First constraints on fuzzy dark matter from Lyman-$\alpha$ forest data and hydrodynamical simulations

TL;DR

This study uses hydrodynamical simulations and Lyman-alpha forest data to set new lower limits on fuzzy dark matter particle masses, challenging their role in solving small-scale structure problems.

Contribution

First to employ hydrodynamical simulations with Lyman-alpha data to constrain fuzzy dark matter particle masses.

Findings

Lower mass limits of 7.1×10⁻²² eV and 14.3×10⁻²² eV from two datasets.

Combined data sets constrain mass to above 20×10⁻²² eV.

Mass range 1-10×10⁻²² eV ruled out at high significance.

Abstract

We present constraints on the masses of extremely light bosons dubbed fuzzy dark matter from Lyman-$\alpha$ forest data. Extremely light bosons with a De Broglie wavelength of $\sim 1$ kpc have been suggested as dark matter candidates that may resolve some of the current small scale problems of the cold dark matter model. For the first time we use hydrodynamical simulations to model the Lyman-$\alpha$ flux power spectrum in these models and compare with the observed flux power spectrum from two different data sets: the XQ-100 and HIRES/MIKE quasar spectra samples. After marginalization over nuisance and physical parameters and with conservative assumptions for the thermal history of the IGM that allow for jumps in the temperature of up to $5000\rm\,K$, XQ-100 provides a lower limit of 7.1$\times 10^{-22}$ eV, HIRES/MIKE returns a stronger limit of 14.3$\times 10^{-22}$ eV, while the combination of both data sets results in a limit of 20 $\times 10^{-22}$ eV (2$\sigma$ C.L.). The limits for the analysis of the combined data sets increases to 37.5$\times 10^{-22}$ eV (2$\sigma$ C.L.) when a smoother thermal history is assumed where the temperature of the IGM evolves as a power-law in redshift. Light boson masses in the range $1-10 \times10^{-22}$ eV are ruled out at high significance by our analysis, casting strong doubts that FDM helps solve the "small scale crisis" of the cold dark matter models.

Tables

Table 1. Table 1: Marginalized constraints at 95 95 95 % (lower limits). The pivot redshifts for different data sets are: z p = 3.6 , 4.5 , 4.2 subscript 𝑧 p 3.6 4.5 4.2 z_{\rm p}=3.6,4.5,4.2 for XQ-100, HIRES/MIKE and combined for reference case, covariance matrix multiplied by 1.3, Planck priors and temperature in redshift bins.

$m_{\mathrm{FDM}}\left[{10}^{-22}\mathrm{eV}\right]$	XQ-100	HIRES/MIKE	Combined
ref.	$4.5$	$16.4$	$37.5$
Cov. $\times$ 1.3	$3.9$	$16.3$	$34.9$
Planck priors	$2.7$	$16.5$	$32.2$
$T_0\left(z\right)$ bins	$7.1$	$14.3$	$20.0$
${\chi }^2/d.o.f.$ (ref.)	$134/124$	$33/40$	$187/173$

Full text

Introduction. Recently, there has been a growing interest in so-called Fuzzy Dark Matter (FDM) models where the dark matter is made of ultra-light bosons. Cosmological and astrophysical consequences have been comprehensively reviewed in (Hu et al., 2000; Matos et al., 2000; Hlozek et al., 2015; Hui et al., 2016; Bernal et al., 2017) highlighting the particle physics motivation (Baldeschi et al., 1983; Preskill et al., 1983; Sikivie and Yang, 2009; Zhang and Chiueh, 2017) for such models, as well as the importance of experimental searches (Sikivie, 1983). A broad variety of astrophysical implications have been investigated in the literature: the halo mass function (Marsh and Silk, 2014), the innermost structure of haloes (Lee and Koh, 1996; Guzmán et al., 1999; Zhang et al., 2016), the dynamical properties of the smallest objects (Calabrese and Spergel, 2016), the linear matter power spectrum (Hu et al., 2000), the development of non-linearities by using N-body simulations (Schive et al., 2014), the abundance of high redshift objects (Menci et al., 2017), the overall impact of FDM on galaxy formation and the reionization history of the Universe, the intergalactic medium (Amendola and Barbieri, 2006; Kim and Marsh, 2016; Marsh, 2016; Hui et al., 2016; Sarkar et al., 2017), pulsar timing and binary pulsars (Blas et al., 2016; Khmelnitsky and Rubakov, 2014), and the properties of our galactic disk (Banik et al., 2017). The general conclusion is that in order to have an appreciable astrophysical impact the mass of ultra-light bosons would have to lie in the range $1-10\times 10^{-22}$ eV, and in this mass range it is indeed possible that some small scale “tensions” of cold dark matter with observations could be alleviated (e.g. (Weinberg et al., 2015) for a review of the small scale “crisis” of cold dark matter).

