum_{k=0}^{m}\frac{(P)_{m}(-1)^{m}}{m!}{m\choose
k}\frac{(Q)_{m}(Q+m)_{k}}{(R-P)_{m}(R-P+m)_{k}}$
$\displaystyle\quad=\frac{\Gamma(R)\Gamma(R-Q-P)}{\Gamma(R-P)\Gamma(R-Q)}\sum_{m=0}^{\infty}\frac{(-1)^{m}(P)_{m}(Q)_{m}}{m!(R-P)_{m}}\
{}_{2}F_{1}\left(-mI,Q+mI;R-P+mI;-1\right).$

This completes the proof. ∎

######  Theorem 2.7.

Let $P,Q$ and $R$ be commuting matrices in $\mathbf{C}^{R\times R}$ such that
$Q,R$ and $R-Q$ are positive stable. Then, for $|z|<1$, the following integral
representation holds true:

$\displaystyle{}_{q+1}F_{q}\left(P,\frac{Q}{q},\frac{Q+I}{q},\dots,\frac{Q+(q-1)I}{q};\frac{R}{q},\frac{R+I}{q},\dots,\frac{R+(q-1)I}{q};z\right)$
$\displaystyle\qquad=\frac{\Gamma(R)}{\Gamma(Q)\Gamma(R-Q)}\sum_{m=0}^{\infty}{R-Q-I\choose
m}\frac{(-1)^{m}}{Q+mI}\
{}_{2}F_{1}\left(P,\frac{Q+mI}{2};\frac{Q+mI}{2}+I;z\right).$ (2.11)

###### Proof.

From Theorem 2.5, we have

$\displaystyle{}_{q+1}F_{q}\left(P,\frac{Q}{q},\frac{Q+I}{q},\dots,\frac{Q+(q-1)I}{q};\frac{R}{q},\frac{R+I}{q},\dots,\frac{R+(q-1)I}{q};z\right)$
$\displaystyle\quad=\frac{\Gamma(R)}{\Gamma(Q)\Gamma(R-Q)}\int_{0}^{1}u^{Q-I}(1-u)^{R-Q-I}(1-zu^{q})^{-P}du$
$\displaystyle\quad=\frac{\Gamma(R)}{\Gamma(Q)\Gamma(R-Q)}\int_{0}^{1}u^{Q-I}\left[\sum_{m=0}^{\infty}{R-Q-I\choose
m}(-u)^{m}\right](1-zu^{q})^{-P}du$
$\displaystyle\quad=\frac{\Gamma(R)}{\Gamma(Q)\Gamma(R-Q)}\sum_{m=0}^{\infty}{R-Q-I\choose
m}(-1)^{m}\int_{0}^{1}u^{Q+(m-1)I}(1-zu^{q})^{-P}du.$

Substitution $s=u^{q}$ in the integral on the right hand side of above
equation yields

$\displaystyle{}_{q+1}F_{q}\left(P,\frac{Q}{q},\frac{Q+I}{q},\dots,\frac{Q+(q-1)I}{q};\frac{R}{q},\frac{R+I}{q},\dots,\frac{R+(q-1)I}{q};z\right)$
$\displaystyle\quad=\frac{\Gamma(R)}{\Gamma(Q)\Gamma(R-Q)}\sum_{m=0}^{\infty}{R-Q-I\choose
m}\frac{(-1)^{m}}{q}\frac{\Gamma\left(\frac{Q+mI}{q}\right)\Gamma(1)}{\Gamma\left(\frac{Q+mI}{q}+I\right)}\
{}_{2}F_{1}\left(P,\frac{Q+mI}{2};\frac{Q+mI}{2}+I;z\right).$

Now using the Pochhammer matrix symbols (1.7) yields the right hand side of
Theorem 2.7. ∎

## Statements & Declarations

### Funding

Not applicable.

### Conflicts of interest/Competing interests

The authors declare that they have no competing interests.

### Authors contributions

Both the authors have equally contributed of reading and writing the
manuscript.

