The low multipoles of holographic cosmology at large flavour number

Holographic cosmology computes the primordial power spectrum from the stress-tensor two-point function of a three-dimensional gauge theory with many matter fields. Its two-loop form fits Planck temperature data at multipoles $\ell\ge30$ , and perturbation theory fails below. When all scalars are conformally coupled and the quartic and Yukawa couplings are negligible (hypotheses (A1)–(A3) of section 2), the leading order in the number of matter fields is a gauge loop whose gluon lines carry chains of matter bubbles. We compute it in closed form up to one numerical function. The logarithm that drives the two-loop running stops growing below a crossover multipole ( $\ell_c\approx3$ for the fit to all multipoles; the logarithm is frozen below $\ell\approx7$ ), and the spectrum at $\ell<30$ depends on the same three parameters as at high $\ell$ . The values fitted to $\ell\ge30$ give an $\ell<30$ spectrum that fits worse than $\Lambda$ CDM, by 18.5 in $-2\ln\mathcal L$ (section 6), so the high multipoles do not predict the low ones in this family.

Positivity of the resummed spectrum at all wavenumbers bounds $N_B=1+\sum_M(1-8\xi_M)^2$ , the gauge field plus the scalars with curvature couplings $\xi_M$ , each weighted by $(1-8\xi_M)^2$ . The bound is $N_B\le0.864\,\beta$ , where $\beta$ is one of the fitted parameters (section 5). For $N_B>1$ the mixed correlators of the gauge and scalar parts of the trace are not computed here, so the bound holds within a model of them, and it extrapolates the resummed form to $q\to0$ , beyond the range in which the resummation is controlled. Over the published one-standard-deviation ranges of $\ln\beta$ it gives $N_B\lesssim1.6$ to $2.9$ , that is a weighted scalar count of at most about $0.6$ to $1.9$ . In the model this excludes the explicit solution with $N_B\approx88$ given with the theory.

A fit of the resummed spectrum to the Planck temperature data at all multipoles, with a compressed likelihood and the background held at its Planck value, is worse than $\Lambda$ CDM by about 5 in $-2\ln\mathcal L$ although it has one more free parameter, and gives a quadrupole $0.57$ times the $\Lambda$ CDM value (section 6). The 18.5 and the 5 come from an approximate likelihood.

The framework is due to McFadden and Skenderis [MS09], [MS10]. Afshordi, Corianò, Delle Rose, Gould and Skenderis [ACDGS] used the two-loop coefficients computed by Corianò, Delle Rose and Skenderis [CDS] to fit Planck 2015 data. They found the fit at all multipoles disfavoured relative to $\Lambda$ CDM with running and attributed this to the breakdown of perturbation theory at $\ell\lesssim30$ ; Afshordi, Gould and Skenderis [AGS] repeated the analysis in more detail. The dressing of a three-dimensional gauge field by many matter fields is standard, and Giombi, Tarnopolsky and Klebanov [GTK] use it to compute the stress-tensor two-point coefficient of conformal QED at the infrared fixed point to next-to-leading order in $1/N$ . What is new here is the crossover function of the trace correlator between the perturbative and infrared regimes (section 3), from which the $\ell<30$ spectrum, the positivity bound and the temperature refit follow.

1. The holographic dictionary and the two-loop spectrum

In holographic cosmology [MS09] the early universe is described by a three-dimensional quantum field theory on the spatial slice, not by quantum fields on an inflating background. The wave function of the universe as a functional of the spatial metric is identified with the generating functional of this dual theory (its partition function with the metric as a source), so correlators of the dual stress tensor $T_{ij}$ determine those of the cosmological perturbations. For the scalar perturbation the operator is the trace $\Theta=T^i{}_i$ , and the power spectrum is, up to a numerical factor, the inverse of the two-point function of $\Theta$ : a large correlator means a small spectrum.

In the model of [ACDGS] the dual theory is SU( $N$ ) Yang–Mills theory in three Euclidean dimensions coupled to $N_\Phi$ real scalars and $N_\psi$ two-component Dirac fermions, all in the adjoint representation. Scalars couple to the background curvature with parameters $\xi_M$ , and $\xi_M=1/8$ is conformal coupling in three dimensions. The Yang–Mills coupling $g_{\rm YM}^2$ has the dimension of a mass, so the theory is super-renormalisable. Its dimensionless coupling at momentum $q$ is the ‘t Hooft coupling $\lambda=g_{\rm YM}^2N/q$ , small at large $q$ (short distances, high multipoles) and growing towards the infrared (low multipoles). The failure of perturbation theory at low $\ell$ is this growth.

Up to local terms the two-point function of the trace has the form $\langle\Theta(q)\Theta(-q)\rangle=N^2q^3f(\lambda)$ , where $q$ is the comoving wavenumber of the cosmological perturbation. The dictionary, in the form used in [ACDGS], gives the curvature power spectrum

\Delta^2(q)=\frac{1}{4\pi^2N^2f(\lambda)}. \tag{1}

In perturbation theory [ACDGS], [CDS],

f(\lambda)=f_0\big[1-f_1\lambda\ln\lambda+f_2\lambda+O(\lambda^2)\big],\qquad f_0=\frac{N_B}{64},\qquad N_B=1+\sum_M(1-8\xi_M)^2. \tag{2}

The “1” in $N_B$ is the gauge field. Each scalar contributes $(1-8\xi_M)^2$ , which vanishes at conformal coupling, and fermions do not contribute to $f_0$ . Without Yukawa couplings the published coefficient is ([ACDGS], eq. (8), with $\Sigma_\Phi=2\sum_M\xi_M^2$ )

