# Stepping into the Forest: Confronting Interacting Radiation Models for the
Hubble Tension with Lyman-$\alpha$ Data

Hengameh Bagherian  Department of Physics, Harvard University, Cambridge, MA
02138, USA  Melissa Joseph  Department of Physics and Astronomy, University of
Utah, Salt Lake City, UT 84112, USA  Martin Schmaltz  Physics Department,
Boston University, Boston, MA 02215, USA  Center for Cosmology and Particle
Physics, Department of Physics, New York University, New York, NY 10003, USA
Institute for Theoretical Physics, Georg-August University Göttingen, 37077
Göttingen, Germany  Eashwar N. Sivarajan  Physics Department, Boston
University, Boston, MA 02215, USA

###### Abstract

Models of interacting dark radiation have been shown to alleviate the Hubble
tension. Extensions incorporating a coupling between dark matter and dark
radiation (DM-DR) have been proposed as combined solutions to both the Hubble
and $S_{8}$ tensions. A key feature of these extended models is a break in the
matter power spectrum (MPS), suppressing power for modes that enter the
horizon before the DM-DR interactions turn off. In scenarios with a massless
mediator, modes that enter before matter-radiation equality get suppressed,
whereas for massive mediators, the break is determined by the mediator mass, a
free parameter. In this work, we test these models against probes of LSS: weak
lensing, CMB lensing, full-shape galaxy clustering, and eBOSS measurements of
the 1D Ly$\alpha\,$forest flux power spectrum. The latter are the most
constraining since they probe small scales where many models predict the
largest deviations. In fact, already within $\Lambda{\rm CDM}\,$, the eBOSS
Ly$\alpha\,$data are in significant tension with Planck CMB data, with the
Ly$\alpha\,$data preferring a steeper slope of the MPS at $k\sim
h\mathrm{Mpc}^{-1}$. We find that the simplest dark radiation models, which
improve the Hubble tension, worsen the fit to the Ly$\alpha\,$data. However,
models with DM-DR interactions can simultaneously address both tensions.

##  I Introduction

The “Hubble tension,” a discordance between different determinations of the
expansion rate of the universe today ($H_{0}\,$), is perhaps the most
significant challenge to the current cosmological model ($\Lambda{\rm
CDM}\,$). Direct local measurements of $H_{0}\,$show approximately 5-10% more
rapid expansion than the values measured from the CMB and from BAO, which both
rely on the $\Lambda{\rm CDM}\,$model. A very simple and sharp formulation of
the Hubble tension, which we will adopt for this paper, is the $\sim 5\sigma$
discrepancy between the local SH0ES [1] determination of $H_{0}\,$using
calibrated supernovae and the $\Lambda{\rm CDM}\,$determination of
$H_{0}\,$from a global fit to Planck CMB data [2]. However, we emphasize that
the tension is robust to replacing the local SH0ES measurement with another
local measurement (in particular TRGB [3, 4]) or replacing the Planck CMB data
in the $\Lambda{\rm CDM}\,$fit with CMB measurements by ACT [5] or SPT [6, 7]
or using BAO [8, 9] instead.

Of the many solutions to the Hubble tension that have been proposed, those
that seek to modify the sound horizon at recombination, $r_{s}$, by modifying
$\Lambda{\rm CDM}\,$at early times before recombination appear particularly
interesting [10]. Such solutions avoid late modifications of the expansion
history which are severely constrained by PANTHEON data [11]. They also
explain why CMB and BAO based measurements which both rely on $r_{s}$ agree
with each other but disagree with direct measurements of $H_{0}\,$. However,
any modifications to $\Lambda{\rm CDM}\,$before recombination are severely
constrained by the high precision of CMB data. In fact, the $H_{0}\,$values
from CMB and supernova measurements are more than 10$\sigma$ apart when
measured in units of the 1$\sigma$ Planck2018 errors. This makes solutions to
the Hubble tension which modify the CMB fit with new early universe physics
challenging.

The angular sound horizon at recombination, $\theta_{s}$, and the redshift of
equality, $z_{\rm eq}$, are especially well determined by the CMB data, and
any new physics must not shift them significantly from their $\Lambda{\rm
CDM}\,$values. The angular sound horizon is determined from the comoving sound
horizon $r_{s}$ as $\theta_{s}=r_{s}/D_{A}$, where
$D_{A}=\int_{0}^{z_{*}}\mathrm{d}z/H\propto H_{0}^{-1}$. Thus a decrease in
$r_{s}$ due to additional energy density beyond $\Lambda{\rm CDM}\,$before
recombination requires a compensating decrease in $D_{A}$ which implies a
larger value for $H_{0}\,$. This is how additional early energy density
increases $H_{0}\,$in CMB fits and alleviates the Hubble tension.111An
analogous argument applies to BAO with the redshift of recombination $z_{*}$
replaced by the redshift of the BAO measurement $z_{\rm BAO}$.

Two well-known classes of models for such additional early energy density are
Early Dark Energy (EDE) [12, 13, 14, 15] and dark radiation ($N_{\rm eff}\,$)
[2]. EDE models introduce additional energy density in the form of a finely-
tuned new fluid whose energy density is time-independent until matter-
radiation equality and then undergoes a transition to rapidly redshift away.

In $N_{\rm eff}\,$models, the additional energy density resides in dark
radiation which does not require such a finely-tuned transition because
radiation energy density automatically redshifts away after matter-radiation
equality. A number of different variants of $N_{\rm eff}\,$models have been
proposed in which the dark radiation may be free-streaming or self-interacting
[16, 17, 18]. Interacting dark radiation sectors can also contain multiple
particle species with different small masses. As the temperature drops below a
mass threshold in such a dark sector, its energy density increases, leading to
a step in $N_{\rm eff}\,$ [19].

Both EDE and $N_{\rm eff}\,$models alleviate the Hubble tension while
maintaining good fits to CMB and BAO data. However, it was found that models
with sufficient EDE to significantly improve the Hubble tension are in
conflict with large scale structure (LSS) data [20, 21]. In particular, the
full-shape MPS measurement from galaxy clustering data from the Baryon
Oscillation Spectroscopic Survey (BOSS) [22, 23] and measurements of the
amplitude of density fluctuations on scales $\sim 8h^{-1}$ Mpc (or
$\sigma_{8}$) extracted from weak lensing surveys [24, 25] constrain the
allowed amount of EDE when included in global fits. In addition, [26] found
that the EDE solution to the Hubble tension significantly worsens the already
poor $\Lambda{\rm CDM}\,$fit to Ly$\alpha\,$data.

In this paper, we investigate how several recently proposed $N_{\rm
eff}\,$solutions to the Hubble tension hold up when confronted with full-shape
LSS, $S_{8}$, and Ly$\alpha\,$data. Among this data, Ly$\alpha\,$is unique as
it probes the smallest scales, which are sensitive to the earliest new physics
affecting the growth of structure and the MPS. The Ly$\alpha\,$forest is
formed when photons from distant quasars are absorbed by neutral hydrogen in
the intergalactic medium at redshifts $2<z<5$, offering insights into matter
clustering at higher redshifts and smaller scales than weak lensing and galaxy
clustering data. The results from such surveys [27, 28] are measurements of
the non-linear MPS for a range of wavenumbers $k$ in bins of redshift, which
are hard to interpret without simulations of non-linear matter clustering and
knowledge of bias parameters. We will use a more user-friendly summary of the
data in the form of a compressed likelihood for the amplitude and slope of the
linear MPS at $k=0.009\text{ (km/s)}^{-1}$ and $z=3$. Interestingly, the two
measurements of the Ly$\alpha\,$forest, eBOSS and XQ-100, show $3-5\sigma$
tensions with $\Lambda{\rm CDM}\,$fit to the CMB, with both preferring a
significantly steeper slope of the MPS at $\sim$ Mpc scales than the best fit
$\Lambda{\rm CDM}\,$model [29].

Figure 1: (Top) Marginalized posterior distributions for
Ly$\alpha\,$likelihood parameters, $\Delta_{L}^{2}$ and $n_{L}$, from
$\Lambda{\rm CDM}\,$model fits to the $\mathcal{D}\,$dataset (blue contours,
dataset $\mathcal{D}\,$is defined in Section III) are shown with
$H_{0}\,$indicated by a color gradient. Solid black contours represent the
$68\%$ and $95\%$ confidence regions from the eBOSS Ly$\alpha\,$likelihood.
(Bottom) Similarly, for the $\Lambda{\rm CDM}\,$+$N_{\rm eff}\,$model,
highlighting the difficulty of solving the Hubble tension while remaining
consistent with the eBOSS Ly$\alpha\,$data.

An analysis of EDE using Ly$\alpha\,$data finds that the parameter space in
EDE models which alleviates the Hubble tension increases the tension with the
Ly$\alpha\,$data [26]. The potential for similar trouble in extra radiation
models is straightforward to understand. In the simplest model of
$\Lambda$CDM+$N_{\rm eff}$ the additional free-streaming radiation (at fixed
$\theta_{s}$ and $z_{eq}$) increases the diffusion damping of the high-$l$
tail of the CMB. To partially compensate this effect a global fit will
increase the primordial tilt $n_{s}$. However, this also increases the slope
of the matter perturbations, making the disagreement with LSS and especially
the Ly$\alpha\,$data worse. The resulting tension is evident in Fig. 1, which
compares the preferred regions of Ly$\alpha\,$data from eBOSS [27] in the
slope $n_{L}$ versus amplitude $\Delta_{L}^{2}$ plane with the posteriors of
$\Lambda{\rm CDM}\,$(top panel) and $\Lambda{\rm CDM}\,$+$N_{\rm eff}$ (bottom
panel) derived from fits to CMB+BAO+Pantheon (uncalibrated supernovae) data.

When observing significant discrepancies between two separate measurements,
caution is warranted in drawing conclusions from a combined analysis. One
plausible explanation is that one of the measurements may have a large hidden
systematic error. Alternatively, the discrepancy could result from large
statistical fluctuations. This type of error typically diminishes with
extended data collection as per the law of large numbers. For the specific
case of a $\sim 5\sigma$ tension between Planck CMB and Ly$\alpha\,$in
$\Lambda{\rm CDM}\,$, a more probable explanation than statistical fluctuation
might be the use of an incorrect cosmological model to fit the data. However,
note that for both $\Lambda{\rm CDM}\,$and $\Lambda{\rm CDM}\,$+$N_{\rm eff}$
the tension with the Ly$\alpha\,$data increases with increasing $H_{0}\,$.

For the remainder of this paper, we assume the datasets are accurate,
unaffected by systematic errors, and we investigate the consistency of models
of interacting dark sectors which significantly lessen the Hubble tension with
the LSS and Ly$\alpha\,$data. Our main results include:

  * •

The simplest models of interacting radiation, with (WZDR) or without (SIDR) a
step, which yield larger values of $H_{0}\,$, align with the $S_{8}$ values
from weak lensing and full-shape LSS data. However, they exacerbate the
existing tension with Ly$\alpha\,$data (see Fig. 3). Future Ly$\alpha\,$data,
if confirming the current preference for a suppressed slope of the MPS, could
rule these models out.

  * •

Models where the interacting dark radiation also couples to dark matter
(SIDR+, WZDR+) show consistency with all LSS data, including Ly$\alpha\,$(see
Fig. 5), while still significantly alleviating the Hubble tension.

##  II Interacting dark radiation models

Models with interacting dark radiation are less constrained by CMB fits ([16])
than models with similar amounts of free-streaming radiation because
interacting radiation is less efficient at erasing small-scale structure.
Therefore, models with interacting radiation can reduce the Hubble tension
better than those with free-streaming radiation (see [19, 30, 31, 32, 33, 34]
for detailed studies).

A significant additional constraint on the allowed amount of extra dark
radiation is provided by BBN. However, for interacting dark radiation, it is
reasonable to assume that the dark sector is populated through thermalization
with SM neutrinos [35, 36, 37, 38]. The constraints from BBN can be completely
bypassed if this thermalization occurs below $\sim$ 100 keV [36, 35,
39].222Combined CMB and BBN constraints on extra radiation present at BBN are
particularly stringent because an effective number of neutrino species $N_{\rm
eff}\,$$>3.043$ during BBN leads to an increased helium abundance. This
increased helium abundance worsens the fit to the CMB for models with extra
radiation by enhancing Silk damping, in addition to the damping of the CMB
tail already caused by the extra radiation. Therefore, this work focuses only
on models in which $N_{\rm eff}\,$is similar to that of $\Lambda{\rm CDM}\,$at
BBN and increases between BBN and the CMB. The mechanism behind this increase,
whether from scattering and mixing with neutrinos or from a late decay, is not
central to our analysis; we simply assume it occurs and establishes the early-
time value of $N_{\rm eff}\,$. In II.1, we provide a brief summary of the
models analyzed in this paper, while II.2 offers more detailed information on
each model. Readers already familiar with these models may choose to skip this
section.

