Scaling transformations and the origins of light relics constraints from cosmic microwave background observations

We use here a family of scaling transformations, that scale key rates in the evolution equations, to analytically understand constraints on light relics from cosmic microwave background (CMB) maps, given cosmological models of varying degrees of complexity. We describe the causes of physical effects that are fundamentally important to the constraining power of the data, with a focus on the two that are most novel. We use as a reference model a cosmological model that admits a scaling transformation that increases light relic energy density while avoiding all of these causes. Constraints on light relics in a given model can then be understood as due to the differences between the given model and the reference model, as long as the additional light relics only interact gravitationally with the Standard Model components. This understanding supports the development of cosmological models that can evade light relics constraints from CMB maps.

Figure 1

A schematic visualization of the relationship in model space between a best-fit $\mathrm{\Lambda }\mathrm{CDM}$ model, the light relics (LR) model of interest, and a reference model used to understand the origin of the spectral differences between $\mathrm{\Lambda }\mathrm{CDM}$ and the LR model of interest. The reference model and the LR model have the same value of $N_{\mathrm{eff}}$ while the $\mathrm{\Lambda }\mathrm{CDM}$ model and the reference model have the same power spectra. Spectral differences between the LR model and the $\mathrm{\Lambda }\mathrm{CDM}$ model are thus the same as those between the LR model and the reference model. Model differences between the LR model and the reference model are fewer in number than are the model differences between the LR model and $\mathrm{\Lambda }\mathrm{CDM}$ and thus understanding the impact of these differences on spectra is easier. Spectral differences can be understood as arising from the constraints on the model space of interest that prevent it from following a FFAT scaling trajectory, and the impact of the resulting differences from FFAT scaling on the spectra.

Figure 2

CMB power spectra comparisons of the best-fit $\mathrm{\Lambda }\mathrm{CDM}(\mathrm{\Sigma }{m}_{\nu }=0)$ model to models scaled from there via FFAT scaling and MWDS scaling by $\lambda =1.1$ to $H_0=74.1\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$. The black dashed lines show the base $\mathrm{\Lambda }\mathrm{CDM}(\mathrm{\Sigma }{m}_{\nu }=0)$ model. The blue solid lines show the CMB power spectra after FFAT scaling. The symbol $Y_{\mathrm{P}}^{\mathrm{BBN}}$ means that the primordial helium fraction is derived from the BBN prediction, while $Y_{\mathrm{P}}^{\text{Sym}}$ means the helium abundances is set by the scaling transformation of Eq. (16). All of the fractional differences in the bottom panels are compared to the $\mathrm{\Lambda }\mathrm{CDM}(\mathrm{\Sigma }{m}_{\nu }=0)$ case. Cosmic variance in individual multipoles is shown as a gray band.

Figure 3

CMB power spectra comparisons of the $\mathrm{MWDS}+Y_{\mathrm{P}}^{\text{Sym}}$ model to several other light relics models. In all models, $\lambda =1.1$. The increase in the effective number of neutrino species due to free-streaming (or fluidlike) species is $\mathrm{\Delta }{N}_{\mathrm{fs}}$ ($\mathrm{\Delta }{N}_{\mathrm{fld}}$). We use the $\mathrm{MWDS}+Y_{\mathrm{P}}^{\text{Sym}}$ model as the reference model, which is indistinguishable from the corresponding FFAT scaling model. The fractional differences in the bottom panels are relative to the reference model. Cosmic variance in individual multipoles is shown as gray bands.

Figure 4

Evolution of ${\mathrm{\Theta }}_0$ and $\psi$ of the scales corresponding to the first and fifth peak in the temperature power spectrum of the $\mathrm{Mix}+Y_{\mathrm{P}}^{\text{Sym}}$ model (in green) and $\mathrm{MWDS}+Y_{\mathrm{P}}^{\text{Sym}}$ model (in red). The top parts of both panels show the amplitude of ${\mathrm{\Theta }}_0$ (dash-dotted lines) and $-\psi$ (dashed lines). The bottom parts of both panels show the difference of ${\mathrm{\Theta }}_0$ (light blue), $\psi$ (orange) and ${\mathrm{\Theta }}_0+\psi$ (dark blue) of the $\mathrm{Mix}+Y_{\mathrm{P}}^{\text{Sym}}$ model compared to the $\mathrm{MWDS}+Y_{\mathrm{P}}^{\text{Sym}}$ model; e.g., the light blue curve in the residual panel is ${\mathrm{\Theta }}_0^{\text{Mix}+Y_{\mathrm{P}}^{\text{Sym}}}-{\mathrm{\Theta }}_0^{\mathrm{MWDS}+Y_{\mathrm{P}}^{\text{Sym}}}$. The solid and dotted vertical lines show the scale factors of matter-radiation equality and last scattering, respectively.