N_Bf_1=-\frac{4}{3\pi^2}\Big(N_\psi-2+2N_\Phi-96\sum_M\xi_M^2\Big),\qquad\text{so that}\quad f_1=-\frac{4}{3\pi^2}\Big(N_\psi+\frac{N_\Phi}{2}-2\Big)\ \text{ at all }\xi_M=\tfrac18. \tag{3}

The spectrum is fitted in the form obtained by truncating $f$ at first order. With the pivot $q_*=0.05\,{\rm Mpc}^{-1}$ and the definitions $g\,q_*=f_1g_{\rm YM}^2N$ and $\ln(1/\beta)=f_2/f_1+\ln\lvert f_1\rvert$ , equation (1) becomes

\Delta^2(q)=\frac{\Delta_0^2}{1+u\ln u+u\ln\beta},\qquad u\equiv\frac{\lvert g\rvert q_*}{q},\qquad\Delta_0^2=\frac{1}{4\pi^2N^2f_0}, \tag{4}

where the fits have $g<0$ , that is $f_1<0$ . The three parameters $(\Delta_0^2,g,\ln\beta)$ replace the amplitude and tilt of $\Lambda$ CDM. With this pivot the range of the ‘t Hooft coupling quoted in [ACDGS] for its explicit solution (section 5), where it is written $g_{\rm eff}^2$ , is reproduced: $3.3\times10^{-4}\le\lambda\le0.41$ between $\ell=2500$ and $\ell=2$ . The fits to Planck 2015 data in [ACDGS], Table 1, give $g=-0.01305^{+0.00452}_{-0.00345}$ and $\ln\beta=1.014^{+0.206}_{-0.272}$ for $\ell\ge30$ , and $g=-0.00703^{+0.00105}_{-0.00167}$ and $\ln\beta=0.877^{+0.186}_{-0.239}$ for all multipoles.

Multipoles and wavenumbers are related by $\ell\approx q\chi_*$ , where $\chi_*=13\,873\,{\rm Mpc}$ is the comoving distance to last scattering at $z_*=1089.9$ for the Planck 2018 parameters of Table 2 of [Planck VI]. At the quadrupole the $\ell\ge30$ fit has $u=4.5$ , so (4) lies outside its range of validity. Extrapolated anyway, it gives a quadrupole $0.32$ times the $\Lambda$ CDM value (section 6 describes the comparison).

2. The expansion parameter is the number of fields

The two-loop term in (2) is large at low $\ell$ although the ‘t Hooft coupling there is small. The tensor-to-scalar ratio of the model is $r=32N_B/N_A$ with $N_A=1+2N_\psi+N_\Phi$ [ACDGS]. If the tensor spectrum is close to a power law at the scales probed by BICEP/Keck, the bound $r<0.036$ [BK18] requires $N_A>889$ when $N_B=1$ ; the tree-level tensor spectrum of the model need not be a power law, and we take $N_A\gtrsim889$ as the regime of interest. With (3), $\lvert f_1\rvert=\tfrac{2}{3\pi^2}(N_A-5)\gtrsim60$ at $\xi_M=\tfrac18$ , and $\lambda=u/\lvert f_1\rvert\le0.08$ at $\ell=2$ for the parameter sets with $\xi_M=\tfrac18$ used below. The two-loop correction is large because $f_1$ counts all the matter fields, while $f_0$ counts only the gauge field.

Each matter field contributes to the one-loop vacuum polarisation of the gluon, and this fixes the relevant limit. For a field of unit charge in three dimensions, a complex scalar and a two-component Dirac fermion each give the transverse self-energy $e^2k/16$ . For the scalar the polarisation tensor has the form $P_{\mu\nu}=(k^2\delta_{\mu\nu}-k_\mu k_\nu)\,p(k)$ , so its trace is $2k^2p$ . In dimensional regularisation the trace is $P_{\mu\mu}=-e^2\int\frac{d^3l}{(2\pi)^3}\frac{(2l+k)^2}{l^2(l+k)^2}$ , and $(2l+k)^2=2l^2+2(l+k)^2-k^2$ leaves $e^2k^2\int d^3l/[(2\pi)^3l^2(l+k)^2]=e^2k/8$ , from the bubble $\int d^3l/[(2\pi)^3l^2(l+k)^2]=1/(8k)$ . Hence $k^2p=e^2k/16$ in magnitude. The fermion trace, with $\gamma_\mu\gamma_\nu\gamma_\mu=-\gamma_\nu$ in three dimensions and ${\rm tr}\,1=2$ , gives the same. For adjoint fields the charge factor is $N$ , and a real adjoint scalar counts half of a complex one. This reproduces the relative weights $1:\tfrac12$ of fermions and scalars in (3). The chain of these bubbles gives the dressed gluon propagator, in a normalisation where the free propagator is $1/k^2$ ,

D(k)=\frac{1}{k^2+q_ck},\qquad q_c=g_{\rm YM}^2N\Big(\frac{N_\Phi}{32}+\frac{N_\psi}{16}\Big)=g_{\rm YM}^2N\,\frac{N_A-1}{32}. \tag{5}

Each bubble is suppressed by $g_{\rm YM}^2$ and enhanced by the number of fields, so at fixed $q_c$ the bubble chain is leading order in $1/N_A$ . We therefore expand in $1/N_A$ at fixed $x\equiv q_c/q$ . With $g\,q_*=f_1g_{\rm YM}^2N$ and (3), $x=\kappa u$ with