###  II.1 Summary of models considered

Figure 2: (Top) This ratio plot compares the MPS of the SIDR+ model to a non-
interacting SIDR model, highlighting how increased interaction strength
$R_{\Gamma}$ enhances MPS damping at scales smaller than $k_{eq}\sim 10{-2}$
h/Mpc. The red line indicates the best-fit parameters of the SIDR+ model to
the $\mathcal{D}HL\,$dataset. (Bottom) This plot contrasts the MPS of the
WZDR+ model with an early transition redshift model, illustrating how $z_{t}$
shifts the suppression scale horizontally in the WZDR+ model. The blue line
represents the WZDR+ model’s best fit to the $\mathcal{D}HL\,$dataset.

  * •

Strongly Interacting Dark Radiation (SIDR): This model posits a single self-
interacting massless particle that becomes thermalized after BBN. It provides
an excellent fit to the CMB data while also accommodating higher values of
$H_{0}\,$[16]. SIDR introduces one additional cosmological parameter compared
to $\Lambda{\rm CDM}\,$: $N_{\rm IR}\,$, which quantifies the amount of extra
dark radiation.

  * •

Wess-Zumino Dark Radiation (WZDR)[19]: WZDR extends SIDR by adding a small
mass to one of the dark radiation particles. As the dark temperature drops
below this mass threshold, the massive particles annihilate away, thereby
increasing the total energy density in the dark radiation relative to a model
without such a step in $N_{\rm eff}\,$. This model is characterized by $N_{\rm
IR}\,$and $z_{t}$, the former corresponds to the abundance of interacting
radiation after the step, while the latter indicates the transition redshift
at which the step occurs, determined by the particle mass.

  * •

SIDR with dark matter - dark radiation interactions (SIDR+)[40, 41, 42]:
Motivated by our result that neither SIDR nor WZDR significantly alleviates
the tension between the CMB-derived MPS slope and the Ly$\alpha\,$data (see
Fig. 3), we explore models with dark matter – dark radiation interactions (DM-
DR). One such model, named SIDR+ for its addition of DM-DR interactions to the
SIDR framework, features self-interacting massless scalars and a Yukawa
coupling to dark matter fermions. SIDR+ introduces another cosmological
parameter, the momentum transfer ratio $R_{\Gamma}$, which is a measure of the
strength of the DM-DR interaction. The top panel of Fig. 2 shows that
increasing $R_{\Gamma}$ smoothly suppresses the MPS at high $k>k_{\rm eq}\,$.

  * •

WZDR with dark matter - dark radiation interactions (WZDR+) [43, 44]: Adding a
coupling between DM-DR in the WZDR model also attenuates the MPS tail; we call
this extension WZDR+. The implementation of WZDR+ is similar to that of SIDR+
with the difference that the dark matter is now only directly coupled to the
massive component of the interacting fluid. Below redshift $z_{t}$, the DM-DR
momentum transfer rate becomes suppressed by the mediator mass, effectively
ending the DM-DR interaction. Thus WZDR+ has three parameters beyond those of
$\Lambda{\rm CDM}\,$: $N_{\rm IR}\,$, $R_{\Gamma}$, and $z_{t}$. The latter
determines the onset of MPS suppression, as shown in the bottom panel of Fig.
2.

###  II.2 Details of models considered

In this Section, we delve into the specifics of how each model addresses the
Hubble and Ly$\alpha\,$tensions and the rationale for introducing each
additional cosmological parameter. We begin with the SIDR model:

  * •

SIDR: SIDR improves on several key issues associated with free-streaming extra
radiation. Free-streaming radiation hinders structure formation due to
enhanced diffusion damping and shifts the CMB peaks to higher multipoles
($\ell$ values) through the neutrino drag effect [45]. Furthermore, CMB modes
entering the horizon during radiation domination experience an increase in
amplitude, known as the radiation driving effect, which is inversely
proportional to $1+\frac{4}{15}\rho_{\rm fs}/\rho_{\rm tot}$ [45, 46, 16].

The presence of free-streaming radiation diminishes this amplitude boost for
$\ell\geq\ell_{\mathrm{eq}}$ by increasing the fraction of free-streaming
species. Consequently, the Planck CMB data places strong constraints on the
amount of free-streaming radiation.

SIDR, a perfect fluid with an equation of state $w=1/3$ and sound speed
$1/\sqrt{3}$, overcomes limitations by using perfect fluid dynamics. This
approach addresses problems related to free-streaming radiation, enabling more
radiation for a better $H_{0}\,$fit to SH0ES while still fitting CMB data
well.

  * •

WZDR: WZDR [19] is a model featuring an interacting dark radiation fluid
composed of two particle species in thermal equilibrium: one, $\phi$, is
massive and annihilates into the other, $\psi$, which is massless, when the
temperature in the sector drops below the mass of $\phi$. A simple realization
of this scenario is provided by the Wess-Zumino model with soft supersymmetry
breaking via the scalar mass $m_{\phi}$. The interactions in WZDR are:

$\mathcal{L}_{\rm{WZDR}}=\lambda\phi\bar{\psi}\psi+\lambda^{2}(\phi^{*}\phi)^{2}\,.$
(1)

The annihilating $\phi$ particles transfer their entropy to the massless
$\psi$ particles, thereby increasing $N_{\rm eff}$ of the combined $\phi-\psi$
fluid. Assuming a sufficiently large $\lambda$, the remaining $\psi$ particles
continue to scatter efficiently via a four-fermi interaction, ensuring the
dark radiation remains a perfect fluid throughout cosmic expansion. The
increase, or step, in $N_{\rm eff}$ is easy to understand because during the
era of $\phi$ particle annihilation, at $z_{t}\,$, the dark radiation sector
is actually a mix of radiation and matter. This mix’s energy density redshifts
more slowly than pure radiation. Additionally, the radiation and matter mix
temporarily reduces the equation of state and sound speed of the combined
fluid below 1/3.

The step in $N_{\rm eff}\,$at $z_{t}$ also introduces a step in the background
expansion rate, $H(z)$, leading to an $\ell$-dependent modulation of CMB
peaks. Modes that enter the horizon before the step experience a slower
expansion rate, causing peaks at higher $\ell$ values to shift to slightly
lower $\ell$, more effectively counteracting the drag effect than SIDR.
Reference [19] demonstrated that a step in $N_{\rm eff}\,$near $z_{t}\sim
10^{4}$ enhances the fit to Planck data and predicts larger values of
$H_{0}\,$from the CMB fit.

  * •

SIDR+: The SIDR+ model differs from SIDR by the additional coupling between
dark matter and dark radiation. This coupling allows momentum transfer between
the two fluids, which produces an effective drag force on the DM fluid that
slows the clustering of matter. Assuming that the momentum transfer rate
scales with temperature as $\Gamma(T)\propto T^{2}$, as is the case in the
SIDR+ model, the ratio $\Gamma/H$ is constant during radiation domination,
which is key for smoothly suppressing the MPS for all momentum modes that
enter the horizon before matter-radiation equality. During matter domination,
the Hubble parameter’s temperature dependence shifts to $H(T)\sim T^{3/2}$,
effectively shutting off DM-DR interactions around equality. Thus, only modes
that crossed the horizon before equality experience slower growth, producing a
break in the MPS at $\sim k_{\rm eq}\,$. Since the impact of the DM-DR
interactions on cosmology takes place during radiation domination, when the
ratio $R_{\Gamma}\equiv\Gamma/H$ is constant, we use $R_{\Gamma}$ during
radiation domination as the additional parameter of SIDR+ . In prior
literature [47, 42, 48, 43], the interaction strength has also been
parameterized with $\Gamma_{0}\,$, the strength of the interaction
extrapolated to today via its $T^{2}$ scaling. The two definitions are related
by $R_{\Gamma}=\Gamma_{0}/(H_{0}\sqrt{\Omega_{r}})\,$.

The break in the MPS, resulting from the suppressed growth of matter
perturbations relative to $\Lambda{\rm CDM}\,$, is analytically approximated
in Eq. 2, with the slope of the decline equal to $-R_{\Gamma}$ [43]:

$\Delta^{2}_{R_{\Gamma}\neq
0}\simeq\begin{cases}\Delta^{2}_{R_{\Gamma}=0}\quad&k\ll k_{\rm eq}\\\
\Delta^{2}_{R_{\Gamma}=0}\left(\frac{k}{k_{\rm
eq}}\right)^{-R_{\Gamma}}\quad&k\gg k_{\rm eq}\end{cases}\,,$ (2)

where $k_{\rm eq}$ is the mode which enters the horizon during matter-
radiation equality.

As mentioned earlier, the increase in $H(z)$ due to the additional radiation
in SIDR (and WZDR) also increase the amount of diffusion damping. In a global
fit, this increased damping can be partially compensated for with a higher
spectral tilt, $n_{s}$. However, this also boosts power in the MPS at smaller
scales $\propto(k/k_{\rm piv})^{\Delta n_{s}}$. In the SIDR+ model this effect
can be balanced by an adjustment in the interaction strength
$R_{\Gamma}\sim\Delta n_{s}$ as per Eq. 2.

This effect is somewhat similar to what happens in $\Lambda{\rm
CDM}\,$extensions with a running spectral tilt, previously analyzed in the
context of eBOSS Ly$\alpha\,$data by Rogers and Poulin [29]. Their analysis
indicates that fitting the Ly$\alpha\,$data requires

$\alpha_{s}\equiv\frac{dn_{s}}{d\ln k}=-0.0108\pm 0.0022\,,$ (3)

which is much larger than expected in slow-roll inflaction for which one
expects $\alpha_{s}=\mathcal{O}(|n_{s}-1|^{2})\approx 0.001$. Moreover, this
degree of running for the tilt also modifies the MPS at scales sensitive to
the CMB data. SIDR+, in contrast, introduces suppression only for $k>k_{\rm
eq}$, leaving CMB observations largely unaffected. Furthermore, our approach
does not face challenges from a model-building perspective since $R_{\Gamma}$
can natually take on values in a broad range. As we will show in subsequent
sections, fits to LSS data prefer $R_{\Gamma}\sim\mathcal{O}(0.05)$.

The inclusion of $R_{\Gamma}$ as an additional parameter enables SIDR+ to
better fit LSS data at small scales, accomodate a higher $H_{0}\,$, and reduce
the $S_{8}$ tension [43] between the Planck CMB and LSS measurements. Fig. 2
(Top) shows how SIDR+ more accurately fits the $\mathcal{D}HL\,$dataset than
SIDR without DM-DR interaction, using $R_{\Gamma}$ to adjust the slope of the
MPS at high $k$.

  * •

WZDR+: WZDR+ adds a Yukawa interaction, $\lambda_{2}\,\phi\bar{\chi}\chi$, of
the scalar $\phi$ to the DM particle $\chi$ to the WZDR Lagrangian 1. The
t-channel $\chi\psi\rightarrow\chi\psi$ process through $\phi$ exchange
dominates over all other DM-DR interactions, with its momentum transfer rate
$\Gamma(T)\propto T^{2}$ when $\phi$ is relativistic, mirroring the SIDR+
scaling during radiation domination. Below temperatures of $m_{\phi}$, the
momentum transfer rate from the four-fermi contact term
$\bar{\psi}\psi\bar{\chi}\chi$ quickly becomes unimportant because it now
scales as $\Gamma(T)\propto T^{6}$.

This means that the momentum exchange shuts off after crossing the $m_{\phi}$
threshold, or equivalently, below the transition redshift $1+z_{t}\equiv
m_{\phi}/T_{d0}$. Consequently, the break in the MPS now occurs at $k_{t}$,
with modes $k>k_{t}$ entering the horizon before the $m_{\phi}$ threshold
being suppressed, while modes $k<k_{t}$ remain unaffected.

It should be noted that in a different version of WZDR+ with alternative
couplings, the momentum transfer could occur between the DM and the massive
radiation particles. In such cases, the exchange rate would decrease
exponentially, $\propto e^{-m_{\phi}/T}$, as the massive particles annihilate
away. This faster decoupling rate only slightly modifies cosmological fits.
For more details on these models and alternative methods of suppressing the
MPS, see [49, 50, 51, 52, 53, 54, 55, 56, 48, 57, 58, 59, 60].

Fig. 2 (Bottom) illustrates how $z_{t}$ affects the MPS in the WZDR+ model,
showing the MPS for different $z_{t}$ values, normalized to a model where
$z_{t}\rightarrow\infty$. Higher $z_{t}$ moves the transition to smaller
scales but keeps larger scales, relevant to CMB data, completely unchanged.
The interaction strength $R_{\Gamma}$ in WZDR+ is similar to SIDR+ (see top
panel). The two additional degrees of freedom enable WZDR+ to align with LSS
data and address the Hubble tension effectively.