The intergalactic medium (IGM) (Meiksin, 2009; McQuinn, 2015) plays a unique role in constraining the (small scale) matter power spectrum, since the low-density, high redshift IGM filaments are particularly sensitive to the small scale properties of dark matter. The main observable manifestation of the IGM, the Lyman-$\alpha$ forest, has provided important constraints on the linear matter power spectrum, especially when combined with cosmic microwave background data (Croft et al., 2002; Zaldarriaga et al., 2003; McDonald, 2003; Viel et al., 2004; McDonald et al., 2005; Seljak et al., 2006a; Lidz et al., 2010). This includes, most notably, the tightest constraints on warm dark matter (WDM) models (Viel et al., 2005, 2013; Seljak et al., 2006b; Baur et al., 2016; Viel et al., 2008; Yeche et al., 2017), upper limits on neutrino masses (Seljak et al., 2006a; Palanque-Delabrouille et al., 2015; Yeche et al., 2017) as well as the recent remarkable discovery of Baryonic Acoustic Oscillations in the transmitted 3D flux (Busca et al., 2013; Slosar et al., 2013). These results, especially those at small scales, are primarily due to the fact that the observed Lyman-$\alpha$ forest flux power spectrum provides a tracer of matter fluctuations on small scales and at high redshifts, where these fluctuations are still in the quasi-linear regime. At present, the tightest limits on the free streaming of WDM, expressed as the equivalent masses of thermal WDM relics, are in the range m${}_{\mathrm{WDM}}>2.3-5$ keV at 2$\sigma$ C.L.. The values at the lower end of this range are probably overly conservative and require the assumption of thermal histories that are likely unphysical (Garzilli et al., 2015).

In FDM models, even though the typical de Broglie scale is rather small ($\sim$ kpc) the effect of FDM on the linear matter power spectrum is noticeable on scales larger than the smallest scales typically constrained by IGM data (Hu et al., 2000). In the absence of fully numerical FDM simulations of the flux power spectrum, it has therefore become common practice to convert the limit on thermal relic WDM models – for which the flux power spectrum has been modelled in considerable detail – into FDM limits by comparing the linear matter power spectrum of WDM and FDM models and using the mass corresponding to $k_{\rm 1/2}$, the wavenumber at which the linear power spectrum departs (i.e. is suppressed) from the corresponding cold dark matter power spectrum by 50%. However, the accuracy of this rather crude mapping can only be checked by performing a full set of hydrodynamical simulations to model the effect of FDM on the properties of the IGM and the Lyman-$\alpha$ forest. Here, we will use such simulations to provide the first constraints on FDM models based on a full modelling of the Lyman-$\alpha$ flux power spectrum and comparison with two high-redshift data sets well suited to probing the small scale matter power spectrum. This will also allow us to check the accuracy of the $k_{\mathrm{1}/2}$ mapping of thermal relic WDM constraints. Our analysis will be quite similar to the one presented in (Viel et al., 2013) and (Iršič et al., 2016), where the flux power spectrum is modelled using a set of hydrodynamical simulations that vary astrophysical and cosmological parameters combined with a Monte Carlo Markov Chain analysis in the multi dimensional parameter space.

Data. The first sample we use is the set of 100 medium resolution, high signal-to-noise QSO spectra of the XQ-100 survey (López et al., 2016), with emission redshifts $3.5<z<4.5$. A more detailed description of the data and the power spectrum measurements of the XQ-100 survey can be found in (Iršič et al., 2016). Here we repeat the most important properties of the data and the derived flux power spectrum. The spectral resolution of the X-shooter spectrograph is 30-50 km/s, depending on wavelength. The flux power spectrum used in the analysis has been calculated for a total of 114 $(k,z)$ data points in the ranges $z=3,3.2,3.4,3.6,3.8,4,4.2$ and 19 bins in $k-$space in the range 0.003-0.057 s/km. We further use the measurements of the flux power spectrum presented in (Viel et al., 2013), at redshift bins $z=4.2,4.6,5.0,5.4$ and in 10 $k-$bins in the range 0.001-0.08 s/km. In this second sample the spectral resolution of the QSO absorption spectra obtained with the MIKE and HIRES spectrographs are about 13.6 and 6.7 km/s, respectively. As in the analysis of (Viel et al., 2013), a conservative cut is imposed on the flux power spectrum obtained from the MIKE and HIRES data, and only the measurements with $k>0.005\;\mathrm{km^{-1}\,s}$ are used to avoid possible systematic uncertainties on large scales due to continuum fitting.