## References

## References

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Bose-Einstein Condensation for Lattice Bosons

Tohru Koma 111Department of Physics, Gakushuin University (retired), Mejiro,
Toshima-ku, Tokyo 171-8588, JAPAN

Abstract: We present a class of models of interacting lattice bosons which
show complete Bose-Einstein condensation for the ground state.

##  1 Introduction

As is well known, it is very difficult to prove the existence of Bose-Einstein
condensation for interacting bosons. Actually, the mathematically rigorous
results are still rare. (See, e.g., [1, 2] and references therein.) In this
paper, we present a class of interacting boson models on the $d$-dimensional
hypercubic lattice for $d\geq 1$. We prove that the models show complete Bose-
Einstein condensation for the ground state. Although the interactions in the
Hamiltonians are not necessarily standard, each of them is finite range or
rapidly decays with the diameter of the support.

The present paper is organized as follows: In the next Sec. 2, we give the
precise definition of the simplest model in our class and the corresponding
theorem with the proof. In Sec. 3, general models in our class are presented,
and we also give an argument about the absence of Bose-Einstein condensation
[3] for certain models at zero temperature. However, the argument is not
necessarily mathematically rigorous.

##  2 Model and Result

We begin with the simplest model in our class. The extension to general models
will be given in the next Sec. 3.

We consider a Bose gas on a $d$-dimensional hypercubic lattice
$\Lambda\subset\mathbb{Z}^{d}$ with the dimension $d\geq 1$. The Hamiltonian
with nearest neighbor hopping and nearest neighbor interactions is given by

$H_{g}^{(N)}:=\frac{1}{2}\sum_{x,y\in\Lambda:|x-y|=1}(a_{x}^{\dagger}-a_{y}^{\dagger})(a_{x}-a_{y})+g\sum_{x,y\in\Lambda:|x-y|=1}[n_{x}(n_{x}-1)+n_{x}n_{y}-a_{x}^{\dagger}n_{x}a_{y}-a_{y}^{\dagger}n_{x}a_{x}],$
(2.1)

where $a_{x}^{\dagger}$ and $a_{x}$ are, respectively, the creation and
annihilation operators at the site $x=(x^{(1)},x^{(2)},\ldots,x^{(d)})$ in the
finite lattice $\Lambda\subset\mathbb{Z}^{d}$; $n_{x}:=a_{x}^{\dagger}a_{x}$
is the number operator at the site $x$, and $g\geq 0$ is the coupling
constant. As usual, the operators, $a_{x}^{\dagger}$ and $a_{x}$, obey the
commutation relations,

$[a_{x},a_{y}^{\dagger}]=\delta_{x,y},\quad[a_{x},a_{y}]=0,\ \ \mbox{and}\ \ \
[a_{x}^{\dagger},a_{y}^{\dagger}]=0.$

We fix the total number of the bosons to $\sum_{x}n_{x}=N$, and we impose the
periodic boundary condition. Then, the Fourier transform of the operator
$a_{x}$ is given by

$\hat{a}_{k}:=\frac{1}{\sqrt{|\Lambda|}}\sum_{x\in\Lambda}e^{-ikx}a_{x},$

where $k:=(k^{(1)},k^{(2)},\ldots,k^{(d)})$ is the momentum, and
$kx=\sum_{i=1}^{d}k^{(i)}x^{(i)}$; $|\Lambda|$ denotes the number of the sites
in the finite lattice $\Lambda$. The inverse is

$a_{x}=\frac{1}{\sqrt{|\Lambda|}}\sum_{k}e^{ikx}\hat{a}_{k}.$ (2.2)

By using this expression, the kinetic term of the Hamiltonian $H_{g}^{(N)}$ is
written

$H_{0}^{(N)}=\sum_{k}\mathcal{E}_{k}\hat{a}_{k}^{\dagger}\hat{a}_{k}$ (2.3)

with the standard dispersion relation,

$\mathcal{E}_{k}:=\sum_{i=1}^{d}(1-\cos k^{(i)}).$ (2.4)

The complete Bose-Einstein condensation state is given by

$\Phi_{\rm
BEC}^{(N)}:=\frac{1}{\sqrt{N!}}(\hat{a}_{0}^{\dagger})^{N}|0\rangle,$ (2.5)

where $|0\rangle$ is the vacuum state, i.e., $\hat{a}_{k}|0\rangle=0$ for all
the momenta $k$.