##  III Data and MCMC

We conduct an MCMC analysis on the models from II using CLASS [61] and
MontePython [62], focusing on specific datasets. We use flat priors for the
six $\Lambda{\rm CDM}\,$parameters $\\{\omega_{b},\,\omega_{\rm
dm},\,\theta_{s},n_{s},A_{s},\,\tau_{\rm reio}\\}$ . For SIDR+, we add flat
priors for the dark radiation $N_{\rm IR}\,$$>0.01$ the DM-DR interaction
$R_{\Gamma}>0$. WZDR+ includes a logarithmic prior on the step’s redshift
$\log_{10}z_{t}\in[4.0,4.6]$. As explained in the previous Section, we assume
that the extra radiation in WZDR+ and SIDR+ occurs post-BBN, therefore leaving
the BBN prediction for the primordial helium abundance $Y_{p}$ as in
$\Lambda{\rm CDM}\,$($N_{\rm eff}\,$= 3.044). For generating figures and
triangle plots, we used GetDist [63].

Our baseline dataset $\boldsymbol{\mathcal{D}}$ comprises:

  * •

Planck 2018 [2]: Includes TT, TE, and EE data for both low-$\ell$ (‘lowl_TT’,
‘lowl_EE’) and high-$\ell$ (‘highl_TTTEEE’) ranges, along with the full set of
nuisance parameters and the Planck lensing likelihood.

  * •

BAO+Pantheon: Combines late-universe constraints with the BAO-only likelihood
from BOSS DR12 ($z=0.38,0.51,0.61$) [64], small-z BAO data from 6dF
($z=0.106$) [65] and MGS ($z=0.15$) [66] catalogs, and the Pantheon supernova
likelihood [11]. We did not include the more recent DES Supernova data [67]
which prefer a high $\Omega_{M}=0.352\pm 0.017$.

  * •

$S_{8}$: This observable is measured through weak lensing and galaxy
clustering, with the former including cosmic shear and galaxy-galaxy lensing.
In this paper, we select the $S_{8}$ likelihood based on a combined cosmic
shear analysis from the Dark Energy Survey (DES Y3) and the Kilo-Degree Survey
(KiDS-1000), with $S_{8}=0.790^{+0.018}_{-0.014}$ [68], implemented as a
Gaussian likelihood. We favor this measurement over one that includes galaxy
clustering data because the EFTofBOSS likelihood which we include in dataset
$\mathcal{D}$ relies on the BOSS DR12 galaxy sample analysis. Focusing solely
on cosmic shear from weak gravitational lensing ensures the $S_{8}$ dataset’s
independence from galaxy clustering information in $\mathcal{D}$. Moreover,
analyzing multiple surveys together provides a more precise $S_{8}$ value.

  * •

EFTofBOSS: Utilizes the Effective Field Theory of Large Scale Structure
(EFTofLSS) [23] for analyzing the SDSS-III BOSS DR12 galaxy sample [64, 69],
incorporating both monopole and quadrupole moments of the galaxy power
spectrum. Our analysis includes the Full Shape of the BOSS galaxy power
spectrum and its cross-correlation with reconstructed BAO parameters [70],
accessible via Pybird [71].333Class-PT is another code that computes the EFT
of LSS for this likelihood, with minor differences in the priors for nuisance
parameters. We find that the selection of priors on these nuisance parameters
can result in significant overall changes to the $\chi^{2}$ values, while only
causing negligible differences in the posteriors for the cosmological
parameters.

  * •

EFTofeBOSS: Analyzes eBOSS DR16 QSO data [72] with EFTofLSS predictions [73,
74]. Covariance matrix derived from mock catalogs [75] using the Extended
Zel’dovic approximation [76]. Available via Pybird.

We refer to the combined datasets of EFTofBOSS and EFTofeBOSS as full-shape.

The dataset $\boldsymbol{\mathcal{H}}$ includes

  * •

SH0ES: Measurement of the supernovae intrinsic magnitude $M_{b}=-19.253\pm
0.027$ by Riess et al. [1], included as a Gaussian likelihood.

The dataset $\boldsymbol{\mathcal{L}}$ originates from the one-dimensional
Ly$\alpha$ forest flux power spectrum, utilizing data from the SDSS DR14 BOSS
and eBOSS surveys [27]. The data spans thirteen redshift bins from $z\approx
2.2$ to $4.6$ and scales from $k\approx 0.001$ to $0.020\text{ (km/s)}^{-1}$.
For ease of comparison to models that modify the MPS, [27] further provides in
Fig. 20 the 2D posterior for the dimensionless amplitude of the linear power
spectrum, $\Delta^{2}_{L}$, and its logarithmic slope, $n_{L}$, at the pivot
wave number $k_{p}=0.009\text{ (km/s)}^{-1}$ and redshift $z_{p}=3$. The
conversion from wavenumbers expressed in velocity units to comoving units is
given by the factor $H(z_{p})/(1+z_{p})$ . We fit a 2D-Gaussian likelihood to
the aforementioned posterior, which yielded:

  * •

$\Delta^{2}_{L}\equiv k^{3}P_{L}/(2\pi^{2})=0.31\pm 0.02$,
$n_{L}\equiv\mathrm{d}\ln{P_{L}}/\mathrm{d}\ln{k}=-2.339\pm 0.006$, with a
correlation coefficient $r=0.50$.

Appendix A supports the 2D-Gaussian likelihood’s applicability to power-law
extensions of $\Lambda{\rm CDM}\,$models, including SIDR+ and WZDR+.

A second Ly$\alpha$ likelihood was derived from observations of QSOs by XQ-100
at redshifts $z\approx 3.5-4.5$ and from MIKE/HIRES quasar samples [28, 77,
78, 79]. This likelihood fits a 2D-Gaussian to the posteriors of
$\Delta^{2}_{L}$ and $n_{L}$ from Fig. A1 of Appendix A in [78], as detailed
in [26]. The XQ-100 likelihood differs from the eBOSS likelihood by covering a
narrower redshift range and focusing on smaller, more non-linear scales. The
results of the two analyses are in some tension with with each other. We
anticipate the upcoming DESI 1D Ly$\alpha\,$flux power spectrum, expected
within the next year, to provide clarity on the situation. In this paper, we
fit only to the eBOSS Ly$\alpha\,$likelihood, though we also present XQ-100
contours for comparative purposes in Fig. 5.

##  IV Results

###  IV.1 The Ly$\alpha\,$analysis

Figure 3: (Top Row) These panels show parameters similar to Fig. 1, adjusted
for SIDR and WZDR models, where the tension between eBOSS $\text{Ly}\alpha$
and $H_{0}\,$remains. (Bottom Row) Similar plots for SIDR+ and WZDR+ models,
which fit within the eBOSS parameter space and are thus compatible with
$\mathcal{D}L\,$ and $\mathcal{D}HL\,$ datasets. For these models, there is no
clear correlation between $H_{0}\,$measurements and $\Delta^{2}_{L}$ or
$n_{L}$.

In this section, we present our full analysis of the extra fluid models SIDR
and WZDR, especially focusing on the DM-DR interaction variants, SIDR+ and
WZDR+.

The immediate question that arises is whether the inclusion of eBOSS
Ly$\alpha\,$data is compatible with any of these four models. To address this,
we fit our models to the $\mathcal{D}\,$dataset and plot the posteriors for
the Ly$\alpha\,$observables $\Delta^{2}_{L}$ and $n_{L}$ in Fig. 3. We find
that, similar to the situation with $\Lambda{\rm CDM}\,$and $N_{\rm
eff}\,$models in Fig. 1, the fit of SIDR and WZDR models to the
$\mathcal{D}\,$dataset is in approximately 5$\sigma$ tension with eBOSS
Ly$\alpha\,$data. Both SIDR and WZDR struggle to accommodate Ly$\alpha\,$data;
in fact, as they attempt to match steeper slopes for the MPS while still
fitting the Planck CMB data, their prediction of $H_{0}\,$decreases,
exacerbating the tension with SH0ES. Fig. 3 further shows that the correlation
between $H_{0}\,$and Ly$\alpha\,$is stronger in the WZDR model than in SIDR.
This suggests more tension in WZDR if a combined fit to $\mathcal{D}HL\,$is
considered.

A naive interpretation might suggest that the Ly$\alpha\,$measurement
challenges the SH0ES $H_{0}\,$values. However, the significant variance
between the two different Ly$\alpha\,$likelihoods (eBOSS and XQ-100) possibly
points to an unaccounted systematic error in one of them, making reliable
interpretation challenging without identifying the error source. To mitigate
the impact of systematics, one approach, as Hill et al. [26] pursued, is to
expand the experimental error margins. A second approach is to take all data
at face value and assume that the discrepancies between measurements are due
to statistical fluctuations. Given the $\sim 5\sigma$ level of discrepancy,
this probability is extremely low. Consequently, we do not prioritize
combining fits from all measurements, including XQ-100 of Ly$\alpha\,$, to
achieve the highest possible accuracy in our statistical analysis. However,
for the interested reader, we have included triangle plots that combine SH0ES
and eBOSS Ly$\alpha\,$data for the SIDR and WZDR models in Appendix B. The
final possibility, and the primary focus of this paper, is that the
Ly$\alpha\,$measurements, as well as the $\mathcal{D}\,$and $H_{0}\,$data, are
correct, and that the discrepancy indicates the presence of new physics not
explained by SIDR or WZDR models.

As previously motivated in Section II.1, this new physics may be a coupling
between the DM and DR. Fig. 3 (bottom row) illustrates how incorporating the
parameter $R_{\Gamma}$ in these models expands the parameter space to
accommodate eBOSS Ly$\alpha$ measurements. Moreover, the color scatter plot
clearly shows the disappearance of the correlation with $H_{0}\,$predictions,
suggesting that the WZDR+ and SIDR+ models can simultaneously provide
satisfactory fits to both $H_{0}\,$and eBOSS Ly$\alpha\,$.

Figure 4: The plot shows the best-fit matter power spectra ratios of selected
models to $\Lambda{\rm CDM}\,$for the $\mathcal{D}HL\,$dataset at $z=0$. The
blue shaded area indicates the wavenumbers to which Planck CMB data is
sensitive. A white square marks the linear eBOSS Ly$\alpha\,$data at $k\approx
1\,h/\text{Mpc}$, with gray error bars for the $1\sigma$ amplitude uncertainty
$\Delta^{2}_{L}$. Tilted cones show the $1\sigma$ uncertainty in $n_{L}$ of
Ly$\alpha\,$, within the $k_{\mathrm{Ly}\alpha}^{\rm
min}<k<k_{\mathrm{Ly}\alpha}^{\rm max}$ range. Adding an interaction strength
parameter significantly improves the fit for SIDR+ and WZDR+ models over SIDR
and WZDR, highlighting the tension of models without DM-DR interaction with
Ly$\alpha\,$data.

Another visual depiction of the Ly$\alpha\,$tension against $\Lambda{\rm
CDM}\,$, SIDR, and WZDR, along with the resolution offered by the WZDR+ and
SIDR+ models, is illustrated in Fig. 4. With the appropriate interaction
strength, these + models smoothly dampen the MPS tail to align with the
Ly$\alpha\,$slope.

Fig. 5 (Top) presents the 2D posteriors of the Ly$\alpha\,$power spectrum
amplitude $\Delta_{L}^{2}$ and slope $n_{L}$ for an SIDR+ model fit to the
$\mathcal{D}\,$dataset, demonstrating how a larger momentum transfer ratio
$R_{\Gamma}$ aligns well with the Ly$\alpha\,$data and remains within the 2D
posterior for a fit that includes Planck CMB. As $R_{\Gamma}$ increases, the
fit points move towards the solid black Gaussian regions of the
Ly$\alpha\,$contours, reflecting the reduction of power at small scales to
match $n_{L}$.

Fig. 5 (Bottom) shows that the WZDR+ model can easily fit either of the
Ly$\alpha\,$measurements (eBOSS and XQ-100) along with the Planck CMB data
from $\mathcal{D}\,$. By introducing the $z_{t}$ parameter, it allows for a
break in the MPS at larger $k_{t}$ creating a flat direction in the parameter
space. Ly$\alpha\,$is the only probe sensitive at small enough scales to
resolve this degeneracy, enabling the model to match either measurement by
adjusting $z_{t}$ and $R_{\Gamma}$, as depicted in Fig. 5 (Bottom).

