A User-Friendly Dark Energy Model Generator
AA User-Friendly Dark Energy Model Generator
Kyle A. Hinton, ∗ Adam Becker, † and Dragan Huterer
1, 3, 4, ‡ Department of Physics, University of Michigan, 450 Church St, Ann Arbor, MI 48109-1040 Freelance, Oakland, CA 94610 Max-Planck-Institut for Astrophysics, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany Excellence Cluster Universe, Technische Universit¨at M¨unchen, Boltzmannstrasse 2, 85748 Garching, Germany
We provide software with a graphical user interface to calculate the phenomenology of a wide classof dark energy models featuring multiple scalar fields and potentials that are arbitrary functionsof exponentials. The user chooses a subclass of models and, if desired, initial conditions, or else arange of initial parameters for Monte Carlo. The code calculates the energy density of componentsin the universe, the equation of state of dark energy, and the linear growth of density perturbations,all as a function of redshift and scale factor. The output also includes an approximate conversioninto the average equation of state, as well as the common ( w , w a ) parametrization. The code isavailable here: http://github.com/kahinton/Dark-Energy-UI-and-MC I. INTRODUCTION
The puzzle of dark energy is one of the most importantoutstanding questions in all of physics. Studying the ob-servational signatures of dark energy – and using thosesignatures to understand the nature of dark energy – hasbeen a highly active area of research since the first signsof dark energy were discovered over 15 years ago. (For areview, see e.g. Frieman et al. [1].) Dark energy modelshave a rich phenomenology, leaving varied fingerprintson the growth of cosmic structure and on geometricalquantities in the universe.We have written a piece of software that makes iteasier to compute quantitative predictions for dark en-ergy models broader than a simple cosmological constant.These models possess a richer phenomenology that canbe tested with ongoing and future experiments. Theyalso complement the choice of dark energy models – andcomputer code – made by others [2]. Our software iscapable of handling a wide class of models with multi-ple scalar fields, and specifically makes two subclassesparticularly easy to calculate with a simple user input.In these models, the user can calculate the expansionhistory and the growth of structure in the linear regime,therefore enabling the user to obtain various implicationsfor observable quantities in cosmology. The code also re-turns an approximate conversion of the equation of stateof dark energy w ( a ) into two commonly used parameters( w , w a ), as well as several other commonly used param-eters. II. DARK ENERGY MODELS
We choose two classes of “assisted dark energy” mod-els, based specifically on the analysis by Ohashi and Tsu- ∗ [email protected] † [email protected] ‡ [email protected] jikawa [3], and originally developed by Refs. [4, 5]. Thestarting assumption is the existence of n scalar fields, anda Lagrangian density that takes the following form L fields = n (cid:88) i =1 X i g ( X i e λ i φ i ) (1)where X i ≡ − g µν ∂ µ φ i ∂ ν φ i / i -thfield, and g ( Y ) is an arbitrary function of its argument Y ≡ X e λφ , where λ is a dimensionless parameter. Theexistence of scaling solutions dictates that this be themost general form of the Lagrangian density for expo-nential potentials [4, 5]. A noteworthy special case of ourclass of models is quintessence with a single field ( n = 1),and a simple exponential potential V ( φ ) = c exp( λφ ), forwhich g ( Y ) = 1 − c/Y . As we explain below, this classof models also includes ghost condensate models [6] fea-turing a background scalar field which, despite having akinetic term of the “wrong” sign, are stable and feasibledue to suitable higher-order kinetic terms.Our code allows a user to test models with an arbitrary FIG. 1. Sample screenshot of the software, showing text in-puts for dark energy model parameters, as well as checkboxesfor selecting the desired plots. a r X i v : . [ a s t r o - ph . C O ] J u l FIG. 2. Example plots generated by our program, showing output for a single model for our quintessence (left panel) or ghostcondensate (right panel) class of models. The user can choose to plot this for any arbitrary choice of model parameters, as afunction of either scale factor a or redshift z . function g ( Y ), for an arbitrary number of fields n . As anexample we also provide two special cases which havebeen run through the Monte Carlo generator, and whoseresults can be