Converting dark matter to dark radiation does not solve cosmological tensions Fiona McCarthy1and J. Colin Hill2 1 1Center for Computational Astrophysics Flatiron Institute New York NY USA 10010

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Converting dark matter to dark radiation does not solve cosmological tensions

Fiona McCarthy1, ∗and J. Colin Hill2, 1

1Center for Computational Astrophysics, Flatiron Institute, New York, NY, USA 10010

2Department of Physics, Columbia University, 538 West 120th Street, New York, NY, USA 10027

(Dated: September 19, 2023)

Tensions between cosmological parameters (in particular the local expansion rate H0and the am-

plitude of matter clustering S8) inferred from low-redshift data and data from the cosmic microwave

background (CMB) and large-scale structure (LSS) experiments have inspired many extensions to

the standard cosmological model, ΛCDM. Models which simultaneously lessen both tensions are of

particular interest. We consider one scenario with the potential for such a resolution, in which some

fraction of the dark matter has converted into dark radiation since the release of the CMB. Such

a scenario encompasses and generalizes the more standard “decaying dark matter” model, allowing

additional ﬂexibility in the rate and time at which the dark matter converts into dark radiation.

In this paper, we constrain this scenario with a focus on exploring whether it can solve (or reduce)

these tensions. We ﬁnd that such a model is eﬀectively ruled out by CMB data, in particular by

the reduced peak-smearing due to CMB lensing on the power spectrum and the excess integrated

Sachs–Wolfe (ISW) signal caused by the additional dark energy density required to preserve ﬂatness

after dark matter conversion into dark radiation. Thus, such a model does not have the power to

reduce these tensions without further modiﬁcations. This conclusion extends and generalizes related

conclusions derived for the standard decaying dark matter model.

I. INTRODUCTION

Within the standard cosmological model, our Universe

comprises cold dark matter (CDM), dark energy (DE) Λ,

a small amount of baryonic matter, and radiation in the

form of photons as well as massive neutrinos. The six pa-

rameters of this ΛCDM model have been constrained to

very high (∼percent) precision by cosmological datasets

such as the anisotropy power spectrum of the cosmic

microwave background (CMB) measured by the Planck

satellite [1]. However, as we have obtained ever more pre-

cise constraints, tensions of varying signiﬁcance between

diﬀerent datasets have emerged. One of the most well-

known of these is the H0tension, the discrepancy of ∼5σ

between certain measurements of the expansion rate of

the Universe today when inferred from diﬀerent datasets.

The H0tension is largely driven by the discrepancy in the

local measurement of the SH0ES collaboration [2–5] and

the model-dependent inference from the CMB (e.g., as

constrained with Planck data) and/or large-scale struc-

ture data; it should be noted that many local measure-

ments of H0are consistent with both Planck and SH0ES,

albeit with larger error bars than the SH0ES measure-

ment [6]. In the coming years, such independent measure-

ments of H0are forecast to get more precise, hopefully

deciding ﬁnally whether the H0tension indeed requires

new physics.

At somewhat less signiﬁcance (∼2−3σ) is the S8

tension, a tension in the amount of clustering of matter

seen between many late-Universe datasets and the CMB

data (see eg [7–12]). While this tension is not yet as

strong as the ∼5σtension seen in some probes of H0, its

∗Electronic address: fmccarthy@ﬂatironinstitute.org

consistency across several datasets is certainly intriguing.

These tensions could either be ﬂuctuations (in the case

of the less signiﬁcant S8tension); caused by experimental

or astrophysical systematic eﬀects; or real hints to new

physics. If we do take the tensions at face value, as in-

dicative of new physics, a simultaneous resolution of the

two would be very compelling. In this work we consider

a scenario with the potential for achieving this goal: a

model which modiﬁes the expansion rate and structure

growth of the Universe by positing that some portion

of the CDM has converted to dark radiation (DR) after

the release of the CMB. This leads to a lower density of

CDM today than predicted by the ΛCDM model ﬁt to

the Planck data, and as a result the DE becomes dom-

inant earlier and leads to more accelerated expansion—

and thus, a higher value of H0. Simultaneously, due

to a) the decay of some of the CDM after recombina-

tion; and b) the free-streaming of the DR decay product

(which suppresses clustering), this model can also lead a

decreased matter power spectrum P(k), and thus a lower

S8—a property that has led to this being suggested as a

potential simultaneous solution of the tensions [13, 14].

