U}^{\prime\prime}_{k}}\,,$ (4.23)

where $\wp=p/k$ and
$-i\mathcal{G}_{k,0}(\eta,\wp)=-i\mathcal{G}_{k,0}[\varphi](\eta,\eta;p)|_{p=k\wp}$
is a solution of (2.43) with $N=a$. For large $k$ the right hand side of the
flow equation (4.23) can be expanded using (3.6). This gives

${\partial}_{k}{\cal
U}_{k}=\frac{\hbar}{(4\pi)^{2}}\bigg{\\{}4k^{3}q_{0}+2k\,g_{1}(\eta)|_{k}+\frac{g_{2}(\eta)|_{k}}{k}+\sum_{n\geq
3}\frac{2\\!-\\!n}{2}\frac{g_{n}(\eta)|_{k}}{k^{2n-3}}\bigg{\\}}\,.$ (4.24)

Here $q_{0},g_{1}(\eta),g_{2}(\eta)$ are as in (S3.Ex31) and we extend the
definition to

$g_{n}(\eta):=(-)^{n}\frac{2}{2-n}\int_{0}^{\infty}d\varrho\,\varrho^{2}\frac{[r(\varrho^{2})-\varrho^{2}r^{\prime}(\varrho^{2})]}{[\varrho^{2}+r(\varrho^{2})]^{1/2}}G_{n}(\eta,\varrho)\,,\quad
n\geq 3\,.$ (4.25)

For all $n\geq 1$ we use $g_{n}(\eta)|_{k}:=g_{n}(\eta)_{W\mapsto{\cal
U}^{\prime\prime}_{k}}$ as a shorthand. As seen in Section 3.1 the
$g_{1},g_{2}$ evaluate up boundary terms to

$\displaystyle g_{1}(\eta)|_{k}$ $\displaystyle\\!=\\!$
$\displaystyle\breve{q}R-q_{1}\big{(}{\cal U}_{k}^{\prime\prime}-R/6\big{)}$
$\displaystyle\\!=\\!$ $\displaystyle-
q_{1}\,\,^{0}\bar{U}^{\prime\prime}_{k}(\varphi)+\big{[}\breve{q}-q_{1}\,\,^{1}\bar{U}^{\prime\prime}_{k}(\varphi)\big{]}R-q_{1}\sum_{n\geq
2}\,\,^{n}\bar{U}^{\prime\prime}_{k}(\varphi)R^{n}\,,$ $\displaystyle
g_{2}(\eta)|_{k}$ $\displaystyle\simeq$ $\displaystyle\frac{1}{2}\big{(}{\cal
U}_{k}^{\prime\prime}-R/6\big{)}^{2}=\frac{1}{2}\,\,^{0}\bar{U}^{\prime\prime}_{k}(\varphi)^{2}+\,\,^{0}\bar{U}^{\prime\prime}_{k}(\varphi)\,\,^{1}\bar{U}_{k}^{\prime\prime}(\varphi)R$
(4.26) $\displaystyle+$ $\displaystyle\sum_{n\geq
2}\Big{[}\frac{1}{2}\sum_{n_{1}+n_{2}=n}\\!^{n_{1}}\bar{U}^{\prime\prime}_{k}(\varphi)\,\,^{n_{2}}\bar{U}^{\prime\prime}_{k}(\varphi)\Big{]}R^{n}\,.$

Here the $\,\,{}^{n}\bar{U}_{k}(\varphi)$ are defined as in (3.25) but refer
to $k$-dependent couplings $\,\,{}^{n}\bar{u}_{2j,k}$, $n,j\geq 0$. Comparing
powers of $R$ one finds

$\displaystyle{\partial}_{k}\\!\,\,^{0}U_{k}$ $\displaystyle\\!=\\!$
$\displaystyle\frac{\hbar}{(4\pi)^{2}}\Big{\\{}4k^{3}q_{0}-2kq_{1}\,\,^{0}\bar{U}^{\prime\prime}_{k}+\frac{1}{2k}\big{(}\\!\\!\,\,^{0}\bar{U}^{\prime\prime}_{k}\big{)}^{2}+O(1/k^{3})\Big{\\}}\,,$
$\displaystyle{\partial}_{k}\\!\,\,^{1}U_{k}$ $\displaystyle\\!=\\!$
$\displaystyle\frac{\hbar}{(4\pi)^{2}}\Big{\\{}2k\big{(}\breve{q}-q_{1}\,\,^{1}\bar{U}^{\prime\prime}_{k}\big{)}+\frac{1}{k}\,\,^{0}\bar{U}^{\prime\prime}_{k}\,\,{}^{1}\bar{U}^{\prime\prime}_{k}+O(1/k^{3})\Big{\\}}\,$
(4.27) $\displaystyle{\partial}_{k}\\!\,\,^{n}U_{k}$ $\displaystyle\\!=\\!$
$\displaystyle\frac{\hbar}{(4\pi)^{2}}\Big{\\{}-2k\,q_{1}\,\,^{n}\bar{U}^{\prime\prime}_{k}+\frac{1}{2k}\sum_{n_{1}+n_{2}=n}\\!^{n_{1}}\bar{U}^{\prime\prime}_{k}(\varphi)\,\,^{n_{2}}\bar{U}^{\prime\prime}_{k}(\varphi)+O(1/k^{3})\Big{\\}},\;\;n\geq
2\,.$

One may check that the transition formulas (4.14) lead to an autonomous system
of flow equations for the dimensionless potentials ${}^{n}V_{k}(\varphi_{0})$,
$n\geq 0$. The subleading terms are determined by $G_{n},n\geq 3$, in the
recursion (3.7). We omit their analysis for now and relate the displayed terms
in (S4.Ex68) to the perturbative analysis of Section 3. Comparison with
(S3.Ex55) exhibits an exact match to the one-loop flows of all potentials.
Hence, the coupling flow determined from the EPA (LABEL:sepa1), (4.23)
coincides – for large enough $k$ – with the one-loop flow of the renormalized
couplings after strict removal of the UV cutoff:

$\,\,^{n}u_{2j,k}=\,\,^{n}u_{2j}(\mu),\;\;n,j\geq
0\,,\quad\mbox{for}\quad\mu=k\quad\mbox{large}$ (4.28)

Underlying this seemingly trivial match are some structural elements [12] that
are easily overlooked:

  * •

The use of a non-minimal subtraction scheme, see (S3.Ex43), which allows one
to treat $\mu$ and $\Lambda$ as an instances of $k$ and the $k=\mu$
quantitites as the renormalized ones.

  * •

The inclusion of all interaction monomials, here (2.2), needed for strict
removal of the UV cutoff. Only then can one make $k<\Lambda$ arbitrarily large
and identify the unstable manifold so as to justify the interpretation of the
couplings as the renormalized ones (as opposed to merely running Wilsonian
parameters).

  * •

Use of the same regulator to set up perturbation theory and the EPA.

The previous comments are not specific to the FL-sFRG. With Euclidean
signature the reintegration of the covariant $O(\hbar)$ FRG,
$2{\partial}_{k}\Gamma_{k,1}[\varphi]={\rm
Tr}[{\partial}_{k}R_{k}(S^{(2)}(\varphi)+R_{k})^{-1}]$, is often used in
combination with the heat kernel expansion to obtain the full renormalized
one-loop effective action at $k=0$. Since the heat kernel expansion (even in
variants where nonlocal terms are included) is an asymptotic expansion
tailored to the ultraviolet, the integration down to $k=0$ is conceptually
unwarranted. Replacing it with an integration from $\mu$ to $\Lambda$ for some
Wilsonian scale $\mu$, it can be seen to produce termwise (precisely) the same
results for the divergent parts as the direct evaluation of
$L_{\Lambda}+L_{1}^{\rm div}$, provided a regularization is used that
facilitates such a comparison. The next step, of attempting to strictly remove
the UV cut-off $\Lambda$, is often omitted. Doing so somewhat trivializes the
notion of one-loop renormalizability, fails to identify the unstable manifold,
and may lead to inconsistent truncations.

In Section 3 of [37] the removal of the UV cutoff is discussed for a scalar
field theory in flat space. Unfortunately, in their Eq. (49) several typos
crept in which obscure the fact that the UV beta functions
$(\Lambda{\partial}/{\partial}\Lambda$ response of the bare couplings) must in
this framework coincide with the IR beta functions ($k{\partial}/{\partial}k$
response of the parameters in the EPA ansatz). In Section 7 of [37] gravity
coupled to a scalar is studied. In their Eq. (180) it is noted that several
terms are generated not present in the original action, but no attempt is made
to absorb them through coupling or field redefinitions.

The EPA flow for scalar QFT on a generic Riemannian manifold has also been
considered in [31, 32, 39, 40], primarily using the heat kernel methodology to
compute the right hand side of the flow equation. In [40] a potential for the
nonminimal coupling proportional to $R$ was included (our
$\,\,{}^{1}U(\varphi)R$ term). Taking into account
${}^{n}\bar{U}_{k}^{\prime\prime}={}^{n}U_{k}^{\prime\prime}-(1/6)\delta_{n,1}$,
the structure of the $n=0,1$ flow equations (S4.Ex68) parallels those in Eqs
(26),(27) of [40] and the modulator independent coefficients agree exactly.
The restriction to linear order in $R$ is seen as truncation, so neither does
the criterion (3.1) enter nor the ensued Ricci tower.

4.3 Infrared regime for SLE based EPA flow

We return to the dimensionless form (LABEL:dimless12) of the EPA flow and seek
to analyze its structure for small $k$. On the right-hand side the dependence
of the Green’s function on ${\cal V}^{\prime\prime}_{k}$ needs to be known. In
the case of a de Sitter background [5, 6, 7], as mentioned after (2.30), the
issue of state dependence of the flow equation does not arise due to the
maximal symmetry of the spacetime. The right-hand side of (LABEL:sepa1) can be
then be determined in closed form for the ‘hockey stick’ FRG modulator, and
(assuming that ${\cal V}^{\prime\prime}_{k}$ remains bounded as $k\to 0$) the
well known infrared enhancement of the Bunch-Davies Green’s function leads to
a “dimensional reduction” in the small $k$ form of the dimensionless EPA flow
equation [7]. For a generic Friedmann-Lemaître background, however, the small
$k$ behavior of the FL-sEPA flow will normally be very different.