\kappa=\frac{q_c}{\lvert g\rvert q_*}=\frac{(N_A-1)/32}{\lvert f_1\rvert}=\frac{3\pi^2}{64}\,\frac{N_A-1}{N_A-5}, \tag{6}

which for $N_A>889$ lies within $0.5\%$ of its limit $3\pi^2/64=0.4626$ . The crossover $x=1$ lies at $\ell_c=\kappa\lvert g\rvert q_*\chi_*$ .

symbol	meaning	size
$q_*$	pivot wavenumber, $0.05\,{\rm Mpc}^{-1}$
$\lambda=g_{\rm YM}^2N/q$	‘t Hooft coupling	small at all observed $\ell$
$u=\lvert g\rvert q_*/q=\lvert f_1\rvert\lambda$	two-loop expansion parameter, eq. (4)	large at low $\ell$
$q_c$	wavenumber at which the bubble term equals the tree term in $D$ , eq. (5)
$x=q_c/q=\kappa u$	variable of the resummation	of order 1 at $\ell_c$
$\kappa=q_c/(\lvert g\rvert q_*)$	ratio of the two scales, eq. (6)	$\approx0.46$

We assume from here on: (A1) all scalars are conformally coupled, $\xi_M=1/8$ , so that $N_B=1$ ; (A2) the Yukawa couplings vanish; (A3) the quartic couplings $\lambda_{\rm quartic}$ are small enough that their own bubble chains, and the quartic term in the trace $\Theta$ itself, can be neglected. At $\xi_M=1/8$ the quartic couplings drop out of the constant term of the two-loop result at the order considered (the constant $a_0$ of [ACDGS] contains them only through $(8\xi_{M_1}-1)(8\xi_{M_2}-1)$ ). A flavour-singlet quartic coupling would nevertheless generate a chain of scalar bubbles at a scale proportional to its coupling times $N_\Phi$ and could alter the low- $\ell$ spectrum; (A3) excludes this.

3. The resummed trace correlator

Under (A1)–(A3) the kinetic terms of the scalars and fermions are conformal, and the parts of $\Theta$ built from them vanish on the equations of motion, so their insertions in correlators produce only contact terms. The quartic term of $\Theta$ does not vanish on the equations of motion; it is neglected under (A3). The trace is then the gauge-field term $\Theta_A=\tfrac14F^a_{\mu\nu}F^a_{\mu\nu}$ in the normalisation of (5), whose free two-point function is the “1” in $N_B$ . At leading order in $1/N_A$ the trace correlator is the one-loop diagram of $\Theta_A$ with both gluon lines replaced by $D$ . That no other diagram contributes at this order rests on power counting and is not proved here. With the propagator (5), each additional gluon exchange or matter loop attached to the two gluon lines costs a factor $g_{\rm YM}^2N/q_c\sim1/N_A$ at every $q$ , including $q\ll q_c$ , where $D\approx1/(q_ck)$ . The pure-gauge two-loop term, the “ $-2$ ” in (3), is of this kind. The counting fails when some scalar has $\xi_M\ne1/8$ : then $\Theta$ has a matter part, the mixed correlators $\langle\Theta_\Phi\Theta_A\rangle$ are of the same order as the gauge loop, and they are not included. This is why the controlled case is $N_B=1$ .

The expansion to first order in $q_c$

The expansion in $q_c$ can be done in closed form, and it fixes the normalisation before the full function is computed.

Proposition 1 (two-loop logarithm). Let $\langle\Theta_A\Theta_A\rangle$ be the trace correlator of one gauge-group component with the propagator (5), normalised so that the free propagator is $1/k^2$ ; at large $N$ the correlator of the theory is $N^2$ times this, which gives $f_0=1/64$ for $N_B=1$ . To first order in $q_c$ its non-analytic part is

\langle\Theta_A\Theta_A\rangle(q)=\frac{q^3}{64}-\frac{q_c\,q^2\ln q}{3\pi^2}+\dots,

where the dots stand for terms analytic in $q$ and terms of order $q_c^2$ . Equivalently, the ratio $\mathcal F=64\langle\Theta_A\Theta_A\rangle/q^3$ obeys $\mathcal F=1+a_1x\ln x+O(x)$ with $a_1=64/(3\pi^2)$ .

Proof. The correlator of $F_{\mu\nu}$ involves only the transverse part of the gluon propagator, so we may use $\langle A_\mu(r)A_\nu(0)\rangle=\delta_{\mu\nu}G(r)$ with $G$ the Fourier transform of $D$ . Expanding $D=1/k^2-q_c/k^3+O(q_c^2)$ and transforming,

G(r)=\frac{1}{4\pi r}+\frac{q_c}{2\pi^2}\ln r+\text{const}+O(q_c^2),

since $\nabla^2\ln r=1/r^2$ and the transform of $1/k$ is $1/(2\pi^2r^2)$ ; the constant drops out of $F_{\mu\nu}$ . Then $\langle F_{\mu\nu}(r)F_{\rho\sigma}(0)\rangle=-(\partial_\mu\partial_\rho G\,\delta_{\nu\sigma}-\partial_\mu\partial_\sigma G\,\delta_{\nu\rho}-\partial_\nu\partial_\rho G\,\delta_{\mu\sigma}+\partial_\nu\partial_\sigma G\,\delta_{\mu\rho})$ , and Wick’s theorem gives $\langle\Theta_A(r)\Theta_A(0)\rangle=\tfrac18\sum\langle F_{\mu\nu}F_{\rho\sigma}\rangle^2$ . For a radial $G$ we have $\partial_\mu\partial_\nu G=(G'/r)\,\delta_{\mu\nu}+(G''-G'/r)\,n_\mu n_\nu$ with $n=\mathbf r/r$ and $\mathbf r$ the separation, and the sum over indices reduces to

\langle\Theta_A(r)\Theta_A(0)\rangle=\Big(G''+\frac{G'}{r}\Big)^2+2\Big(\frac{G'}{r}\Big)^2. \tag{7}

For the $G$ above, $G''+G'/r=1/(4\pi r^3)$ with no $q_c$ term, and $G'/r=-1/(4\pi r^3)+q_c/(2\pi^2r^2)$ . Therefore

\langle\Theta_A(r)\Theta_A(0)\rangle=\frac{3}{16\pi^2r^6}-\frac{q_c}{2\pi^3r^5}+O(q_c^2).