Figure 5

CMB power spectra comparisons of free-streaming and fluid light relics models without scattering rate scaling to the $\mathrm{Mix}+Y_{\mathrm{P}}^{\text{Sym}}$ model and $\mathrm{MWDS}+Y_{\mathrm{P}}^{\text{Sym}}$ model. In all models, $\lambda =1.1$. The $\mathrm{MWDS}+Y_{\mathrm{P}}^{\text{Sym}}$ CMB power spectra are shown in solid blue lines. The black dashed lines show the same spectra of the $\mathrm{Mix}+Y_{\mathrm{P}}^{\text{Sym}}$ model as in Fig. 3. The increase in the effective number of neutrino species due to free-streaming (or fluidlike) species is $\mathrm{\Delta }{N}_{\mathrm{fs}}$ ($\mathrm{\Delta }{N}_{\mathrm{fld}}$). By $Y_{\mathrm{P}}^{\mathrm{BBN}}$ we mean that the primordial helium fraction is derived from BBN prediction. All of the fractional differences in the bottom panels are relative to the reference model, $\mathrm{MWDS}+Y_{\mathrm{P}}^{\text{Sym}}$. Cosmic variance in individual multipoles is shown as gray bands.

Figure 6

CMB power spectra of MWDS models with and without fixing the recombination history. The CMB spectra of the FFAT scaling model are shown in blue solid lines. The $\mathrm{MWDS}+Y_{\mathrm{P}}^{\text{Sym}}$ model with fixed ionization history is shown in black dashed lines. The $\mathrm{MWDS}+Y_{\mathrm{P}}^{\text{Sym}}$ model with physical recombination process is shown in red dash-dotted lines.

Figure 7

Similar to Fig. 6, but showing power spectra for a $\mathrm{\Lambda }\mathrm{CDM}$ model with massive neutrinos and power spectra for MWDS models scaled from this $\mathrm{\Lambda }\mathrm{CDM}$ model, with and without fixing $x_{\mathrm{e}}(z)$ and scaling the neutrino mass.

Figure 8

Constraints on the effective number of neutrino species ($N_{\mathrm{eff}}$) and the Hubble constant ($H_0$). The free-streaming species, fluidlike species, and dark photons all contribute to $N_{\mathrm{eff}}=N_{\mathrm{fs}}+N_{\mathrm{fld}}+N_{\gamma }^D$. In black are the constraints given the model with only free-streaming species and BBN-predicted $Y_{\mathrm{P}}$ [$N_{\mathrm{fs}}+\text{Recfast}+\mathrm{\Sigma }{m}_{\nu }(=0.06\mathrm{eV})$], while the constraints given the model with only free-streaming species and free $Y_{\mathrm{P}}$ are shown in gray [$N_{\mathrm{fs}}+Y_{\mathrm{P}}+\text{Recfast}+\mathrm{\Sigma }{m}_{\nu }(=0.06\mathrm{eV})$]. Results given the model with free $N_{\mathrm{fs}}$, $N_{\mathrm{fld}}$ and $Y_{\mathrm{P}}$ are shown in blue [$N_{\mathrm{fs}}+N_{\mathrm{fld}}+Y_{\mathrm{P}}+\text{Recfast}+\mathrm{\Sigma }{m}_{\nu }(=0.06\mathrm{eV})$]. The green curves show the constraints of the $\mathrm{MWDS}+Y_{\mathrm{P}}+\text{Recfast}+\mathrm{\Sigma }{m}_{\nu }(=0.06\mathrm{eV})$ model. In these models, the primordial helium abundance, $Y_{\mathrm{P}}$, is also a free parameter. The total mass of neutrinos, $\mathrm{\Sigma }{m}_{\nu }$ is set to 0.06 eV in all three models. The gray band in the left two panels shows the $H_0$ measurement from [117].

Figure 9

Constraints in the $N_{\mathrm{eff}}-Y_{\mathrm{P}}$ and $N_{\mathrm{eff}}-H_0$ planes from a variety of datasets. The black contour lines indicate the 68% and 95% credible regions from CMB and BAO data given the $\mathrm{MWDS}+Y_{\mathrm{P}}$ model. In the right panel are also samples from that posterior probability distribution color coded by the age of the Universe at $z=0$. In the left panel the green contours, so tight in one direction that they appear collapsed down to a green curve, are the 68% and 95% credible regions for BBN-consistent helium for the same model and datasets. Also shown in the left panel are $Y_{\mathrm{P}}$ constraints from [14], inferred from extragalactic regions of metal-poor ionized gas, and constraints on $N_{\mathrm{eff}}$ derived from inferences of helium and deuterium and BBN predictions [118]. The right edge of the shaded region in the right panel is the 95% confidence upper limit on $H_0$ from Vagnozzi et al. [119] based on inferred ages of old astrophysical objects over a range of redshifts. It is cosmology model dependent, but in a way that makes it appropriate for our application here, as we describe in the text.

Figure 10

Fractional contributions of different source terms to the difference in the unlensed CMB power spectrum between the $\mathrm{Mix}+Y_{\mathrm{P}}^{\text{Sym}}$ model and the FFAT scaling model. The gray vertical lines show the peak locations in CMB power spectrum and the black vertical lines show the trough locations.

Figure 11

Residuals of $\mathrm{Mix}+Y_{\mathrm{P}}^{\text{Sym}}$ model with respect to $\mathrm{MWDS}+Y_{\mathrm{P}}^{\text{Sym}}$ model at the time of last scattering as a function of $k/h$. The baryon-to-photon density ratio factor, $R$, at last scattering is 0.61.

Figure 12

Fractional changes of the lensed and unlensed power spectra from $\mathrm{Mix}+Y_{\mathrm{P}}^{\text{Sym}}$ model to the FFAT scaling model.