accessed with very simple inputs in thegraphical user interface (GUI): g ( Y i ) = 1 − c i /Y i (quintessence) (2) g ( Y i ) = − c i Y i (ghost condensate) . (3)In each case, the temporal evolution of all quantities ismost easily tracked in terms of scaled variables x i ≡ ˙ φ i √ H ; y i ≡ e − λ i φ i / √ H ; u ≡ √ ρ r √ H (4)where ρ r is the radiation energy density. The evolutionequations, expressed in terms of the time variable N ≡ ln( a ), are [3] dx i dN = x i (cid:34) n (cid:88) i =1 x i g ( Y i ) + u − √ λ i x i (cid:35) + √ A ( Y i ) (cid:104) λ i Ω φ i − √ { g ( Y i ) + Y i g (cid:48) ( Y i ) } x i (cid:105) ,dy i dN = y i (cid:34) n (cid:88) i =1 x i g ( Y i ) + u − √ λ i x i (cid:35) , (5) dudN = u (cid:34) − n (cid:88) i =1 x i g ( Y i ) + u (cid:35) . (6)We numerically evolve these equations. Then all phys-ical quantities of interest can be found, for example theenergy densities of matter and dark energyΩ DE = n (cid:88) i =1 x i [ g ( Y i ) + 2 Y i g (cid:48) ( Y i )] (7)and the dark energy equation of state w DE ≡ (cid:80) ni =1 p φ i (cid:80) ni =1 ρ φ i = (cid:80) ni =1 x i g ( Y i ) (cid:80) ni =1 x i [ g ( Y i ) + 2 Y i g (cid:48) ( Y i )] (8) where the time-dependent parameter Y i can be calcu-lated via Y i = x i /y i . Note that for quintessence, thefunctional form of g ( Y i ) in Eq. (2) implies that the po-tential is the sum of exponential potentials for each field, V ( φ ) = (cid:80) i c i exp( λ i φ i ). III. USING THE PROVIDED SOFTWARE
Our code can be found at the followingGitHub repository: http://github.com/kahinton/Dark-Energy-UI-and-MC . The code gives the userseveral tools for testing and analyzing various modelsof dark energy. The first step in being able to usethese tools is to run a model through our Monte Carlogenerator. The user must supply several inputs to theMC generator: a specific function g ( Y ), as describedabove; a name to differentiate the model from others;and the number of different initial conditions to test themodel over. The provided name will be used to labelthis model in the user interface. The user must alsospecify the values for several parameters governing eachmodel. Specifically these parameters are the numberof fields that will be acting as dark energy, n , as wellas the range for the initial conditions of x i , y i , c i , and u . In our code we adopt a simplification suggested in[3], which assumes that a single field will act to assistinflation in the early universe before becoming small,while the remaining fields will be small in the earlyuniverse before becoming the dominant form of energyat late times. The “assisted inflation” field will haveone set of unique initial conditions, while the remaining n − w DE as closeto − x to x n are equal. To ensure that a scalar field FIG. 3. Left panel: the first two principal components used to convert from w ( z ) to w ave and ( w , w a ); note that w ave requiresonly the first principal component. The components are based on the NASA/DOE Figure of Merit Science Working Groupanalysis [7]; see Appendix for details. Right panel: a sample scatterplot made from the user interface. Here one can see theresults of w and w a for 10,000 pre-supplied ghost condensate models, case with 15 fields. dominated solution is produced, the values for λ i arechosen automatically, according to the inequalities (seeEqs. (29) and (32) in [3]) λ p ,X > . λ − n p ,X < , (9)Here p is the Lagrangian density, and p ,X ≡ ∂p/∂X .When the code is run, for each specific test, the initialconditions for each parameter are chosen from a randomflat distribution over the selected range. When all ofthe tests have finished running the model will be addedto the user interface automatically. As an example ofwhat is output to the user interface, we have includedthe results of tests of quintessence and ghost condensatemodels having 5, 10, 15, and 20 fields.The user interface also contains two optional tools.The first of these allows the user to run a single instanceof any model having been run through the Monte Carlogenerator. Each of the required parameters are assigneda set of fiducial values that depend on which model is cho-sen; however, the range over which the model has beentested is also supplied, allowing the user to experimentwith values other than those provided. Once the user se-lects values for these parameters, they can choose to plotvarious quantities as a function of either scale factor a orredshift z . These quantities include: • Energy densities in units of critical (radiationΩ R ( a ), matter Ω M ( a ), dark energy