Meanwhile, within the SIDR+ model, which breaks the MPS around $k_{\rm eq}$,
fitting the central values of XQ-100’s higher $\Delta^{2}_{L}$ and steeper
$n_{L}$ significantly degrades the CMB and full-shape fits. Therefore, SIDR+
can only accommodate the central values of the eBOSS measurement.

Figure 5: (Top) Marginalized posterior distributions for
Ly$\alpha\,$parameters from SIDR+ fits to $\mathcal{D}\,$(silver contours),
with $R_{\Gamma}$ shown as a color gradient. Solid and dashed black contours
indicate $68\%$ and $95\%$ confidence regions from eBOSS and XQ-100
Ly$\alpha\,$data, respectively (see Appendix A). The analysis suggests eBOSS
data alignment requires non-zero $R_{\Gamma}$, with SIDR+ struggling to fit
XQ-100 data. (Bottom) Similarly for WZDR+. Note that the transition redshift
$z_{t}$ opened up a degeneracy direction with increasing $z_{t}$ and
$R_{\Gamma}$ that allows good matches to either eBOSS or XQ-100 data. This
degeneracy is fixed by including eBOSS data into the analysis likelihood (see
Fig. 6).

To quantify this statement and determine the data’s preference for one model
over another, we calculate the Gaussian Tensions (GT) in Table 1. GT measures
the difference in the mean values of the posteriors in units of total
$\sigma$, which is the quadrature sum of each standard deviation. The ability
of WZDR+ to fit either of the Ly$\alpha\,$datasets is now evident in Table 1,
where WZDR+ shows no tension with Ly$\alpha\,$data in the case of both eBOSS
and XQ-100 likelihoods, while a $\sim 4\sigma$ tension persists for SIDR+ with
the XQ-100 measurement.

| $\Lambda{\rm CDM}\,$ | WZDR+ | SIDR+ | WZDR |  SIDR
---|---|---|---|---|---
$GT$ \- eBOSS | $4.9\sigma$ | $0.58\sigma$ | $1.3\sigma$ | $5.0\sigma$ |  $4.9\sigma$
$GT$ \- XQ-100 | $4.5\sigma$ | $1.0\sigma$ | $3.9\sigma$ | $4.6\sigma$ |  $4.5\sigma$
Table 1: Gaussian Tension between direct measurements of Ly$\alpha\,$and from
$\Lambda{\rm CDM}\,$, WZDR+, SIDR+, WZDR and SIDR fit to $\mathcal{D}\,$.

Models WZDR and SIDR, without the DM-DR interactions, fail to suppress the MPS
at Ly$\alpha\,$scales while fitting the CMB. Fitting these to
$\mathcal{D}L\,$and comparing to $\mathcal{D}\,$, we observe a
$\Delta\chi^{2}_{\rm CMB}\sim 7$, while still being in strong disagreement
with eBOSS Ly$\alpha\,$. The full set of $\chi^{2}$ values are provided for
the interested reader in Table 8 in Appendix B. For SIDR+ and WZDR+,
$\Delta\chi^{2}_{\rm CMB}$ improves to $\sim 5.6$ and $\sim 0.4$,
respectively. SIDR+ worsens the CMB $\chi^{2}$ in a combined fit that includes
Ly$\alpha\,$, by suppressing the MPS near $k\sim k_{\rm eq}$, to which the CMB
data are still sensitive. The WZDR+ model avoids this problem by shifting the
break in the MPS to smaller scales. Both + models significantly improve the
Ly$\alpha\,$fit over their non-interacting counterparts.

We also compare SIDR+ and WZDR+ model parameter posteriors across four
datasets in Fig.6. The $H_{0}\,$\- $N_{\rm IR}\,$panels show a clear
preference for non-zero $N_{\rm IR}\,$in $\mathcal{D}H\,$and
$\mathcal{D}HL\,$to match SH0ES measurements. Since $N_{\rm IR}\,$and
$R_{\Gamma}$ are uncorrelated, adjusting $R_{\Gamma}$ to fit
$\mathcal{D}L\,$and $\mathcal{D}HL\,$is straightforward. The inclusion of
$S_{8}$ and MPS data in $\mathcal{D}\,$indicates a preference for non-zero
$R_{\Gamma}$ (see Tables 3 and 3). Furthermore, a preference for
$\mathrm{log}\,z_{t}\sim-4.2$ in WZDR+ across all datasets, as shown in Table
3 and discussed in previous works [19, 43], may suggest that new physics is
active at this scale. This preference, potentially due to adjustments in CMB
phase shifts and the break in the MPS, is remarkably consistent across
datasets, with $\mathcal{D}\,$values staying within $1\sigma$ of those from
$\mathcal{D}L\,$or $\mathcal{D}HL\,$.

$R_{\Gamma}$ | $\mathcal{D}\,$ |  $\mathcal{D}H\,$ |  $\mathcal{D}L\,$ | $\mathcal{D}HL\,$
---|---|---|---|---
Bestfit | 0.0032 |  0.0082 |  0.052 | 0.055
Mean | $\,0.0226_{-0.0054~{}~{}}^{+0.0226~{}~{}}$ |  $\,0.03_{-0.01}^{+0.03}$ |  $\,0.05_{-0.01}^{+0.01}$ | $\,0.056_{-0.009}^{+0.011}$
Table 2: Interaction strength parameter for SIDR+ in different datasets.

$\mathrm{log}~{}z_{t}$ | $\mathcal{D}\,$ | $\mathcal{D}H\,$ | $\mathcal{D}L\,$ | $\mathcal{D}HL\,$
---|---|---|---|---
Bestfit | -4.20 | -4.29 | -4.20 | -4.26
Mean | -$4.46_{-0.19}^{+0.29}$ | -$4.29_{-0.12}^{+0.09}$ | -$4.33_{-0.07}^{+0.33}$ | -$4.25_{-0.12}^{+0.15}$
|  |  |  |
$R_{\Gamma}$ | $\mathcal{D}\,$ | $\mathcal{D}H\,$ | $\mathcal{D}L\,$ | $\mathcal{D}HL\,$
Bestfit | 0.067 | 0.11 | 0.052 | 0.070
Mean | $0.22_{-0.22}^{+0.04}$ | $0.15_{-0.11}^{+0.07}$ | $0.054_{-0.011}^{+0.010}$ | $0.07_{-0.01}^{+0.01}$
Table 3: Step and interaction strength parameters for WZDR+ in different
datasets.

Figure 6: (Left) Triangle plot of additional cosmological parameters for the
SIDR+ model, along with SIDR+’s predictions for the Ly$\alpha\,$slope and
amplitude, and $H_{0}\,$. The grey bands display the latest SH0ES measurement,
and the solid black ellipses correspond to the $68\%$ and $95\%$ confidence
levels from the eBOSS Ly$\alpha\,$measurement. (Right) The same plot for the
WZDR+ model.

###  IV.2 $S_{8}$ and full-shape analysis

An analysis of the full-shape data and its constraining power on the WZDR+
model has been conducted in [44, 80]. The main finding by Schöneberg et al.
was that the full-shape data imposes stronger constraints on the allowed
interaction strength $R_{\Gamma}$, when combined with a measurement of
$S_{8}=0.769^{+0.016}_{-0.012}$ [25, 24] that is in strong tension with
Planck. This result suggests that the full-shape data disfavors stronger
interaction strengths or, correspondingly, steeper slopes that arise from
their chosen $S_{8}$ likelihood, resulting in an upward shift of $+0.7\sigma$
on the $S_{8}$ tension. Our analysis is consistent with their findings;
however, with the updated cosmic shear analysis of
$S_{8}=0.790^{+0.018}_{-0.014}$ [68], there is no tension with the full-shape
data in the DM-DR interacting models, as indicated by the $\chi^{2}$ values in
Table 8.

##  V Discussion

Models which contain dark radiation beyond the Standard Model neutrinos
predict a shorter “standard ruler”, i.e. sound horizon, as a solution to the
Hubble tension. We compare such models to LSS data, and find that all the
models considered, $\Lambda{\rm CDM}\,$, SIDR, WZDR, SIDR+, and WZDR+ are in
good agreement with CMB lensing, weak lensing, and the full-shape galaxy power
spectrum. However, the situation with Ly$\alpha\,$data is more complex.

The Ly$\alpha\,$forest is unique in that it probes shorter co-moving length
scales than the CMB and other LSS data. It is sensitive to physical processes
that acted at an earlier time in the history of the universe, when the wave
modes measured in the CMB were still ”frozen” outside the horizon. Therefore,
Ly$\alpha\,$data are particularly powerful in constraining models that impact
the growth of structure at early times during radiation domination. This is
the period where so-called ”early universe” solutions to the Hubble tension
introduce new physics. Interestingly, and perhaps relatedly, the eBOSS 1D
Ly$\alpha\,$flux power spectrum measurements are also in $\sim 5\sigma$
tension with $\Lambda{\rm CDM}\,$predictions when fit to the CMB [29]. [26]
found that this tension becomes worse in models with EDE designed to solve the
Hubble tension.

We find a similar situation for the SIDR and WZDR dark radiation models. Their
predictions for the MPS at $\sim h\mathrm{Mpc}^{-1}$ scales are very similar
to those of $\Lambda{\rm CDM}\,$, but we also find a positive correlation
between increasing values of $H_{0}\,$, aimed at resolving the Hubble tension,
and an increase of the amplitude and slope of the MPS. Thus in the region of
parameter space where the SIDR and WZDR models significantly improve the
Hubble tension they worsen the fit to eBOSS Ly$\alpha\,$data.

The tension between eBOSS Ly$\alpha\,$data and the predictions from
$\Lambda{\rm CDM}\,$, SIDR, and WZDR fit to CMB and BAO data allows three
possible interpretations: i. The Ly$\alpha\,$data or its interpretation as a
measurement of the linear MPS may have a significant, hitherto unidentified
systematic error, which is at the root of the tension. If this is the case, we
do not have much to say, since we do not know either the magnitude or
direction of this ”unknown unknown.” ii. It is possible, although highly
unlikely, that the $\sim 5\sigma$ disagreement is the result of a statistical
fluctuation. In that case, the correlation of higher values of $H_{0}\,$with
an increased MPS worsens the Hubble tension within $\Lambda{\rm CDM}\,$and
also strongly disfavors SIDR and WZDR as solutions to the tension. iii. If the
tension is neither due to a statistical fluctuation nor a systematic error, it
calls for new physics that suppresses the MPS at small scales. SIDR+ and WZDR+
are two possible models in which such a suppression naturally occurs due to
the scattering of dark radiation off dark matter. In particular, as was first
shown in [40], if the momentum transfer rate between dark matter and dark
fluid scales proportional to $T^{2}$, like the Hubble expansion rate, then a
smooth suppression of the MPS at small scales results. This suppression takes
the form of a break in the power of the k-dependence:

$\Delta^{2}_{R_{\Gamma}\neq
0}\simeq\begin{cases}\Delta^{2}_{R_{\Gamma}=0}\quad&k\ll k_{\rm break}\\\
\Delta^{2}_{R_{\Gamma}=0}\left(\frac{k}{k_{\rm
break}}\right)^{-R_{\Gamma}}\quad&k\gg k_{\rm break}\end{cases}\,,$ (4)

where the exponent $R_{\Gamma}$ is related to the strength of DM-DR
scattering, and $k_{\rm break}$ is either $k_{\rm eq}$ in the case of SIDR+ or
the larger transition wave number $k_{t}$ in WZDR+. A good fit to all data,
including the Ly$\alpha\,$forest, requires $R_{\Gamma}\sim 0.05$.

Given the discrepancy between the measurements from eBOSS and XQ-100, it may
be premature to claim evidence for new physics. See also [81] for a recent
reanalysis of eBOSS data based on new simulations, which points to a tension
between different redshift bins of the eBOSS data. We therefore look forward
to the release of results from the Ly$\alpha\,$1D flux power spectrum measured
from the DESI year 1 dataset, which is currently being analyzed. The increased
resolution of the DESI spectrograph, along with new synthetic data from
improved simulations and careful analysis as demonstrated with the DESI Early
Data Release [82, 83], promises to make this a particularly exciting result.
It will not only directly improve our knowledge of the MPS at
$h\mathrm{Mpc}^{-1}$ scales but also shed indirect light on the Hubble tension
by distinguishing between “early universe models.” As we showed in this paper,
models with a DM-DR interaction such as SIDR+ and WZDR+ provide an excellent
fit to CMB, BAO, uncalibrated supernova, and large-scale structure data
including Ly$\alpha\,$, while slicing the Hubble tension in half.