In three dimensions the Fourier transform of $r^{-2a}$ is $\pi^{3/2}2^{3-2a}\Gamma(\tfrac32-a)\Gamma(a)^{-1}q^{2a-3}$ . At $a=3$ this is $\pi^2q^3/12$ , so the first term gives $q^3/64$ . At $a=\tfrac52+\epsilon$ we have $\Gamma(\tfrac32-a)=\Gamma(-1-\epsilon)=1/\epsilon+O(1)$ and $q^{2a-3}=q^2(1+2\epsilon\ln q+\dots)$ , so the transform is $\pi^{3/2}2^{-2}\Gamma(\tfrac52)^{-1}q^2(1/\epsilon+2\ln q)+O(\epsilon)=\tfrac{2\pi}{3}q^2\ln q$ plus a local $q^2$ term. The second term therefore gives $-q_cq^2\ln q/(3\pi^2)$ plus a local term. At fixed $q_c$ , $x\ln q=x\ln q_c-x\ln x$ , and the first piece is analytic in $q$ . Dividing by $q^3/64$ gives $a_1=64/(3\pi^2)$ .

The first term reproduces the normalisation $f_0=1/64$ per unit of $N_B$ . For the logarithm, translate back to (2): with $x=\kappa\lambda\lvert f_1\rvert$ , the term $a_1x\ln x$ contributes $-f_1\lambda\ln\lambda$ with $-f_1=a_1\,q_c/(g_{\rm YM}^2N)=\tfrac{4}{3\pi^2}(N_\psi+N_\Phi/2)$ . This is the matter-dependent part of the published coefficient (3) at $\xi_M=\tfrac18$ ; the pure-gauge $-2$ of (3), of relative order $1/N_A$ , is not reproduced. Here $a_1$ comes from the gauge loop alone and the bubble coefficients $\tfrac1{16}$ and $\tfrac1{32}$ from the one-loop vacuum polarisation. At $\xi_M=1/8$ the matter-dependent part of the two-loop logarithm is therefore the self-energy insertion.

For a scalar with general $\xi$ the published coefficient in (3) is $2-96\xi^2$ per scalar, against the dressing value $\tfrac12$ . The difference $\tfrac32(1-64\xi^2)=12(1-8\xi)(\xi+\tfrac18)$ vanishes only at $\xi=\tfrac18$ ; it is $\tfrac32$ at $\xi=0$ , where the dressing accounts for a quarter of the published coefficient, and $-0.19$ at $\xi=0.1327$ . The remainder comes from diagrams that mix $\Theta_\Phi$ and $\Theta_A$ . The check concerns only the first order in $x$ , while the low- $\ell$ spectrum depends on $x\gtrsim1$ , where the full function is needed.

The contact term

The full calculation produces a term independent of $q$ , which in $\mathcal F$ would appear as a multiple of $x^3$ and would dominate at small $q$ . It does not enter the power spectrum, which is computed from the spin-0 form factor $\mathcal B(q)$ in the decomposition $\langle T_{ij}T_{kl}\rangle=\mathcal A(q)\Pi_{ijkl}+\mathcal B(q)\pi_{ij}\pi_{kl}$ of the transverse part of the correlator ([CDS], eq. (5.3)). Here $\pi_{ij}=\delta_{ij}-q_iq_j/q^2$ and $\Pi_{ijkl}=\tfrac12(\pi_{ik}\pi_{jl}+\pi_{il}\pi_{jk}-\pi_{ij}\pi_{kl})$ ; away from contact terms $\langle\Theta\Theta\rangle=4\mathcal B$ .

Lemma 1 (no constant in $\mathcal B$ ). A term in $\langle T_{ij}T_{kl}\rangle$ independent of $q$ does not contribute to $\mathcal B$ .

Proof. A term independent of $q$ is local and has no derivatives. By diffeomorphism invariance the only such term in the generating functional is $c\int d^3x\sqrt{\gamma}$ , with $\gamma_{ij}$ the background metric of the dual theory. Its second variation at flat space is $c[\tfrac14\delta_{ij}\delta_{kl}-\tfrac14(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})]$ . Projecting each index with $\pi$ gives $c[\tfrac14\pi_{ij}\pi_{kl}-\tfrac14(\pi_{ik}\pi_{jl}+\pi_{il}\pi_{jk})]=-\tfrac{c}{2}\Pi_{ijkl}+0\cdot\pi_{ij}\pi_{kl}$ , so its transverse part lies entirely in $\mathcal A$ .

The constant is present in the full trace $\delta_{ij}\delta_{kl}\langle T_{ij}T_{kl}\rangle$ , with weight $\tfrac34c$ , where it belongs to the Ward identity involving $\langle T_{ij}\rangle$ . In the loop calculation below we subtract it, which is the choice $\mathcal B(0)=0$ of subtraction scheme.