In analyzing the small $k$ form of the EPA flow (LABEL:sepa1), we seek to keep
the scale factor $a(t)$ and the modulator function (S2.Ex12) generic, and may
allow for a time dependent $\varphi_{0}(t)$. Then even the modulator
independent basic wave equation can not be solved in terms of known special
functions. As argued before, the inevitable state dependence of the small $k$
flow is a major difference to the large $k$ regime. Rather than restricting
attention to some drastic specializations and truncations we aim at developing
an analytically controllable series expansion of the Green’s function for
small $k$. Inevitably, this requires a choice of underlying Hadamard state,
and we shall focus on the States of Low Energy (SLE) described in Section 2.4.
With the assumption that ${\cal V}^{\prime\prime}_{k}$ remains bounded as
$k\rightarrow 0$, an analytically controllable small $k$ expansion turns out
to be feasible.

The role of $p$ as an expansion parameter in (S2.Ex26) is played by $k$ after
transition to dimensionless variables. Using (4.16) the dimensionless form of
the basic wave equation reads

$\big{\\{}(a^{d}{N}^{-1}{\partial}_{t})^{2}+k^{2}a^{2d}[\wp^{2}/a^{2}+r(\wp^{2}/a^{2})+{\cal
V}^{\prime\prime}_{k}]\big{\\}}S_{k}(t,\wp)=0\,.$ (4.29)

An arbitrary Wronskian normalized solution of (4.29) is used as input for the
construction of the SLE solution $T_{k}^{\rm SLE}[S_{k}](t,k\wp)$, see (2.49).
Comparing with (2.48) and the wave equation based on it, one is lead to the
identifications $\omega_{0}(t)\mapsto 0$ and

$\displaystyle p\mapsto k\,,\quad w_{2,p}(t)\mapsto a(t)^{2d}\big{[}{\cal
V}^{\prime\prime}_{k}+\wp^{2}/a^{2}+r(\wp^{2}/a^{2})\big{]}=:w_{2}({\cal
V}_{k}^{\prime\prime},\wp)\,,\quad$ (4.30)

We relegate the justification of this substitution rule to Appendix C and note
here its important consequence: the coincidence limit
$kG_{k}[\varphi_{0}](t,t;\wp)|=ik|T_{k}^{\rm SLE}(t,k\wp)|^{2}$ of the
intractable (dimensionless) Green’s function admits a convergent series
expansion in powers of $k^{2}$. The coefficients can be computed analytically
from (2.55) – (S2.Ex28) in combination with (LABEL:sleepa0). We thus define

$\displaystyle\varepsilon_{n,\wp}[{\cal V}^{\prime\prime}_{k}]$
$\displaystyle:=$
$\displaystyle\varepsilon_{n}\big{|}_{\omega_{2}^{2}=w_{2}({\cal
V}^{\prime\prime}_{k},\wp)}$ $\displaystyle J_{n,\wp}[{\cal
V}^{\prime\prime}_{k}](t)$ $\displaystyle:=$ $\displaystyle
J_{n}(t)\big{|}_{\omega_{2}^{2}=w_{2}({\cal
V}^{\prime\prime}_{k},\wp)}\,,\quad n\geq 0\,,$
$\displaystyle\bar{a}_{\wp}[{\cal V}^{\prime\prime}_{k}]$ $\displaystyle:=$
$\displaystyle\frac{J_{0,\wp}}{\varepsilon_{1,\wp}}=\left(\dfrac{\int\\!dt{n}(t)f(t)}{\int\\!dt{n}(t)a(t)^{2d}f(t)\big{[}{\cal
V}^{\prime\prime}_{k}+\wp^{2}/a^{2}+r(\wp^{2}/a^{2})\big{]}}\right)^{1/2}\\!\\!.$
(4.31)

The trivial $J_{0}=J_{0,\wp}=\frac{1}{2}\int\\!dt\,{n}(t)f(t)$ could be
normalized to some value, but is kept for dimensionality reasons. Then,
$\bar{a}_{\wp}$ is dimensionless while $J_{n,\wp}$ and $\varepsilon_{n,\wp}$
have mass dimension $-(2n\\!+\\!1)$ and $-n$, respectively. In this notation

$|T_{k}^{\rm SLE}(t,k\wp)|^{2}=\frac{\bar{a}_{\wp}[{\cal
V}^{\prime\prime}_{k}]}{2k}\bigg{\\{}1+\bigg{(}\frac{J_{1,\wp}[{\cal
V}^{\prime\prime}_{k}](t)}{J_{0}}-\frac{\bar{a}_{\wp}[{\cal
V}^{\prime\prime}_{k}]^{2}\,\varepsilon_{2,\wp}^{2}[{\cal
V}^{\prime\prime}_{k}]}{2J_{0}^{2}}\bigg{)}k^{2}+O(k^{4})\bigg{\\}}\,.$ (4.32)

This has to be inserted into the projected FL-sEPA equation (LABEL:dimless12).
From now on we choose the weight in the orthonormal basis proportional to the
window function of the SLE, specifically

$w(t)=\kappa(0)n(t)a(t)^{2d}f(t)\,,$ (4.33)

where $n(t)=N(t)a(t)^{-d}$ and the mass scale $\kappa(0)$ simply accounts for
dimensions, see the paragraph preceding (4.4). Further, we take $\varphi_{0}$
to be time independent. Due to the orthonormality (4.6), in the denominator of
$\bar{a}_{\wp}[{\cal V}^{\prime\prime}_{k}]$ then only the $l=0$ term in the
decomposition ${\cal V}_{k}^{\prime\prime}(\varphi_{0},t)=\sum_{l\geq
0}{}_{l}{\cal V}^{\prime\prime}_{k}(\varphi_{0})\mathfrak{p}_{l}(t)$
contributes:

$\displaystyle\bar{a}_{\wp}[{\cal V}^{\prime\prime}_{k}]$
$\displaystyle\\!=\\!$
$\displaystyle\bigg{(}\frac{2J_{0}\kappa(0)\mathfrak{p}_{0}}{{}_{0}{\cal
V}_{k}^{\prime\prime}(\varphi_{0})+{}_{0}\rho(\wp)}\bigg{)}^{1/2}\,,$
$\displaystyle{}_{l}\rho(\wp)$ $\displaystyle:=$
$\displaystyle\int\\!dtw(t)\mathfrak{p}_{l}(t)\big{[}\wp^{2}/a(t)^{2}+r(\wp^{2}/a(t)^{2})\big{]}\,.$
(4.34)

In turn, this suggests to define for a function $h(t,\wp)$

$\displaystyle{}_{l}Q_{m}(h|V):=\frac{\hbar}{c_{0}(4\pi)^{d/2}\Gamma(d/2)}\int_{0}^{\infty}\\!d\wp\wp^{d-1}\bigg{(}\frac{2J_{0}\kappa(0)\mathfrak{p}_{0}}{V+{}_{0}\rho(\wp)}\bigg{)}^{1/2+m}$
(4.35)
$\displaystyle\quad\quad\times\int\\!dt\,w(t)\mathfrak{p}_{l}(t)\big{[}r(\wp^{2}/a(t)^{2})-(\wp^{2}/a(t)^{2})r^{\prime}(\wp^{2}/a(t)^{2})\big{]}h(t,\wp)\,,\quad
l,m\geq 0\,.$

All terms on the right hand side of (LABEL:dimless12) can be expressed in
terms of these averages. We arrive at the SLE based FL-sEPA flow equation:

$\displaystyle k\frac{d}{dk}\,{}_{l}\\!{\cal
V}_{k}(\varphi_{0})+(d\\!+\\!1)\,{}_{l}\\!{\cal
V}_{k}(\varphi_{0})-\frac{d\\!-\\!1}{2}\varphi_{0}\frac{{\partial}}{{\partial}\varphi_{0}}\,{}_{l}\\!{\cal
V}_{k}(\varphi_{0})$ $\displaystyle={}_{l}Q_{0}\big{(}1|\,{}_{0}{\cal
V}_{k}^{\prime\prime}\big{)}-\frac{k^{2}}{2J_{0}^{2}}\;{}_{l}Q_{1}\big{(}1|{}_{0}{\cal
V}_{k}^{\prime\prime}\big{)}\sum_{l_{1},l_{2}\geq 0}{}_{l_{1}}\\!{\cal
V}_{k}^{\prime\prime}(\varphi_{0}){}_{l_{2}}\\!{\cal
V}_{k}^{\prime\prime}(\varphi_{0})E_{l_{1},l_{2}}$ (4.36)
$\displaystyle\quad+\frac{k^{2}}{J_{0}}\sum_{l_{1}\geq 0}{}_{l_{1}}\\!{\cal
V}_{k}^{\prime\prime}(\varphi_{0})\bigg{(}{}_{l}Q_{0}\big{(}\jmath_{l_{1}}|{}_{0}{\cal
V}_{k}^{\prime\prime}\big{)}-\frac{1}{2J_{0}}\,{}_{l}Q_{1}\big{(}E_{l_{1}}|{}_{0}{\cal
V}_{k}^{\prime\prime}\big{)}\bigg{)}$
$\displaystyle\quad+\frac{k^{2}}{J_{0}}\bigg{(}{}_{l}Q_{0}\big{(}\jmath|{}_{0}{\cal
V}_{k}^{\prime\prime}\big{)}-\frac{1}{2J_{0}}{}_{l}Q_{1}\big{(}E|{}_{0}{\cal
V}_{k}^{\prime\prime}\big{)}\bigg{)}+O(k^{4})\,.$

The functions $E_{l,l^{\prime}},E_{l}(\wp),E(\wp)$ and
$\jmath_{l}(t),\jmath(t,\wp)$ are collected in Appendix C. Equation
(LABEL:sleepa3) is the main result of this section.

Remarks:

  * (i)

Both sides of (LABEL:sleepa3) depend on the choice of window function $f$. The
support of $f$ will be chosen in accordance with the cosmological period where
one seeks to inject information about the primordial vacuum state. Since
general relativity demands a pre-inflationary period, locating $f$ immediately
after a quantum gravity epoch is a natural choice.

  * (ii)

The scale factor $a$ can be chosen to model a realistic expansion history. For
example, one can have a pre-inflationary period of non-accelerated expansion
followed by a de Sitter like acceleration, etc. This avoids the technically
clumsy matching of piecewise powerlike $a$’s.

  * (iii)

The flow equation (LABEL:sleepa3) is non-autonomous, i.e. $k$ occurs
explicitly in the dimensionless formulation. The $O(k^{4})$ and higher order
corrections refer to (2.55), (2.56) and can be computed systematically. This
feature differs from the UV flow (S4.Ex68) where the same substitution (4.12),
(4.13) leads to an autonomous system of flow equations for the dimensionless
potentials ${}^{n}V_{k}(\varphi_{0})$, $n\geq 0$. In Appendix B we analyze the
instantaneous limit of (LABEL:sleepa3) where again an autonomous system of
flow equations for the ${}^{n}V_{k}(\varphi_{0})$ arises. The non-autonomous
character of the (LABEL:sleepa3) therefore originates in the temporal
averaging (and the ensued temporal nonlocality) needed to render the
underlying state (of low energy) mathematically and physically well-defined
[28, 17].