In Ref. [13], a phenomological, model-independent pre-

scription for this conversion of CDM to DR was intro-

duced; this includes (and generalizes) a decaying CDM

(DCDM) scenario (e.g., [15–17]) in which some fraction

of the CDM is unstable with a cosmological-scale lifetime

τand decays into DR. The amount of DCDM has been

constrained using CMB, BAO, and LSS data (e.g., [16–

27]), and also investigated as a solution to the Hubble

and/or S8tensions (e.g., [19, 28–31]).

The model of Ref. [13] allows for a time-dependent con-

version of some fraction of the CDM to DR, occurring at

an arbitrary cosmological time (set by a parameter of the

theory); this should be contrasted with the exponential

conversion of CDM to DR at the cosmological time corre-

arXiv:2210.14339v3 [astro-ph.CO] 18 Sep 2023

sponding to its lifetime τin the DCDM scenario. It was

noted explicitly in Ref. [13] that this model can result in

a higher H0as well as a lower value of S8than in ΛCDM

and thus has the potential to allow for the simultaneous

resolution of the H0tension and the S8tension.

In this work, we constrain this DM→DR model, and

compare it to ΛCDM using a Bayesian approach to in-

vestigate if it can indeed solve the Hubble or S8tensions.

We ﬁnd that, for CMB data, it is not preferred over

ΛCDM, and that even when the SH0ES H0constraint is

included in the analysis, the amount of CDM that con-

verts to DR is constrained such that H0does not signiﬁ-

cantly increase relative to ΛCDM, while S8also remains

nearly the same. We conclude that a model in which

some fraction of the DM has converted to DR since re-

combination will not solve the cosmological concordance

problem, unless other modiﬁcations are also considered,

such as changes to the equation(s) of state of the species

involved or additional (self)-interactions. In the course

of our investigation, we explain the origin of these con-

straints in detail and correct various aspects of earlier

implementations of this scenario. Our modiﬁed Boltz-

mann code is publicly available1.

An outline of this paper is as follows. In Section II,

we discuss the H0and S8tensions. In Section III, we

outline the theory of the DM→DR model, including the

modiﬁcations to the homogeneous Universe and the per-

turbation structure. In Section IV, we describe the data

products and likelihoods used in our analysis. In Sec-

tion V, we present our results. In Section VI, we discuss

our results and conclude.

II. COSMOLOGICAL TENSIONS

A. The H0tension

Assuming ΛCDM, the Planck CMB data predict H0=

67.4±0.5 km/s/Mpc [32]. This is derived from the di-

rect measurement of the angular size of the acoustic

scale in the CMB power spectrum. Some local mea-

surements, which measure H0directly by constructing

a distance-redshift relation (the [cosmic] “distance lad-

der”), are in tension with this result, e.g., the most re-

cent measurement from the SH0ES collaboration, H0=

73.04 ±1.04 km/s/Mpc [33].

If this tension is not due to experimental or astro-

physical systematics, one of these inferences is incorrect,

and the tension can be taken as an indicator of new

physics. The direct measurement is (in principle) model-

independent, and thus we should address the modeling

that predicts H0from the directly-constrained acoustic

scale in the CMB. This calculation of H0relies on our

model of the expansion history of the Universe since the

1https://github.com/fmccarthy/class_DMDR

CMB was released in the early Universe (at “recombina-

tion”), when the Universe was very young; and our model

of the sound horizon at recombination. Solutions to the

Hubble tension must modify (at least) one of these mod-

els, while remaining consistent with the Planck data. For

a recent review of the proposed models to alleviate this

tension, see [34].