The full crossover

The dressed propagator has a positive spectral representation over a mass $m$ ,

D(k)=\int_0^\infty dm\,\frac{\rho(m)}{k^2+m^2},\qquad\rho(m)=\frac{2}{\pi}\,\frac{q_c}{m^2+q_c^2},

as follows from $\int_0^\infty dm\,[(m^2+a^2)(m^2+k^2)]^{-1}=\pi/[2ak(a+k)]$ with $a=q_c$ . This reduces the dressed loop to a superposition, over masses $m_1,m_2$ , of loops with two massive vector propagators. For gluon momenta $k$ and $p=q-k$ the field-strength correlators are $\langle F_{\mu\nu}F_{\rho\sigma}\rangle(k)\propto k_\mu k_\rho\delta_{\nu\sigma}-k_\mu k_\sigma\delta_{\nu\rho}-k_\nu k_\rho\delta_{\mu\sigma}+k_\nu k_\sigma\delta_{\mu\rho}$ , and the product of the two, summed over indices in $d$ dimensions, is $4[(d-2)(k\cdot p)^2+k^2p^2]$ . With $d=3$ and $k\cdot p=\tfrac12(q^2-k^2-p^2)$ the numerator is

\mathcal N=4\big[(k\cdot p)^2+k^2p^2\big]=(q^2-k^2-p^2)^2+4k^2p^2=q^4+k^4+p^4-2q^2(k^2+p^2)+6k^2p^2.

Write $D_1=k^2+m_1^2$ and $D_2=p^2+m_2^2$ , so that $k^2=D_1-m_1^2$ and $p^2=D_2-m_2^2$ . The part of $\mathcal N$ that contains neither $D_1$ nor $D_2$ is its value at $k^2=-m_1^2$ , $p^2=-m_2^2$ , namely $(q^2+m_1^2+m_2^2)^2+4m_1^2m_2^2$ . Every other term cancels one propagator and leaves a one-propagator integral, which is a polynomial in $q^2$ of degree at most one; such terms are local and are removed by the subtractions below. The remaining integral is the three-dimensional bubble

\int\frac{d^3k}{(2\pi)^3}\frac{1}{(k^2+m_1^2)(p^2+m_2^2)}=\frac{\arctan(q/(m_1+m_2))}{4\pi q}.

With the prefactor $\tfrac18$ from Wick’s theorem, $\langle\Theta_A\Theta_A\rangle=\frac{1}{32\pi}\iint\rho\rho\,J$ plus polynomial terms. Dividing by $q^3/64$ ,

\mathcal F(q)=\frac{2}{\pi q^3}\int_0^\infty\!\!\int_0^\infty dm_1dm_2\,\rho(m_1)\rho(m_2)\Big[J(q)-J_0-q^2J_2\Big],\quad J=\big[(q^2+m_1^2+m_2^2)^2+4m_1^2m_2^2\big]\frac{\arctan\frac{q}{m_1+m_2}}{q}, \tag{8}

where $J_0$ and $J_2$ are the coefficients of $q^0$ and $q^2$ in the expansion of $J$ about $q=0$ . As $q_c\to0$ , $\rho$ tends to a delta function at $m=0$ and $\mathcal F\to1$ . The first subtraction implements $\mathcal B(0)=0$ ; the second removes a local term $\propto q_cq^2$ , whose coefficient is a logarithmically divergent renormalisation constant. The subtracted integrand is $O(q^4)$ at small $q$ and $O(1/m)$ at large masses, and the integral converges absolutely.

We evaluated (8) by quadrature and, independently, in position space. In the latter, with $q_c=1$ , the exact propagator is $G(r)=[\mathrm{Ci}(r)\sin r-(\mathrm{Si}(r)-\pi/2)\cos r]/(2\pi^2r)$ , the correlator is (7), and

\mathcal F=(256\pi/q^3)\int_0^\infty r^2\langle\Theta_A\Theta_A\rangle(r)\,[\sin(qr)/(qr)-1+(qr)^2/6]\,dr,

where the last two terms in the bracket are the same two subtractions. The two evaluations agree for $0.1\le x\le100$ .

$x$	$10^{-3}$	$10^{-2}$	$0.1$	$1$	$10$	$10^2$	$10^4$
$\mathcal F(x)$	0.98704	0.91958	0.64439	0.19920	0.026346	0.0027272	$2.7378\times10^{-5}$

At small $x$ we write $\mathcal F=1+x(a_1\ln x+b)+O(x^2\ln^2x)$ . A numerical fit over $10^{-6}\le x\le10^{-3}$ gives $b=1.9814$ , which agrees with $\tfrac{11}{12}a_1$ . Over $10^{-4}\le x\le10^4$ the function decreases monotonically from 1 and stays positive, and $x\mathcal F(x)$ approaches $0.2738$ at $x=10^4$ .

This large- $x$ tail is a local term. Once $r\gg1/q_c$ the correlator falls as $r^{-8}$ instead of $r^{-6}$ , so $\mathcal F$ has no $q^3$ term at $q\ll q_c$ , and the first term of its expansion in $q$ is the analytic $q^4$ term with coefficient $(32\pi/15)\int_0^\infty r^6\langle\Theta_A\Theta_A\rangle\,dr=0.2738$ in units $q_c=1$ . A finite curvature-squared counterterm adds an arbitrary multiple of this term, so its coefficient depends on the scheme.

Beyond the tail, the first non-analytic term is $q^5/q_c^2$ , that is $\mathcal F\propto x^{-2}$ , which is subleading to the tail. In this scheme the large- $x$ expansion has neither a constant nor a term linear in $x$ .