  * (iv)

The small $k$ expansion (4.32) entails that the flow equation specializes
correctly to the spatial EPA flow in Minkowski space. Since ${n}\equiv 1=a$
one has $\Delta_{p}(t,t^{\prime})=p^{-1}\sin p(t-t^{\prime})$. Then (S2.Ex24)
reduces to $J_{p}(t)=J_{0}$, ${\cal E}_{p}^{\rm SLE}=p\varepsilon_{1}$ and
only the leading term on the right hand side of (4.32) remains. For a constant
$\varphi_{0}$ the window function drops out in $\bar{a}_{\wp}[{\cal
V}^{\prime\prime}_{k}]$ and the corresponding flow equation reduces to (B.5),
as required.

Importantly, (LABEL:sleepa3) has a well-defined IR fixed point equation

$(d\\!+\\!1){}_{l}{\cal
V}_{*}-\frac{d\\!-\\!1}{2}\varphi_{0}\frac{{\partial}}{{\partial}\varphi_{0}}{}_{l}{\cal
V}_{*}={}_{l}Q_{0}\big{(}1|\,{}_{0}{\cal V}^{\prime\prime}_{*}\big{)}\,,$
(4.37)

which resembles that in Minkowski space but refers to a curvature dependent
generalized potential, ${\cal V}_{*}={\cal V}_{*}(\varphi_{0},t)=\sum_{l\geq
0}{}_{l}{\cal V}_{*}(\varphi_{0})\mathfrak{p}_{l}(t)$. To proceed, we insert
power series expansions

$\displaystyle{}_{l}{\cal V}_{\ast}(\varphi_{0})=\sum_{j\geq
0}{}_{l}v_{*,2j}\frac{\varphi_{0}^{2j}}{(2j)!}\,,\quad{}_{0}{\cal
V}^{\prime\prime}_{\ast}(\varphi_{0})={}_{0}v_{*,2}+\sum_{j\geq
1}{}_{l}v_{*,2j+2}\frac{\varphi_{0}^{2j}}{(2j)!}\,,$ (4.38)

and expand ${}_{l}Q_{0}(1|\,{}_{0}\\!{\cal V}_{*}^{\prime\prime})$ in powers
of the field. This gives

$\displaystyle\\!\\!\\!(d\\!+\\!1){}_{l}v_{*,0}$ $\displaystyle\\!=\\!$
$\displaystyle\,{}_{l}Q_{0}\big{(}1|{}_{0}v_{*,2}\big{)}\,,$ (4.39)
$\displaystyle\\!\\!\\![d\\!+\\!1-j(d\\!-\\!1)]{}_{l}v_{*,2j}$
$\displaystyle\\!=\\!$ $\displaystyle\sum_{m=0}^{j}{-1/2\choose
m}\frac{{}_{l}Q_{m}\big{(}1|{}_{0}v_{*,2}\big{)}}{(2J_{0}\mathfrak{p}_{0})^{m}}$
$\displaystyle\times$ $\displaystyle\sum_{\scriptstyle
j_{1}+\ldots+j_{m}=j\atop\scriptstyle j_{1},\ldots,j_{m}\geq 1}{2j\choose
2j_{1},\ldots,2j_{m}}{}_{0}v_{*,2j_{1}+2}\ldots{}_{0}v_{*,2j_{1}+2}\,.$

The $l=0$ instance of the second equation is a coupled system for the
${}_{0}v_{*,2j}$, $j\geq 0$. Assuming that a solution has been found all other
${}_{l}v_{*,2j},l\geq 1,j\neq 1$ are determined by them. The same holds for
$d>3$ for $j=1$, while for $d=3$ and $j=1$ the coefficient of ${}_{l}v_{*,4}$
vanishes and the ${}_{0}v_{,2j}$ are further constrained. With the possible
exception of ${}_{l}v_{*,4}$ the couplings associated with the temporally
averaged potential ${}_{0}{\cal V}_{*}(\varphi_{0})$ therefore determine all
other ${}_{l}{\cal V}_{*}(\varphi_{0})$, $l\geq 0$. Although time dependent,
the fixed point solution ${\cal V}_{*}(\varphi,t)$ is therefore essentially
determined by its temporal average.

The derivation of the $O(k^{2})$ correction terms in (LABEL:sleepa3) is
relegated to Appendix C. The further analysis of (LABEL:sleepa3) will begin
with a linearization around the fixed point solutions(s) and then use a
shooting technique to numerically relate the infrared couplings
${}_{l}v_{2j,k}$, $l\geq 0$, to the ultraviolet ones ${}^{n}v_{2j,k}$, $n\geq
0$, using the respective flow equations. For length reasons a detailed
investigation will have to be relegated to a separate paper. To the best of
our knowledge the flow (LABEL:sleepa3) is the first result that allows one to
explore the consequences of the Hadamard property for the non-perturbative
infrared dynamics of quantum fields on a Friedmann-Lemaître background with a
realistic expansion history.

5\. Conclusions

The main take away from the results obtained here is the inevitable state
dependence of a Lorentzian signature FRG on a generic foliated background. At
present, no selection criteria for physically viable vacuum like states are
known beyond perturbation theory. The advocated correspondence between vacuum-
like states and a spatial FRG may in fact help to develop such criteria. The
resulting interplay between state selection, Wilsonian perturbation theory,
and a non-perturbative spatial FRG flow has been summarized in Figure 1.
Details were elaborated for Friedmann-Lemaître backgrounds with States of Low
Energy as the Hadamard states of choice. In particular, systematic expansions
for the infrared and ultraviolet regime of the associated effective potential
flow equations have been presented. For length reasons, a detailed
quantitative investigation of the ensued flows has to be relegated to a
separate study.

There are many threads of research to be taken up from here. Most immediate is
the non-perturbative flow of the scalar power spectrum. A case study in [19]
suggests that the States of Low Energy are viable candidates for pre-
inflationary vacua. In particular, the low multipole moments in the CMB
spectrum are suppressed in a natural way. The approximately scale invariant
power spectrum arises in a cross over region, which in a spatial FRG setting
has to be explored numerically. On a de Sitter background Stochastic Inflation
provides an alternative framework to capture the non-perturbative inflaton
dynamics and has some interface with the FRG flow [5, 6, 7]. Since general
relativity demands a pre-inflationary (kinetic energy dominated) period, the
generalization of this interplay to Friedmann-Lemaître backgrounds other than
de Sitter is an important desideratum. A third thread is the extension to
other types of cosmological backgrounds. Since the framework of Figure 1
requires some degree of computational control over the Hadamard states in
question, the choices are presently limited. The States of Low Energy can
however be extended to spatially inhomogeneous generalized FRW geometries
[20]. Another option are temporally averaged Sorkin-Johnston states [21, 22].
Eventually, one would like to go beyond the Effective Potential Approximation,
for which the spatial hopping expansion [10, 11] provides a starting point.

A spatial FRG should also be the appropriate tool to investigate Quantum
Gravity at strong coupling. This takes foliated geometries as basic as in
[38], though with a different dynamics that underlies an associated Anti-
Newtonian expansion [11], antipodal to perturbation theory.

Acknowledgements: This research was partially supported by PittPACC. R.B. also
acknowledges support through a Andrew P. Mellon fellowship from the University
of Pittsburgh. We also wish to thank the referee for an unusually thorough
reading of the manuscript, which lead to a variety of improvements.

A: Regulator dependence of $\Gamma_{k,1}^{\rm div}$ coefficients

In this appendix we present a proof of the Lemma entering (LABEL:Bcoeff1),
(3.13) together with explicit formulas for the regulator dependent
coefficients $\breve{q},q_{1},b_{1},b_{2},b_{3}$. We begin by formulating

Lemma A: Let $r(x)$ be a generic FRG regulator function as defined in
(S2.Ex12).

  1. (a)

The regulator dependent coefficients $B_{1},B_{2},B_{3}$ always satisfy the
ratio

$\displaystyle B_{1}:B_{2}:B_{3}$ $\displaystyle\\!=\\!$
$\displaystyle-1:4:3\,,$ (A.1)

and hence can be expressed in terms of a single regulator dependent constant
$b_{1}$,

$\displaystyle B_{1}=-6b_{1}\,,\quad B_{2}=24b_{1}\,,\quad B_{3}=18b_{1}\,.$
(A.2)

  2. (b)

$B_{4}+36b_{1}+B_{5}=0$.

  3. (c)

$B_{6}=B_{7}=B_{8}$.

Before proceeding to the proof, we note the explicit expressions for
$B_{1},\ldots,B_{8}$ (obtained with Mathematica):