Direct measurements of H0: the cosmic distance ladder

We can measure H0today by directly measuring the

apparent recession velocity and distance to distant ob-

jects. While velocity can be measured directly by mea-

suring the redshift of spectra of objects, distance requires

the use of a “standard candle” of known intrinsic bright-

ness along with the distance-luminosity relation. Type

Ia Supernovae (SNe) can be used as a standardizable

candle to measure H0; however, the normalization of

their brightness is not known absolutely, and so they

can only constrain the relative evolution of cosmological

distance, H(z)/H0. To constrain their intrinsic bright-

ness, we need to know the absolute distance to some of

the SNe; we measure this by using other standard can-

dles, such as cepheids, which are known to obey a tight

period-luminosity relation [35]. In turn, the cepheid in-

trinsic brightness is measured by taking parallax mea-

surements of the nearest cepheids, in particular those

in nearby galaxies. Thus we have the cosmic distance

ladder: the parallax measurements of nearby cepheids

are used to calibrate the more distant cepheids, which

in turn are used to calibrate the nearby SNe; using this

calibration, these and the more distance SNe are used to

measure H0.

The SH0ES collaboration uses this approach to mea-

sure H0directly as H0= 73.04 ±1.04 km/s/Mpc [33].

Other methods include using tip of the red giant branch

(TRGB) stars instead of cepheids to calibrate the SNe;

these measurements are in less tension with Planck, ﬁnd-

ing H0= 69.8±1.9 km/s/Mpc [6].

Inference of H0from the CMB

We infer H0from the angular scale θsimprinted on the

CMB by baryonic acoustic oscillations (BAOs). θsis a

projection of the physical sound horizon at recombination

rs(z⋆), according to

θs=rs(z⋆)

DA(z⋆),(1)

where DA(z⋆) is the comoving angular diameter distance

to the surface where the CMB was released at redshift z⋆

(the “surface of last scattering”).

The sound horizon rs(z⋆), the distance a sound wave

could travel in the time between the beginning of the

Universe and recombination, is given by the integral over

comoving distance multiplied by the sound speed cs(z):

rs(z⋆) = Z∞

z⋆

H(z)cs(z); (2)

the comoving distance is given by2

DA(z⋆) = Zz⋆

H(z).(3)

In these distance integrals, H(z) accounts for the geom-

etry of the expanding Universe. Its form depends on the

density of the Universe via the Friedmann equation:

H(z) = r8πG

3ρ(z) (4)

where ρ(z) is the energy density of the Universe at red-

shift z. Within ΛCDM, the form of ρ(z) is speciﬁed ex-

plicitly, and thus so is the z-dependence of H(z):

ρ(z)ΛCDM =ρm(z) + ργ(z) + ρν(z) + ρΛ(5)

=ρ0

m(1 + z)3+ρ0

γ(1 + z)4+ρν(z) + ρΛ,(6)

where the subscripts {m, γ, ν, Λ}refer to the components

of the Universe within ΛCDM: matter m(including CDM

and baryons); radiation γ(including photons and mass-

less neutrinos); massive neutrinos ν; and the cosmolog-

ical constant Λ, respectively; the values of ρimust be

measured to fully characterize ρ(z)ΛCDM .

ρ0

mand ρ0

γare the densities of matter and radiation to-

day; ρ0

mis constrained indirectly from the CMB, which

most directly constrains ρm(z≈z⋆), by assuming the

standard evolution of matter ρm(z)∝(1 + z)3.ρ0

is constrained from the monopole temperature of the

CMB [36]. ρν(z), the evolution of the neutrino den-

sity, is also constrained from the CMB; the form of

its evolution ρν(z) depends on the neutrino mass but

is speciﬁed within ΛCDM. The only remaining compo-

nent of the density is ρΛ. However, the CMB directly

constrains θs; and the physics of the sound speed cs(z)

are well understood within ΛCDM. Thus, the CMB data

along with Equation (1) specify ρΛ; as such, H(z), in-

cluding its value today H0≡H(z= 0), is fully spec-

iﬁed by the CMB within ΛCDM, although it is useful

to break the “geometric degeneracy” [37] in the ﬁt to

CMB data using an external probe of the matter den-

sity, such as BAO or CMB lensing data. Planck ﬁnds

H0= 67.4±0.5 km/s/Mpc [32]; similar inferences which

use the BAO scale of galaxy surveys (as opposed to the

CMB) are in agreement with this, with the DES survey

combined with BOSS BAO data and BBN data ﬁnding

H0= 67.4±1.2 km/s/Mpc [38].

