(chap:16)= # Quantum questions in cosmology :::{admonition} Chapter opening :class: chapter-opening The CMB is the closest astronomical object to "the whole Universe as an optical experiment," but it is not a quantum light source that can be prepared again and again in the laboratory. The observer receives a sky map with frequency, polarization, angular position, and noise weights, and then estimates angular power spectra, correlation functions, non-Gaussianity, and polarization rotation. Quantum language enters through primordial fluctuations, mode amplification during inflation, two-mode squeezing, and decoherence. Data language uses $\mu{\rm K}$-level temperature fluctuations, $E/B$ polarization, $C_\ell$, $f_{\rm NL}$, the tensor-to-scalar ratio $r$, and systematic-error matrices. COBE/FIRAS, COBE/DMR, WMAP/Planck, BICEP/Keck, and CMB birefringence analyses set the observational scales for these quantities {cite:p}`1967PhRvL..19.1199S,1990ApJ...354L..37M,1992ApJ...396L...1S,1994ApJ...420..439M,1996ApJ...473..576F,2009ApJ...707..916F,2020A&A...641A...1P,2020A&A...641A...5P,2020A&A...641A...6P,2020A&A...641A..10P,2021PhRvL.127o1301A`. ::: ## What kind of light field is the CMB? The mean CMB is very close to a blackbody with temperature ```{math} T_0 = 2.7255\pm0.0006\,{\rm K}. ``` The Planck distribution gives the mean photon occupation number and specific intensity in each frequency mode, ```{math} :label: eq:ch16-planck-spectrum n_\nu = \frac{1}{\exp(h\nu/k_{\rm B}T_0)-1}, \qquad I_\nu = \frac{2h\nu^3}{c^2}n_\nu . ``` $\nu$ is in Hz, and the SI units of $I_\nu$ are ${\rm W\,m^{-2}\,Hz^{-1}\,sr^{-1}}$; CMB papers often convert this to ${\rm MJy\,sr^{-1}}$. At $30\,{\rm GHz}$, $h\nu/k_{\rm B}T_0\simeq0.53$ and $n_\nu\sim1.4$, so low-frequency channels are still close to a classical thermal field. At $150\,{\rm GHz}$, $n_\nu\simeq0.07$, below one photon per mode, although the total photon count is large because the telescope bandwidth and angular resolution element contain many modes. The blackbody intensity peaks near $160\,{\rm GHz}$, which is one reason Planck HFI and many ground-based CMB experiments use channels near $90/150/220\,{\rm GHz}$. FIRAS constrained the blackbody spectrum so tightly that early energy injection, spectral distortions, and nonthermal backgrounds are all small. Modern CMB cosmology therefore treats the mean spectrum as a known background and puts most of the information in angular fluctuations and polarization {cite:p}`1994ApJ...420..439M,1996ApJ...473..576F,2009ApJ...707..916F`. ```{figure} ../_static/figures/generated/chapter_16/ch16_cmb_blackbody.png :name: fig:ch16-cmb-blackbody :width: 94.0% The mean CMB light field is specified by a blackbody spectrum and a mode occupation number. The left panel writes the $T_0=2.7255\,{\rm K}$ Planck spectrum in ${\rm MJy\,sr^{-1}}$, peaking near $160\,{\rm GHz}$. The right panel shows the mean photon number per mode at different frequencies: low-frequency channels have high occupation, while high-frequency channels move toward n_ν < 1. ``` Angular fluctuations are expanded as ```{math} :label: eq:ch16-temperature-expansion T(\hat{\bf n}) = T_0\,[1+\Theta(\hat{\bf n})], \qquad \Theta(\hat{\bf n}) = \sum_{\ell m}a_{\ell m}^{T}Y_{\ell m}(\hat{\bf n}). ``` $\hat{\bf n}$ is sky direction, and $a_{\ell m}^T$ is the dimensionless temperature-fluctuation coefficient. In maps, $T_0\Theta$ is usually quoted in $\mu{\rm K}$. COBE/DMR first detected all-sky structure at $\Delta T/T\sim10^{-5}$. Modern Planck maps decompose this signal out to $\ell\sim2500$. Multipole $\ell$ and angular scale are roughly related by $\theta\simeq180^\circ/\ell$: $\ell\simeq2$ is half the sky, $\ell\simeq200$ is about a degree, and $\ell\simeq2000$ is several