$\displaystyle B_{1}$ $\displaystyle\\!=\\!$
$\displaystyle-\int_{0}^{\infty}\\!d\varrho\,\varrho^{2}\frac{\big{[}r(\varrho^{2})-\varrho^{2}r^{(1)}(\varrho^{2})\big{]}^{2}}{4\big{[}\varrho^{2}+r(\varrho^{2})\big{]}^{7/2}}\,,$
$\displaystyle B_{2}$ $\displaystyle\\!=\\!$
$\displaystyle-\int_{0}^{\infty}\\!d\varrho\,\varrho^{2}\frac{\big{[}r(\varrho^{2})-\varrho^{2}r^{(1)}(\varrho^{2})\big{]}^{2}}{2\big{[}\varrho^{2}+r(\varrho^{2})\big{]}^{9/2}}\Big{\\{}\varrho^{4}r^{(2)}(\varrho^{2})\big{(}\varrho^{2}+r(\varrho^{2})\big{)}$
$\displaystyle-$ $\displaystyle
4\big{(}r(\varrho^{2})-\varrho^{2}r^{(1)}(\varrho^{2})\big{)}\big{(}5r(\varrho^{2})-\varrho^{2}(2+7r^{(1)}(\varrho^{2}))\big{)}\Big{\\}}\,,$
$\displaystyle 4B_{3}$ $\displaystyle\\!=\\!$ $\displaystyle 3B_{2}$
$\displaystyle B_{4}$ $\displaystyle\\!=\\!$
$\displaystyle\frac{3}{8}\int_{0}^{\infty}\\!d\varrho\,\varrho^{2}\frac{\big{[}r(\varrho^{2})-\varrho^{2}r^{(1)}(\varrho^{2})\big{]}^{2}}{\big{[}\varrho^{2}+r(\varrho^{2})\big{]}^{9/2}}\Big{\\{}-35r(\varrho^{2})^{3}$
$\displaystyle\\!+\\!$
$\displaystyle\\!\\!\varrho^{2}r(\varrho^{2})^{2}\Big{[}42+147r^{(1)}(\varrho^{2})+108\varrho^{2}r^{(2)}(\varrho^{2})+16\varrho^{4}r^{(3)}(\varrho^{2})\Big{]}$
$\displaystyle\\!+\\!$
$\displaystyle\\!\\!\varrho^{6}\Big{[}42r^{(1)}(\varrho^{2})^{2}+77r^{(1)}(\varrho)^{3}+24\varrho^{2}r^{(2)}(\varrho^{2})-84\varrho^{2}r^{(1)}(\varrho^{2})r^{(2)}(\varrho^{2})+16\varrho^{4}r^{(3)}(\varrho^{2})\Big{]}$
$\displaystyle\\!+\\!$
$\displaystyle\\!\\!\varrho^{4}r(\varrho^{2})\Big{[}132\varrho^{2}r^{(2)}(\varrho^{2})-21r^{(1)}(\varrho^{2})\big{(}4+9r^{(1)}(\varrho^{2})+4\varrho^{2}r^{(2)}(\varrho^{2})\big{)}+32\varrho^{2}r^{(3)}(\varrho^{2})\Big{]}\Big{\\}}\,.$
$\displaystyle B_{5}$ $\displaystyle\\!=\\!$
$\displaystyle-\int_{0}^{\infty}\\!d\varrho\,\varrho^{2}\frac{r(\varrho^{2})-\varrho^{2}r^{(1)}(\varrho^{2})}{32\big{[}\varrho^{2}+r(\varrho^{2})\big{]}^{9/2}}\Big{\\{}-315r(\varrho^{2})^{4}$
$\displaystyle\\!+\\!$
$\displaystyle\\!\\!4\varrho^{2}r(\varrho^{2})^{3}\Big{[}189+504r^{(1)}(\varrho^{2})+606\varrho^{2}r^{(2)}(\varrho^{2})+192\varrho^{4}r^{(3)}(\varrho^{2})+16\varrho^{6}r^{(4)}(\varrho^{2})\Big{]}$
$\displaystyle\\!+\\!$
$\displaystyle\\!\\!\varrho^{8}\Big{[}-924r^{(1)}(\varrho^{2})^{3}-1155r^{(1)}(\varrho^{2})^{4}+240\varrho^{2}r^{(2)}(\varrho^{2})+84r^{(1)}(\varrho^{2})^{2}\big{(}22\varrho
r^{(2)}(\varrho^{2})-1\big{)}$ $\displaystyle\\!-\\!$
$\displaystyle\\!\\!112r^{(1)}(\varrho^{2})\big{(}3\varrho^{2}r^{(2)}(\varrho^{2})+4\varrho^{4}r^{(3)}(\varrho^{2})\big{)}$
$\displaystyle\\!+\\!$
$\displaystyle\\!\\!16\varrho^{4}\big{(}4\varrho^{2}r^{(4)}(\varrho^{2})+20r^{(3)}(\varrho^{2})-21r^{(2)}(\varrho^{2})^{2}\big{)}\Big{]}$
$\displaystyle\\!+\\!$
$\displaystyle\\!\\!4\varrho^{6}r(\varrho^{2})\Big{[}924r^{(1)}(\varrho^{2})^{3}+264\varrho^{2}r^{(2)}(\varrho^{2})+21r^{(1)}(\varrho^{2})^{2}\big{(}31+22\varrho^{2}r^{(2)}(\varrho^{2})\big{)}$
$\displaystyle\\!-\\!$
$\displaystyle\\!\\!14r^{(1)}(\varrho^{2})\big{(}16\varrho^{4}r^{(3)}(\varrho^{2})+78\varrho^{2}r^{(2)}(\varrho^{2})-3\big{)}$
$\displaystyle\\!+\\!$ $\displaystyle\\!\\!8\varrho^{2}\big{(}6\varrho
r^{(4)}(\varrho^{2})+44r^{(3)}(\varrho^{2})-21r^{(2)}(\varrho^{2})\big{)}\Big{]}$
$\displaystyle\\!+\\!$ $\displaystyle
2\varrho^{2}r(\varrho^{2})^{2}\Big{[}-2121r^{(1)}(\varrho^{2})^{2}+6\big{(}270\varrho^{2}r^{(2)}(\varrho^{2})-7\big{)}$
$\displaystyle\\!-\\!$
$\displaystyle\\!\\!14r^{(1)}(\varrho^{2})\big{(}16\varrho^{4}r^{(3)}(\varrho^{2})+144\varrho^{2}r^{(2)}(\varrho^{2})+87\big{)}$
$\displaystyle\\!+\\!$
$\displaystyle\\!\\!8\varrho^{2}\big{(}12\varrho^{2}r^{(4)}(\varrho^{2})+116r^{(3)}(\varrho^{2})-21r^{(2)}(\varrho^{2})^{2}\big{)}\Big{]}\Big{\\}}\,,$
$\displaystyle B_{6}$ $\displaystyle\\!=\\!$
$\displaystyle\int_{0}^{\infty}\\!d\varrho\,\varrho^{2}\frac{r(\varrho^{2})-\varrho^{2}r^{(1)}(\varrho^{2})}{2\big{[}\varrho^{2}+r(\varrho^{2})\big{]}^{7/2}}\Big{\\{}-3r(\varrho^{2})+\varrho^{2}\big{(}2+5r^{(1)}(\varrho^{2})(\varrho^{2})\big{)}\Big{\\}}\,,$
$\displaystyle B_{7}$ $\displaystyle\\!=\\!$
$\displaystyle\int_{0}^{\infty}\\!d\varrho\,\varrho^{2}\frac{r(\varrho^{2})-\varrho^{2}r^{(1)}(\varrho^{2})}{4\big{[}\varrho^{2}+r(\varrho^{2})\big{]}^{9/2}}\Big{\\{}-9r(\varrho^{2})^{2}+2\varrho^{2}r(\varrho^{2})\Big{[}11+20r^{(1)}(\varrho^{2})+10\varrho^{2}r^{(2)}(\varrho^{2})\Big{]}\quad$
$\displaystyle\\!+\\!$
$\displaystyle\varrho^{4}\Big{[}20\varrho^{2}r^{(2)}(\varrho^{2})-5r^{(1)}(\varrho^{2})\big{(}6+7r^{(1)}(\varrho^{2})\big{)}-4\Big{]}\Big{\\}}\,,$
$\displaystyle B_{8}$ $\displaystyle\\!=\\!$ $\displaystyle B_{6}\,.$ (A.3)

Note that we use the notation $r^{(n)}(x):=\frac{d^{n}}{dx^{n}}r(x)$, $n\geq
0$, throughout.

Proof of Lemma A.

  1. (a)

The ratio $B_{2}:B_{3}=4:3$ is clear from (A1.Ex79), so it remains to show

$\displaystyle 4B_{1}+B_{2}=0\,.$ (A.4)

Indeed, it follows by direct computation that

$\displaystyle 4B_{1}+B_{2}$ $\displaystyle\\!=\\!$
$\displaystyle\frac{1}{2}\int_{0}^{\infty}\\!d\varrho\,\frac{{\partial}}{{\partial}\varrho}\bigg{\\{}\frac{\varrho^{3}r(\varrho^{2})^{2}-2\varrho^{5}r(\varrho^{2})r^{(1)}(\varrho^{2})+\varrho^{7}r^{(1)}(\varrho^{2})^{2}}{\big{[}\varrho^{2}+r(\varrho^{2})\big{]}^{7/2}}\bigg{\\}}$
(A.5) $\displaystyle\\!=\\!$
$\displaystyle\lim_{\varrho\to\infty}\frac{\varrho^{3}r(\varrho^{2})^{2}-2\varrho^{5}r(\varrho^{2})r^{(1)}(\varrho^{2})+\varrho^{7}r^{(1)}(\varrho^{2})^{2}}{2\big{[}\varrho^{2}+r(\varrho^{2})\big{]}^{7/2}}\,.$

By (S2.Ex12) $r(\varrho^{2})$ and its derivatives approach zero as
$\varrho\to\infty$ faster than any polynomial in $\varrho$. Thus (A.5)
vanishes, i.e. $4B_{1}+B_{2}=0$, establishing Lemma A($a$).

  2. (b)

Similarly, one finds

$\displaystyle
B_{4}+36b_{1}+B_{5}=\int_{0}^{\infty}\\!d\varrho\,\,\frac{{\partial}}{{\partial}\varrho}\bigg{\\{}\frac{\varrho^{3}}{32\big{[}\varrho^{2}+r(\varrho^{2})\big{]}^{11/2}}\Big{[}5r(\varrho^{2})^{4}$
$\displaystyle\makebox[28.45274pt]{
}-4\varrho^{2}r(\varrho^{2})^{3}\big{(}22+27r^{(1)}(\varrho^{2})+36\varrho^{2}r^{(2)}(\varrho^{2})+8\varrho^{4}r^{(3)}(\varrho^{2})\big{)}$
$\displaystyle\makebox[28.45274pt]{
}+\varrho^{8}\Big{(}112r^{(1)}(\varrho^{2})^{3}+105r^{(1)}(\varrho^{2})^{4}-16\varrho^{4}r^{(2)}(\varrho^{2})^{2}$
$\displaystyle\makebox[28.45274pt]{
}+4r^{(1)}(\varrho^{2})^{2}[3-28\varrho^{2}r^{(2)}(\varrho^{2})]+32\varrho^{2}r^{(1)}(\varrho^{2})[r^{(2)}(\varrho^{2})+\varrho^{2}r^{(3)}(\varrho^{2})]\Big{)}$
$\displaystyle\makebox[28.45274pt]{
}+2\varrho^{4}r(\varrho^{2})^{2}\Big{(}6+153r^{(1)}(\varrho^{2})-88\varrho^{2}r^{(2)}(\varrho^{2})$
$\displaystyle\makebox[28.45274pt]{
}-8\varrho^{2}[r^{(2)}(\varrho^{2})^{2}+4r^{(3)}(\varrho^{2})]+16r^{(1)}(\varrho^{2})[9+8\varrho^{2}r^{(2)}(\varrho^{2})+\varrho^{4}r^{(3)}(\varrho^{2})]\Big{)}$
$\displaystyle\makebox[28.45274pt]{
}-4\varrho^{6}r(\varrho^{2})\Big{(}77r^{(1)}(\varrho^{2})^{3}+2r^{(1)}(\varrho^{2})^{2}[39+14\varrho^{2}r^{(2)}(\varrho^{2})]$
$\displaystyle\makebox[28.45274pt]{
}+8\varrho^{2}r^{(2)}(\varrho^{2})(\varrho^{2})+8\varrho^{2}[r^{(2)}(\varrho^{2})^{2}+r^{(3)}(\varrho^{2})]$
$\displaystyle\makebox[28.45274pt]{
}-2r^{(1)}(\varrho^{2})[8\varrho^{2}r^{(3)}(\varrho^{2})+36\varrho^{2}r^{(2)}(\varrho^{2})-3]\Big{)}\Big{]}\bigg{\\}}\,,$
(A.6)

which vanishes for a generic regulator function $r$ along the same lines as
(A.5). This establishes Lemma A($b$).