Our result is:

###### Theorem 2.1

The complete Bose-Einstein condensation state $\Phi_{\rm BEC}^{(N)}$ is the
unique ground state of the Hamiltonian $H_{g}^{(N)}$ of (2.1) for any coupling
constant $g\geq 0$ in any dimensions $d\geq 1$.

Proof:  The summand of the interaction term of the Hamiltonian $H_{g}^{(N)}$
of (2.1) is written

$n_{x}(n_{x}-1)+n_{x}n_{y}-a_{x}^{\dagger}n_{x}a_{y}-a_{y}^{\dagger}n_{x}a_{x}=(a_{x}^{\dagger}-a_{y}^{\dagger})a_{x}^{\dagger}a_{x}(a_{x}-a_{y}).$
(2.6)

One notices that this right-hand side is positive. Using the expression (2.2)
of the operator $a_{x}$ in terms of $\hat{a}_{k}$, one has

$a_{x}-a_{y}=\frac{1}{\sqrt{|\Lambda|}}\sum_{k}(e^{ikx}-e^{iky})\hat{a}_{k}=\frac{1}{\sqrt{|\Lambda|}}\sum_{k\neq
0}(e^{ikx}-e^{iky})\hat{a}_{k}.$ (2.7)

Namely, the two contributions of the zero momentum mode $\hat{a}_{0}$ cancel
with each other in the sum. Therefore, one has

$(a_{x}-a_{y})\Phi_{\rm BEC}^{(N)}=0.$ (2.8)

Combining this with (2.6), the summand of the interaction term also
annihilates the state $\Phi_{\rm BEC}^{(N)}$, i.e.,

$[n_{x}(n_{x}-1)+n_{x}n_{y}-a_{x}^{\dagger}n_{x}a_{y}-a_{y}^{\dagger}n_{x}a_{x}]\Phi_{\rm
BEC}^{(N)}=0.$ (2.9)

Immediately,

$\langle\Phi_{\rm BEC}^{(N)},H_{\rm int}^{(N)}\Phi_{\rm BEC}^{(N)}\rangle=0,$
(2.10)

where we have written $H_{\rm int}^{(N)}:=H_{g}^{(N)}-H_{0}^{(N)}$ for the
interaction term of the Hamiltonian $H_{g}^{(N)}$. This yields

$\langle\Phi_{\rm BEC}^{(N)},H_{g}^{(N)}\Phi_{\rm BEC}^{(N)}\rangle=0$ (2.11)

for the total Hamiltonian $H_{g}^{(N)}$ as well. We write $\Phi_{\rm
GS}^{(N)}$ for the ground state of $H_{g}^{(N)}$. Then, from this result
(2.11), we have

$0=\langle\Phi_{\rm BEC}^{(N)},H_{g}^{(N)}\Phi_{\rm
BEC}^{(N)}\rangle\geq\langle\Phi_{\rm GS}^{(N)},H_{g}^{(N)}\Phi_{\rm
GS}^{(N)}\rangle\geq\sum_{k}\mathcal{E}_{k}\langle\Phi_{\rm
GS}^{(N)},\hat{a}_{k}^{\dagger}\hat{a}_{k}\Phi_{\rm GS}^{(N)}\rangle\geq 0,$
(2.12)

where we have used the positivity $H_{\rm int}^{(N)}\geq 0$ which is obtained
from the positivity (2.6). This implies

$\langle\Phi_{\rm GS}^{(N)},\hat{a}_{k}^{\dagger}\hat{a}_{k}\Phi_{\rm
GS}^{(N)}\rangle=0$

for all $k\neq 0$. Thus, we obtain $\Phi_{\rm GS}^{(N)}=\Phi_{\rm BEC}^{(N)}$,
and the ground state is unique.

Remark: The method in the above proof is similar to that in [4]. Namely, both
of kinetic and interaction terms in a Hamiltonian annihilate a state.