arcminutes. For an isotropic Gaussian sky, the angular power-spectrum estimator is ```{math} :label: eq:ch16-cl-estimator \widehat C_\ell^{TT} = \frac{1}{2\ell+1} \sum_{m=-\ell}^{\ell} |a_{\ell m}^{T}|^2, \qquad D_\ell^{TT} = \frac{\ell(\ell+1)}{2\pi}C_\ell^{TT}. ``` The units of $C_\ell$ depend on whether $a_{\ell m}$ has been multiplied by temperature. If $a_{\ell m}$ is in $\mu{\rm K}$, then $C_\ell$ and $D_\ell$ are in $\mu{\rm K}^2$. The first acoustic peak of $D_\ell^{TT}$ is at $\ell\simeq220$, with height about $5\times10^3\,\mu{\rm K}^2$. The peaks come from acoustic oscillations in the photon-baryon fluid before recombination. Baryon density changes the heights of compression peaks. Dark-matter density changes gravitational-potential decay. The angular-diameter distance projects the physical sound horizon into angular scale. In the six-parameter flat $\Lambda{\rm CDM}$ model, Planck 2018 gives typical precision around $\Omega_bh^2=0.0224$, $\Omega_ch^2=0.120$, $n_s=0.965$, $\tau=0.054$, $H_0=67.4\,{\rm km\,s^{-1}\,Mpc^{-1}}$, $\Omega_m=0.315$, and $\sigma_8=0.811$ {cite:p}`1992ApJ...396L...1S,1996ApJ...469..437S,2020A&A...641A...5P,2020A&A...641A...6P,2021A&A...652C...4P`. Even with no instrumental noise, the full sky gives only $2\ell+1$ values of $m$ for each $\ell$. For a Gaussian sky, the cosmic variance is ```{math} :label: eq:ch16-cosmic-variance \frac{\sigma(C_\ell)}{C_\ell} = \sqrt{\frac{2}{(2\ell+1)f_{\rm sky}}}, ``` where $f_{\rm sky}$ is the effective sky fraction. On the full sky, the relative uncertainty is about $63\%$ at $\ell=2$, $7\%$ at $\ell=200$, and $2.2\%$ at $\ell=2000$. Masking the Galactic plane and strong foreground regions gives $f_{\rm sky}<1$, increasing the uncertainty. Low $\ell$ anomalies, a low quadrupole, or hemispherical asymmetry therefore have to be judged inside the covariance of a finite number of modes, not only by how conspicuous one point looks on a plot. ```{figure} ../_static/figures/generated/chapter_16/ch16_cmb_power_spectrum.png :name: fig:ch16-cmb-power :width: 94.0% The angular power spectrum compresses sky fluctuations into multipole space. The left panel sketches the acoustic peaks and damping tail, with the blue band showing the scale of full-sky cosmic variance. The right panel plots $\sqrt{2/(2\ell+1)}$ alone, showing that large-angle uncertainty is limited by the number of available sky modes, not by telescope sensitivity. ``` Polarization maps add two Stokes parameters to temperature. Combining $Q$ and $U$ into a spin-2 field, ```{math} :label: eq:ch16-eb-definition (Q\pm iU)(\hat{\bf n}) = \sum_{\ell m} a_{\pm2,\ell m}\,{}_{\pm2}Y_{\ell m}(\hat{\bf n}), \qquad a_{\ell m}^{E} = -\frac{1}{2}(a_{2,\ell m}+a_{-2,\ell m}), \quad a_{\ell m}^{B} = \frac{i}{2}(a_{2,\ell m}-a_{-2,\ell m}). ``` The $E$ mode transforms like a scalar under parity, and the $B$ mode like a pseudoscalar. Scalar density perturbations produce $T$ and $E$ at linear order but no primordial $B$. Tensor perturbations, weak lensing, polarization- angle rotation, Galactic dust, and synchrotron emission can all produce $B$. A $B$-mode detection therefore has to be interpreted with frequency separation, delensing, angle calibration, and foreground modeling before it can point to primordial gravitational waves. The full-sky $E/B$ decomposition was developed in the classic work of Zaldarriaga and Seljak and of Kamionkowski, Kosowsky, and Stebbins. BICEP/Keck constraints later turned that mathematical decomposition into the main observational channel for primordial gravitational wave searches {cite:p}`1997PhRvD..55.1830Z,1997PhRvL..78.2054S,1997PhRvL..78.2058K,2021PhRvL.127o1301A`. ## Inflationary fluctuations as squeezed states Inflationary quantum fluctuations are usually not written