Compared to XQ-100, the HIRES/MIKE sample has the advantage of probing smaller scales and higher redshift. There is a small redshift overlap between the two samples at $z=4.2$. Since the thermal broadening (measured in km/s) of Lyman-$\alpha$ forest lines is roughly constant with redshift, the presence of a cutoff in the matter power spectrum due to free-streaming becomes more prominent in velocity space at high redshift due to the $H(z)/(1+z))$ scaling between the fixed comoving length scale set by the free-streaming length and the corresponding velocity scale. Moreover, the 1D power spectrum is more sensitive to the presence of a cutoff compared to the 3D power spectrum.

Simulations. Similarly as in (Viel et al., 2013) and (Iršič et al., 2016), we model the flux power spectrum using a set of hydrodynamical simulations performed with the GADGET-3 code, a modified version of the publicly available Gadget-2 code (Springel, 2005). A simplified star formation criterion is applied for which gas particles above an overdensity 1000, and below T$=10^{5}$ K are converted into stars (e.g. (Bolton et al., 2016)). The reference model simulation has a box length of 20 Mpc$/h$ with $2\times 768^{3}$ and (cold) dark matter particles (gravitational softening of 1 comov. kpc$/h$) in a flat $\Lambda$CDM Universe with cosmological parameters $\Omega_{\rm m}=0.301$, $\Omega_{\rm b}=0.0457$, $n_{\rm s}=0.961$, $H_{0}=70.2\;\mathrm{km\,s^{-1}\,Mpc^{-1}}$ and $\sigma_{8}=0.829$ in agreement with (Planck Collaboration et al., 2016). Three different WDM models with masses $m_{\rm WDM}=2,3,4\;\mathrm{keV}$ have also been simulated in order to obtain WDM constraints.

We simulate 5 different FDM models using the transfer function provided by (Hu et al., 2000) with light axion masses mFDM of 1, 4, 5.7, 15.7 and 30 $\times 10^{-22}$ eV. These values have roughly the same k1/2 as models corresponding to thermal WDM relic masses of 1, 1.73, 2, 3, 3.87 keV, covering the range relevant for the “small scale crisis” of cold dark matter. These models were also simulated using the axionCAMB code (Hlozek et al., 2015) to obtain the linear transfer function, finding negligible impact on the simulated flux power. The corresponding $\Lambda$CDM model is also simulated along with a range of IGM thermal histories and cosmological parameters. In Fig. 1 we show the linear, non-linear and flux power spectrum at $z=5.4$ for WDM and FDM models that have the same k1/2: non-linearities erase some of the information contained in the linear power spectrum. Note that the 1D flux power is much more sensitive to the cutoff. The maximum wavenumbers at which the flux power spectrum is measured by HIRES/MIKE and XQ-100 are represented by the horizontal arrows.

It has been noted before (e.g. (Schive et al., 2016)) that for the analysis of the Lyman-$\alpha$  it is sufficient to use the appropriate transfer function without modelling the full quantum effects below the de-Broglie wavelength of the FDM particle. This hypothesis is supported by the fact that the quantum pressure starts to dominate over gravity on scales smaller than the FDM Jeans scale ($k>k_{J}$) (Schive et al., 2016). The FDM Jeans scale increases with cosmic time, and also increases with the mass of the FDM particle. For the largest redshift, $z=5.4$, and smallest mass in our simulations, $m_{\rm FDM}=1\times 10^{-22}\;\mathrm{eV}$, the FDM Jeans scale is $64.7\,h\,\mathrm{Mpc^{-1}}$, which corresponds to scales smaller than the scale probed by our data ($k_{\mathrm{max}}=12.7\,h\,\mathrm{Mpc^{-1}}$, for HIRES at $z=5.4$). 111Furthermore, the growth rate ratio $\xi$, as defined in (Schive et al., 2016), is $\xi>0.995$ for scales $k<12.7\,h\,\mathrm{Mpc^{-1}}$ for the redshift range considered in this paper. The value of $\xi$ decreases as we approach the FDM Jeans scale, with $\xi=0.55$ at $k=40\,h\,\mathrm{Mpc^{-1}}$. The effect of quantum pressure term should thus have a negligible effect on the structure formation relevant for the Lyman-$\alpha$  forest.