  3. (c)

As above, direct computation yields

$\displaystyle B_{6}-B_{7}$ $\displaystyle\\!=\\!$
$\displaystyle-\int_{0}^{\infty}\\!d\varrho\,\frac{{\partial}}{{\partial}\varrho}\bigg{\\{}\frac{5\varrho^{3}\big{[}r(\varrho^{2})-\varrho^{2}r^{(1)}(\varrho^{2})\big{]}^{2}}{4\big{[}\varrho^{2}+r(\varrho^{2})\big{]}^{7/2}}\bigg{\\}}=0\,.$
(A.7)

Together with (A1.Ex79) this completes our proof of Lemma A.

∎

We end this appendix with a summary of the expressions for the regulator
dependent coefficients in (LABEL:Bcoeff1), (3.13).

$\displaystyle\breve{q}$ $\displaystyle\\!=\\!$
$\displaystyle\frac{1}{12}\int_{0}^{\infty}\\!d\varrho\,\varrho^{2}\frac{\big{[}r(\varrho^{2})-\varrho^{2}r^{(1)}(\varrho^{2})\big{]}^{2}}{\big{[}\varrho^{2}+r(\varrho^{2})\big{]}^{5/2}}\,,\quad
q_{1}=\int_{0}^{\infty}\\!d\varrho\,\varrho^{2}\frac{r(\varrho^{2})-\varrho^{2}r^{(1)}(\varrho^{2})}{\big{[}\varrho^{2}+r(\varrho^{2})\big{]}^{3/2}}\,,$
$\displaystyle b_{1}$ $\displaystyle\\!=\\!$
$\displaystyle\frac{3}{2}\int_{0}^{\infty}\\!d\varrho\,\varrho^{2}\frac{\big{[}r(\varrho^{2})-\varrho^{2}r^{(1)}(\varrho^{2})\big{]}^{2}}{\big{[}\varrho^{2}+r(\varrho^{2})\big{]}^{7/2}}\,,\quad\quad\;b_{2}=-\frac{B_{5}}{12}\,,$
$\displaystyle b_{3}$ $\displaystyle\\!=\\!$
$\displaystyle\frac{1}{2}\int_{0}^{\infty}\\!d\varrho\,\varrho^{2}\frac{r(\varrho^{2})-\varrho^{2}r^{(1)}(\varrho^{2})}{\big{[}\varrho^{2}+r(\varrho^{2})\big{]}^{7/2}}\Big{\\{}\\!-3r(\varrho^{2})+\varrho^{2}\big{(}2+5r^{(1)}(\varrho^{2})(\varrho^{2})\big{)}\Big{\\}}\,.$
(A.8)

B: Spatial EPA in Minkowski space and SLE instantaneous limit

The analysis of the spatial EPA (LABEL:sepa1) is normally hampered by the
difficulty of gaining analytical control over the Green’s function
$G_{k}[\varphi](t,t;p)$. There are two instructive exceptions which we present
in this appendix: Minkowski space and the “instantaneous limit” of the version
of (LABEL:sepa1) based on a State of Low Energy (SLE), see Section 2.4.

In Minkowski space $a={N}=1$, the generalized potential reduces to the
standard one ${\cal U}_{k}(\varphi,R)=\,\,^{0}U_{k}(\varphi)=:U_{k}(\varphi)$,
and we may take the field $\varphi$ to be constant. In line with the general
discussion we choose the Feynman type solution of the Green’s function
relation in (LABEL:sepa1), viz.

$G_{k}[\varphi](t,t^{\prime};p)=i\dfrac{e^{-i|t-t^{\prime}|\sqrt{p^{2}+R_{k}(p)+U_{k}^{\prime\prime}(\varphi)}}}{2\sqrt{p^{2}+R_{k}(p)+U^{\prime\prime}_{k}(\varphi)}}\,.$
(B.1)

This corresponds to (2.34) with a Wronskian normalized positive frequency
solution of the homogeneous wave equation. The spatial EPA flow equation then
reads

${\partial}_{k}U_{k}=\frac{\hbar}{4}\int\\!\frac{d^{d}p}{(2\pi)^{d}}\dfrac{{\partial}_{k}R_{k}(p)}{\sqrt{p^{2}+R_{k}(p)+U^{\prime\prime}_{k}}}\,.$
(B.2)

We note that no contour deformation or other indirect reference to a Euclidean
time or momentum regime enters in arriving at (B.1), (B.2). Other Minkowski
space EPA flow equations can be found in [8, 9]. After transition to radial
momenta $|p|$ and $\wp=|p|/k$ one has

${\partial}_{k}U_{k}=\hbar\frac{k^{d}}{(4\pi)^{d/2}\Gamma(d/2)}\int_{0}^{\infty}\\!d\wp\,\wp^{d-1}\dfrac{r(\wp^{2})-\wp^{2}r^{\prime}(\wp^{2})}{\sqrt{\wp^{2}+r(\wp^{2})+U_{k}^{\prime\prime}/k^{2}}}\,,$
(B.3)

using ${\rm vol}(S^{d-1})=2\pi^{d/2}/\Gamma(d/2)$ for the volume of the
$S^{d-1}$ sphere. The transition to a dimensionless formulation is effected by

$U_{k}(\varphi)=c_{0}k^{1+d}V_{k}(\varphi_{0})\Big{|}_{\varphi_{0}=c_{0}^{-1/2}k^{-\frac{d-1}{2}}\varphi}\,,$
(B.4)

where $c_{0}>0$ a fudge factor to be adjusted later on. In terms of
$V_{k}(\varphi_{0})$ the spatial EPA flow equation reads

$k{\partial}_{k}V_{k}(\varphi_{0})+(1\\!+\\!d)V_{k}(\varphi_{0})-\frac{d\\!-\\!1}{2}\varphi_{0}V^{\prime}_{k}(\varphi_{0})=\hbar
q_{0}\big{(}V_{k}^{\prime\prime}(\varphi_{0})\big{)}\,.$ (B.5)

For re-use later on we defined here

$q_{\nu}(v):=\frac{1}{c_{0}(4\pi)^{d/2}\Gamma(d/2)}\int_{0}^{\infty}\\!d\wp\,\wp^{d-1}\frac{r(\wp^{2})-\wp^{2}r^{\prime}(\wp^{2})}{(\wp^{2}+r(\wp^{2})+v)^{1/2+\nu}}\,,$
(B.6)

with $\nu\in\mathbb{R}$, $v\geq 0$. Following the general discussion in
Section 4 we use (B.3) for the large $k$ expansion and (B.5) to study the
fixed point regime.

For the large $k$ expansion the dimensionful potential should be treated as
$k$ independent, with $U^{\prime\prime}\mapsto U_{k}^{\prime\prime}$ re-
substituted after expansion. This gives

${\partial}_{k}U_{k}=\hbar c_{0}\sum_{n\geq 0}{-1/2\choose
n}q_{n}(0)\Big{(}\frac{U_{k}^{\prime\prime}}{k^{2}}\Big{)}^{n}\,.$ (B.7)

The coefficients are just the binomial ones for which we note ${-1/2\choose
n}=(-)^{n}\frac{(2n\\!-\\!1)!!}{2^{n}n!}$. Comparing with the general large
$k$ expansion (4.24), (4.25) one can read off

$G_{n}=\frac{(2n\\!-\\!1)!!}{2^{n}(\wp^{2}+r(\wp^{2}))^{n}}\frac{(U^{\prime\prime})^{n}}{n!}\,,\quad
n\geq 0\,.$ (B.8)

For constant $\varphi$ the $(U^{\prime\prime})^{n}/n!$ are in fact the
conventionally normalized heat kernel coefficients of the Schrödinger operator
${\partial}_{t}^{2}+U^{\prime\prime}(\varphi)$. For $d\\!=\\!3$ the $n=0,1,2$
terms match those for $\,\,{}^{0}U_{k}$ in (S4.Ex68), using
$4\pi^{2}c_{0}q_{0}(0)=q_{0}$, $4\pi^{2}c_{0}q_{1}(0)=q_{1}$,
$4\pi^{2}c_{0}q_{2}(0)=1/3$.

The fixed point regime of the flow is explored via the dimensionless flow
equation (B.5). Its right hand side can be expressed as
$q_{0}(V^{\prime\prime}_{k})$. Typically, $V^{\prime\prime}$ has a field
independent part $v_{2}$ which one will split off when comparing powers of
$\varphi_{0}^{2}$ later on. The relevant expansion then is

$q_{0}(v_{2}+\Delta V^{\prime\prime})=\sum_{n\geq 0}{-1/2\choose
n}q_{n}(v_{2})(\Delta V^{\prime\prime})^{n},$ (B.9)

where $\Delta V^{\prime\prime}=\sum_{j\geq
1}(\varphi_{0}^{2j}/(2j)!)v_{2j+2}$. Expanding around a non-zero $v_{2}$ gives
a better approximation to the exact $q_{0}(V^{\prime\prime})$ than the
$q_{n}(0)$’s entering (B.7). Note that neither (B.7) nor (B.9) explore the
large field regime of the nonlinear flow (B.3) or (B.5), which would amount to
an expansion in inverse powers of $U^{\prime\prime}_{k}$ or
$V^{\prime\prime}_{k}$. This reflects the choice of a potential Ansatz in
positive (even) powers of the fields.