in the language of individual photons. They are written as modes of gauge-invariant fields. For the scalar curvature perturbation $\mathcal R$, define the Mukhanov--Sasaki variable ```{math} :label: eq:ch16-ms-variable v({\bf x},\eta)=z(\eta)\mathcal R({\bf x},\eta), \qquad z=\frac{a\dot\phi}{H}, ``` where $\eta$ is conformal time, $a$ is the scale factor, $\phi$ is the background field driving inflation, and $H=\dot a/a$. In the quadratic action, each Fourier mode is approximately a harmonic oscillator with time-dependent frequency: ```{math} :label: eq:ch16-ms-equation v_k''+ \left(k^2-\frac{z''}{z}\right)v_k = 0 . ``` $k$ is comoving wavenumber, often in ${\rm Mpc^{-1}}$, and primes denote derivatives with respect to $\eta$. Well inside the horizon, $k\gg aH$, the equation reduces to $v_k''+k^2v_k=0$, and the Bunch--Davies vacuum is ```{math} v_k\simeq \frac{e^{-ik\eta}}{\sqrt{2k}}. ``` After the mode crosses the horizon at $k\simeq aH$, the $z''/z$ term dominates, the growing solution freezes, the decaying solution rapidly becomes small, and $\mathcal R_k=v_k/z$ becomes nearly constant. Here "freezing" means that one quadrature in phase space is squeezed strongly, leaving a random amplitude that later controls the density perturbation. It does not mean that the mode literally stops evolving {cite:p}`1988JETP...67.1297M,1990PhRvD..42.3413G,1996CQGra..13..377P`. The power spectrum is defined by ```{math} :label: eq:ch16-primordial-spectrum \langle \mathcal R_{\bf k}\mathcal R_{{\bf k}'} \rangle = (2\pi)^3\delta^{(3)}({\bf k}+{\bf k}') \frac{2\pi^2}{k^3} \mathcal P_{\mathcal R}(k), \qquad \mathcal P_{\mathcal R}(k) = A_s \left(\frac{k}{k_*}\right)^{n_s-1}. ``` $A_s$ is dimensionless. Planck often uses the pivot $k_*=0.05\,{\rm Mpc^{-1}}$ and finds $\ln(10^{10}A_s)\simeq3.04$, or $A_s\simeq2.1\times10^{-9}$. Exact scale invariance would be $n_s=1$. Planck's $n_s\simeq0.965$ is a red tilt, meaning that long-wavelength modes are slightly stronger. This primordial curvature power of $10^{-9}$ is processed by radiation transfer functions, baryon acoustic oscillations, the finite thickness of the recombination surface, lensing, and smoothing before appearing as today's $\Delta T/T\sim10^{-5}$ sky fluctuations {cite:p}`2020A&A...641A...6P,2020A&A...641A..10P`. In quantum-optics language, inflation evolves the ${\bf k}$ and $-{\bf k}$ modes from vacuum into a two-mode squeezed state, ```{math} :label: eq:ch16-squeezed-state |\psi\rangle = \prod_{\bf k} \exp\!\left[ r_k e^{-2i\varphi_k}a_{\bf k}a_{-{\bf k}} - r_k e^{2i\varphi_k}a_{\bf k}^\dagger a_{-{\bf k}}^\dagger \right] |0\rangle . ``` $r_k$ is the squeezing parameter and $\varphi_k$ is the squeezing angle. The mean occupation number is $N_k=\sinh^2r_k$; long after horizon exit, $r_k$ is large and $N_k$ can grow exponentially. This occupation number refers to quantum excitations of the early field mode, not to the CMB photons counted by a telescope. For modes that re-enter the horizon after $50$--60 e-folds, the decaying quadrature is already tiny. The Wigner function is a long, thin ellipse, and later acoustic oscillations inherit a fixed phase. This fixed phase explains the sequence of acoustic peaks in the CMB $TT$, $TE$, and $EE$ spectra rather than a random-phase ripple pattern {cite:p}`1990PhRvD..42.3413G,1996CQGra..13..377P,2006CQGra..23.2317P`. ```{figure} ../_static/figures/generated/chapter_16/ch16_squeezed_modes.png :name: fig:ch16-squeezed :width: 94.0% Mode amplification during inflation can be viewed as two-mode squeezing. The left panel shows k/(aH) falling exponentially after horizon exit, while the decaying quadrature is squeezed. The right panel shows phase- space ellipses: as r_k grows, one quadrature stretches and the other narrows. The