##  3 Extension to a Class of Models and Discussion

In this section, we extend the model of (2.1) and Theorem 2.1 to general
models. We also discuss certain interactions for which the absence of Bose-
Einstein condensation can be expected at zero temperature.

A straightforward extension of (2.6) is as follows: Consider an interaction
Hamiltonian,

$H_{\rm int}^{(N)}=g\sum_{x\in\Lambda}C_{x}^{\dagger}A_{x}C_{x},$ (3.1)

where $A_{x}$ is a positive local operator and $C_{x}$ is a local operator
which is written in the form,

$C_{x}=\sum_{j}\alpha_{j}a_{x_{j}},$ (3.2)

with coefficients $\alpha_{j}$ which satisfy $\sum_{j}\alpha_{j}=0$.

We can also choose $C_{x}$ to be

$C_{x}=\sum_{y\in\Lambda}K(x-y)a_{y},$ (3.3)

where the function $K(x-y)$ is given by the Fourier transform of $\hat{K}(k)$,
i.e.,

$K(x-y)=\frac{1}{|\Lambda|}\sum_{k}\hat{K}(k)e^{ik(x-y)}.$ (3.4)

When the function $\hat{K}(k)$ is sufficiently many times differentiable and
satisfies $\hat{K}(0)=0$, the operator $C_{x}$ is local and annihilates the
state $\Phi_{\rm BEC}^{(N)}$. Actually, one has

$C_{x}=\frac{1}{\sqrt{|\Lambda|}}\sum_{k}\hat{K}(k)\hat{a}_{k}e^{ikx}.$ (3.5)

Conversely, when the function $\hat{K}(k)$ has a narrow distribution with a
high peak at $k=0$, we can expect the absence of Bose-Einstein condensation
for the zero momentum mode. In fact, one has

$C_{x}=\frac{1}{\sqrt{|\Lambda|}}\hat{K}(0)\hat{a}_{0}+\frac{1}{\sqrt{|\Lambda|}}\sum_{k\neq
0}\hat{K}(k)\hat{a}_{k}e^{ikx}.$ (3.6)

If we choose $A_{x}=a_{x}^{\dagger}a_{x}$ for the operator $A_{x}$ in (3.1),
then the contribution of the zero momentum mode in (3.1) is written

$\sum_{x}\frac{1}{|\Lambda|}(\hat{K}(0))^{2}\hat{a}_{0}^{\dagger}A_{x}\hat{a}_{0}=\sum_{x}\frac{1}{|\Lambda|}(\hat{K}(0))^{2}\hat{a}_{0}^{\dagger}a_{x}^{\dagger}a_{x}\hat{a}_{0}=\frac{N}{|\Lambda|}(\hat{K}(0))^{2}\hat{a}_{0}^{\dagger}\hat{a}_{0},$
(3.7)

where we have used $\sum_{x}a_{x}^{\dagger}a_{x}=N$. This right-hand side
raises the energy by order of the volume $|\Lambda|$ with the large amplitude
$(\hat{K}(0))^{2}$ for a fixed particle density $N/|\Lambda|$ when the zero
momentum mode exhibits a long-range order. Therefore, the long-range order of
the zero momentum mode is expected to be suppressed for a large
$\hat{K}(0)>0$.

When $\hat{K}(k)=1$ for all $k$, one has $K(x-y)=\delta_{x,y}$ from (3.4).
Then, we have

$H_{\rm int}^{(N)}=g\sum_{x\in\Lambda}n_{x}(n_{x}-1)$ (3.8)

for $A_{x}=a_{x}^{\dagger}a_{x}$, from (3.1) and (3.3). This is one of the
standard interactions [5]. However, it seems to be very difficult to prove the
existence of Bose-Einstein condensation in this case because the function
$\hat{K}(k)=1$ is an intermediate function in the above class.

Thus, in general, whether Bose-Einstein condensation occurs or not is expected
to depend on interactions of models. In particular, it may depend on the
dimension [3] of the lattice.

Acknowledgements: I would like to thank Akinori Tanaka for helpful comments.

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