Comparing in (B.5) powers of $\varphi_{0}^{2}$ results in a ‘upward’ coupled
system of flow equations for the dimensionless couplings. Setting $\hbar=1$
the first few read

$\displaystyle k{\partial}_{k}v_{k,0}$ $\displaystyle\\!=\\!$
$\displaystyle-(d+1)v_{k,0}+q_{0}(v_{k,2})\,,$ $\displaystyle
k{\partial}_{k}v_{k,2}$ $\displaystyle\\!=\\!$
$\displaystyle-2v_{k,2}-\frac{1}{2}q_{1}(v_{k,2})v_{k,4}\,,$ $\displaystyle
k{\partial}_{k}v_{k,4}$ $\displaystyle\\!=\\!$
$\displaystyle(d-3)v_{k,4}-\frac{1}{2}q_{1}(v_{k,2})v_{k,6}+\frac{9}{4}q_{2}(v_{k,2})v_{k,4}^{4}\,.$
(B.10)

The further analysis now proceeds along the familiar lines: Truncating the
system (B.1) at some order $N$ (i.e. $v_{k,2j}\equiv 0$, $j\geq N$) results in
a closed coupled system of ordinary differential equations which can be
integrated numerically. The fixed point equations are coupled algebraic
equations for which one finds many solutions. In $d=2$ only two of them have a
stability matrix with a single negative eigenvalue, however. These are the
Gaussian fixed point and the Minkowski space counterpart of the Wilson-Fisher
fixed point. For $d\geq 3$ only the Gaussian fixed point has a stability
matrix with a single negative eigenvalue. Linearizing the flow around the
chosen fixed point sets the target for the numerical shooting technique. In
the ultraviolet boundary conditions are set by the expansion of (B.7). The
quantitative results so obtained do not differ significantly from those
obtained with Euclidean signature and the standard EPA flow, see e.g. [1] and
references therein. This validates (B.2) as a viable Minkowski space EPA flow
equation.

A flow equation of the form (B.2) also arises in the instantaneous limit of
the SLE based version of (LABEL:sepa1). Recall that the equal time limit of
the Green’s function $G_{k}[\varphi](t,t;p)$ entering (LABEL:sepa1) is then
given by $i|T^{\rm SLE}[S_{k}](t,p)|^{2}$, where the fiducial solution
$S_{k}(t,p)$ is an arbitrary solution of (LABEL:FLhad2). The modulus square
can be expressed as in (2.53) and involves a window function $f(t)$ of compact
support in $[t_{i},t_{f}]$. Although this quantity is amenable to analytical
control, it is fairly complicated. A remarkable simplification occurs when the
window function $f$ of the SLE is increasingly centered around an instant
$t_{0}\in(t_{i},t_{f})$, giving
$f(t)^{2}\rightarrow{n}(t)^{-1}\delta(t\\!-\\!t_{0})$ in a formal limit. If
the underlying second order operator is of Schrödinger type (i.e. without
first order term) the limit of the modulus square of the SLE solution equals
$1/(2\sqrt{\mbox{full potential}})$, see [19]. In the situation at hand this
gives

$\big{|}T^{\rm SLE}[S_{k}](t,k\wp)\big{|}^{2}\ \longrightarrow\
\frac{1}{2ka(t_{0})^{d}\sqrt{\wp^{2}/a(t_{0})^{2}+r(\wp^{2}/a(t_{0}^{2}))+{\cal
U}^{\prime\prime}(\varphi,R(t_{0}))/k^{2}}}\,,$ (B.11)

with $\wp=p/k$. Let us stress that the instantaneous limit ruins the Hadamard
property and is in general neither mathematically nor physically viable [28].
Another caveat is that the operations “taking the instantaneous limit” and
“selecting a time function through a choice of ${n}$” do not commute because
the (non-instantaneous) Schwarzian terms arising from the elimination of the
first order term are missed.

It is nevertheless instructive to see that the limit produces a flow equation
of the form (B.3) which is formally applicable to generic Friedmann-Lemaître
spacetimes merely by replacing $\wp$ in the integrand by $\wp/a(t_{0})$ and
$U_{k}$ by ${\cal U}_{k}=\sum_{n\geq 0}{}^{n}U_{k}(\varphi)R(t_{0})^{n}$. The
explicit time dependence of the SLE EPA flow disappears and the projection
onto the orthonormal basis (4.4) is both unnecessary and tautological.
Generally, if in (4.8) the weight $w(t)$ approaches $\delta(t\\!-\\!t_{0})$,
the coefficients $h_{l}$ approach $\mathfrak{p}_{l}(t_{0})h(t_{0})$, and on
account of the completeness relation the series resums to
$h(t)=h(t_{0})\delta(t\\!-\\!t_{0})$. In the expansion (4.32) one finds from
the definitions (S2.Ex27), (2.59) that $J_{1,\wp}[\overline{{\cal
V}}^{\prime\prime}_{k}](t_{0})\rightarrow 0$ and
$\varepsilon_{2,\wp}^{2}[\overline{{\cal V}}^{\prime\prime}_{k}]\rightarrow 0$
in the instantaneous limit. The right hand side (4.32) therefore reduces to
(B.11).

In order to compare with the results of Section 3 we fix conformal time and
consider the SLE solution $\chi_{k}^{\rm SLE}(\eta,p)$ associated with the
(homogeneous) wave equation (LABEL:fl1l3). The shifted potential is
$W(\eta)-\frac{d-1}{4d}R(\eta)$, and we write $\overline{{\cal
V}}^{\prime\prime}(\varphi_{0}(\eta),\mathfrak{r}_{k}(\eta))$ for its
dimensionless counterpart, in parallel to (3.25). In both cases, as in (B.11),
we omit the subscript $k$ referring to the dynamical $k$ dependence. The
instantaneous limit of the SLE solution in conformal time and $d=3$ then reads

$|\chi^{\rm SLE}_{k}(\eta_{0},k\wp)|^{2}\ \longrightarrow\
\frac{1}{2ka(\eta_{0})\sqrt{\overline{{\cal
V}}^{\prime\prime}(\varphi_{0}(\eta_{0}),\mathfrak{r}_{k}(\eta_{0}))+\wp^{2}/a(\eta_{0})^{2}+r(\wp^{2}/a(\eta_{0}^{2}))}}\,,$
(B.12)

with $\eta_{0}\in(\eta_{i},\eta_{f})$. Restoring the subscript and
abbreviating momentarily ${\cal V}_{k}(\eta_{0}):={\cal
V}_{k}(\varphi_{0}(\eta_{0}),\mathfrak{r}_{k}(\eta_{0}))$, etc., this results
in

$\displaystyle k{\partial}_{k}{\cal V}_{k}(\eta_{0})+(1\\!+\\!d){\cal
V}_{k}(\eta_{0})-k{\partial}_{k}\ln\kappa(k)^{2}\;\mathfrak{r}_{k}(\eta_{0})\frac{{\partial}{\cal
V}_{k}(\eta_{0})}{{\partial}\mathfrak{r}_{k}(\eta_{0})}-\frac{d\\!-\\!1}{2}\varphi_{0}(\eta_{0})\frac{{\partial}{\cal
V}_{k}(\eta_{0})}{{\partial}\varphi_{0}(\eta_{0})}$
$\displaystyle\quad=\frac{\hbar}{c_{0}(4\pi)^{d/2}\Gamma(d/2)}\int_{0}^{\infty}\\!d\varrho\varrho^{d-1}\dfrac{r(\varrho^{2})-\varrho^{2}r^{\prime}(\varrho^{2})}{\sqrt{\overline{{\cal
V}}_{k}^{\prime\prime}(\eta_{0})+\varrho^{2}+r(\varrho^{2})}}\,.$ (B.13)

The right hand side coincides with that of the spatial EPA flow equation in
Minkowski space (B.5) up to the replacement of
$V_{k}^{\prime\prime}(\varphi_{0})={}^{0}V^{\prime\prime}_{k}(\varphi_{0})$ by
the generalized potential $\overline{{\cal V}}_{k}^{\prime\prime}(\eta_{0})$
depending also on the Ricci scalar
$\mathfrak{r}_{k}(\eta_{0})=R(\eta_{0})/\kappa(k)^{2}$. Upon expansion of
(LABEL:sleinst2) in powers of $\mathfrak{r}_{k}(\eta_{0})$ one obtains (B.5)
at lowest order and (with $\hbar=1$)

$\displaystyle
k{\partial}_{k}{}^{1}V_{k}(\eta_{0})+\big{(}1\\!+\\!d-k{\partial}_{k}\ln\kappa(k)^{2}\big{)}{}^{1}V_{k}(\eta_{0})-\frac{d\\!-\\!1}{2}\varphi_{0}(\eta_{0})\frac{{\partial}\,{}^{1}V_{k}(\eta_{0})}{{\partial}\varphi_{0}(\eta_{0})}$
$\displaystyle\quad=-\frac{1}{2}q_{1}\big{(}{}^{0}V_{k}^{\prime\prime}(\eta_{0})\big{)}{}^{1}\overline{V}^{\prime\prime}_{k}(\eta_{0})\,,$
$\displaystyle
k{\partial}_{k}{}^{2}V_{k}(\eta_{0})+\big{(}1\\!+\\!d-2k{\partial}_{k}\ln\kappa(k)^{2}\big{)}{}^{2}V_{k}(\eta_{0})-\frac{d\\!-\\!1}{2}\varphi_{0}(\eta_{0})\frac{{\partial}\,{}^{2}V_{k}(\eta_{0})}{{\partial}\varphi_{0}(\eta_{0})}$
$\displaystyle\quad=-\frac{1}{2}q_{1}\big{(}{}^{0}V_{k}^{\prime\prime}(\eta_{0})\big{)}{}^{2}V^{\prime\prime}(\eta_{0})+\frac{3}{8}q_{2}\big{(}{}^{0}V_{k}^{\prime\prime}(\eta_{0})\big{)}\big{(}{}^{1}\overline{V}_{k}^{\prime\prime}(\eta_{0})\big{)}^{2}\,,$
(B.14)

at the next two orders, with $q_{\nu}(v)$ from (B.6). Here
${}^{n}\overline{V}_{k}^{\prime\prime}(\varphi_{0})={}^{n}V_{k}(\varphi_{0})-\frac{d-1}{4d}\delta_{n,1}$.

It is instructive to compare the UV flow (S4.Ex68) with the instantaneous
limit flow (LABEL:sleinst3). The flow equations (LABEL:sleinst2),
(LABEL:sleinst3) are not limited to large $k$ but an expansion in powers of
the potential can be compared with the dimensionless form of (S4.Ex68). Since
$\kappa(k)\sim k$ for large $k$ the latter is obtained via the transition
relations ${}^{n}U_{k}(\varphi)=c_{0}k^{d+1-2n}\;{}^{n}V_{k}(\varphi_{0})$.
For the comparison of the right hand sides we note
$q_{\nu}(v)=q_{\nu}(0)-\frac{1}{2}(2\nu\\!+\\!1)vq_{\nu+1}(0)+\frac{1}{8}(2\nu\\!+\\!1)(2\nu\\!+\\!3)v^{2}q_{\nu+2}(0)+O(v^{3})$,
in the notation (B.6). Using this in the $d=3$ version of (LABEL:sleinst3)
together with $1/[(4\pi)^{d/2}\Gamma(d/2)]\big{|}_{d=3}=4/(4\pi)^{2}$,
$4\pi^{2}c_{0}q_{0}(0)=q_{0}$, $4\pi^{2}c_{0}q_{1}(0)=q_{1}$,
$4\pi^{2}c_{0}q_{2}(0)=1/3$, one finds for $n=0,2$ an exact match, while for
$n=1$ the $\breve{q}$ term in (S4.Ex68) is missed. Since the latter arises
from differentiating the time dependence in the modulator, it is unsurprising
that this term is not seen in the instantaneous limit.