H0inferences using the cosmic “inverse” distance lad-

der (see, e.g. [39, 40]), wherein an absolute SNIa lumi-

nosity calibration is determined directly from the BAO

2We work in units with the speed of light c= 1.

scale (by comparing directly luminosity and the BAO

angular distance measurement at the same redshift),

are also not in tension with Planck, with [40] ﬁnding

H0= 69.71 ±1.28 for an analysis in which the SNe were

calibrated from the BAO angular sound horizon (which

itself was calibrated from the CMB angular sound hori-

zon). Such methods disfavor models that modify cosmic

evolution after recombination to attempt to increase H0,

as they depend only on the fact that the BAO scale is

the same at z≈1100 and at low redshifts. However, the

error bars are large enough that some potential wiggle

room remains.

To infer a diﬀerent value of H0, there are three options:

modify the pre-recombination sound speed; modify ρ(z)

before recombination; or modify ρ(z) after recombination

(or some combination of these). In this work, we focus on

the modiﬁcation of ρ(z) after recombination, in particu-

lar by modifying the evolution of the CDM density. We

allow some component of the CDM to convert into dark

radiation (DR), and thus modify the form of ρm(z) while

also adding a new component ρDR(z) to Equation (5).

This can lead to a diﬀerent value of ρΛ, as well as a dif-

ferent value of ρ0

m, a diﬀerent z-evolution H(z), and a

diﬀerent value of H0today.

B. The S8tension

The S8parameter is deﬁned as

S8≡σ8Ωm

0.30.5

.(7)

Ωm≡ρ0

m/ρ0

cr is the density of matter today as a fraction

of the critical density ρ0

cr and σ8measures the rms ampli-

tude of linear matter density ﬂuctuations over a sphere

of radius R= 8 Mpc/h at z= 0:

(σ8)2=1

2π2Zdk

kW2(kR)k3P(k),(8)

where P(k) is the linear matter power spectrum today

and W(kR) is a spherical top-hat ﬁlter of radius R=

8Mpc/h.

There is a slight tension emerging between S8as mea-

sured from late-Universe datasets and indirecty inferred

the CMB; i.e., by constraining the ΛCDM parameters

from the CMB and calculating the resulting S8. In

particular, weak lensing surveys such as KIDS measure

S8= 0.759 ±0.024 [8]; clustering surveys analyses such

as BOSS also consistently ﬁnd low S8[10, 11]. DES mea-

sures S8= 0.776 ±0.017 from galaxy-galaxy lensing [9]

(the combined analysis of the clustering of foreground

galaxies and lensing of background galaxies). These num-

bers should be compared to the indirect Planck con-

straint, S8= 0.834 ±0.016 from the primary CMB [32].

While CMB lensing from Planck alone is not in ten-

sion with respect to the primary CMB constraints, its

low-zcontribution, measured through cross-correlation

with the unWISE galaxy sample (galaxies at redshifts at

around z∼1−2) gives S8= 0.784 ±0.015 [12], an in-

teresting addition to the low-S8measurements due to its

complementary systematics.

III. THEORY

A. Background cosmology

We consider ΛCDM modiﬁed by the addition of an

extra dark matter (DM) component, such that the total

background DM density evolves as [13]

ρDM (a) = ρ0

(a/a0)3

1 + ζ



1−(a/a0)κ

1 + a/a0

atκ



,(9)

where ρ0

DM is the total DM density today, ais the scale

factor, and a0is a reference scale factor. This modiﬁca-

tion to ΛCDM is fully characterized by three parameters:

ζ, κ, and at.ζdescribes the amount of DM that converts

into DR—in particular, the comoving DM density de-

creases by a factor of (1 + ζ) between a→0 and a0;κ

characterizes the rate of the DM→DR conversion; and at

sets the timescale for the conversion to occur. For transi-

tions long before a0(i.e., for which (a0/at)κ≪a0), all of

the DM remaining at a0evolves as normal and ρDM (a)

can be split neatly into a “converting” and a standard

component (which decays as a−3) by inspection. For

transitions that are not complete at a0(or for transi-

tions in the future relative to a0), this is not the case, as

some of the contribution to ρ0

DM will decay; however, it

is possible to reparameterize Equation (9) by redeﬁning

the reference scale factor a0such that (a0/at)κ≪a0

in such a way that there is a well-deﬁned split into the

converting and standard component. In any case, the ﬁ-

nal comoving DM density (a3ρDM (a→ ∞)) is given by

ρ0

(1/a0)3(1 −aκ

tζ). The requirement that the DM density

always be positive thus gives a constraint on the param-

eters: we require

ζ≤1

aκ

.(10)

This also allows for the case that all DM is of the

converting type and will eventually convert, in which

case the inequality in Equation (10) is saturated and

a3ρDM (a→ ∞) = 0.