observed information is the long-axis degree of freedom that later behaves as a classical random amplitude. ``` Decoherence turns the "pure squeezed state" into an effective state that can be described by a classical random field. The environment can be unobserved short-wavelength modes, scalar-tensor interactions, plasma degrees of freedom before recombination, the lensing potential, or foreground and instrumental modes marginalized over in data analysis. For an observed variable $q_k$, the reduced density matrix can be approximated as ```{math} :label: eq:ch16-decoherence-density \rho_{\rm red}(q,q') = \rho_0(q,q') \exp\!\left[-\Gamma_k(q-q')^2\right], ``` where $\Gamma_k$ is the decoherence strength, with units determined by the normalization of $q_k$. If $\Gamma_k(\Delta q)^2\gg1$, interference between different amplitude branches is suppressed, and power spectra and correlation functions can be computed as for a classical random field. The "measurement problem" in the early Universe remains conceptually debated, but for observables such as $C_\ell$, bispectra, and lensing reconstruction, the surviving phase coherence is not enough to appear as a laboratory-style Bell test or a reproducibly prepared nonclassical state {cite:p}`1996CQGra..13..377P,2006CQGra..23.2317P`. ## Non-Gaussianity and tensor modes If $\mathcal R$ is exactly Gaussian, all information is in the two-point function and all connected three-point and higher-point functions vanish. Inflationary interactions, nontrivial sound speed, multifield transfer, excited initial states, and sharp features can produce non-Gaussianity. The three-point function is written ```{math} :label: eq:ch16-bispectrum \langle \mathcal R_{{\bf k}_1} \mathcal R_{{\bf k}_2} \mathcal R_{{\bf k}_3} \rangle = (2\pi)^3 \delta^{(3)}({\bf k}_1+{\bf k}_2+{\bf k}_3) B_{\mathcal R}(k_1,k_2,k_3). ``` The three $k$'s form a triangle. The local shape is strongest in squeezed triangles, $k_1\ll k_2\simeq k_3$. The equilateral shape is strongest when all sides are equal. The orthogonal shape is an approximately orthogonal template built to separate different interactions. The local ansatz is often written ```{math} :label: eq:ch16-local-fnl \mathcal R({\bf x}) = \mathcal R_g({\bf x}) + \frac{3}{5}f_{\rm NL}^{\rm local} \left[ \mathcal R_g^2({\bf x}) - \langle\mathcal R_g^2\rangle \right], ``` where $\mathcal R_g$ is the Gaussian part and $f_{\rm NL}$ is dimensionless. Planck 2018 temperature plus polarization gives $f_{\rm NL}^{\rm local}=-0.9\pm5.1$, $f_{\rm NL}^{\rm equil}=-26\pm47$, and $f_{\rm NL}^{\rm orth}=-38\pm24$ at 68% C.L.; no standard template is significantly detected. The squeezed-limit consistency relation for single- field slow-roll inflation predicts a small local non-Gaussianity, of order $1-n_s$. Maldacena's cubic action calculation is the standard source of this result {cite:p}`2003JHEP...05..013M,2020A&A...641A...9P,2020A&A...641A..10P`. ```{figure} ../_static/figures/generated/chapter_16/ch16_non_gaussian_templates.png :name: fig:ch16-nongaussian :width: 94.0% Non-Gaussianity is specified by both triangle configuration and amplitude. The left panel compares local, equilateral, and orthogonal template shapes in a k₁ = k₂ slice; the local shape grows in the squeezed limit, while the equilateral shape is largest near equal side lengths. The right panel shows representative Planck 2018 constraints on the three common $f_{\rm NL}$ parameters, all consistent with zero. ``` Tensor perturbations are the most direct CMB target for a quantum gravitational background. Define ```{math} :label: eq:ch16-tensor-r r(k_*)= \frac{\mathcal P_t(k_*)}{\mathcal P_{\mathcal R}(k_*)}, \qquad \mathcal P_t(k) = A_t\left(\frac{k}{k_*}\right)^{n_t}. ``` $r$ is dimensionless. The