C: Subleading small $k^{2}$ terms in SLE based EPA flow

Here we justify the substitution recipe (LABEL:sleepa0) and use it to compute
the explicit form of the $O(k^{2})$ corrections displayed in (LABEL:sleepa3).

As seen in Appendix B, in the instantaneous limit $|T^{\rm
SLE}_{k}(t_{0},k\wp)|^{2}$ is a shifted inverse square root in the
(generalized) potential ${\cal U}_{k}^{\prime\prime}$ and it is plain that the
substitutions (4.14) or (4.16) transitioning to the dimensionless formulation
do not affect the dependence on the re-interpreted (generalized) potential. In
the non-instantaneous case, however, the dimensionful formulation
(LABEL:sepa1) does not allow one to isolate the ${\cal U}^{\prime\prime}_{k}$
dependence of the Green’s function explicitly, not even in a series expansion
in powers of $k^{2}$. This is because in an expansion like (C.3) below the
lowest order Green’s function of
$({n}^{-1}{\partial}_{t})^{2}+\omega_{0}(t)^{2}$ needed to solve the recursion
cannot be obtained explicitly for generic $\omega_{0}(t)^{2}=a(t)^{2d}W(t)$.
In contrast, the iterative solution of (4.21) for the power series expansion
of the dimensionless Green’s function $G_{k}[\varphi_{0}](t,t^{\prime};\wp)$
is in itself straightforward, as the only a Green’s function for the trivial
$({n}^{-1}{\partial}_{t})^{2}$ is needed. For the SLE this $\omega_{0}\equiv
0$ expansion can be implemented and leads to the explicit result (4.32). The
gap to the dimensionful expansion is bridged by the following

Lemma C: The SLE Green’s function $kG_{k}[\varphi](t,t;p)$ in (LABEL:sepa1)
with dimensionful generalized potential ${\cal U}^{\prime\prime}(\varphi,t)$
expanded in powers of $k^{2}$ has upon substitution of
$\varphi=\varphi[\varphi_{0}]$, $p=k\wp$, ${\cal
U}^{\prime\prime}(\varphi,t)=k^{2}{\cal V}^{\prime\prime}(\varphi_{0},t)$, and
re-expansion in powers of $k^{2}$ (with ${\cal
V}^{\prime\prime}(\varphi_{0},t)$ held fixed) the same coefficients as the
directly expanded dimensionless $kG_{k}[\varphi_{0}](t,t;\wp)$ leading to
(4.32).

Remarks:

  * (i)

A result of this form does not hold for the large $k$ expansion (2.44). One
way of seeing this is by noting that for $\omega_{0}\equiv 0$ the coefficients
$\bar{G}_{n}$ vanish for constant $w_{2}$. Since constant $w_{2}$ occurs in
Minkowski space (for constant field) this clashes with the direct expansion in
the dimensionful formulation, where the coefficients (B.8) arise. The origin
of the clash is that upon substitution ${\cal
U}^{\prime\prime}(\varphi,R)\mapsto k^{2}{\cal
V}^{\prime\prime}(\varphi_{0},\mathfrak{r}_{0})$ the $\bar{G}_{n}$ are
$O(k^{2n})$, which invalidates the organizing principle in (B.8).

  * (ii)

A simple toy model may illustrate the point of the argument. Consider the
following three series (where ${\partial}^{2}$, $W$, $V$, $r$ are suggestive
notations for positive numbers)

$\displaystyle\frac{1}{{\partial}^{2}+W+k^{2}r}=\sum_{n\geq
0}(-)^{n}\frac{r^{n}k^{2n}}{({\partial}^{2}+W)^{n+1}}\,,$
$\displaystyle\frac{1}{({\partial}^{2}+W)^{n+1}}\Big{|}_{W\rightarrow
k^{2}V}=\sum_{j\geq 0}{-n\\!-\\!1\choose
j}\frac{V^{j}k^{2j}}{({\partial}^{2})^{n+1+j}}\,,$
$\displaystyle\frac{1}{{\partial}^{2}+k^{2}(r+V)}=\sum_{n\geq
0}(-)^{n}\frac{(V+r)^{n}k^{2n}}{({\partial}^{2})^{n+1}}\,.$ (C.1)

Inserting the first two series into each other (re-)produces the third, on
account of $\sum_{j=0}^{n}(-)^{n-j}{-n-1+j\choose
j}V^{j}r^{n-j}=(-)^{n}(V+r)^{n}$.

Proof of Lemma C. We apply an abstracted version of (LABEL:sleepa5) to the
fiducial solutions entering the SLE. To this end, we search for a fiducial
solution $S_{k}(t,\wp)$ of the homogeneous wave equation (LABEL:FLhad2) for
fixed $\wp=p/k$ in the form of a power series ansatz

$S_{k}(t,\wp)=S_{0}(t)+\sum_{n\geq
1}S_{n}(t,\wp)\,k^{2n}\,,\quad\wp=p/k\;\;\mbox{fixed}\,.$ (C.2)

Inserted into (LABEL:FLhad2) one obtains the recursive system

$\displaystyle\big{[}(a^{d}{N}^{-1}{\partial}_{t})^{2}+a^{2d}W\big{]}S_{0}(t)$
$\displaystyle\\!=\\!$ $\displaystyle 0\,,$ (C.3)
$\displaystyle\big{[}(a^{d}{N}^{-1}{\partial}_{t})^{2}+a^{2d}W\big{]}S_{n}(t,\wp)$
$\displaystyle\\!=\\!$
$\displaystyle-a^{2d-2}\Big{(}\wp^{2}+a^{2}r(\wp^{2}/a^{2})\Big{)}S_{n-1}(t,\wp)\,,\quad
n\geq 1\,,$

with $W={\cal U}^{\prime\prime}$ is as in (S2.Ex15). The coefficients $S_{n}$
are in addition slightly constrained by the expansion of the Wronskian
condition. Since an SLE does not depend on the choice of fiducial solution we
may begin with a fiducial solution of the dimensionful wave equation whose
initial data are $k,\wp$ independent. The recursion relations (C.3) can then
be solved unambiguously in terms of the (universal) retarded Green’s functions
for $({n}^{-1}{\partial}_{t})^{2}+a^{2d}{\cal U}^{\prime\prime}$, $n=Na^{-d}$
[19]. The resulting series has finite radius of convergence and thus yields an
exact solutions for $k<k_{*}$. Since its initial data are $k$ independent one
can appeal to a standard result on ordinary differential equations (see e.g.
[24], Chapter 5.2) to conclude that the dependence of the exact solution on
${\cal U}^{\prime\prime}$’s overall scale is smooth. The same holds for the
Green’s function of $({n}^{-1}{\partial}_{t})^{2}+a^{2d}{\cal
U}^{\prime\prime}$. Replacing now ${\cal U}^{\prime\prime}$ by $k^{2}{\cal
V}^{\prime\prime}$ the basic Green’s function needed for the iteration as well
as the exact solution can be re-expanded in powers of $k^{2}$. The re-expanded
series is a solution of the $\omega_{0}^{2}=a^{2d}W\equiv 0$ recursion
relations uniquely characterized by the initial data induced by re-expanding
the exact solution, and the trivial $\omega_{0}^{2}\equiv 0$ retarded Green’s
function. It also has finite radius of convergence $k<k_{\ast,0}$, see [19].
Within the (nonempty) intersection of the intervals of convergence the two-
step expanded solution and the solution of the $\omega_{0}^{2}\equiv 0$
recursion will coincide. This justifies the substitution recipe
(LABEL:sleepa0) for the fiducial solutions.

Either of these expansions can now be inserted into the SLE solution
$T_{k}^{\rm SLE}[S_{k}](t,p)$, viewed as a functional of $S_{k}$. The
functionals also carry an explicit $k,\wp$ dependence entering via the ${\cal
E},{\cal D}$ from (S2.Ex23). Since analyticity in $k^{2}$ is maintained in the
substitution process the explicit $k$ dependence allows for a power series
expansion in $k^{2}$. The same holds for the $k^{2}$ dependence carried by the
two distinct series realizations for the fiducial solution. The respective
expansions are of the form (2.55) (with $k$ playing the role $p$) but the
$\varepsilon_{0}=0$ version must by the previous step be the re-expansion of
the $\varepsilon_{0}>0$ version. This justifies the substitution recipe
(LABEL:sleepa0) for $|T_{k}^{\rm SLE}[S_{k}](t,p)|^{2}$, constructed from the
above specific fiducial solution $S_{k}(t,p)$. Since $|T_{k}^{\rm
SLE}[S_{k}](t,p)|^{2}$ does not depend on the choice of $S_{k}$ the result
follows. ∎

Application of Lemma C yields (4.32) so that the small $k$ regime of the SLE
based FL-sFRG flow equation (LABEL:dimless12) is governed by the coefficients
$J_{1,\wp}[{\cal V}^{\prime\prime}](t)$ and $\varepsilon_{2,\wp}^{2}[{\cal
V}^{\prime\prime}]$, where ${\cal V}^{\prime\prime}(t)$ is short for ${\cal
V}^{\prime\prime}(\varphi_{0},t)$. Using the averages (LABEL:sleepa4) the
right hand side of (LABEL:dimless12) becomes

$\displaystyle\mbox{RHS
of}\;(\ref{sleepa3})={}_{l}Q_{0}\big{(}1|\,{}_{0}\\!{\cal
V}_{k}^{\prime\prime}\big{)}+\frac{k^{2}}{J_{0}}\,{}_{l}Q_{0}\big{(}J_{1,\cdot}|\,{}_{0}\\!{\cal
V}_{k}^{\prime\prime}\big{)}-\frac{k^{2}}{2J_{0}^{2}}\,{}_{l}Q_{1}\big{(}\,\varepsilon_{2,\cdot}^{2}[{\cal
V}^{\prime\prime}_{k}]|\,{}_{0}\\!{\cal V}_{k}^{\prime\prime}\big{)}\,.$ (C.4)