Hereafter, we will always take a0to be the scale fac-

tor today and set a0= 1. Note that, in this case, ζ

describes the fraction of the original DM that has con-

verted by today, but not the total fraction of DM that

will eventually convert, unless the transition is in the

past: (1/at)κ≪1. In this parametrization it is evident

that scenarios in which the transition is yet to begin are

degenerate with ΛCDM as they demand ζ→0.

We consider the case where the DM converts into a

dark radiation (DR) particle, whose background energy

density evolves as a−4. The conservation of energy de-

mands that

dt a3ρDM =−1

dt a4ρDR,(11)

which allows us to explicitly write the DR energy density

ρDR as [13]

ρDR(a) = ζρ0

(1 + aκ

(aκ+aκ

t)×

(aκ+aκ

t)2F11,1

κ; 1 + 1

κ;−a

atκ−aκ

t,(12)

where 2F1(b, c;d;z) is the hypergeometric function.

The ansatz in Equation 9 encompasses and general-

izes the standard decaying DM model, in which a sub-

component of the DM exponentially decays with lifetime

τ. Such a model is accurately captured by setting κ= 2

in Equation 9 and setting atsuch that H(at)≈Γ, where

Γ is the DM decay rate. As pointed out in Ref. [13], a

model in which a sub-component of the DM undergoes

Sommerfeld-enhanced annihilation can be accurately rep-

resented by setting κ= 1. This approach thus naturally

encompasses a wide range of possible scenarios in which

DM converts to DR, in a relatively model-independent

manner.

Impact on the expansion of the Universe

This modiﬁcation to the evolution of the background

density of the Universe directly changes the evolution

of the Hubble rate H(a). In order to compare with

ΛCDM, we must think about what parameters should

remain ﬁxed. In our ΛCDM plots in Figures 1 and 2,

we ﬁx the background cosmological parameters to the

best-ﬁt values from the Planck ﬁt to the CMB alone

(TT-EE-TE): thus for the ΛCDM case we take {100θs=

1.040909,Ωbh2= 0.022383,ΩCDM h2= 0.12011}, where

θsis the angular size of the acoustic scale at last scatter-

ing, Ωbh2is the physical density of baryons today, and

ΩCDM h2is the physical density of CDM today (these

quantities are directly constrained by the CMB).

For the modiﬁed DM case, we must consider that the

CMB does not directly constrain the density of CDM to-

day, but instead the density of CDM when it was released

at z=z⋆≈1100, the redshift of the surface of last scat-

tering. Thus we modify ΩCDM h2in the DM→DR plots

to demand that the matter density at the redshift of last

scattering matches the constraint from the CMB; this

results in the relation

ΩCDM h2ΛCDM =

ΩCDM h2DM→DR 1 + ζ1−aκ

⋆

1+(a⋆

at)κ(13)

where a⋆=1

1+z⋆was the scale factor at the time of

last scattering. Note that ΩCDM h2DM→DR includes

the density both of the converting part and the non-

converting part of the DM.

The evolution of the resulting DM and DR densities

are shown in Figure 1, for some choices of the DM→DR

parameters. Note that because we hold θsﬁxed in each

case, it is not immediately straightforward to calculate

the evolution of these densities directly, as one must de-

duce H0(more generally H(z), in particular by ﬁnding

the dark energy density ρΛrequired to make the Uni-

verse ﬂat) appropriately. In the plots in Figures 1 and 2,

we have deduced H0using the “shooting” method imple-

mented in CLASS [41].

In Figure 2, we show on the left how these density

evolutions lead to an earlier redshift of Λ-matter equality,

and thus result in a higher value of H0today, as is shown

on the right.