##  VI Notes added

As this work was being prepared the DESI BAO measurement [9] was announced,
finding $\Omega_{M}=0.295\pm 0.015$ in $\Lambda{\rm CDM}\,$. As this is in
excellent agreement with our fits (See Tables 5, 6,and 7 in Appendix B) and
the models we studied do not modify the late time expansion history we do not
expect this measurement to significantly impact our results. In addition, an
EFTofLSS analysis of the BOSS and eBOSS Ly$\alpha\,$data [84] appeared. This
work conservatively focuses on a subset of the data and includes a number of
bias parameters to reduce astrophysical model dependence. [84] perform a fit
to $\Lambda{\rm CDM}\,$with all parameters except $\sigma_{8}$ fixed from a
Planck2018 and determines $\sigma_{8}=0.841\pm 0.017$, consistent with Planck
and higher than other direct LSS probes. We also find agreement with
$\Lambda{\rm CDM}\,$when fitting only the overall amplitude (and not the
slope) of the MPS with the $\Delta_{L}^{2}+n_{L}$ likelihood. It would be
interesting to see if the EFT of Ly$\alpha\,$can be used to determine the more
model-independent slope and amplitude of the linear MPS at
$h\mathrm{Mpc}^{-1}$.

Acknowledgments We thank Andreu Font Ribera and Satya Gontcho A Gontcho for
useful discussions on the validity of approaches to fit the eBOSS
Ly$\alpha\,$data. H.B. is supported by the DOE Grant DE-SC0013607 and NASA
Grant 80NSSC20K0506. The work of M.S. and E.N.S. is supported by DOE under
Award DE-SC0015845. M.S. thanks the CCPP at NYU and the Physics Department at
Göttingen University for their hospitality and support. Our MCMC runs were
performed on the Shared Computing Cluster, which is administered by Boston
University’s Research Computing Services.