pivot is often $0.002$ or $0.05\,{\rm Mpc^{-1}}$, and values with different pivots should not be mixed. Single-field slow-roll inflation gives the approximate consistency relation $n_t\simeq-r/8$. Converting $r$ into an inflationary energy scale gives ```{math} :label: eq:ch16-inflation-scale V_*^{1/4} \simeq 1.06\times10^{16}\,{\rm GeV} \left(\frac{r}{0.01}\right)^{1/4}, ``` assuming Einstein gravity, standard slow roll, and a single energy scale. Planck 2018 alone gives $r_{0.002}<0.10$ at 95% C.L. Combining with BK15 $B$-mode polarization gives $r_{0.002}<0.056$, corresponding to $V_*^{1/4}<1.6\times10^{16}\,{\rm GeV}$. The BICEP/Keck 2018 season combined with Planck and WMAP gives $r_{0.05}<0.036$ at 95% C.L. and shows that ideal simulations without lensing or dust would substantially reduce $\sigma(r)$. Current constraints are limited by both lensing $B$ modes and Galactic dust {cite:p}`2020A&A...641A..10P,2021PhRvL.127o1301A,2019JCAP...02..056A`. ```{figure} ../_static/figures/generated/chapter_16/ch16_bmode_constraints.png :name: fig:ch16-bmode :width: 94.0% Tensor modes are constrained mainly through B-mode polarization. The left panel compares lensing B modes with primordial tensor B-mode scales for different r. The recombination bump near ℓ ∼ 80 is the main target of ground-based experiments, while the low-ℓ reionization bump requires large-area control. The right panel shows BICEP/Keck combined constraints improving from r < 0.12 to r_0.05 < 0.036. ``` ## CMB polarization birefringence Chapter {ref}`chap:15` already wrote the polarization rotation caused by an axion-like field in terms of Stokes quantities and $E/B$ mixing; see Eqs. {eq}`eq:ch15-qu-rotation` and {eq}`eq:ch15-cmb-eb-tb`. In a CMB analysis, the task is to separate a tiny global angle $\beta$ from detector angle calibration, Galactic dust, synchrotron radiation, and map-making residuals. $\beta$ is inserted into trigonometric functions in radians; $0.35^\circ=6.1\times10^{-3}\,{\rm rad}$, so in the small-angle limit $EB/(EE-BB)\simeq2\beta\simeq1.2\%$. This percent-level signal is large enough to leave a statistical trace in Planck-quality all-sky data, and small enough that absolute polarization-angle calibration, dust $EB$, bandpass mismatch, or beam leakage can imitate it {cite:p}`2020PhRvL.125v1301M,2022PhRvD.106f3503E,2022PhRvL.128i1302D,2022NatRP...4..452K`. ```{figure} ../_static/figures/generated/chapter_16/ch16_birefringence_eb.png :name: fig:ch16-birefringence :width: 94.0% CMB birefringence places polarization-angle error and cosmological signal in the same plane. The left panel shows the small-angle leakage from β into EB and induced BB, with the blue band marking the scale of β = 0.35 ± 0.14^∘. The right panel shows the degeneracy direction between detector angle α and cosmic angle β: using only the CMB mainly measures α + β, while foreground frequency and multipole information helps separate them. ``` Minami and Komatsu's Planck 2018 analysis used the different frequency dependence of CMB and Galactic foreground polarization to estimate detector miscalibration $\alpha_\nu$ and cosmic birefringence $\beta$ together. They found $\beta=0.35\pm0.14^\circ$ and separated a $0.28^\circ$ ground-based angle-calibration systematic from the final error. Later WMAP+Planck PR4 work combined LFI/HFI data from $23$ to $353\,{\rm GHz}$ with a dust $EB$ model, giving a nearly all-sky value $\beta=0.342^{+0.094}_{-0.091}{}^\circ$, about $3.6\sigma$. Intrinsic polarized dust $EB$, low-frequency synchrotron $EB$, and independent angle calibration remain the main limitations. If this signal is interpreted as an axion-like field, the rotation angle is ```{math} :label: eq:ch16-axion-birefringence \beta = \frac{1}{2}g_{\phi\gamma} [\phi(t_0)-\phi(t_{\rm