We vary the thermal history by modifying the photo-heating rates in the simulations as in (Bolton et al., 2008). The low density IGM ($\Delta=1+\delta<10$) is well described by a power-law temperature-density relation, $T=T_{0}\Delta^{\gamma-1}$. We consider a range of values for the temperature at mean density $T_{0}$ and the slope of the $T-\rho$ relation, $\gamma$, based on the previous analysis of the Lyman-$\alpha$  forest and recent observations (Becker et al., 2011). As in (Iršič et al., 2016) these consist of a set of three different temperatures at mean density, $T_{0}(z=3.6)=7200,11000,14800\;\mathrm{K}$, which evolve with redshift, as well as a set of three values of the slope of the $T-\rho$ relation: $\gamma(z=3.6)=1.0,1.3,1.5$. The reference thermal history assumes $(T_{0}(z=3.6),\gamma(z=3.6))=(11000\;\mathrm{K},1.5)$.

Following again (Iršič et al., 2016) we use two parameters describing cosmology, $\sigma_{8}$ and $n_{\rm eff}=d\ln{P_{\rm m}(k)}/d\ln{k}$, evaluated at $k=0.005\;\mathrm{km^{-1}\,s}$. Five different values are considered for both $\sigma_{8}=0.754,0.804,0.829,0.854,0.904$, and $n_{\rm eff}=-2.3474,-2.3274,-2.3074,-2.2874,-2.2674$. The reference model has $(\sigma_{8},n_{\rm eff},n_{\rm s})=(0.829,-2.3074,0.961)$. We also vary the redshift of reionization $z_{\rm rei}$ which is chosen to be $z_{\rm rei}=9$ for the reference model as well as $z_{\rm rei}=7,15$ for two additional models (sup, ). The last parameter ($f_{\rm UV}$) characterizes the effect of Ultraviolet (UV) background fluctuations. An extreme model dominated by QSOs has been chosen with a strong scale dependence at higher redshift and towards large scales. The mean flux is also varied a-posteriori through rescaling the effective optical depth, $\tau_{\rm eff}=-\ln{\bar{F}}$. We use three different values $(0.8,1,1.2)\times\tau_{\rm obs,eff}$, with the observed value of $\tau_{\rm obs,eff}$ chosen to be those of the SDSS-III/BOSS measurements (Palanque-Delabrouille et al., 2013).

Method. Using the models of the flux obtained from the simulations we establish a grid of points for each redshift, in the parameter space of $({\bar{F}}(z),T_{0}(z),\gamma(z),\sigma_{8},z_{\rm rei},n_{\rm eff},f_{\rm UV},m_{\rm FDM})$. We then perform a linear interpolation between the grid points in this multidimensional parameter space, to obtain predictions of flux power for the desired models. The interpolation is performed for $P_{\rm F}(k,z)$, directly, rather than for ratios of flux power w.r.t. the corresponding $\Lambda CDM$ simulation as was done in Viel et al. (2013). Parameter constraints are then obtained with a Monte Carlo Markov Chain code that explores the likelihood space until convergence is reached.

The redshift evolution of the IGM parameters $T_{0}$ and $\gamma$ are modelled as power-laws for our reference analysis: $T_{0}(z)=T_{0}^{A}\left[(1+z)/(1+z_{\rm p})\right]^{T_{0}^{S}}$ and $\gamma(z)=\gamma^{A}\left[(1+z)/(1+z_{\rm p})\right]^{\gamma^{S}}$. The pivot redshift is different for each data set and roughly corresponds to the redshift at which most of the data lies ($z_{\rm p}=3.6,4.5,4.2$ for XQ-100, HIRES/MIKE and the combined analysis, respectively).