The subtraction in (8) is arbitrary up to a term linear in $x$ , since that term is the local counterterm $\propto q_cq^2$ , and this term is absorbed in the fitted $\ln\beta$ in section 4. What remains predictive is the shape of $\mathcal F$ . A counterterm adds $c\,x$ to $\mathcal F$ at all $x$ . It shifts the small- $x$ slope $b$ to $b+c$ , and it gives the coefficient $b_\infty$ of the term linear in $x$ at large $x$ (the tail $0.2738/x$ is a different term; $b_\infty=0$ in the scheme of (8)) the value $c$ . The difference $b-b_\infty$ is therefore scheme independent, and so is

x_*=e^{-(b-b_\infty)/a_1}=e^{-11/12}=0.39985. \tag{9}

Near $x=x_*$ the logarithm in $\mathcal F=1+a_1x\ln(x/x_*)+\dots$ stops growing.

4. The resummed spectrum

The physical $f$ is $f_0\mathcal F$ plus the local term, that is, plus a multiple of $u$ . Matching to the fitted form (4) at small $u$ , where $\mathcal F(\kappa u)=1+u\ln u+u\ln\kappa+(b/a_1)u+\dots$ for $N_B=1$ (using $a_1\kappa=1$ ), fixes the local term to $u\ln(\beta x_*/\kappa)$ . Hence, for $N_B=1$ and $\kappa$ at its limit (6) for large $N_A$ ,

\frac{f}{f_0}=\mathcal F(\kappa u)+u\ln\frac{\beta x_*}{\kappa},\qquad\kappa=\frac1{a_1}=\frac{3\pi^2}{64}. \tag{10}

For $u\ll1$ , (10) reduces to $1+u\ln u+u\ln\beta$ up to $O(u^2\ln^2u)$ . For $u\gg1$ , $f/f_0\to u\ln(\beta x_*/\kappa)$ : the logarithm $\ln(\beta u)$ of (4) is frozen at its value at $x=x_*$ , the scale $q=q_c/x_*\approx2.5\,q_c$ , and the tree term has switched off. The low- $\ell$ spectrum is therefore a function of $(\Delta_0^2,g,\ln\beta)$ alone, the parameters that describe the high multipoles. The colour number $N$ and the split between $N_\Phi$ and $N_\psi$ drop out at leading order. The values fitted to $\ell\ge30$ do not describe $\ell<30$ (section 6).

For $N_B>1$ we use a model in which the extra non-conformal scalars contribute only to the tree term and the gauge bubble carries the whole two-loop logarithm. Keeping the coefficient of $u\ln u$ equal to 1, as in (4), requires replacing $\kappa$ by $\kappa_B\equiv N_B/a_1$ , since the gauge loop enters with weight $1/N_B$ and $a_1\kappa_B=N_B$ . This equals $\kappa$ at $N_B=1$ , and the same matching gives

\frac{f}{f_0}=1+\frac{\mathcal F(\kappa_Bu)-1}{N_B}+u\ln\frac{\beta x_*}{\kappa_B}. \tag{11}

This omits the mixed diagrams of section 3, so (11) is a model of the theory at $N_B>1$ that is not derived from it.

5. Positivity and the weighted number of non-conformal scalars

A power spectrum is positive, so $f$ must be positive at every $q$ , including scales far outside the observed range.

Proposition 2 (positivity bound within the model (11)). Let $N_B\ge1$ and assume, as found numerically in section 3, that $\mathcal F>0$ and that $\mathcal F$ is bounded on $x>0$ . Then the spectrum (11) is positive for all $u>0$ if and only if $\kappa_B\le\beta x_*$ , that is,

N_B\le a_1x_*\,\beta=0.864\,\beta. \tag{12}

Proof. Each term of $f/f_0=(1-1/N_B)+\mathcal F(\kappa_Bu)/N_B+u\ln(\beta x_*/\kappa_B)$ is non-negative when $\ln(\beta x_*/\kappa_B)\ge0$ , and the second is strictly positive, so $f>0$ . If $\ln(\beta x_*/\kappa_B)<0$ , the last term tends to $-\infty$ as $u\to\infty$ while the others stay bounded, so $f<0$ at small enough $q$ .

The resummation is controlled for $\lambda\lesssim1$ , that is $u\lesssim\lvert f_1\rvert$ , with $\lvert f_1\rvert\gtrsim60$ for $N_A\gtrsim889$ . For the fitted parameters this corresponds to $\ell\gtrsim0.1$ . The condition “positive for all $q$ ” extrapolates (11) to $u\to\infty$ , beyond that range, and for $N_B$ close to the bound the sign change would lie at arbitrarily large $u$ . Equation (12) is a condition on the extrapolated form; a non-perturbative treatment at $\lambda\gtrsim1$ could change it. The finite-range statements below do not use the extrapolation.

For $N_B=1$ , (12) requires $\ln\beta\ge\ln(1/0.864)=0.146$ , which every published fit satisfies. For larger $N_B$ , (12) holds within the model (11). Since $N_B-1$ is the weighted count of non-conformal scalars, the bound on that count is $0.864\beta-1$ . For several values of $\ln\beta$ :

$\ln\beta$	0.638	0.877	1.014	1.112	1.22	2.0	3.5
$N_{B,\max}$	1.64	2.08	2.38	2.63	2.93	6.4	28.6
$N_{B,\max}-1$	0.64	1.08	1.38	1.63	1.93	5.4	27.6

The first and fifth entries are the lower and upper ends of the one-standard-deviation ranges of the two fits of [ACDGS] combined ( $0.877-0.239$ and $1.014+0.206$ ), so over those ranges $N_B\le1.6$ to $2.9$ . The second and third are the central values of those fits, the fourth is the all- $\ell$ refit of section 6, and $3.5$ is the upper end of the prior range used in [AGS]. A violation, when it occurs, can lie outside the observed range: at the parameters of the refit, $N_B=3$ makes $f$ negative only below $\ell\approx1.2$ , $N_B=4$ below $\ell\approx3.5$ and $N_B=6$ below $\ell\approx6.2$ .