LSS})], ``` where $g_{\phi\gamma}$ is in ${\rm GeV^{-1}}$, and $\phi$ is the effective field-value difference between the last-scattering surface and today. This expression assumes the rotation is nearly isotropic, frequency independent, and caused by propagation rather than intrinsic $EB$ at emission. If $\beta(\nu)\propto\nu^{-2}$, the signal looks more like Faraday rotation. If $\beta$ changes with sky region, redshift, or observing time, the analysis has to move from one global angle to direction-dependent or time-dependent correlation functions {cite:p}`2020PhRvL.125v1301M,2022PhRvL.128i1302D,2022NatRP...4..452K`. ## Interface with optical quantum astronomy CMB analysis and the optical quantum astronomy of earlier chapters share the language of correlation functions, but the observables mean different things. In laboratory or stellar HBT work, one often uses ```{math} g^{(2)}(\tau) = \frac{\langle I(t)I(t+\tau)\rangle}{\langle I\rangle^2} ``` to describe intensity fluctuations in a detector time series. The basic CMB two-point function lives on the sphere: ```{math} :label: eq:ch16-cmb-correlator \langle X_{\ell m}Y_{\ell' m'}^* \rangle = C_\ell^{XY}\delta_{\ell\ell'}\delta_{mm'}, \qquad X,Y\in\{T,E,B,\phi_{\rm lens}\}. ``` For $g^{(2)}$, the time delay and bandwidth are chosen by the instrument. For $C_\ell$, the number of modes is set by the Universe; the observer can only change the sky mask, frequency combination, beam, noise weighting, and foreground model. The CMB cannot be modulated, prepared repeatedly, or put into a new quantum state like a laboratory system. It provides a projection of early field-mode statistics into $T/E/B$, lensing, and higher-order correlations. At map level, frequency maps, polarization maps, and external tracers enter one data vector ${\bf d}$: ```{math} :label: eq:ch16-cmb-likelihood {\bf d} = {\bf P} [ {\bf s}_{\rm CMB}({\boldsymbol\theta}) + {\bf s}_{\rm fg}({\boldsymbol\eta}) ] + {\bf n}, \qquad -2\ln\mathcal L = ({\bf d}-{\bf m})^{\mathsf T} C^{-1} ({\bf d}-{\bf m}) + \ln\det C . ``` ${\bf P}$ contains beam, mask, scan strategy, bandpass, and map-making. ${\boldsymbol\theta}$ contains cosmological parameters such as $A_s,n_s,\Omega_bh^2,\Omega_ch^2,\tau,r,\beta$. ${\boldsymbol\eta}$ contains foreground parameters such as dust temperature, spectral index, synchrotron spectral index, and spatial decorrelation. $C$ includes instrument noise, sample variance, foreground residuals, and calibration uncertainties. The differences between Planck, Simons Observatory, and future CMB-S4/LiteBIRD-like experiments are mainly in ${\bf P}$, $C$, frequency coverage, angular resolution, and controllable systematics {cite:p}`2016A&A...594A..13P,2016A&A...594A..20P,2017MNRAS.472.1195C,2019JCAP...02..056A,2020A&A...641A...6P`. Units and pivots are the easiest places to get confused when reading CMB papers. $T_0$ is in K, fluctuation maps are often in $\mu{\rm K}$, and the primordial $\mathcal P_{\mathcal R}$ is dimensionless. $A_s\sim2.1\times 10^{-9}$ cannot be compared directly with $D_\ell^{TT}\sim5000\,\mu{\rm K}^2$; radiation transfer functions sit between them. $k$ is in ${\rm Mpc^{-1}}$, while $\ell$ is an angular mode number; roughly, $\ell\simeq kD_A(z_*)$, where $D_A(z_*)$ is the comoving angular-diameter distance to last scattering. The pivot for $r$ is not standardized. $r_{0.002}$ and $r_{0.05}$ are not the same number, especially if $n_t$ is allowed to vary. Quantum-network telescopes bring the discussion back toward controllable experimental systems. The CMB is the opposite limit: the initial state cannot be remade and the light path cannot be reset, but sky correlation functions preserve the statistical trace of early-Universe quantum fluctuations.