Results. In Figure 2 we show the main result of this letter: the marginalized 1D likelihood for 1/mFDM. For our reference analysis, in which the temperature evolution is parameterized as a power-law at different pivot redshifts, XQ-100 returns an upper limit of $4.6\times 10^{-22}$ eV, HIRES/MIKE gives 16.4$\times 10^{-22}$ eV, while the combination of the two data sets results in a considerable improvement to 37.5$\times 10^{-22}$ eV (2$\sigma$ C.L.). These numbers become 2.7, 16.5, 32.2 $\times 10^{-22}$ eV, for XQ-100, HIRES/MIKE and both data sets, when using the following Planck priors on n${}_{\rm eff}=-2.307\pm 0.01$ and $\sigma_{8}=0.829\pm 0.01$ (1$\sigma$ Gaussian priors). The improvement in the joint constraints is due to the fact that when combining the two data sets the thermal evolution of the IGM is assumed to be one power-law for both $T_{0}(z)$ and $\gamma(z)$ for the full redshift range of the combined data sets. If we drop the assumption of a power-law evolution and we let the temperature vary independently in each redshift bin, with a maximum jump $\Delta T=5000\,$K in bins that are separated by $\Delta z=0.2$, we obtain 7.1, 14.3, 20.0 $\times 10^{-22}$ eV for XQ-100, HIRES/MIKE and both combined. We regard this result as the most conservative, since sudden jumps of temperature are not physically plausible in this redshift range (e.g. (Puchwein et al., 2015; Upton Sanderbeck et al., 2016)).

Increasing the covariance matrix by a multiplicative factor 1.3 in order to better represent a possible underestimation of the errors does not affect the results appreciably. In Tab. 1 we summarize the results including the $\chi^{2}/d.o.f.$ for the reference case, which appear to be very reasonable in all cases. For the combined analysis the other parameters lie within the following 2$\sigma$ C.L. ranges at $z_{\rm p}=4.2$: $\sigma_{8}=[0.83,0.95]$, $n_{\rm eff}=[-2.43,-2.31]$, $T^{A}(z_{\rm p})\;\mathrm{[10^{4}\,K]}=[0.71,1.06]$, $T^{S}(z_{\rm p})=[-3.37,-0.80]$, $\gamma^{A}(z_{\rm p})=[1.27,1.69]$, $\gamma^{S}(z_{\rm p})=[-0.11,1.82]$, $z_{\rm rei}=[6.27,13.62]$, $f_{\rm UV}=[0.04,0.94]$.

We have verified the constraints obtained by considering additional simulations with $m_{\rm FDM}$ of 30 $\times 10^{-22}$ eV.

In Figure 3 we illustrate the relation between WDM and FDM constraints. Due to the lack of the detailed modelling of the flux power spectrum for FDM models using hydrodynamical simulation so far, it has been usually assumed that one can map thermal relic WDM masses into FDM constraints by identifying the corresponding k1/2 values for the linear power spectra, i.e. the wavenumber at which the power reaches 50% of the $\Lambda$CDM linear power spectrum. Figure 3 shows that this is not a good approximation. For XQ-100 only, i.e. for a WDM constraint of 1.34 keV, the corresponding FDM limit is slightly above the k1/2 curve, which remains a reasonable approximation. The HIRES/MIKE data only, however, gives a WDM lower limit of 4.7 keV that translates into a FDM limit which is much weaker than the one that would be obtained using the k1/2 mapping. The same holds for the combined analysis. For these higher thermal relic WDM masses the WDM constraints are better mapped into FDM constrains by using k0.75 rather than k1/2, where k0.75 is defined as the wavenumber at which the power reaches 75% of the $\Lambda$CDM linear power spectrum. The reason for this is that as the free-streaming cut-off moves to smaller scales for larger particle masses, the scales affected by the free-streaming cut-off become more non-linear.

We have also checked that similar conclusions are reached when considering the 1D linear power spectrum, which is the quantity really probed by the flux power spectrum (rather than the 3D power spectrum). The different data sets are obviously constraining linear power at different $(k,z)$ values. Non-linearities will develop differently in WDM and FDM models and this makes the mapping between the two scenarios not straightforward. Full non-linear hydrodynamic simulations and detailed modelling of the flux power spectrum is required for accurate constraints on FDM models. WDM models retain more small scale power at $k>k_{0.75}$ compared to the corresponding FDM modelq that has instead a more prominent knee at $k<k_{0.75}$. These differences partially compensate in terms of non linear 1D flux power in a non-trivial way that depends on how non-linear the matter power spectrum is at the free-streaming scale. Similar to the shape of the WDM cutoff, the FDM cutoff in the flux power spectrum appears to be rather distinctive with no significant degeneracies with the other parameters in the analysis.