## References

  * Riess  _et al._ [2021] A. G. Riess _et al._ , A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km/s/Mpc Uncertainty from the Hubble Space Telescope and the SH0ES Team, (2021), arXiv:2112.04510 [astro-ph.CO] .
  * Aghanim  _et al._ [2020] N. Aghanim _et al._ (Planck), Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO] .
  * Freedman  _et al._ [2019] W. L. Freedman _et al._ , The Carnegie-Chicago Hubble Program. VIII. An Independent Determination of the Hubble Constant Based on the Tip of the Red Giant Branch 10.3847/1538-4357/ab2f73 (2019), arXiv:1907.05922 [astro-ph.CO] .
  * Scolnic  _et al._ [2023] D. Scolnic, A. G. Riess, J. Wu, S. Li, G. S. Anand, R. Beaton, S. Casertano, R. I. Anderson, S. Dhawan, and X. Ke, CATS: The Hubble Constant from Standardized TRGB and Type Ia Supernova Measurements, Astrophys. J. Lett. 954, L31 (2023), arXiv:2304.06693 [astro-ph.CO] .
  * Aiola  _et al._ [2020] S. Aiola _et al._ (ACT), The Atacama Cosmology Telescope: DR4 Maps and Cosmological Parameters, JCAP 12, 047, arXiv:2007.07288 [astro-ph.CO] .
  * Dutcher  _et al._ [2021] D. Dutcher _et al._ (SPT-3G), Measurements of the E-mode polarization and temperature-E-mode correlation of the CMB from SPT-3G 2018 data, Phys. Rev. D 104, 022003 (2021), arXiv:2101.01684 [astro-ph.CO] .
  * Balkenhol  _et al._ [2022] L. Balkenhol _et al._ (-3G), A Measurement of the CMB Temperature Power Spectrum and Constraints on Cosmology from the SPT-3G 2018 TT/TE/EE Data Set, (2022), arXiv:2212.05642 [astro-ph.CO] .
  * Schöneberg  _et al._ [2019] N. Schöneberg, J. Lesgourgues, and D. C. Hooper, The BAO+BBN take on the Hubble tension, JCAP 10, 029, arXiv:1907.11594 [astro-ph.CO] .
  * Adame  _et al._ [2024] A. G. Adame _et al._ (DESI), DESI 2024 VI: Cosmological Constraints from the Measurements of Baryon Acoustic Oscillations, (2024), arXiv:2404.03002 [astro-ph.CO] .
  * Aylor  _et al._ [2019] K. Aylor, M. Joy, L. Knox, M. Millea, S. Raghunathan, and W. L. K. Wu, Sounds Discordant: Classical Distance Ladder & $\Lambda$CDM -based Determinations of the Cosmological Sound Horizon, Astrophys. J. 874, 4 (2019), arXiv:1811.00537 [astro-ph.CO] .
  * Scolnic  _et al._ [2018] D. M. Scolnic _et al._ (Pan-STARRS1), The Complete Light-curve Sample of Spectroscopically Confirmed SNe Ia from Pan-STARRS1 and Cosmological Constraints from the Combined Pantheon Sample, Astrophys. J. 859, 101 (2018), arXiv:1710.00845 [astro-ph.CO] .
  * Poulin  _et al._ [2019] V. Poulin, T. L. Smith, T. Karwal, and M. Kamionkowski, Early Dark Energy Can Resolve The Hubble Tension, Phys. Rev. Lett. 122, 221301 (2019), arXiv:1811.04083 [astro-ph.CO] .
  * Agrawal  _et al._ [2023] P. Agrawal, F.-Y. Cyr-Racine, D. Pinner, and L. Randall, Rock ‘n’ roll solutions to the Hubble tension, Phys. Dark Univ. 42, 101347 (2023), arXiv:1904.01016 [astro-ph.CO] .
  * Lin  _et al._ [2019] M.-X. Lin, G. Benevento, W. Hu, and M. Raveri, Acoustic Dark Energy: Potential Conversion of the Hubble Tension, Phys. Rev. D 100, 063542 (2019), arXiv:1905.12618 [astro-ph.CO] .
  * Poulin  _et al._ [2023] V. Poulin, T. L. Smith, and T. Karwal, The Ups and Downs of Early Dark Energy solutions to the Hubble tension: A review of models, hints and constraints circa 2023, Phys. Dark Univ. 42, 101348 (2023), arXiv:2302.09032 [astro-ph.CO] .
  * Blinov and Marques-Tavares [2020] N. Blinov and G. Marques-Tavares, Interacting radiation after Planck and its implications for the Hubble Tension, JCAP 09, 029, arXiv:2003.08387 [astro-ph.CO] .
  * Brust  _et al._ [2017] C. Brust, Y. Cui, and K. Sigurdson, Cosmological Constraints on Interacting Light Particles, JCAP 08, 020, arXiv:1703.10732 [astro-ph.CO] .
  * Escudero and Witte [2021] M. Escudero and S. J. Witte, The hubble tension as a hint of leptogenesis and neutrino mass generation, Eur. Phys. J. C 81, 515 (2021), arXiv:2103.03249 [hep-ph] .
  * Aloni  _et al._ [2022] D. Aloni, A. Berlin, M. Joseph, M. Schmaltz, and N. Weiner, A Step in understanding the Hubble tension, Phys. Rev. D 105, 123516 (2022), arXiv:2111.00014 [astro-ph.CO] .
  * Hill  _et al._ [2020] J. C. Hill, E. McDonough, M. W. Toomey, and S. Alexander, Early dark energy does not restore cosmological concordance, Phys. Rev. D 102, 043507 (2020), arXiv:2003.07355 [astro-ph.CO] .
  * Ivanov  _et al._ [2020a] M. M. Ivanov, E. McDonough, J. C. Hill, M. Simonović, M. W. Toomey, S. Alexander, and M. Zaldarriaga, Constraining Early Dark Energy with Large-Scale Structure, Phys. Rev. D 102, 103502 (2020a), arXiv:2006.11235 [astro-ph.CO] .
  * Ivanov  _et al._ [2020b] M. M. Ivanov, M. Simonović, and M. Zaldarriaga, Cosmological Parameters from the BOSS Galaxy Power Spectrum, JCAP 05, 042, arXiv:1909.05277 [astro-ph.CO] .
  * Zhang  _et al._ [2022] P. Zhang, G. D’Amico, L. Senatore, C. Zhao, and Y. Cai, BOSS Correlation Function analysis from the Effective Field Theory of Large-Scale Structure, JCAP 02 (02), 036, arXiv:2110.07539 [astro-ph.CO] .
  * Abbott  _et al._ [2022] T. M. C. Abbott _et al._ (DES), Dark Energy Survey Year 3 results: Cosmological constraints from galaxy clustering and weak lensing, Phys. Rev. D 105, 023520 (2022), arXiv:2105.13549 [astro-ph.CO] .
  * Heymans  _et al._ [2021] C. Heymans _et al._ , KiDS-1000 Cosmology: Multi-probe weak gravitational lensing and spectroscopic galaxy clustering constraints, Astron. Astrophys. 646, A140 (2021), arXiv:2007.15632 [astro-ph.CO] .
  * Goldstein  _et al._ [2023] S. Goldstein, J. C. Hill, V. Iršič, and B. D. Sherwin, Canonical Hubble-Tension-Resolving Early Dark Energy Cosmologies are Inconsistent with the Lyman-$\alpha$ Forest, (2023), arXiv:2303.00746 [astro-ph.CO] .
  * Chabanier  _et al._ [2019] S. Chabanier _et al._ (eBOSS), The one-dimensional power spectrum from the SDSS DR14 Ly$\alpha$ forests, JCAP 07, 017, arXiv:1812.03554 [astro-ph.CO] .
  * Iršič  _et al._ [2017] V. Iršič _et al._ , The Lyman $\alpha$ forest power spectrum from the XQ-100 Legacy Survey, Mon. Not. Roy. Astron. Soc. 466, 4332 (2017), arXiv:1702.01761 [astro-ph.CO] .
  * Rogers and Poulin [2023] K. K. Rogers and V. Poulin, $5\sigma$ tension between Planck cosmic microwave background and eBOSS Lyman-alpha forest and constraints on physics beyond $\Lambda$CDM, (2023), arXiv:2311.16377 [astro-ph.CO] .
  * Brinckmann  _et al._ [2023] T. Brinckmann, J. H. Chang, P. Du, and M. LoVerde, Confronting interacting dark radiation scenarios with cosmological data, Phys. Rev. D 107, 123517 (2023), arXiv:2212.13264 [astro-ph.CO] .
  * Allali  _et al._ [2024a] I. J. Allali, D. Aloni, and N. Schöneberg, Cosmological probes of Dark Radiation from Neutrino Mixing, (2024a), arXiv:2404.16822 [astro-ph.CO] .
  * Allali  _et al._ [2024b] I. J. Allali, A. Notari, and F. Rompineve, Dark Radiation with Baryon Acoustic Oscillations from DESI 2024 and the $H_{0}$ tension, (2024b), arXiv:2404.15220 [astro-ph.CO] .
  * Ghosh  _et al._ [2022] S. Ghosh, S. Kumar, and Y. Tsai, Free-streaming and coupled dark radiation isocurvature perturbations: constraints and application to the Hubble tension, JCAP 05 (05), 014, arXiv:2107.09076 [astro-ph.CO] .
  * Ghosh  _et al._ [2024] S. Ghosh, D. W. R. Ho, and Y. Tsai, Dark Matter-Radiation Scattering Enhances CMB Phase Shift through Dark Matter-loading, (2024), arXiv:2405.08064 [astro-ph.CO] .
  * Aloni  _et al._ [2023] D. Aloni, M. Joseph, M. Schmaltz, and N. Weiner, Dark Radiation from Neutrino Mixing after Big Bang Nucleosynthesis, Phys. Rev. Lett. 131, 221001 (2023), arXiv:2301.10792 [astro-ph.CO] .
  * Berlin and Blinov [2018] A. Berlin and N. Blinov, Thermal Dark Matter Below an MeV, Phys. Rev. Lett. 120, 021801 (2018), arXiv:1706.07046 [hep-ph] .
  * Berlin and Blinov [2019] A. Berlin and N. Blinov, Thermal neutrino portal to sub-MeV dark matter, Phys. Rev. D 99, 095030 (2019), arXiv:1807.04282 [hep-ph] .
  * Berlin  _et al._ [2019] A. Berlin, N. Blinov, and S. W. Li, Dark Sector Equilibration During Nucleosynthesis, Phys. Rev. D 100, 015038 (2019), arXiv:1904.04256 [hep-ph] .
  * Giovanetti  _et al._ [2024] C. Giovanetti, M. Schmaltz, and N. Weiner, Neutrino-Dark Sector Equilibration and Primordial Element Abundances, (2024), arXiv:2402.10264 [hep-ph] .
  * Buen-Abad  _et al._ [2015] M. A. Buen-Abad, G. Marques-Tavares, and M. Schmaltz, Non-Abelian dark matter and dark radiation, Phys. Rev. D 92, 023531 (2015), arXiv:1505.03542 [hep-ph] .
  * Lesgourgues  _et al._ [2016] J. Lesgourgues, G. Marques-Tavares, and M. Schmaltz, Evidence for dark matter interactions in cosmological precision data?, JCAP 02, 037, arXiv:1507.04351 [astro-ph.CO] .
  * Pan  _et al._ [2018] Z. Pan, M. Kaplinghat, and L. Knox, Searching for Signatures of Dark matter-Dark Radiation Interaction in Observations of Large-scale Structure, Phys. Rev. D 97, 103531 (2018), arXiv:1801.07348 [astro-ph.CO] .
  * Joseph  _et al._ [2023] M. Joseph, D. Aloni, M. Schmaltz, E. N. Sivarajan, and N. Weiner, A Step in understanding the S8 tension, Phys. Rev. D 108, 023520 (2023), arXiv:2207.03500 [astro-ph.CO] .
  * Schöneberg  _et al._ [2023] N. Schöneberg, G. Franco Abellán, T. Simon, A. Bartlett, Y. Patel, and T. L. Smith, The weak, the strong and the ugly – A comparative analysis of interacting stepped dark radiation, (2023), arXiv:2306.12469 [astro-ph.CO] .
  * Bashinsky and Seljak [2004] S. Bashinsky and U. Seljak, Neutrino perturbations in CMB anisotropy and matter clustering, Phys. Rev. D 69, 083002 (2004), arXiv:astro-ph/0310198 .
  * Ma and Bertschinger [1995] C.-P. Ma and E. Bertschinger, Cosmological perturbation theory in the synchronous and conformal Newtonian gauges, Astrophys. J. 455, 7 (1995), arXiv:astro-ph/9506072 .
  * Buen-Abad  _et al._ [2018] M. A. Buen-Abad, M. Schmaltz, J. Lesgourgues, and T. Brinckmann, Interacting Dark Sector and Precision Cosmology, JCAP 01, 008, arXiv:1708.09406 [astro-ph.CO] .
  * Archidiacono  _et al._ [2019] M. Archidiacono, D. C. Hooper, R. Murgia, S. Bohr, J. Lesgourgues, and M. Viel, Constraining Dark Matter-Dark Radiation interactions with CMB, BAO, and Lyman-$\alpha$, JCAP 10, 055, arXiv:1907.01496 [astro-ph.CO] .
  * Buen-Abad  _et al._ [2023a] M. A. Buen-Abad, Z. Chacko, C. Kilic, G. Marques-Tavares, and T. Youn, Stepped partially acoustic dark matter, large scale structure, and the Hubble tension, JHEP 06, 012, arXiv:2208.05984 [hep-ph] .
  * Buen-Abad  _et al._ [2023b] M. A. Buen-Abad, Z. Chacko, C. Kilic, G. Marques-Tavares, and T. Youn, Stepped partially acoustic dark matter: likelihood analysis and cosmological tensions, JCAP 11, 005, arXiv:2306.01844 [astro-ph.CO] .
  * Chacko  _et al._ [2016] Z. Chacko, Y. Cui, S. Hong, T. Okui, and Y. Tsai, Partially Acoustic Dark Matter, Interacting Dark Radiation, and Large Scale Structure, JHEP 12, 108, arXiv:1609.03569 [astro-ph.CO] .
  * Archidiacono  _et al._ [2017] M. Archidiacono, S. Bohr, S. Hannestad, J. H. Jørgensen, and J. Lesgourgues, Linear scale bounds on dark matter–dark radiation interactions and connection with the small scale crisis of cold dark matter, JCAP 11, 010, arXiv:1706.06870 [astro-ph.CO] .
  * Chacko  _et al._ [2018] Z. Chacko, D. Curtin, M. Geller, and Y. Tsai, Cosmological Signatures of a Mirror Twin Higgs, JHEP 09, 163, arXiv:1803.03263 [hep-ph] .
  * Garny  _et al._ [2018] M. Garny, T. Konstandin, L. Sagunski, and S. Tulin, Lyman-$\alpha$ forest constraints on interacting dark sectors, JCAP 09, 011, arXiv:1805.12203 [astro-ph.CO] .
  * Raveri  _et al._ [2017] M. Raveri, W. Hu, T. Hoffman, and L.-T. Wang, Partially Acoustic Dark Matter Cosmology and Cosmological Constraints, Phys. Rev. D 96, 103501 (2017), arXiv:1709.04877 [astro-ph.CO] .
  * Rubira  _et al._ [2023] H. Rubira, A. Mazoun, and M. Garny, Full-shape BOSS constraints on dark matter interacting with dark radiation and lifting the S8 tension, JCAP 01, 034, arXiv:2209.03974 [astro-ph.CO] .
  * Choi  _et al._ [2021] G. Choi, T. T. Yanagida, and N. Yokozaki, A model of interacting dark matter and dark radiation for H0 and $\sigma_{8}$ tensions, JHEP 01, 127, arXiv:2010.06892 [hep-ph] .
  * Lehmann  _et al._ [2023] B. V. Lehmann, L. Morrison, S. Profumo, and N. Smyth, Kinetic recoupling of dark matter, (2023), arXiv:2310.20513 [hep-ph] .
  * Mazoun  _et al._ [2024] A. Mazoun, S. Bocquet, M. Garny, J. J. Mohr, H. Rubira, and S. M. L. Vogt, Probing interacting dark sector models with future weak lensing-informed galaxy cluster abundance constraints from SPT-3G and CMB-S4, Phys. Rev. D 109, 063536 (2024), arXiv:2312.17622 [astro-ph.CO] .
  * Zu  _et al._ [2023] L. Zu, C. Zhang, H.-Z. Chen, W. Wang, Y.-L. S. Tsai, Y. Tsai, W. Luo, and Y.-Z. Fan, Exploring mirror twin Higgs cosmology with present and future weak lensing surveys, JCAP 08, 023, arXiv:2304.06308 [astro-ph.CO] .
  * Blas  _et al._ [2011] D. Blas, J. Lesgourgues, and T. Tram, The Cosmic Linear Anisotropy Solving System (CLASS) II: Approximation schemes, JCAP 07 (07), 034, arXiv:1104.2933 [astro-ph.CO] .
  * Brinckmann and Lesgourgues [2019] T. Brinckmann and J. Lesgourgues, MontePython 3: boosted MCMC sampler and other features, Phys. Dark Univ. 24, 100260 (2019), arXiv:1804.07261 [astro-ph.CO] .
  * Lewis [2019] A. Lewis, GetDist: a Python package for analysing Monte Carlo samples, (2019), arXiv:1910.13970 [astro-ph.IM] .
  * Alam  _et al._ [2017] S. Alam _et al._ (BOSS), The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological analysis of the DR12 galaxy sample, Mon. Not. Roy. Astron. Soc. 470, 2617 (2017), arXiv:1607.03155 [astro-ph.CO] .
  * Beutler  _et al._ [2011] F. Beutler, C. Blake, M. Colless, D. H. Jones, L. Staveley-Smith, L. Campbell, Q. Parker, W. Saunders, and F. Watson, The 6dF Galaxy Survey: Baryon Acoustic Oscillations and the Local Hubble Constant, Mon. Not. Roy. Astron. Soc. 416, 3017 (2011), arXiv:1106.3366 [astro-ph.CO] .
  * Ross  _et al._ [2015] A. J. Ross, L. Samushia, C. Howlett, W. J. Percival, A. Burden, and M. Manera, The clustering of the SDSS DR7 main Galaxy sample – I. A 4 per cent distance measure at $z=0.15$, Mon. Not. Roy. Astron. Soc. 449, 835 (2015), arXiv:1409.3242 [astro-ph.CO] .
  * Abbott  _et al._ [2024] T. M. C. Abbott _et al._ (DES), The Dark Energy Survey: Cosmology Results With ~1500 New High-redshift Type Ia Supernovae Using The Full 5-year Dataset, (2024), arXiv:2401.02929 [astro-ph.CO] .
  * Abbott  _et al._ [2023] T. M. C. Abbott _et al._ (Kilo-Degree Survey, Dark Energy Survey), DES Y3 \+ KiDS-1000: Consistent cosmology combining cosmic shear surveys, Open J. Astrophys. 6, 2305.17173 (2023), arXiv:2305.17173 [astro-ph.CO] .
  * Kitaura  _et al._ [2016] F.-S. Kitaura _et al._ , The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: mock galaxy catalogues for the BOSS Final Data Release, Mon. Not. Roy. Astron. Soc. 456, 4156 (2016), arXiv:1509.06400 [astro-ph.CO] .
  * Gil-Marín  _et al._ [2016] H. Gil-Marín _et al._ (BOSS), The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: BAO measurement from the LOS-dependent power spectrum of DR12 BOSS galaxies, Mon. Not. Roy. Astron. Soc. 460, 4210 (2016), arXiv:1509.06373 [astro-ph.CO] .
  * D’Amico  _et al._ [2021] G. D’Amico, L. Senatore, and P. Zhang, Limits on $w$CDM from the EFTofLSS with the PyBird code, JCAP 01, 006, arXiv:2003.07956 [astro-ph.CO] .
  * Alam  _et al._ [2021] S. Alam _et al._ (eBOSS), Completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: Cosmological implications from two decades of spectroscopic surveys at the Apache Point Observatory, Phys. Rev. D 103, 083533 (2021), arXiv:2007.08991 [astro-ph.CO] .
  * Beutler and McDonald [2021] F. Beutler and P. McDonald, Unified galaxy power spectrum measurements from 6dFGS, BOSS, and eBOSS, JCAP 11, 031, arXiv:2106.06324 [astro-ph.CO] .
  * Simon  _et al._ [2023] T. Simon, P. Zhang, and V. Poulin, Cosmological inference from the EFTofLSS: the eBOSS QSO full-shape analysis, JCAP 07, 041, arXiv:2210.14931 [astro-ph.CO] .
  * Ross  _et al._ [2020] A. J. Ross _et al._ (eBOSS), The Completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: Large-scale structure catalogues for cosmological analysis, Mon. Not. Roy. Astron. Soc. 498, 2354 (2020), arXiv:2007.09000 [astro-ph.CO] .
  * Chuang  _et al._ [2015] C.-H. Chuang, F.-S. Kitaura, F. Prada, C. Zhao, and G. Yepes, EZmocks: extending the Zel’dovich approximation to generate mock galaxy catalogues with accurate clustering statistics, Mon. Not. Roy. Astron. Soc. 446, 2621 (2015), arXiv:1409.1124 [astro-ph.CO] .
  * López  _et al._ [2016] S. López, V. D’Odorico, S. Ellison, G. Becker, L. Christensen, G. Cupani, K. Denney, I. Pâris, G. Worseck, T. Berg, _et al._ , Xq-100: A legacy survey of one hundred 3.5 $\lesssim$ z $\lesssim$ 4.5 quasars observed with vlt/x-shooter, Astronomy & Astrophysics 594, A91 (2016), arXiv:1607.08776 [astro-ph.GA] .
  * Esposito  _et al._ [2022] M. Esposito, V. Iršič, M. Costanzi, S. Borgani, A. Saro, and M. Viel, Weighing cosmic structures with clusters of galaxies and the intergalactic medium, Mon. Not. Roy. Astron. Soc. 515, 857 (2022), arXiv:2202.00974 [astro-ph.CO] .
  * Viel  _et al._ [2013] M. Viel, G. D. Becker, J. S. Bolton, and M. G. Haehnelt, Warm dark matter as a solution to the small scale crisis: New constraints from high redshift Lyman-$\alpha$ forest data, Phys. Rev. D 88, 043502 (2013), arXiv:1306.2314 [astro-ph.CO] .
  * Allali  _et al._ [2023] I. J. Allali, F. Rompineve, and M. P. Hertzberg, Dark sectors with mass thresholds face cosmological datasets, Phys. Rev. D 108, 023527 (2023), arXiv:2305.14166 [astro-ph.CO] .
  * Fernandez  _et al._ [2023] M. A. Fernandez, S. Bird, and M.-F. Ho, Cosmological Constraints from the eBOSS Lyman-$\alpha$ Forest using the PRIYA Simulations, (2023), arXiv:2309.03943 [astro-ph.CO] .
  * Ravoux  _et al._ [2023] C. Ravoux _et al._ (DESI), The Dark Energy Spectroscopic Instrument: one-dimensional power spectrum from first Ly $\alpha$ forest samples with Fast Fourier Transform, Mon. Not. Roy. Astron. Soc. 526, 5118 (2023), arXiv:2306.06311 [astro-ph.CO] .
  * Karaçaylı  _et al._ [2024] N. G. Karaçaylı _et al._ , Optimal 1D Ly$\alpha$ Forest Power Spectrum Estimation – III. DESI early data, Mon. Not. Roy. Astron. Soc. 528, 3941 (2024), arXiv:2306.06316 [astro-ph.CO] .
  * Ivanov  _et al._ [2024] M. M. Ivanov, M. W. Toomey, and N. G. Karaçaylı, Fundamental physics with the Lyman-alpha forest: constraints on the growth of structure and neutrino masses from SDSS with effective field theory, (2024), arXiv:2405.13208 [astro-ph.CO] .
  * Hooper  _et al._ [2022] D. C. Hooper, N. Schöneberg, R. Murgia, M. Archidiacono, J. Lesgourgues, and M. Viel, One likelihood to bind them all: Lyman-$\alpha$ constraints on non-standard dark matter, JCAP 10, 032, arXiv:2206.08188 [astro-ph.CO] .
  * Borde  _et al._ [2014] A. Borde, N. Palanque-Delabrouille, G. Rossi, M. Viel, J. S. Bolton, C. Yèche, J.-M. LeGoff, and J. Rich, New approach for precise computation of Lyman-$\alpha$ forest power spectrum with hydrodynamical simulations, JCAP 07, 005, arXiv:1401.6472 [astro-ph.CO] .