B. Perturbative cosmology

The perturbations to the homogeneous background,

which give rise to the CMB and the clustering of mat-

ter today, are evolved with the linearized perturbed

Einstein–Boltzmann equations. This is done numeri-

cally, usually with an Einstein–Boltzmann solver such as

CLASS3[41] or CAMB4[42]. In our implementation of the

DM→DR dark matter model, we modify CLASS.

Both the DM and the DR perturbations must be

evolved correctly, even though the DR perturbations can-

not be detected directly: they interact gravitationally

with the perturbations to the Universe’s spacetime met-

ric, and thus indirectly with the measureable perturba-

tions of interest, in particular the DM perturbations and

the perturbations to the photon ﬂuid (which we observe

as the CMB). Following Ref. [43], in this Subsection we

present the Boltzmann equations for the DM and DR.

Formally, the ﬂuids obey the Boltzmann equation

dfi

dt =Qi(14)

where fiis the distribution function of either DM or DR,

and Qiis the appropriate collision term. For the system

as a whole, there are no collisions (i.e., PiQi= 0) and

so QDR =−QDM ≡ Q. The 0th-order, momentum-

integrated Boltzmann equation is

dt a3ρ(0)

DM =−1

dt a4ρ(0)

DR=−Q(0),(15)

where superscript (0) refers to 0th-order (background)

quantities; and so we see (from comparison with the

3https://lesgourg.github.io/class public/class.html

4https://camb.info/

derivative of Equation (9)) that the 0th-order collision

term is

Q(0) =Hκρ0

DM ζ

a3





aκ+a

atκ

1 + a

atκ2



.(16)

The perturbations are evolved with the 1st-order Boltz-

mann equation; to evolve these we need to specify the

perturbation to Q, ie Q(1). We follow the “minimal op-

tion” of Ref. [13]:

Q(1) =Q(0)δDM (17)

where δDM ≡δρDM

ρDM (0) is the dimensionless perturbation

to the DM density, with δρDM ≡ρ(1)

DM the dimensionful

perturbation. The exact speciﬁcation of Q(1) is model-

dependent, but any change from the ansatz in Equa-

tion (17) must be proportional to Q(0), as emphasized

in Ref. [13]. Since this is already tightly constrained (see

below) solely by the evolution of background densities,

any correction to this assumption will have a negligible

impact on our results. It is also worth noting that in the

standard decaying DM scenario Equation (17) is exact,

and even in a Sommerfeld-enhanced annihilation scenario

the corrections to it are negligible [13].

In the synchronous gauge, the perturbed FRW

(Friedmann–Robertson–Walker) metric is

ds2=gµν dxµdxν=a2−dτ2+ (δij +hij )dxidxj,

(18)

where τis conformal time; δij is the Kronecker delta in

three (spatial) dimensions; and hij are the synchronous

metric perturbations (with three-dimensional trace h≡

Pihii). In this gauge the Boltzmann equations for the

DM→DR model are5:

δ′

DM =−h′

2; (19)

θ′

DM =−HθDM ,(20)

with prime (′) denoting diﬀerentiation with respect to

conformal time and H ≡ a′

The DR ﬁeld is deﬁned in terms of its perturbed phase-

space distribution

fDR(⃗x, ⃗p, τ) = f(0)

DR(p, τ) (1 + ΨDR (⃗x, ⃗p, τ)) (21)

where f(0)

DR(p, τ) is the background phase-space distribu-

tion and ΨDR (⃗x, ⃗p, τ) is its perturbation. ΨDR is ex-

panded over the Legendre polynomials Pℓ(µ) in moments

5Due to the “minimal” assumption for Q, the Boltzmann equa-

tions for the perturbations to the component of the DM that

converts into DR coincide with those for the perturbations to

the component that undergoes standard evolution, and so we

retain the very general subscript DM here.

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ConvertingdarkmattertodarkradiationdoesnotsolvecosmologicaltensionsFionaMcCarthy1,∗andJ.ColinHill2,11CenterforComputationalAstrophysics,FlatironInstitute,NewYork,NY,USA100102DepartmentofPhysics,ColumbiaUniversity,538West120thStreet,NewYork,NY,USA10027(Dated:September19,2023)Tensionsbetweencosmologic...

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Converting dark matter to dark radiation does not solve cosmological tensions Fiona McCarthy1and J. Colin Hill2 1 1Center for Computational Astrophysics Flatiron Institute New York NY USA 10010

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