Conclusions. We have presented constraints on FDM models based on detailed modelling of the Lyman-$\alpha$ forest 1D flux power spectrum and high resolution data at intermediate and high redshifts with hydrodynamical simulations. These are the first constraints that incorporate the effect of the relevant IGM physics, including thermal and pressure smoothing on the non-linear evolution of the flux power spectrum on the relevant scales. Our final, conservative lower limit from a joint analysis of the intermediate and high-resolution Lyman-$\alpha$ forest data, m${}_{\mathrm{FDM}}>20\times 10^{-22}$ eV (2$\sigma$ C.L.), was obtained with conservative assumptions for the thermal history of the IGM that allow for (unphysical) sudden jumps of the IGM temperature up to 5000K. This lower limit for the mass of ultra-light bosons strengthens by about a further factor two if we assume a smoother thermal history of the IGM. Our analysis appears to close the window of FDM models with significant astrophysical implications, in particular for alleviating the tension between observations and theoretical predictions of cold dark matter models on small scales. 222After this paper was completed an analysis of IGM data has been performed by Armengaud et al. (2017) reaching similar conclusions with slightly weaker constraints on FDM mass.

Supplemental material

The effect of pressure smoothing

The gas physics imprints two distinct scales on the distribution of the flux. The first is the thermal Doppler broadening, which smoothes the optical depth along the line of sight and this effect depends on the temperature at a given redshift. The second effect is from Jeans thermal pressure smoothing which affects the gas distribution relative to the 3D dark matter distribution. Because it takes some time for the gas to respond to the changes in temperature, this effect depends on the thermal history evolution at earlier times.

Similarly to pressure smoothing, additional smoothing due to FDM model also affects the 3D distribution. Thus a certain degree of degeneracy is expected.

In the analysis presented in this paper we have parametrised the effects of thermal physics by three main observables ($T_{0}$,$\gamma$ and $z_{\rm rei}$), with further parameterisation of the redshift evolution of $T_{0}$ and $\gamma$ (see main text for details). The pressure smoothing scale would thus mostly differ from the Doppler broadening scale due to the value of the redshift of reionisation ($z_{\rm rei}$).

Due to the different manifestation of the smoothing (1D vs 3D) between the thermal parameters, as well as different redshift evolution between thermal smoothing scales and FDM smoothing scale, degeneracies in the parameter space can be broken. This is shown in Fig. 4. The left hand panel shows that the temperature ($T_{0}$ at a chosen redshift) is degenerate with the FDM particle mass, particularly when estimating the parameters in the lower redshift range (blue contours - XQ-100 data set). The degeneracy is broken efficiently when moving to higher redshift range (red contours - HIRES/MIKE data) and almost gone when including all the data (green contours).

Moreover, the right hand side of Fig. 4 shows the degeneracy between the $m_{\rm FDM}$ and $z_{\rm rei}$ parameters. It is clear that when performing the analysis on higher redshift data, and increasing the redshift range, the degeneracy between these two parameters is almost broken. This indicates the importance of the different redshift evolution of the smoothing scales, for estimating the parameter values.

To further support this, we have estimated the thermal Jeans pressure scale following (Gnedin and Hui, 1998). As was shown in (Gnedin and Hui, 1998), the pressure smoothing scale (in $\mathrm{Mpc}$/h) will always be smaller than the instantaneous Jeans scale. Moreover, a more recent analysis on the simulations has shown that the true pressure smoothing scale (at redshift concerning the Lyman-$\alpha$  forest) lies somewhere between the filtering and Jeans smoothing scale (Kulkarni et al., 2015).

As was already discussed, the degeneracy between $z_{\rm rei}$ and $m_{\rm FDM}$ is very small, thus the expected difference between degeneracies in $m_{\rm FDM}$ vs $\lambda_{J}$ plane will be small. This is illustrated in Fig. 5. The figure further shows (in the plane of the smoothing scales) that while degeneracy exists - either there is more thermal smoothing and less FDM smoothing or viceversa - it is severely broken when using higher redshift data, and when extending the redshift range of the analysis considered.

The best-fit FDM model

The Fig. 6 shows the data sets used in the analysis along with the best-fit model of the power-spectrum, when performing the analysis on the combined data set using the REF prior model. For comparison the plot also shows the power spectrum where the parameters were kept fixed at best-fit values, except the FDM particle mass was decreased down to $10\times\mathrm{10^{-22}eV}$ to guide the eye. The plot nicely illustrates that the shape of the cutoff is the most constraining effect when considering FDM models.