In view of the linearity of the ${}_{l}Q_{m}(h|V)$ in the first argument we
prepare expansions for $J_{1,\wp}(t)$ and $\varepsilon_{2,\wp}^{2}$. To this
end the expansion of the commutator function $\Delta_{1,\wp}$ and its
derivatives is needed. In parallel to (S4.Ex70) we write

$\displaystyle{n}(t)^{-1}{\partial}_{t}\Delta_{1,\wp}[{\cal
V}^{\prime\prime}](t^{\prime},t)$ $\displaystyle\\!=\\!$
$\displaystyle\int_{t^{\prime}}^{t}\\!ds\,{n}(s)a(s)^{2d}\big{[}{\cal
V}^{\prime\prime}(s)+\mathfrak{r}(\wp/a(s))\big{]}\Delta_{0}(s,t^{\prime})\,,$
$\displaystyle\Big{[}{n}(t)^{-1}{n}(t^{\prime})^{-1}{\partial}_{t^{\prime}}{\partial}_{t}\Delta_{1,\wp}[{\cal
V}^{\prime\prime}](t^{\prime},t)\Big{]}^{2}$ $\displaystyle\\!=\\!$
$\displaystyle\int_{t^{\prime}}^{t}\\!dsds^{\prime}{n}(s){n}(s^{\prime})a(s)^{2d}a(s^{\prime})^{2d}$
$\displaystyle\times$ $\displaystyle\big{[}{\cal
V}^{\prime\prime}(s)+\mathfrak{r}(\wp/a(s))\big{]}\big{[}{\cal
V}^{\prime\prime}(s^{\prime})+\mathfrak{r}(\wp/a(s^{\prime}))\big{]}\,,$

with the abbreviation
$\mathfrak{r}(\wp/a(t)):=\wp^{2}/a(t)^{2}+r(\wp^{2}/a(t)^{2})$. Using
(S2.Ex27), (S2.Ex28), (LABEL:sleepa0), these enter

$\displaystyle J_{1,\wp}[{\cal V}^{\prime\prime}](t)$ $\displaystyle\\!=\\!$
$\displaystyle\frac{1}{2}\int\\!dt^{\prime}{n}(t^{\prime})f(t^{\prime})\Big{\\{}2{n}(t^{\prime})^{-1}{\partial}_{t^{\prime}}\Delta_{1,\wp}[{\cal
V}^{\prime\prime}](t^{\prime},t)$ (C.6) $\displaystyle+$
$\displaystyle\Delta_{0}(t,t^{\prime})^{2}a(t^{\prime})^{2d}\big{[}{\cal
V}^{\prime\prime}(t^{\prime})+\mathfrak{r}\big{(}\wp/a(t^{\prime})\big{)}\big{]}\Big{\\}}\,,$

and

$\displaystyle\varepsilon_{2,\wp}^{2}[{\cal V}^{\prime\prime}]=\frac{1}{8}\int
dtdt^{\prime}\,{n}(t){n}(t^{\prime})f(t)f(t^{\prime})\Big{\\{}\big{(}{n}(t)^{-1}{n}(t^{\prime})^{-1}{\partial}_{t}{\partial}_{t^{\prime}}\Delta_{1,\wp}[{\cal
V}^{\prime\prime}](t,t^{\prime})\big{)}^{2}$
$\displaystyle\quad-4a(t^{\prime})^{2d}\big{[}{\cal
V}^{\prime\prime}(t^{\prime})+\mathfrak{r}\big{(}\wp/a(t^{\prime})\big{)}\big{]}{n}(t)^{-1}{\partial}_{t}\Delta_{1,\wp}[{\cal
V}^{\prime\prime}](t^{\prime},t)$ (C.7)
$\displaystyle\quad+a(t)^{2d}a(t^{\prime})^{2d}\Delta_{0}(t,t^{\prime})^{2}\big{[}{\cal
V}^{\prime\prime}(t)+\mathfrak{r}\big{(}\wp/a(t)\big{)}\big{]}\big{[}{\cal
V}^{\prime\prime}(t^{\prime})+\mathfrak{r}\big{(}\wp/a(t^{\prime})\big{)}\big{]}\Big{\\}}\,.$

Next, we expand ${\cal V}^{\prime\prime}(t)={\cal
V}^{\prime\prime}(\varphi_{0},t)=\sum_{l\geq 0}{}_{l}{\cal
V}^{\prime\prime}(\varphi_{0})\mathfrak{p}_{l}(t)$ and insert into (C.6). This
gives

$\displaystyle J_{1,\wp}[{\cal V}^{\prime\prime}](t)\\!=\\!\sum_{l\geq
0}{}_{l}{\cal V}^{\prime\prime}(\varphi)\jmath_{l}(t)+\jmath(t,\wp)\,,$
$\displaystyle\quad\jmath_{l}(t)\\!=\\!\int\\!dt^{\prime}n(t^{\prime})f(t^{\prime})\Big{\\{}\int_{t^{\prime}}^{t}\\!dsn(s)a(s)^{2d}\,\mathfrak{p}_{l}(s)\Delta_{0}(s,t^{\prime})+\frac{1}{2}a(t^{\prime})^{2d}\Delta_{0}(t,t^{\prime})^{2}\,\mathfrak{p}_{l}(t^{\prime})\Big{\\}}\,,$
(C.8)
$\displaystyle\jmath(t,\wp)\\!=\\!\int\\!\\!dt^{\prime}n(t^{\prime})f(t^{\prime})\Big{\\{}\\!\int_{t^{\prime}}^{t}\\!\\!dsn(s)a(s)^{2d}\,\mathfrak{r}\big{(}\wp/a(s)\big{)}\Delta_{0}(s,t^{\prime})+\frac{1}{2}a(t^{\prime})^{2d}\Delta_{0}(t,t^{\prime})^{2}\,\mathfrak{r}\big{(}\wp/a(t^{\prime})\big{)}\Big{\\}}.$

Finally,

$\displaystyle{\cal E}_{2,\wp}^{2}[{\cal V}^{\prime\prime}]\\!\\!$
$\displaystyle\\!=\\!$ $\displaystyle\\!\\!\sum_{l,l^{\prime}\geq
0}{}_{l}{\cal V}^{\prime\prime}(\varphi_{0})\;{}_{l^{\prime}}{\cal
V}^{\prime\prime}(\varphi_{0})E_{l,l^{\prime}}+\sum_{l\geq 0}{}_{l}{\cal
V}^{\prime\prime}(\varphi_{0})E_{l}(\wp)+E(\wp)\,,$ (C.9)
$\displaystyle\\!E_{l,l^{\prime}}\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!\frac{1}{8}\int\\!dtdt^{\prime}n(t)f(t)n(t^{\prime})f(t^{\prime})\Big{\\{}\int_{t^{\prime}}^{t}\\!dsn(s)a(s)^{2d}\,\mathfrak{p}_{l}(s)\int_{t^{\prime}}^{t}\\!ds^{\prime}n(s^{\prime})a(s^{\prime})^{2d}\,\mathfrak{p}_{l^{\prime}}(s^{\prime})$
$\displaystyle\\!-\\!$
$\displaystyle\\!4a(t^{\prime})^{2d}\mathfrak{p}_{l^{\prime}}(t^{\prime})\int_{t^{\prime}}^{t}\\!dsa(s)^{2d}\,\mathfrak{p}_{l}(s)\Delta_{0}(s,t^{\prime})+a(t)^{2d}a(t^{\prime})^{2d}\Delta_{0}(t,t^{\prime})^{2}\,\mathfrak{p}_{l}(t)\mathfrak{p}_{l^{\prime}}(t^{\prime})\Big{\\}}\,,$
$\displaystyle\\!E_{l}(\wp)\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\frac{1}{4}\int\\!dtdt^{\prime}n(t)f(t)n(t^{\prime})f(t^{\prime})\Big{\\{}\int_{t^{\prime}}^{t}\\!dsn(s)a(s)^{2d}\,\mathfrak{p}_{l}(s)\\!\int_{t^{\prime}}^{t}\\!ds^{\prime}n(s^{\prime})a(s^{\prime})^{2d}\,\mathfrak{r}\big{(}\wp/a(s^{\prime})\big{)}$
$\displaystyle\\!-\\!$
$\displaystyle\\!2a(t^{\prime})^{2d}\,\mathfrak{p}_{l}(t^{\prime})\int_{t^{\prime}}^{t}\\!dsn(s)a(s)^{2d}\,\mathfrak{r}\big{(}\wp/a(s)\big{)}\Delta_{0}(s,t^{\prime})$
$\displaystyle\\!-\\!$
$\displaystyle\\!2a(t^{\prime})^{2d}\,\mathfrak{p}_{l}(t^{\prime})\mathfrak{r}\big{(}\wp/a(t^{\prime})\big{)}\int_{t^{\prime}}^{t}\\!dsn(s)a(s)^{2d}\,\mathfrak{p}_{l}(s)\Delta_{0}(s,t^{\prime})$
$\displaystyle\\!+\\!$
$\displaystyle\\!a(t)^{2d}a(t^{\prime})^{2d}\Delta_{0}(t,t^{\prime})^{2}\,\mathfrak{p}_{l}(t)\,\mathfrak{r}\big{(}\wp/a(t^{\prime})\big{)}\Big{\\}}\,,$
$\displaystyle\\!E(\wp)\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\frac{1}{8}\int\\!dtdt^{\prime}n(t)f(t)n(t^{\prime})f(t^{\prime})\Big{\\{}\Big{[}\int_{t^{\prime}}^{t}\\!dsn(s)a(s)^{2d}\mathfrak{r}\big{(}\wp/a(s)\big{)}\Big{]}^{2}$
(C.10) $\displaystyle\\!-\\!$
$\displaystyle\\!4a(t^{\prime})^{2d}\,\mathfrak{r}(\wp/a(t^{\prime})\big{)}\int_{t^{\prime}}^{t}\\!dsn(s)a(s)^{2d}\mathfrak{r}\big{(}\wp/a(s)\big{)}\Delta_{0}(s,t^{\prime})$
$\displaystyle\\!+\\!$
$\displaystyle\\!a(t)^{2d}a(t^{\prime})^{2d}\Delta_{0}(t,t^{\prime})^{2}\,\mathfrak{r}\big{(}\wp/a(t)\big{)}\mathfrak{r}\big{(}\wp/a(t^{\prime})\big{)}\Big{\\}}\,.$

These expansions are now inserted into (C.4) and yield the explicit form of
(LABEL:sleepa3)’s right hand side.

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