The explicit solution of [ACDGS] takes $N_\Phi$ scalars with a common $\xi$ chosen so that $r=0.12$ , no fermions and a quartic coupling $\lambda_{\rm quartic}=1$ , and finds from the observed $\Delta_0^2$ and $\ln\beta$ the values $N=2995$ and $N_\Phi=23\,255$ . For large $N_\Phi$ the tensor-to-scalar ratio is $r=32(1-8\xi)^2$ , so $r=0.12$ means $\lvert1-8\xi\rvert=0.0612$ ( $\xi\approx0.1327$ ). Then $N_B=1+23\,255\times0.0612^2=88.1$ and, from (3), $f_1=-11.08$ , as quoted there ( $\approx-11$ ). The exact relation $r=32N_B/N_A$ with $r=0.12$ gives $N_B=87.2$ instead; the conclusions below do not change. Within the model (11), positivity requires $\ln\beta\ge\ln(88.1/0.864)=4.62$ , far outside the fitted range and outside the prior range $-0.2\le\ln\beta\le3.5$ used in [AGS].

The model understates the coupling of this solution: equation (6) gives $\kappa=(N_\Phi/32)/\lvert f_1\rvert=65.6$ , against $\kappa_B=88.1/a_1=40.8$ assumed in (11). Independently of (11), the scale $q_c=g_{\rm YM}^2N\,N_\Phi/32$ of the gluon self-energy does not depend on $\xi$ , so the gauge loop of this solution is dressed for $x\gtrsim1$ , that is for $\ell\lesssim\kappa\lvert g\rvert q_*\chi_*\approx590$ with $\lvert g\rvert=0.01305$ . The two-loop form on which the parameters of the solution were fitted does not apply to the gauge loop below that multipole. If the tensor spectrum is close to a power law at the BICEP/Keck scales, the ratio $r=0.12$ of this solution is also above the bound $r<0.036$ [BK18].

6. Comparison with Planck temperature data

This section is numerical and approximate; only differences between models are meaningful.

The data are the Planck 2018 temperature power spectrum [Planck V]: for $2\le\ell\le29$ the Commander estimates $\hat D_\ell$ (from the component-separated low-multipole temperature map), and for $30\le\ell\le800$ the unbinned estimates with their per-multipole errors. The background is fixed at the base $\Lambda$ CDM values of [Planck VI], Table 2 (TT,TE,EE+lowE+lensing): $\Omega_bh^2=0.02237$ , $\Omega_ch^2=0.1200$ , $H_0=67.36$ , $\tau=0.0544$ , with massive neutrinos of $0.06\,{\rm eV}$ . The $C_\ell$ are computed from the line-of-sight transfer functions of a Boltzmann code, which are linear in the primordial spectrum, and lensing is included as the $\Lambda$ CDM ratio of lensed to unlensed spectra.

For $\ell\le29$ the likelihood is the full-sky form for a single multipole, scaled by a sky fraction $f_{\rm sky}=0.86$ , $-2\ln\mathcal L=\sum_\ell(2\ell+1)f_{\rm sky}\,(\hat C_\ell/C_\ell-\ln(\hat C_\ell/C_\ell)-1)$ , which approximates the Planck low- $\ell$ likelihood; for $30\le\ell\le800$ it is a diagonal $\chi^2$ with the published errors. $\Lambda$ CDM has $(A_s,n_s)$ free, and the holographic models have $(\Delta_0^2,g,\ln\beta)$ free with positivity at all $q$ imposed. In the table, $\delta\chi^2$ is the value of $-2\ln\mathcal L$ at the best-fit parameters of each row minus that of the $\Lambda$ CDM best fit ( $n_s=0.9642$ ), summed over $2\le\ell\le800$ ; the brackets split the sum at $\ell=30$ .

model	$g$	$\ln\beta$	$\delta\chi^2$ total [ $\ell<30$ , $\ell\ge30$ ]	$C_2/C_2^{\Lambda\rm CDM}$
resummed (10), $N_B=1$	$-0.00914$	1.112	$+5.0$ [ $+2.1$ , $+2.9$ ]	0.565
resummed (11), $N_B=2$	$-0.01238$	1.326	$+3.2$ [ $+1.1$ , $+2.1$ ]	0.591
two-loop form (4), all $\ell$	$-0.00648$	0.832	$+7.6$ [ $+3.2$ , $+4.4$ ]	0.534
resummed, $N_B=1$ , fitted to $\ell\ge30$ only	$-0.01744$	1.284	$+19.8$ [ $+18.5$ , $+1.3$ ]	0.359

The $N_B=2$ row requires one non-conformally coupled scalar with $(1-8\xi)^2=1$ , for which the resummation is incomplete, and is shown for comparison only. Using instead the published $\ell\ge30$ values of $g$ and $\ln\beta$ and fitting only the amplitude, the two-loop form worsens the $\ell<30$ fit by $+23.6$ with $C_2/C_2^{\Lambda\rm CDM}=0.32$ , and the resummed form by $+4.6$ with $0.51$ . The two forms differ by more than 10% of the two-loop spectrum below $\ell\approx33$ , so the $\ell\ge30$ fit has to be redone with the resummed spectrum, as in the last row.