##  Appendix A Details of Ly$\alpha\,$likelihood

Figure 7: (Top) Ratio of the MPS for the SIDR+ best fit to the
$\mathcal{D}L\,$dataset compared to the corresponding $\Lambda{\rm
CDM}\,$avatar, with parameters detailed in Table 4. The shaded cones and the
height of the shaded rectangle represent the errors in the compressed eBOSS
Ly$\alpha\,$measurement, normalized to the underlying $\Lambda{\rm
CDM}\,$avatar. The width of the shaded rectangle indicates the scales included
in the measurement. (Bottom) A similar plot for the WZDR+ model.

In this Appendix, we detail the derivation of the compressed eBOSS
Ly$\alpha\,$likelihood and address concerns regarding its applicability to
$\Lambda{\rm CDM}\,$extensions, specifically SIDR+ and WZDR+. Note that the
shape of our MPS suppression does not fit the parameterizations considered in
[85].

To extract information from data influenced by non-linear physics, the eBOSS
collaboration [27] matched the observed 1D Ly$\alpha\,$flux power spectrum
along the line of sight up to second order by conducting hydrodynamical
simulations within the $\Lambda{\rm CDM}\,$cosmology. This involved varying
$H_{0}\,$, $\Omega_{M}$, $n_{s}$, and $\sigma_{8}$ [86]. Additionally, the
effects of intergalactic medium (IGM) parameters, such as the temperature at
mean density ($T_{0}$) and the spectral index ($\gamma$) at $z=3$, which
defines the temperature-density relation $T=T_{0}\delta^{\gamma-1}$, were
examined.

Readers may be concerned that extracting linear parameters from experimental
data, which is sensitive to full non-linear effects, has been conducted within
a $\Lambda{\rm CDM}\,$cosmology framework. However, Pedersen et al.
demonstrate that $\Delta^{2}_{L}$ and $n_{L}$ posteriors are model-independent
across a range of models that smoothly modify the MPS at early times without
altering the late-time growth of structure from $\Lambda{\rm CDM}\,$.

In Fig. 7, we show that, within the measurement’s error bounds, the slope and
amplitude of the SIDR+ and WZDR+ models can be approximated by a $\Lambda{\rm
CDM}\,$model (referred to as avatars). Error bounds are depicted as shaded
cones for the slope and shaded rectangles for the amplitude, with the widths
of the rectangles indicating the $k$ scales measured by eBOSS Ly$\alpha$. The
red and blue lines represent the best fits for SIDR+ and WZDR+ to
$\mathcal{D}L\,$, normalized to $\Lambda{\rm CDM}\,$avatars. These avatars
only differ from a $\mathcal{D}\,$best fit in the $A_{s}$ and $n_{s}$ values,
which are adjusted to match the MPS of SIDR+ and WZDR+ best fits to
$\mathcal{D}L\,$, respectively.

The selected $A_{s}$ and $n_{s}$ values for these avatars are detailed in
Table 4. The agreement of the $\Lambda{\rm CDM}\,$power spectrum with both
SIDR+ and WZDR+ models is within $0.1\%$ for the eBOSS Ly$\alpha$ sensitive
regions, indicating that WZDR+ and SIDR+ share the same linear MPS as
$\Lambda{\rm CDM}\,$at these scales. Additionally, they exhibit identical late
universe parameters, such as $H_{0}\,$and $\omega_{m}$, to those of the
$\Lambda{\rm CDM}\,$avatars.

| $ln10^{10}A_{s}$ | $n_{s}$
---|---|---
$\Lambda{\rm CDM}\,$: $\mathcal{D}$ | 3.046 | 0.968
$\Lambda{\rm CDM}\,$: SIDR+ Avatar | 3.016 | 0.935
$\Lambda{\rm CDM}\,$: WZDR+ Avatar | 3.059 | 0.933
Table 4: Cosmological parameter values for the $\Lambda{\rm CDM}\,$avatars
used in Fig. 7 that differ from those of a best-fit $\Lambda{\rm CDM}\,$to
$\mathcal{D}\,$.

Given their shared linear MPS and late universe physics, including
$\omega_{\rm cdm}$, $\omega_{b}$, and $H_{0}\,$, SIDR+ and WZDR+ undergo the
same growth. Consequently, the results from N-body simulations are applicable
to both SIDR+ and WZDR+ models as well.

##  Appendix B Triangle Plots and Parameter Values

This Section presents the full set of marginalized posteriors for $\Lambda{\rm
CDM}\,$parameters and additional parameters of the models SIDR(+) and WZDR(+).
Figs. 8, 9, 10, 11 and 12 show comparisons of SIDR, WZDR, SIDR+, WZDR+, and
$\Lambda{\rm CDM}\,$fits to datasets $\mathcal{D}\,$, $\mathcal{D}H\,$,
$\mathcal{D}L\,$, and $\mathcal{D}HL\,$. Meanwhile, 13 compares the + model’s
fits to $\Lambda{\rm CDM}\,$for datasets $\mathcal{D}\,$, $\mathcal{D}H\,$,
$\mathcal{D}L\,$, and $\mathcal{D}HL\,$.

Tables 5, 6, and 7 show marginalized 1D posterior means and $68\%$ credible
intervals for cosmological parameters across the $\Lambda{\rm CDM}\,$, SIDR+,
and WZDR+ models, evaluated for various datasets respectively. Table 8 incudes
the $\chi^{2}$ for fits of all models to all the datasets.