Ω DE ( a )); • Equation of state of dark energy w ( a ); • Linear growth of density fluctuations D ( a ) (we plotthe quantity D ( a ) /a ).In order to provide the growth of linear perturbations,concurrently with the other equations we also evolve D (cid:48)(cid:48) + D (cid:48) ( H (cid:48) /H +2) − / H /H ) Ω M (1+ z ) D = 0 (10) where D ( a ) ≡ ( δρ/ρ ) / ( δρ/ρ ) a =1 is the linear growth rateof matter perturbations, and where the primes indicatea derivative with respect to ln( a ). The quantity D/a for the selected model will immediately be displayed onscreen for the user to examine.The second option available to the user is to examinethe results from the Monte Carlo generator. Specifically,this tool allows the user to generate two-dimensional scat-terplots on the fly for further analysis – for example,to see what range of phenomenological outcomes is pro-duced by a given model or class or models. Each point ina given scatterplot corresponds to an output of a singlemodel. The output can be any of the following parame-ters or functions: • Initial conditions from each model including x , x , y , y , c , c , λ , λ , u , and n ; • Energy densities in matter and dark energy todayΩ M and Ω DE , as well as the equation of state today w DE ( a = 1). • Effective “average” equation of state w ave , calcu-lated from the first principal component, as de-scribed in the Appendix. • Effective values for the parameters w and w a basedon parametrization [8] w ( a ) = w + w a (1 − a ), de-rived from the first two principal components asdescribed in the Appendix.An example of the Monte Carlo output showing the w − w a plane is shown in the right panel of Fig. 3. ACKNOWLEDGMENTS
We thank Eric Linder for useful comments. Our workhas been supported by NSF under contract AST-0807564and DOE under contract DE-FG02-95ER40899.
Appendix: Principal components, w ave , and ( w , w a ) Here we explain in more detail how to calculate theaforementioned quantities w , w a , and w p from the prin-cipal components of the equation of state, following[9, 10]. First, we formally expand the equation of statein terms of principal components1 + w ( a ) = (cid:88) i α i e i ( a ) , (11)where α i is the coefficient of the i -th principal component e i ( a ). The idea is to convert the first principal compo-nent α into the averaged value of the equation of state w ave , and first two principal components ( α , α ) into( w , w a ). This approach is justified because the first fewprincipal components carry essentially all the necessaryinformation about the effects of dark energy dynamics onthe expansion of the universe on observable scales.The principal components e i can be determined for anygiven dataset; here we use the Figure of Merit ScienceWorking Group’s (FoMSWG) publicly available code [7],and a combination of Planck+BAO+SN+WL availablein their code. Note that, while the component shapes— especially the all-important peak value of e ( a ) thatshows the temporal epoch of maximum sensitivity to theequation of state — depend on the cosmological probe aswell as specifications of a given experiment, once we com-bine the probes the pull of different probes is expectedto average out, leading to a fixed set of shapes.While the normalization of the e i ( a ) is arbitrary inprinciple, the FoMSWG principal components that weuse are normalized as (cid:82) e i ( a ) da = 1. The coefficients α i can be obtained as α i = (cid:90) [1 + w ( a )] e i ( a ) da (12)where w ( a ) is the actual equation of state of the theoret-ical dark energy model we are studying. The final step is converting the first principal compo-nent into w ave , and the first two into w and w a ; we dothis via [10] 1 + w ave = α β (average w ) (13)and 1 + w ≡ α ( γ − β ) + α ( β − γ ) β γ − β γ w a ≡ α β − α β β γ − β γ (14)where α i are defined in Eq. (12), and β i ≡ (cid:90) e i ( a ) da ; γ i ≡ (cid:90) a e i ( a ) da (15)Equations (13) and (14) are now our definitions of theparameters w ave and ( w , w a ), respectively, given a darkenergy history w ( z ) which determines the α i . Previouswork [10] confirms that the two-parameter equation ofstate closely follows the true w ( z ) over the redshift rangemost effectively probed by the data.Note too that the constraint w ≥ −
1, which followsfrom w ( z ) ≥ −
1, is not strictly obeyed by w obtained inthis way since w and w a are now essentially a fit to thedark energy equation of state history. In other words,the derived parameter w can be slightly smaller than −
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