For $N_B=1$ the refit has $q_c=\kappa\lvert g\rvert q_*=2.11\times10^{-4}\,{\rm Mpc}^{-1}$ , that is $\ell_c=2.9$ , and the predicted temperature spectrum relative to $\Lambda$ CDM at $\ell=2,\dots,8$ is

0.565,\ 0.623,\ 0.681,\ 0.734,\ 0.781,\ 0.821,\ 0.855.

The observed quadrupole is $0.22$ of the $\Lambda$ CDM value and $0.39$ of the resummed prediction, so the model moves towards the data without reaching it. The cost, $\delta\chi^2\approx5$ , is shared between low and high multipoles ( $+2.1$ and $+2.9$ ). The resummed model has one more free parameter than $\Lambda$ CDM and is nevertheless worse by about 5. It is not nested in $\Lambda$ CDM, so $\delta\chi^2$ measures the relative quality of two fits and is not a significance of preference.

The last row of the table tests whether the parameters fitted to the high multipoles, inserted in the resummed spectrum of section 4, also give the low- $\ell$ spectrum. They do not: parameters fitted to $\ell\ge30$ alone cost only $+1.3$ on $\ell\ge30$ , but give a quadrupole ratio $0.36$ and worsen the fit at $\ell<30$ by $18.5$ , so the suppression is overshot. The values fitted to $\ell\ge30$ are not those that fit all multipoles.

The total $\delta\chi^2$ at $N_B=1$ , minimised over the other two parameters at each fixed $\ln\beta$ , is

$\ln\beta$	0.5	0.7	0.85	0.95	1.112	1.25	1.4	1.6
$\delta\chi^2$	14.8	10.9	7.8	6.1	5.0	5.8	7.9	11.9

It gives $\ln\beta=1.11^{+0.16}_{-0.15}$ at one unit above the minimum, so the bound (12) is $2.6$ ( $2.3$ to $3.1$ ) there, and the weighted scalar count is at most $1.6$ ( $1.3$ to $2.1$ ).

The background was held fixed, and $(\Omega_bh^2,\Omega_ch^2)$ are partly degenerate with the spectral shape at $\ell>30$ , so freeing them could change the gap. A fit to the full Planck temperature and polarisation likelihood with the background free is needed to turn these numbers into a significance.

7. The running of the tilt at the pivot

The form (4) predicts relations among the logarithmic derivatives of $\ln\Delta^2$ with respect to $\ln q$ : the tilt $n_s-1$ , its running $\alpha_s$ and the running of the running $\varsigma$ . They bear on the high multipoles, where $u\ll1$ , and test (4) for any parameters.

Remark 1 (relation among the runnings). To first order in $u$ , the spectrum (4) has $n_s-1=-u(L-1)$ , $\alpha_s=u(L-2)$ and $\varsigma=-u(L-3)$ , with $L=\ln\big(q/(\beta\lvert g\rvert q_*)\big)$ , and hence

\varsigma+2\alpha_s+(n_s-1)=0 \tag{13}

wherever the first-order form holds, for all values of $g$ and $\beta$ .

Proof. To first order, $\ln\Delta^2=\ln\Delta_0^2-u\ln(\beta u)=\ln\Delta_0^2+uL$ . Write $\partial=d/d\ln q$ . Then $\partial u=-u$ and $\partial L=1$ , so $(\partial+1)u=0$ and $(\partial+1)(uL)=u$ , hence $(\partial+1)^2(uL)=0$ . The left side of (13) is $(\partial^3+2\partial^2+\partial)\ln\Delta^2=\partial(\partial+1)^2\ln\Delta^2$ , which vanishes because $(\partial+1)^2$ annihilates $uL$ and $\partial$ annihilates the constant. The individual expressions follow by differentiating $uL$ three times.

At the pivot, finite differences of $\ln\Delta^2$ for the full forms give $n_s-1=-0.0247$ , $\alpha_s=+0.0161$ and $\varsigma=-0.0077$ for the resummed spectrum (10) at the $N_B=1$ refit. For (10) at the published $\ell\ge30$ values they give $-0.0323$ , $+0.0202$ and $-0.0084$ , and for the two-loop form (4) at those values $-0.0317$ , $+0.0191$ and $-0.0062$ . The pivot lies at $x\approx0.004$ , where the full forms differ from the first-order form by terms of higher order in $u$ , so (13) holds there up to such terms. The ratio $\alpha_s/(n_s-1)$ is $-(L-2)/(L-1)$ at first order, $-0.57$ to $-0.61$ for $L=3.3$ to $3.6$ , and $-0.60$ to $-0.65$ from the full forms. The value $-1$ quoted in [AGS] is the limit $L\to\infty$ . A measurement of $(n_s,\alpha_s,\varsigma)$ tests (13) wherever the first-order form holds.

8. Mixed diagrams, power counting and the likelihood

The mixed correlators $\langle\Theta_\Phi\Theta_A\rangle$ at $N_B>1$ , once computed, would replace the model (11) and fix $N_{B,\max}$ , which could move in either direction.

A diagram-by-diagram power counting to all orders would settle section 3. If the counting holds, the corrections to the gauge loop are of relative order $1/N_A\le1/889\approx1.1\times10^{-3}$ at $N_B=1$ .

A closed-form evaluation of (8) would turn the value $b=\tfrac{11}{12}a_1$ , and hence $x_*=e^{-11/12}$ , into a derived constant; none is known. The positivity of $\mathcal F$ and the boundedness of $x\mathcal F$ are established numerically for $10^{-4}\le x\le10^{4}$ and are not proved.

Data from ACT and SPT at high $\ell$ would test the form (4) beyond the multipoles used here.

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