Figure 8: Comparison of SIDR model fits to four datasets, with light and dark shaded regions representing $68\%$ and $95\%$ confidence levels, respectively. The marginalized posteriors for $\Delta_{L}^{2}$ and $n_{L}$ are juxtaposed with the 2D Ly$\alpha\,$likelihood (black contours). Figure 9: Comparison of WZDR model fits to four datasets, with light and dark shaded regions representing $68\%$ and $95\%$ confidence levels, respectively. The marginalized posteriors for $\Delta_{L}^{2}$ and $n_{L}$ are juxtaposed with the 2D Ly$\alpha\,$likelihood (black contours). Figure 10: Comparison of SIDR+ model fits to four datasets, with light and dark shaded regions representing $68\%$ and $95\%$ confidence levels, respectively. The marginalized posteriors for $\Delta_{L}^{2}$ and $n_{L}$ are juxtaposed with the 2D Ly$\alpha\,$likelihood (black contours). Figure 11: Comparison of WZDR+ model fits to four datasets, with light and dark shaded regions representing $68\%$ and $95\%$ confidence levels, respectively. The marginalized posteriors for $\Delta_{L}^{2}$ and $n_{L}$ are juxtaposed with the 2D Ly$\alpha\,$likelihood (black contours). Figure 12: Comparison of $\Lambda{\rm CDM}\,$model fits to four datasets, with light and dark shaded regions representing $68\%$ and $95\%$ confidence levels, respectively. The marginalized posteriors for $\Delta_{L}^{2}$ and $n_{L}$ are juxtaposed with the 2D Ly$\alpha\,$likelihood (black contours). Figure 13: Comparison of $\Lambda{\rm CDM}\,$model parameters from $\mathcal{D}HL\,$dataset with interacting DM-DR models: WZDR+ and SIDR+ . | $\mathcal{D}\,$ | $\mathcal{D}H\,$ | $\mathcal{D}L\,$ | $\mathcal{D}HL\,$
---|---|---|---|---
Param | best-fit | mean$\pm\sigma$ | best-fit | mean$\pm\sigma$ | best-fit | mean$\pm\sigma$ | best-fit | mean$\pm\sigma$
$100~{}\omega{}_{b}$ | 2.246 | $2.245_{-0.014}^{+0.014}$ | 2.263 | $2.262_{-0.013}^{+0.013}$ | 2.243 | $2.238_{-0.013}^{+0.013}$ | 2.255 | $2.256_{-0.014}^{+0.013}$
$100\,\theta{}_{s}$ | 1.042 | $1.042_{-0.00026}^{+0.00029}$ | 1.042 | $1.042_{-0.00030}^{+0.00030}$ | 1.042 | $1.042_{-0.00030}^{+0.00029}$ | 1.042 | $1.042_{-0.00028}^{+0.00031}$
$ln10^{10}A_{s}$ | 3.046 | $3.045_{-0.015}^{+0.013}$ | 3.055 | $3.056_{-0.017}^{+0.014}$ | 3.046 | $3.041_{-0.014}^{+0.013}$ | 3.051 | $3.052_{-0.013}^{+0.013}$
$n_{s}$ | 0.968 | $0.9669_{-0.0039}^{+0.0035}$ | 0.972 | $0.9712_{-0.0037}^{+0.0036}$ | 0.958 | $0.9578_{-0.0033}^{+0.0036}$ | 0.961 | $0.9614_{-0.0035}^{+0.0032}$
$\tau{}_{\rm reio}$ | 0.057 | $0.0564_{-0.0075}^{+0.0067}$ | 0.063 | $0.0627_{-0.0089}^{+0.0070}$ | 0.054 | $0.0531_{-0.0070}^{+0.0071}$ | 0.059 | $0.0584_{-0.0065}^{+0.0067}$
$\omega{}_{\rm cdm}$ | 0.119 | $0.1186_{-0.0009}^{+0.0009}$ | 0.117 | $0.1172_{-0.0009}^{+0.0008}$ | 0.120 | $0.1194_{-0.0008}^{+0.0009}$ | 0.118 | $0.1179_{-0.0009}^{+0.0007}$
$\Omega{}_{m}$ | 0.308 | $0.307_{-0.005}^{+0.005}$ | 0.308 | $0.2978_{-0.0050}^{+0.0047}$ | 0.313 | $0.3123_{-0.0051}^{+0.0050}$ | 0.303 | $0.3023_{-0.0052}^{+0.0042}$
$H_{0}\,$ | 67.87 | $67.94_{-0.40}^{+0.39}$ | 68.70 | $68.68_{-0.38}^{+0.40}$ | 67.52 | $67.55_{-0.38}^{+0.37}$ | 68.26 | $68.33_{-0.33}^{+0.40}$
$\sigma_{8}$ | 0.808 | $0.8068_{-0.0057}^{+0.0051}$ | 0.807 | $0.8070_{-0.0062}^{+0.0056}$ | 0.808 | $0.8055_{-0.0058}^{+0.0049}$ | 0.806 | $0.8051_{-0.0056}^{+0.0052}$
$n_{L}$ | -2.303 | $-2.304_{-0.0035}^{+0.0029}$ | -2.301 | $-2.302_{-0.0034}^{+0.0029}$ | -2.311 | $-2.312_{-0.0027}^{+0.0032}$ | -2.311 | $-2.311_{-0.0027}^{+0.0030}$
$\Delta_{L}^{2}$ | 0.345 | $0.3428_{-0.0054}^{+0.0055}$ | 0.342 | $0.3413_{-0.0059}^{+0.0054}$ | 0.341 | $0.3383_{-0.0054}^{+0.0053}$ | 0.336 | $0.3357_{-0.0052}^{+0.0055}$
Table 5: Parameter values from $\Lambda{\rm CDM}\,$fits. | $\mathcal{D}\,$ | $\mathcal{D}H\,$ | $\mathcal{D}L\,$ | $\mathcal{D}HL\,$
---|---|---|---|---
Param | best-fit | mean$\pm\sigma$ | best-fit | mean$\pm\sigma$ | best-fit | mean$\pm\sigma$ | best-fit | mean$\pm\sigma$
$100~{}\omega{}_{b}$ | 2.252 | $2.256_{-0.017}^{+0.017}$ | 2.291 | $2.288_{-0.016}^{+0.014}$ | 2.239 | $2.240_{-0.019}^{+0.014}$ | 2.283 | $2.280_{-0.013}^{+0.015}$
$100\,\theta{}_{s}$ | 1.0420 | $1.042_{-0.0003}^{+0.0003}$ | 1.0424 | $1.043_{-0.00028}^{+0.00033}$ | 1.0420 | $1.042_{-0.00031}^{+0.00041}$ | 1.0427 | $1.043_{-0.0003}^{+0.00028}$
$ln10^{10}A_{s}$ | 3.044 | $3.046_{-0.015}^{+0.016}$ | 3.042 | $3.045_{-0.016}^{+0.015}$ | 3.055 | $3.053_{-0.015}^{+0.013}$ | 3.054 | $3.054_{-0.017}^{+0.013}$
$n_{s}$ | 0.9681 | $0.9686_{-0.0040}^{+0.0038}$ | 0.9713 | $0.9714_{-0.0038}^{+0.0038}$ | 0.9672 | $0.9663_{-0.0034}^{+0.0042}$ | 0.9718 | $0.9711_{-0.0036}^{+0.0035}$
$\tau{}_{\rm reio}$ | 0.0559 | $0.0574_{-0.0078}^{+0.0079}$ | 0.0599 | $0.0610_{-0.0084}^{+0.0072}$ | 0.0578 | $0.0583_{-0.0079}^{+0.0062}$ | 0.0630 | $0.0638_{-0.0085}^{+0.0073}$
$N_{\rm IR}$ | 0.092 | $0.312_{-0.220}^{+0.057}$ | 0.71 | $0.78_{-0.17}^{+0.17}$ | 0.013 | $2.47_{-0.23}^{+0.08}$ | 0.87 | $0.90_{-0.19}^{+0.17}$
$R_{\Gamma}$ | 0.0032 | $0.0226_{-0.0226}^{+0.0054}$ | 0.0082 | $0.03_{-0.03}^{+0.01}$ | 0.052 | $0.05_{-0.01}^{+0.01}$ | 0.055 | $0.056_{-0.011}^{+0.009}$
$\omega_{\rm idm}$ | 0.1198 | $0.123_{-0.003}^{+0.0016}$ | 0.1276 | $0.129_{-0.003}^{+0.0028}$ | 0.1196 | $0.124_{-0.004}^{+0.002}$ | 0.1316 | $0.1319_{-0.0028}^{+0.0029}$
$\Omega{}_{m}$ | 0.3061 | $0.3059_{-0.0071}^{+0.0062}$ | 0.2941 | $0.2965_{-0.0058}^{+0.0056}$ | 0.3124 | $0.3134_{-0.0057}^{+0.0054}$ | 0.3006 | $0.3011_{-0.0054}^{+0.0052}$
$H_{0}\,$ | 68.38 | $69.25_{-1.10}^{+0.78}$ | 71.71 | $71.80_{-0.85}^{+0.78}$ | 67.60 | $68.43_{-1.30}^{+0.66}$ | 71.84 | $71.86_{-0.90}^{+0.81}$
$\sigma_{8}$ | 0.8066 | $0.8016_{-0.0082}^{+0.0110}$ | 0.8144 | $0.807_{-0.008}^{+0.010}$ | 0.7864 | $0.7885_{-0.0070}^{+0.0072}$ | 0.7969 | $0.7969_{-0.0071}^{+0.0075}$
$n_{L}$ | -2.305 | $-2.316_{-0.006}^{+0.013}$ | -2.306 | $-2.317_{-0.008}^{+0.013}$ | -2.339 | $-2.3360_{-0.0063}^{+0.0049}$ | -2.334 | $-2.3350_{-0.0048}^{+0.0047}$
$\Delta_{L}^{2}$ | 0.342 | $0.3312_{-0.0092}^{+0.0150}$ | 0.345 | $0.334_{-0.011}^{+0.014}$ | 0.304 | $0.3104_{-0.0087}^{+0.0076}$ | 0.317 | $0.3162_{-0.0076}^{+0.0070}$
Table 6: Parameter values from SIDR+ fits. Note that $R_{\Gamma}\sim 0.05$ is equivalent to $10^{7}\Gamma_{0}\sim 1.2$ in the alternative formulation of the interaction strength parameter. | $\mathcal{D}\,$ | $\mathcal{D}H\,$ | $\mathcal{D}L\,$ | $\mathcal{D}HL\,$
---|---|---|---|---
Param | best-fit | mean$\pm\sigma$ | best-fit | mean$\pm\sigma$ | best-fit | mean$\pm\sigma$ | best-fit | mean$\pm\sigma$
$100~{}\omega{}_{b}$ | 2.250 | $2.256_{-0.016}^{+0.016}$ | 2.283 | $2.283_{-0.016}^{+0.017}$ | 2.249 | $2.255_{-0.017}^{+0.015}$ | 2.281 | $2.277_{-0.016}^{+0.015}$
$100*\,\theta{}_{s}$ | 1.0422 | $1.042_{-0.00046}^{+0.00036}$ | 1.0433 | $1.043_{-0.00037}^{+0.00039}$ | 1.0422 | $1.042_{-0.00039}^{+0.00039}$ | 1.0432 | $1.043_{-0.00036}^{+0.00037}$
$ln10^{10}A_{s}$ | 3.049 | $3.051_{-0.015}^{+0.015}$ | 3.057 | $3.059_{-0.016}^{+0.016}$ | 3.048 | $3.045_{-0.014}^{+0.014}$ | 3.050 | $3.049_{-0.014}^{+0.015}$
$n_{s}$ | 0.9699 | $0.9717_{-0.0060}^{+0.0048}$ | 0.9838 | $0.9835_{-0.0050}^{+0.0049}$ | 0.9700 | $0.9717_{-0.0052}^{+0.0047}$ | 0.9829 | $0.9806_{-0.0050}^{+0.0052}$
$\tau{}_{\rm reio}$ | 0.0567 | $0.0574_{-0.0075}^{+0.0078}$ | 0.0601 | $0.0610_{-0.0083}^{+0.0074}$ | 0.0567 | $0.0557_{-00.0076}^{+0.0069}$ | 0.0574 | $0.0575_{-0.0075}^{+0.0068}$
$N_{\rm IR}$ | 0.12 | $0.19_{-0.18}^{+0.05}$ | 0.64 | $0.63_{-0.17}^{+0.13}$ | 0.12 | $0.21_{-0.19}^{+0.05}$ | 0.64 | $0.59_{-0.15}^{+0.17}$
$\mathrm{log}\,z_{t}$ | -4.20 | $-4.46_{-0.19}^{+0.29}$ | -4.29 | $-4.29_{-0.12}^{+0.09}$ | -4.20 | $-4.33_{-0.07}^{+0.33}$ | -4.26 | $-4.25_{-0.12}^{+0.15}$
$R_{\Gamma}$ | 0.067 | $0.22_{-0.22}^{+0.04}$ | 0.11 | $0.15_{-0.11}^{+0.07}$ | 0.052 | $0.054_{-0.011}^{+0.010}$ | 0.070 | $0.07_{-0.01}^{+0.01}$
$\omega{}_{\rm idm}$ | 0.121 | $0.123_{-0.003}^{+0.002}$ | 0.129 | $0.129_{-0.003}^{+0.003}$ | 0.121 | $0.122_{-0.003}^{+0.002}$ | 0.129 | $0.128_{-0.003}^{+0.003}$
$\Omega{}_{m}$ | 0.308 | $0.308_{-0.007}^{+0.006}$ | 0.295 | $0.297_{-0.006}^{+0.006}$ | 0.307 | $0.305_{-0.006}^{+0.006}$ | 0.294 | $0.295_{-0.005}^{+0.005}$
$H_{0}\,$ | 68.42 | $68.84_{-1.10}^{+0.71}$ | 71.91 | $71.75_{-0.91}^{+0.77}$ | 68.55 | $69.11_{-1.20}^{+0.69}$ | 71.97 | $71.63_{-0.89}^{+0.97}$
$\sigma_{8}$ | 0.7997 | $0.793_{-0.012}^{+0.015}$ | 0.8103 | $0.805_{-0.013}^{+0.014}$ | 0.8016 | $0.804_{-0.007}^{+0.007}$ | 0.8143 | $0.812_{-0.007}^{+0.007}$
$n_{L}$ | -2.348 | $-2.448_{-0.037}^{+0.120}$ | -2.368 | $-2.393_{-0.044}^{+0.077}$ | -2.338 | $-2.338_{-0.006}^{+0.006}$ | -2.338 | $-2.338_{-0.006}^{+0.006}$
$\Delta_{L}^{2}$ | 0.313 | $0.273_{-0.035}^{+0.050}$ | 0.310 | $0.296_{-0.035}^{+0.041}$ | 0.320 | $0.322_{-0.008}^{+0.007}$ | 0.328 | $0.327_{-0.008}^{+0.007}$
Table 7: Parameter values from WZDR+ fits. Datasets | Model | $\chi^{2}_{\rm Tot.}$ | $\chi^{2}_{\rm CMB}$ | $\chi^{2}_{\rm Pantheon}$ | $\chi^{2}_{\rm BAO}$ | $\chi^{2}_{\rm Pl.Lensing}$ | $\chi^{2}_{S_{8}}$ | $\chi^{2}_{\rm SH0ES}$ | $\chi^{2}_{\mathrm{full-shape}}$ | $\chi^{2}_{\mathrm{Ly}\alpha}$
---|---|---|---|---|---|---|---|---|---|---
| $\Lambda{\rm CDM}\,$ | 4075.9 | 2769.4 | 1026.6 | 1.4 | 9.2 | 2.5 | - | 266.8 | -
| SIDR | 4073.3 | 2767.7 | 1025.7 | 1.5 | 9.3 | 2.2 | - | 266.8 | -
$\mathcal{D}\,$ | WZDR | 4072.7 | 2766.7 | 1025.7 | 1.7 | 9.5 | 2.2 | - | 266.9 | -
| SIDR+ | 4073.4 | 2768.2 | 1025.9 | 1.5 | 9.4 | 1.9 | - | 266.5 | -
| WZDR+ | 4071.1 | 2767.2 | 1025.9 | 1.4 | 9.2 | 1.3 | - | 266.0 | -
| $\Lambda{\rm CDM}\,$ | 4102.8 | 2772.7 | 1025.6 | 2.3 | 9.9 | 0.5 | 26.1 | 265.6 | -
| SIDR | 4082.4 | 2773.0 | 1025.7 | 2.8 | 10.3 | 0.8 | 2.8 | 267.0 | -
$\mathcal{D}H\,$ | WZDR | 4080.4 | 2768.7 | 1025.8 | 3.0 | 10.4 | 1.8 | 2.2 | 268.4 | -
| SIDR+ | 4082.6 | 2773.7 | 1025.7 | 2.6 | 10.1 | 0.8 | 2.9 | 266.7 | -
| WZDR+ | 4078.8 | 2770.4 | 1025.7 | 2.6 | 10.0 | 0.6 | 2.1 | 267.3 | -
| $\Lambda{\rm CDM}\,$ | 4103.7 | 2774.5 | 1026.2 | 1.1 | 8.6 | 3.8 | - | 267.4 | 22.1
| SIDR | 4101.4 | 2774.2 | 1026.0 | 1.1 | 8.9 | 3.0 | - | 267.5 | 20.7
$\mathcal{D}L\,$ | WZDR | 4101.1 | 2775.0 | 1025.9 | 1.3 | 9.0 | 2.7 | - | 266.8 | 20.3
| SIDR+ | 4072.2 | 2769.3 | 1026.0 | 1.1 | 9.3 | 0.5 | - | 265.9 | 0.1
| WZDR+ | 4071.7 | 2767.6 | 1025.8 | 1.5 | 9.5 | 1.3 | - | 265.9 | 0.3
| $\Lambda{\rm CDM}\,$ | 4138.2 | 2779.4 | 1025.7 | 1.7 | 9.3 | 1.2 | 31.2 | 265.6 | 24.0
| SIDR | 4111.7 | 2781.5 | 1025.8 | 2.3 | 9.5 | 1.6 | 2.7 | 266.6 | 21.6
$\mathcal{D}HL\,$ | WZDR | 4115.7 | 2781.1 | 1025.7 | 2.0 | 9.2 | 2.1 | 9.0 | 266.8 | 19.8
| SIDR+ | 4087.9 | 2780.1 | 1025.6 | 2.0 | 10.5 | 0.2 | 2.1 | 266.6 | 0.9
| WZDR+ | 4080.0 | 2769.9 | 1025.7 | 2.8 | 10.5 | 0.8 | 2.0 | 267.5 | 0.9
Table 8: Minimum $\chi^{2}$ for all datasets and cosmological models.
Ly$\alpha\,$refers to the eBOSS Lyman-alpha forest likelihood.